ABSTRACT Title of dissertation: TURBULENT SHEAR FLOW IN A RAPIDLY ROTATING SPHERICAL ANNULUS Daniel S. Zimmerman, Doctor of Philosophy, 2010 Dissertation directed by: Professor Daniel P. Lathrop Department of Physics This dissertation presents experimental measurements of torque, wall shear stress, pressure, and velocity in the boundary-driven turbulent ow of water between concentric, independently rotating spheres, commonly known as spherical Couette ow. The spheres? radius ratio is 0.35, geometrically similar to that of Earth?s core. The measurements are performed at unprecedented Reynolds number for this geometry, as high as fty-six million. The role of rapid overall rotation on the turbulence is investigated. A number of di erent turbulent ow states are possible, selected by the Rossby number, a dimensionless measure of the di erential rotation. In certain ranges of the Rossby number near state borders, bistable co-existence of states is possible. In these ranges the ow undergoes intermittent transitions between neighboring states. At xed Rossby number, the ow properties vary with Reynolds number in a way similar to that of other turbulent ows. At most parameters investigated, the large scales of the turbulent ow are characterized by system-wide spatial and temporal correlations that co-exist with intense broadband velocity uctuations. Some of these wave-like motions are iden- ti able as inertial modes. All waves are consistent with slowly drifting large scale patterns of vorticity, which include Rossby waves and inertial modes as a subset. The observed waves are generally very energetic, and imply signi cant inhomogene- ity in the turbulent ow. Increasing rapidity of rotation as the Ekman number is lowered intensi es those waves identi ed as inertial modes with respect to other velocity uctuations. The turbulent scaling of the torque on inner sphere is a focus of this disserta- tion. The Rossby-number dependence of the torque is complicated. We normalize the torque at a given Reynolds number in the rotating states by that when the outer sphere is stationary. We nd that this normalized quantity can be considered a Rossby-dependent friction factor that expresses the e ect of the self-organized ow geometry on the turbulent drag. We predict that this Rossby-dependence will change considerably in di erent physical geometries, but should be an important quantity in expressing the parameter dependence of other rapidly rotating shear ows. TURBULENT SHEAR FLOW IN A RAPIDLY ROTATING SPHERICAL ANNULUS by Daniel S. Zimmerman Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial ful llment of the requirements for the degree of Doctor of Philosophy 2010 Advisory Committee: Professor Daniel P. Lathrop, Chair/Advisor Professor Peter Olson Professor William Dorland Professor Saswata Hier-Majumder Professor James Duncan c Copyright by Daniel S. Zimmerman 2010 Dedication Dedicated to my wife, Jess, who has accompanied me through the latter half of my extended graduate career with patience, encouragement, and interest in my work. ii Acknowledgments First and foremost, I must acknowledge our core experimental team, consisting of my advisor, Dan Lathrop, my fellow graduate student Santiago Andr es Triana, and technical supervisor Don Martin, who has most certainly earned his uno cial lab title of \Ub ertechnician." A well known turbulence experimentalist, upon visiting our lab and seeing our experiment for the rst time, asked, before anything else, how large our technical sta was. At this time the permanent technical sta for the three meter experiment then consisted solely of the four of us. We have spent years together, starting when the experimental space was full of other experiments. We have brainstormed, sketched, engineered, and constructed together, and we are all rightfully proud of the machine we?ve built. However, while each of us has assumed many roles, we could not do it ourselves. I?d like to acknowledge Jim Weldon, a long time acquaintance of Dan Lathrop?s and indispensable engineering consult on the design of rapidly rotating equipment, and Allen Selz of Pressure Sciences Incorporated in Richardson, TX, whose stress analysis of the outer sphere gives us con dence that we may safely reach our design speed goals. Furthermore, I would like to acknowledge the ne fabrication job done on the outer sphere by Central Fabricators in Cincinnati, OH, and the e orts of the other machine and fabrication shops, especially Motion Systems in Warren, MI who built our large, custom pulley and Chesapeake Machine in Baltimore, MD who were responsible for the fabrication of many large parts. I?d like to acknowledge the great crew of young scientists that Dan Lathrop iii has assembled here at the University of Maryland over the years. I have enjoyed working with and learning from Woodrow Shew, Dan Sisan, Barbara Brawn, Nicolas Mujica and others on various smaller experiments. I?d like to especially thank Doug Kelley and Matthew Adams, who are respectively the previous and current researchers on the 60 cm experiment , without whom the insights from magnetic data would not be possible, and Matt Paoletti who was undertaking a similar program of torque measurement in Taylor-Couette ow, and with whom I had several helpful discussions. Thanks to IREAP technician Jay Pyle for additional technical support and machining advice, and also to Dr. John Rodgers, who provides much insight into the issues of taking good data in a noisy world. I?d like to acknowledge all of those who have tirelessly supported my scienti c leanings throughout my life, especially my mother Nancy and her late father, John D. Sipple. When I was young, he gave me a hefty science encyclopedia. This single volume was nearly more than I could lift at the time, probably nine inches thick, and bound with steel bolts. Perhaps that in uenced my eventual choice of Ph.D. project. And last, I cannot thank my wife Jess enough. Two years ago, when we were married after several years together, we requested that the guests take it easy with physical gifts since \Dan was going to get his Ph.D. soon," and we expected to relocating. Jess has been patient, understanding, encouraging, and constantly interested in what I do, helping me through the inevitable frustrations of this long project. iv CONTENTS Table of Contents : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii List of Tables : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : viii List of Figures : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xiii List of Abbreviations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xiv 1. Introduction and Theoretical Background : : : : : : : : : : : : : : : : : : 1 1.1 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 The Problem With Turbulence . . . . . . . . . . . . . . . . . 1 1.1.2 Homogeneity, Isotropy, and Universality of Statistics . . . . . 3 1.2 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Rotating Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Two-Dimensionality . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.3 Inertial Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.4 Inertial Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.5 Rossby Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.6 Ekman Layers: Pumping and Stability . . . . . . . . . . . . . 23 1.4 Rotating Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4.1 Rotating vs. 2D Turbulence . . . . . . . . . . . . . . . . . . . 28 1.4.2 Inertial Waves in Turbulence: Linear Propagation . . . . . . . 30 1.4.3 Inertial Waves in Turbulence: Nonlinear Interaction . . . . . . 36 1.4.4 Inertial Modes and Rossby Waves in Turbulence . . . . . . . . 38 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2. Motivation and Prior Work : : : : : : : : : : : : : : : : : : : : : : : : : : 44 2.1 Earth?s Dynamo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2 Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.3 Prior work on Spherical Couette . . . . . . . . . . . . . . . . . . . . . 57 2.3.1 Di erential Rotation vs. Convection . . . . . . . . . . . . . . 57 2.3.2 Non-Rotating Laminar States . . . . . . . . . . . . . . . . . . 59 2.3.3 Turbulent Equatorial Jet . . . . . . . . . . . . . . . . . . . . . 63 2.3.4 Stewartson Layer and Instabilities . . . . . . . . . . . . . . . . 65 2.3.5 Turbulent Sodium Spherical Couette . . . . . . . . . . . . . . 66 3. Experimental Apparatus : : : : : : : : : : : : : : : : : : : : : : : : : : : : 69 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 Design Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2.1 Dimensionless Parameter Goals . . . . . . . . . . . . . . . . . 70 3.2.2 Experiment Limitations . . . . . . . . . . . . . . . . . . . . . 77 3.3 Machinery Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3.1 Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3.2 Lab Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.3.3 Motor Drive System . . . . . . . . . . . . . . . . . . . . . . . 89 3.4 Mechanical Engineering . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.4.1 Support Frame . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.4.2 Static Stress Analysis . . . . . . . . . . . . . . . . . . . . . . . 101 3.4.3 Vibrational Mode Analysis . . . . . . . . . . . . . . . . . . . . 102 3.4.4 Additional Contributions . . . . . . . . . . . . . . . . . . . . . 108 3.5 Instrumentation and Control . . . . . . . . . . . . . . . . . . . . . . . 114 3.5.1 Data and Control Overview . . . . . . . . . . . . . . . . . . . 116 3.5.2 Rotating Instrumentation . . . . . . . . . . . . . . . . . . . . 119 3.5.3 Speed Encoders . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.5.4 Wall Shear Stress Array . . . . . . . . . . . . . . . . . . . . . 128 3.5.5 Torque Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.6 Sixty Centimeter Apparatus . . . . . . . . . . . . . . . . . . . . . . . 142 4. Results I: Turbulent Flow States : : : : : : : : : : : : : : : : : : : : : : : 146 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.2 Ro =1, Outer Sphere Stationary . . . . . . . . . . . . . . . . . . . . 149 4.2.1 Flow Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.2.2 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 4.3 Ro < 0, Outer Sphere Rotating . . . . . . . . . . . . . . . . . . . . . 169 4.3.1 Flow Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.3.2 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 4.4 Ro > 0, Outer Sphere Rotating . . . . . . . . . . . . . . . . . . . . . 179 4.4.1 Narrow Range, Ro < 5 . . . . . . . . . . . . . . . . . . . . . . 179 4.4.2 Wide View, 0 < Ro < 62 . . . . . . . . . . . . . . . . . . . . . 183 4.4.3 Wide Ro > 0 Torque and Shear . . . . . . . . . . . . . . . . . 189 5. Results II: H/L Bistability in Detail : : : : : : : : : : : : : : : : : : : : : : 196 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.2 Torque-Derived Measurements . . . . . . . . . . . . . . . . . . . . . . 197 5.2.1 Torque Revisited . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.2.2 Net Angular Momentum . . . . . . . . . . . . . . . . . . . . . 204 5.3 Direct Flow Measurements . . . . . . . . . . . . . . . . . . . . . . . . 208 5.3.1 Mean Flow Measurements . . . . . . . . . . . . . . . . . . . . 208 5.3.2 Flow Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 212 5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 vi 5.5 Dynamical Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 5.6 Magnetic Field Measurements . . . . . . . . . . . . . . . . . . . . . . 230 6. Turbulent Scaling : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 238 6.1 Torque Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 6.1.1 Torque Collapses . . . . . . . . . . . . . . . . . . . . . . . . . 238 6.1.2 Constant Rossby Number . . . . . . . . . . . . . . . . . . . . 248 6.2 Fluctuation Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6.2.1 Ro = +2:15, Bistable L State . . . . . . . . . . . . . . . . . . 253 6.2.2 Ro = 2:15, Inertial Mode Dominated . . . . . . . . . . . . . 257 6.3 Inertial Modes? Role . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 7. Summary, Speculation, and Future Work : : : : : : : : : : : : : : : : : : : 269 7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 7.2 On Extreme Inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . 272 7.3 Potential Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 7.3.1 Atmospheric Dynamics . . . . . . . . . . . . . . . . . . . . . . 277 7.3.2 Dynamo Prognosis . . . . . . . . . . . . . . . . . . . . . . . . 279 7.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Appendix 286 A. Vorticity of Inertial Modes in a Sphere : : : : : : : : : : : : : : : : : : : : 287 B. Short Form Experimental Overview : : : : : : : : : : : : : : : : : : : : : : 290 C. Mechanical Drawings : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 295 D. Modal and Stress Analysis Directory : : : : : : : : : : : : : : : : : : : : : 296 E. Instrumentation Information : : : : : : : : : : : : : : : : : : : : : : : : : : 309 E.1 Torque Transducer Schematics . . . . . . . . . . . . . . . . . . . . . . 309 E.2 Resonant UVP Transformer . . . . . . . . . . . . . . . . . . . . . . . 313 E.3 CTA Circuit Board and Layout . . . . . . . . . . . . . . . . . . . . . 316 Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 320 vii LIST OF TABLES 2.1 Comparison of dimensionless parameters of Earth?s atmosphere and experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.1 Design goal dimensionless parameters vs. Earth?s core . . . . . . . . . 76 3.2 Beam sectional properties for frame FEM . . . . . . . . . . . . . . . . 96 4.1 Accessible experimental dimensionless parameters . . . . . . . . . . . 148 5.1 Statistics of the interval between high torque onsets, Ro = 2:13, E = 2:1 10 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 D.1 List of nal-stage Algor models . . . . . . . . . . . . . . . . . . . . . 296 LIST OF FIGURES 1.1 Plane inertial wave in an unbounded uid . . . . . . . . . . . . . . . 12 1.2 Depiction of an inertial eigenmode of the full sphere . . . . . . . . . . 16 1.3 Schematic of -e ect dependence of Rossby wave propagation in a spherical annulus and schematic of Ekman pumping in the same . . . 22 1.4 Isolated three-dimensional vortex evolution under the linear propa- gation of inertial waves . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.5 Inertial mode fronts in the transient evolution of turbulence in a for- merly quiescent rotating tank . . . . . . . . . . . . . . . . . . . . . . 34 1.6 Inertial mode induced magnetic eld in spherical Couette and schematic of inertial modes in rotating grid turbulence . . . . . . . . . . . . . . 39 2.1 Earth?s interior and magnetic eld . . . . . . . . . . . . . . . . . . . . 46 2.2 Earth?s general circulation . . . . . . . . . . . . . . . . . . . . . . . . 53 2.3 Equatorial jet in non-rotating spherical Couette ow and Stewartson layer in low Ro rotating ow . . . . . . . . . . . . . . . . . . . . . . . 62 3.1 Brief schematic overview of the experiment . . . . . . . . . . . . . . . 72 3.2 Schematic cross section of bottom base and vessel . . . . . . . . . . . 81 3.2 Schematic cross section, continued from prior page . . . . . . . . . . . 82 3.3 Annotated photograph, experiment exterior . . . . . . . . . . . . . . 86 3.4 Schematic top view of the experiment . . . . . . . . . . . . . . . . . . 88 3.5 Schematic of a beam element . . . . . . . . . . . . . . . . . . . . . . 92 3.6 Beam cross sections used in frame FEM . . . . . . . . . . . . . . . . 95 3.7 Wireframe frame drawing . . . . . . . . . . . . . . . . . . . . . . . . 97 3.8 Four bottom-edge constraint cases discussed in the text . . . . . . . . 99 3.9 Sphere mass model for FEM frame analysis . . . . . . . . . . . . . . . 100 3.10 Static stress analysis, 10,000lb. (44kN) force applied to the motor subframe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.11 Lowest vibrational mode, fullest edge constraints . . . . . . . . . . . . 105 3.12 Lowest vibrational mode, single edge constraints . . . . . . . . . . . . 106 3.13 Spectrum of the thirty lowest modes for various edge constraints . . . 107 3.14 Large ring pulley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.15 Inner sphere stress analysis . . . . . . . . . . . . . . . . . . . . . . . . 113 3.16 Instrumentation and computer block diagram . . . . . . . . . . . . . 116 3.17 Rotating frame instrumentation block diagram . . . . . . . . . . . . . 120 3.18 Rotating DAQ computer and power conversion . . . . . . . . . . . . . 124 3.19 Encoder block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.20 Wall shear stress sensor and array layout . . . . . . . . . . . . . . . . 131 3.21 Constant temperature anemometer schematic . . . . . . . . . . . . . 133 3.22 Wireless torque transducer annotated photograph . . . . . . . . . . . 138 3.23 Wireless torque transducer block diagram . . . . . . . . . . . . . . . . 140 3.24 60cm apparatus with magnetic sensor arrays . . . . . . . . . . . . . . 144 4.1 Time series of dimensionless pressure, Ro =1 . . . . . . . . . . . . . 151 4.2 Power spectra of wall shear stress (three meter experiment) and in- duced magnetic eld (60 cm experiment), Ro =1 . . . . . . . . . . . 153 4.3 Spacetime diagram of the induced eld from the 60 cm experiment, Ro =1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.4 Probability distribution of lag angle across the equator for turbulent wave state, Ro =1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 x 4.5 Probability distribution of wall shear stress, Ro =1 . . . . . . . . . 159 4.6 Large vortices in the bulk interacting with the Ekman layer on the inner sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.7 Dimensionless torque vs. Reynolds number, Ro =1 . . . . . . . . . 167 4.8 Wall pressure spectrogram, Ro < 0, E = 1:6 10 7 . . . . . . . . . . 170 4.9 Wall shear stress power spectrum, Ro = 1:8, E = 1:6 10 7 . . . . 172 4.10 Wall shear stress probability distribution, Ro = 1:8, E = 1:6 10 7 174 4.11 Wall shear stress time series, Ro = 1:8, E = 1:6 10 7 . . . . . 175 4.12 Dimensionless torque and torque ratio, Ro < 0, E = 1:6 10 7 . . . . 178 4.13 Pressure spectrogram, Ro > 0, E = 1:6 10 7 . . . . . . . . . . . . . 180 4.14 Dimensionless torque and azimuthal velocity time series, Ro = 2:3, E = 1:6 10 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 4.15 Dimensionless torque and compensated torque, Ro > 0, E = 5:3 10 7184 4.16 Wall shear stress spectra, wide positive Ro, 1:8 < Ro < 62 . . . . . . 186 4.17 Typical time series of torque and wall shear stress, Ro = 4:2 and Ro = 6:3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 4.18 Compensated torque, wide positive Ro, 1:8 < Ro < 62 . . . . . . . . . 191 4.19 Shear concentration factor, Ro > 0 . . . . . . . . . . . . . . . . . . . 193 5.1 Dimensionless torque vs. Reynolds number, o = 0 . . . . . . . . . . 198 5.2 Dimensionless torque vs. Rossby number, E = 2:1 10 7 . . . . . . 199 5.3 Time series of the torque on the inner sphere, Ro = 2:13, E = 2:1 10 7201 5.4 Probability density of the dimensionless torque conditioned on torque state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.5 Probability of residence in the high and low torque states as a function of Ro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.6 Angular momentum, inner, outer, and net torque for Ro = 2:13 and E = 2:1 10 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 xi 5.7 Velocity, wall shear stress, and torque as a function of time . . . . . . 210 5.8 Probability density of the dimensionless velocity conditioned on torque state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.9 Probability density of the dimensionless wall shear stress conditioned on torque state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 5.10 Wall shear stress power spectra conditioned on torque state . . . . . . 215 5.11 Spectrogram of wall pressure as a function of Ro, E = 1:3 10 7 . . 218 5.12 Schematic sketch of possible mean ow states in the rst torque switching regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 5.13 Jet velocity time average suggesting Ekman pumping . . . . . . . . . 226 5.14 Time delay embedding of slow velocity functions as a function of Ro . 229 5.15 Normalized torque and g31 Gauss coe cient, Ro = 2:7, E = 3:2 10 7232 5.16 Torque state conditioned Gauss coe cient spectra, Ro = 2:7, E = 3:2 10 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 6.1 G=G1 vs. Ro, 0:5 < Ro < 4:5, 1:6 106 < Re < 3:5 107, with ts . 242 6.2 G=G1 vs. Ro, 0:5 < Ro < 4:5, 1:6 106 < Re < 3:5 107, with t . 244 6.3 G=G1 vs. Ro, 4:5 < Ro < 1, with ts . . . . . . . . . . . . . . . . 247 6.4 G=G1, Ro = 2:15;1, 5 106 < Re < 5 107 . . . . . . . . . . . . 249 6.5 G=Re2, Ro = 2:15;1, 5 106 < Re < 5 107 . . . . . . . . . . . . 251 6.6 Collapsed wall shear stress spectra, L state, Ro = +2:15, 5 106 < Re < 5 107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.7 Scaling of power spectral density in selected bands, L state, Ro = +2:15, 5 106 < Re < 5 107 . . . . . . . . . . . . . . . . . . . . . . 256 6.8 Collapsed wall shear stress spectra, inertial mode state, Ro = 2:15, 5 106 < Re < 5 107 . . . . . . . . . . . . . . . . . . . . . . . . . . 258 6.9 Scaling of power spectral density in selected bands, Ro = 2:15, 5 106 < Re < 5 107 . . . . . . . . . . . . . . . . . . . . . . . . . . 260 xii 6.10 Band-pass ltered inertial mode velocity time series, Ro = 2:15, E = 1:1 10 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 6.11 Probability distribution of modulated inertial mode envelope, Ro = 2:15, E = 1:1 10 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 264 6.12 Velocity power spectrum of inertial mode state, Ro = 2:15, E = 1:1 10 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 7.1 Pair of turbulent mixers with weak interaction . . . . . . . . . . . . . 274 7.2 Helicity density of (4, 1, 0.612) inertial mode . . . . . . . . . . . . . . 281 B.1 Schematic view of the experiment for short form overview . . . . . . . 291 E.1 Wireless torque transducer main schematic . . . . . . . . . . . . . . . 310 E.2 Wireless torque transducer voltage regulator and ltering . . . . . . . 311 E.3 Wireless torque sensor DC-DC converter . . . . . . . . . . . . . . . . 312 E.4 Resonant ultrasound coupling rings . . . . . . . . . . . . . . . . . . . 315 xiii List of Abbreviations Coriolis force dependence with latitude in the linearized vorticity equation Ekman boundary layer thickness Energy dissipation Vertical vorticity Radius ratio, ri=ro m Magnetic di usivity c Critical latitude Magnetic permeability m Micrometer Kinematic viscosity Fluid density Electrical conductivity, (Dimensionless half frequency, App. A) u Standard deviation of u, similar for other quantities , w Wall shear stress 1 Predicted shear stress as a function of Re when Ro =1 Cylindrical azimuthal coordinate ! Angular frequency (dimensionless in Ch. 1) !lab Angular frequency measured from lab (nonrotating) frame !0 Dimensionless angular frequency (Ch. 6) Characteristic angular velocity (In electronics context, Ohms) Angular velocity vector o Outer sphere rotation rate i Inner sphere rotation rate (s) Angular velocity as a function of cylindrical radius @t Partial derivative with respect to time 2D two-dimensional 3D three-dimensional AC Alternating current ADC Analog to digital converter AISI American Iron and Steel Institute B \Bursty" torque state label B Magnetic eld, vector B0 Applied magnetic eld Bind Induced magnetic eld cg Wave group velocity c Wave phase speed cm Centimeters C Capacitance CTA Constant temperature anemometry, Constant temperature anemometer xiv DAQ Data acquisition dB Decibel DC Direct current E Ekman number, herein based on gap width ? ER Ekman number based on outer sphere radius ro E(k) Energy density in wavenumber space E \Ekman layer modi ed" torque state f Frequency fc Filter cuto frequency, usually -3dB fo Natural resonant frequency F Force, vector g31 l=3 m=1 Gauss coe cient, similar for other l and m G Dimensionless torque on the inner sphere Go Dimensionless torque on the outer sphere G1 Dimensionless torque as a function of Re with Ro =1 Gnet Net torque, G+Go GHz Gigahertz h Helicity density H \High torque" state Hz Hertz in. Inches I Moment of inertia k Vector wavenumber k Wavenumber magnitude kz Z-component of wavenumber kf Forcing wavenumber kg Kilogram kHz Kilohertz kN Kilonewton kPa Kilopascal kW Kilowatt l Polar wavenumber, spherical harmonic degree ? Spherical Couette uid gap width, ro ri L Generic characteristic length, angular momentum, inductance L \Low torque" state LL \Lower than low" torque state lb. Pounds LED Light emitting diode m Meters m Azimuthal wavenumber, spherical harmonic order xv MHz Megahertz MOSFET Metal oxide silicon eld e ect transistor N Newton NOAA National Oceanic and Atmospheric Administration OSHA Occupational Safety and Health Administration P Pressure p Dimensionless reduced pressure, incl. centrifugal pressure gradient. P!0 Dimensionless spectral power Pa Pascal Pm Magnetic Prandtl number PSD Power Spectral Density psi Pounds per square inch r Position vector (also x) Re Reynolds number Re Reynolds number based on Ekman boundary layer thickness Rei Reynolds number based on inner sphere radius and angular speed ri Inner sphere radius Rm Magnetic Reynolds Number Ro Rossby number Rojet Zonal ow mean Rossby number RoR Rayleigh line Rossby number ro Outer sphere radius RF Radio frequency RMS Root mean square RTD Resistive temperature detector s Cylindrical radial coordinate S(!) Power Spectral Density S 0(!0) Dimensionless Power Spectral Density SPI Serial peripheral interface T Dimensional torque TTL Transistor-transistor logic u Velocity vector u0 Velocity normalized by outer sphere tangential velocity u Azimuthal velocity ue Envelope velocity (Ch. 6) us Cylindrical radial velocity uw Band pass ltered inertial wave velocity uz Vertical velocity uZ Zonal velocity U Characteristic velocity xvi U0 Inertial wave amplitude vector USB Universal serial bus UVP Ultrasound velocity pro le V Volume V Volt y0 Dimensionless distance along ultrasound beam z Cylindrical vertical coordinate xvii 1. INTRODUCTION AND THEORETICAL BACKGROUND 1.1 Turbulence 1.1.1 The Problem With Turbulence The spatiotemporal complexity of the motion of owing uids has provided both intriguing possibilities and vexing di culties for scientists and mathematicians for hundreds of years. Posing the problem, though, is simple enough. Assuming a continuum of in nitesimal parcels of uid, conservation of mass requires a continuity equation for the uid density as the uid moves in space with velocity u, @ @t +r u = 0: (1.1) Assuming that the uid ow is incompressible reduces the above to r u = 0. Con- servation of momentum density u; is described by the Navier-Stokes momentum equation with the pressure P and kinematic viscosity : @u @t + (u r)u = rP + r2u + F: (1.2) This partial di erential equation describes uid acceleration due to advection, pres- sure forces, viscous drag, and external forces F. If we adopt a velocity scale U , length scale L, pressure scale U2, and time scale L=U , we may recast Eq. 1.2 (without external forces) in dimensionless form: @u @t + (u r)u = rP + 1 Re r2u: (1.3) In the above, all quantities are dimensionless; dimensional quantities are formed by multiplying the dimensionless variables by the chosen scales. The sole adjustable dimensionless parameter, the Reynolds number, Re = UL ; (1.4) describes the relative importance of inertia to viscous drag. In the experiments described herein, we are interested in high Reynolds number ows, with Reynolds number ranging from a quarter million up to sixty million. Our knowledge of the correct equations of motion1 does not imply knowledge of all possible solutions. Non-trivial exact nonlinear solutions to Eq. 1.3 are hard to come by. Some of the most interesting and important uid motions in nature and industry, those of turbulent uid ow at high Reynolds number, certainly are not described by known solutions. As computing power has grown, it has become practical to numerically seek closely approximate solutions to Eq. 1.3 for turbulent ows. It is now possible to use a ne enough grid in space and time to resolve the wide range of spatial and temporal scales in somewhat high Reynolds number turbulence for short times. Direct numerical simulation, as this procedure is called, provides a important tool to test theories of turbulence and to closely predict weakly 1 Certainty on this point requires that the thermodynamic degrees of freedom completely decor- relate with the hydrodynamic uctuations. This may not be the case [1, 2], but this is far beyond the scope of this dissertation. 2 turbulent ow in some useful situations. However, even the state of the computational art at the time of this writing has not obviated the need for a predictive theory of turbulent motions. Many geophysical and astrophysical applications will remain beyond direct numerical simulation for many decades at least. And it is only fairly recently that we can compute turbulent velocity elds at all. Modern turbulence theory has likely been shaped by the impossibility of computing solutions to Eq. 1.3. Much of the productive theory of the 20th century starts from a statistical description of turbulent motions and correlations rather than an attempt to predict the uid motions in any detail. 1.1.2 Homogeneity, Isotropy, and Universality of Statistics In 1941, A.N. Kolmogorov advanced a statistical theory of uid turbulence with important and far-reaching predictions about the properties of three-dimensional, spatially homogeneous and isotropic ows. His predictions, often called K41 theory, still inform our understanding of turbulent ows nearly 70 years later. This is due in part to the success of the predictions and in part to their elegance. The ease of the dimensional analysis needed to re-derive some of Kolmogorov?s predictions and the conceptual simplicity of a local-in-wavenumber \cascade" of turbulent kinetic energy provides a convenient entry point into a study of the theory of turbulence. One such result predicts the distribution of the uid speci c kinetic energy 12 u u across the wide range of spatial scales in a turbulent ow. Starting from the as- sumption that the only quantities important for su ciently high Reynolds number turbulent ow are the energy dissipation per unit mass per unit volume and the 3 spatial wavenumber k, one can write down the energy spectrum for an intermediate range of scales (the inertial range) on dimensional grounds: E(k) = C 2=3 k 5=3: (1.5) The role of the assumptions of homogeneity and isotropy in the above result must not be neglected. Any statistical theory of the distribution of energy across the spa- tial scales of a turbulent ow must consider an ensemble average over all possible realizations of the turbulent velocity eld. The assumption of spatial homogeneity (and large extent) allows the replacement of this ensemble average with a spatial av- erage, assuming, in a sense, that everything that can happen will happen somewhere in the ow: 1 2 hu ui = 1 2V Z u(x) u(x) d3x: (1.6) If a system is homogenous throughout space and in nitely extended it is also reason- able to use a spectral decomposition of the above, with an energy spectrum E(k). With the assumption of isotropy, it becomes reasonable to expect that the proper- ties of the turbulence at scales of size 2 =k to depend only on the magnitude of the wavenumber. In an anistropic ow, one must consider the distribution of energy in di erent directions in wavenumber space and how the equilibrium transfer among the di erent directions might proceed. The restrictions of homogeneity and isotropy can never be precisely true for real ows, which are often bounded and usually subject to inhomogeneous and anisotropic forcing. Flow near boundaries and energy-injecting stirrers can be dif- ferent from that in regions that are more remote. Isotropy is easily broken: large 4 scale shear, uid buoyancy e ects, magnetic elds, and overall rotation all impart direction on the uid as a whole. Such e ects are extremely important in geophysi- cal, astrophysical, and industrial applications, and the particular physics of rotating ow will be discussed in detail later in this dissertation. There is an often expressed hope that we may nd a region of the uid where we observe only the generic properties of turbulence if we look far enough from walls or stirrers. But as we perform increasingly more sophisticated investigations, it is clear that the in uence of the \non-universal large scales"[3] persist in the statistics of the even the smallest scales of the ow. Turbulence that is forced inhomogenously or anisotropically does not appear to homogenize and isotropize at any scales. Systems in nature that provide homogenous, isotropic forcing are rare, so homogenous, isotropic turbulence may also be quite rare. This is not diminish the value of the large body of existing research on homo- geneous, isotropic turbulence. Nor is it to suggest that the problem of anisotropic and inhomogeneous turbulence is a neglected area of study. It is merely that our ability to predict the turbulent motion and associated material transport in many important ows must extend to anisotropic and inhomogeneous systems with re- alistic forcing, boundary conditions, and body forces. It is not only the common behaviors of turbulent uid motions that are of interest. We also need to understand the possible non-universal behaviors of entire systems. 5 1.2 Outline of the Dissertation This dissertation will focus on a series of experimental measurements of the anisotropic and inhomogeneous turbulent shear ow between concentric, rapidly ro- tating spherical boundaries. At approximately 3 m outer diameter and 1 m inner diameter, the University of Maryland three-meter device is by far the largest spher- ical Couette apparatus ever built. A substantial fraction of the device was designed and built by the author. This device clearly demonstrates the strong in uence of the di erential and overall rotation, boundary geometry, and boundary layer ow on high Reynolds number turbulence. In the remainder of Chapter 1, we will re- view some of the relevant concepts and literature on the e ects of rotation, shear, boundary geometry, and boundary layers on turbulent ows. In Chapter 2, we will motivate our interest in rapidly rotating turbulent ow in the context of geophysics, and brie y review a few relevant aspects of the prior work in spherical Couette ow. Chapter 3 will describe the experimental apparatus in detail. It will also discuss the design process in the context of our motivations, and detail some of the engineering and instrumentation contributions of the author. Chapter 4 will present a broad view of turbulent ow states observed throughout the parameter range explored in the experimental campaign in water. Chapter 5 will focus in detail on one particular turbulent ow transition and attendant bistable behavior in the transition region. Chapter 6 will discuss scaling of torque and turbulent uctuations as a function of the relevant dimensionless parameters, and will propose a particular picture of the overall dynamics based on these scalings. Chapter 7 will summarize the dissertation 6 and advance some speculative ideas regarding our observations. Finally, we will look at the prognosis for our original experimental goal of generating self-excited magnetic elds in an Earthlike geometry, and propose a few ideas for future ex- perimental work to further illuminate the ndings of this dissertation and test the predictions we make herein. 1.3 Rotating Fluid Flow 1.3.1 Equations of Motion Fluid ow with signi cant overall rotation can often be more conveniently described in a frame rotating with an angular velocity, = z^. The choice of +z^ for the overall rotation vector is a typical convention. The equations of motion in the rotating frame can be deduced by noting that @tu = @2t r and replacing @t with @t + . The result is: @u @t + (u r)u + 2 u + r = 1 rP + r2u: (1.7) The term r is the centrifugal force. A static pressure gradient set up when the uid is retained by walls balances part of the centrifugal force, keeping uid from being ung outward. As this is a prerequisite for studying contained uid, the static pressure gradient that holds the uid in the container is often implicit in the \reduced pressure" p = P + 12 2r2. Di erentially rotating uid regions, like the ow near di erentially rotating walls, may feel a centrifugal force that is out of balance with the gradient of the pressure, and in this case, the centrifugal term will result in ow. The term 2 u is the Coriolis force and is responsible for many 7 interesting e ects in rotating uid ow. The velocity eld u = r is a solution to the nonlinear equations of motion in the inertial frame. In the rotating frame, this corresponds to u = 0, and any small perturbation to the solid body state is subject only to pressure force, the Coriolis force, and viscous force. The pressure force is conservative, the Coriolis force can do no work and the viscous drag is dissipative, so the fate of otherwise unforced perturbations is to decay. Solid body rotation is, therefore, an energy stable steady state. Furthermore, it satis es the boundary conditions on any rigidly rotating set of bounding surfaces, independent of shape. In the absence of additional forces, ow in any rotating container will eventually decay to solid body rotation. It is clear from Eq. 1.7 and Eq. 1.3 that the relative strength of the Coriolis force is what separates rotating ow from nonrotating ow. We can make this clearer by writing Eq. 1.7 in dimensionless form. In a frame rotating with angular velocity = z^, we adopt a time scale of 1= , a length scale L, a relative velocity scale in the rotating frame U , and a pressure scale LU . This pressure scale will also be used as the expected wall shear stress scale in Chapter 6. These choices make the Coriolis term of order unity, and transform Eq. 1.7 to @u @t +Ro (u r)u + 2z^ u = rp+ Er2u (1.8) We now have two dimensionless parameters. The Rossby number, Ro = U L ; (1.9) characterizes the importance of nonlinear interactions of velocities in the rotating 8 frame relative to the Coriolis force. The Ekman number, de ned as E = L2 ; (1.10) is the dimensionless ratio of the viscous drag force to the Coriolis force. Note that with these de nitions we can still de ne a Reynolds number dependent on the two dimensionless parameters, Re = Ro E = U L2 L = UL : (1.11) Geophysical and astrophysical work will often adopt Ro and E, as does prior work on rapidly rotating spherical Couette ow. Work on general rotating turbulence, both in experiment and numerics, seems to prefer Ro and Re, perhaps to compare to nonrotating turbulence at the same Reynolds number. The use of Ro and Re will turn out to be a natural choice for certain aspects of this work. However, where use of Ro and Re is not particularly called for, Ro and E will be used. Fluid ow is considered rapidly rotating when E is low and Ro is not too large. If Ro is so large that the nonlinear term in Eq. 1.8 dominates the Coriolis term everywhere in the ow at all times, we would expect the dynamics to reduce to those described by Eq. 1.3 with Re = Ro=E. This is a regime of essentially non-rotating turbulence. When E is small and Ro is extremely small, a variety of interesting behaviors can be noted by looking at linear or weakly nonlinear perturbations to rapid solid body rotation. Flows in this regime expose some of the phenomena that seem to be important to rapidly rotating turbulence. When Ro is increased and nonlinearity becomes signi cant, it is important to look not only at the dimensionless prefactors of the terms but their behavior in space 9 and time. When a system-size length scale is chosen in Eq. 1.9, the Coriolis force is important at larger, slower scales in the ow, where weak large-scale velocity gradients and small accelerations ensure that the Coriolis term dominates. With steep velocity gradients and large accelerations due to pressure forces, the Coriolis force can be negligible. A ready example is found in Earth?s atmosphere. Large scale weather systems, like tropical cyclones and hurricanes, are shaped by the Coriolis force but small-scale ordinary thunderstorms and tornadoes with similar2 planetary Rossby number are not. 1.3.2 Two-Dimensionality One of the most well-appreciated e ects of rapid rotation is a tendency for the ow to become apparently two-dimensional, with uid motions in columns aligned with the axis of rotation. This e ect is easily demonstrated even in relatively small tanks of water rotating once every few seconds, with Ekman number perhaps 10 4 or 10 5, so it is a popular classroom demonstration. The Taylor-Proudman theorem states that steady ow at low Rossby number will be two-dimensional. Mathemat- ically, steady ow @tu = 0 in the limit Ro;E ! 0 requires that the Coriolis force exactly balance with the gradient of the reduced pressure: 2z^ u = rp: (1.12) Taking the curl of the above equation and invoking incompressibility, we arrive at (z^ r)u = 0: (1.13) 2 The non-tornadic wind speed record of about 113 m/s was set in 1996 in Australia by Tropical Cyclone Olivia [4]. This corresponds to a planetary Rossby number of about 0.25. 10 The gradient of the velocity along the rotation axis (in this case, z^) must be zero for steady ow. Often Eq. 1.13 is usefully used in an approximate sense for slow ow where @tu << 0. Care must be taken in conceptually applying this theorem to ows where accelerations and nonlinearity are not negligible. In Sec. 1.4.1 we will further discuss the role of the Taylor-Proudman theorem in turbulent ows, where @tu is non-negligible. 1.3.3 Inertial Waves The Taylor-Proudman theorem is arrived at by neglecting uid acceleration in the Navier-Stokes equations. Keeping the acceleration in the limit Ro;E ! 0 exposes other interesting linear dynamics. In this case, @u @t + 2z^ u = rp: (1.14) We may again take the curl of the above to eliminate the gradient of pressure, and after some manipulation we arrive at @ @t r u = 2(z^ r)u: (1.15) In an unbounded uid, it is reasonable to insert a plane wave ansatz into Eq. 1.15, u = U0ei(k r !t) (1.16) with U0 a constant vector. The dispersion relation for plane inertial waves that can be deduced from Eq. 1.15 is ! = 2kz jkj = 2 cos ; (1.17) 11 where is the angle of the wavevector from the vertical and ! is a dimensionless frequency in the interval (0; 2). The dimensional frequency is ! . At dimensional frequencies exceeding 2 , the governing equation changes character and no wave motions are possible. Plane inertial waves are circularly polarized transverse waves as depicted schematically in Fig. 1.1. 0 U0 ?/2 ? 3?/2 2? k z Fig. 1.1: Plane inertial wave in an unbounded uid. Planes of constant phase extend in nitely far transverse to the wavevector k. In each plane, uid moves with speed U0, with direction changing as the wave passes. As the wave passes a point in space, the velocity vector rotates at a frequency given by Eq. 1.17 and uid parcels trace out circular paths in planes perpendicular to k. Unlike many common waves, the wave frequency of plane inertial waves does not depend on the wavelength. The dispersion relation of Eq. 1.17 depends only on the 12 direction of wave propagation, not the magnitude of the wavevector. Furthermore, the group velocity of these waves [5], cg = k (2z^ k) k3 ; (1.18) is perpendicular to the phase velocity. The independence of wave frequency and wavelength allows the construction of purely monochromatic wavepackets, localized in space and consisting only of uid motions at a single frequency. Since energy propagates in the direction of the group velocity, it does not spread in the direction of the wavevector. A typ- ical observation of \plane" inertial waves, then, generally consists of a frequency- monochromatic packet with only a few crests and troughs in the direction of k with a dominant wavelength of about the size of the forcing device. Experiments have used a small vertically oscillating disc in a rotating cylindrical tank to excite a con- ical wavepacket. The localized forcing radiates a superposition of plane waves with horizontal wavevectors in all directions, forming a conical layer, across which several wave crests and troughs are observed. Older examples as well as a full treatment of inertial waves can be found in Greenspan?s 1968 monograph [6]. Messio et al. [5] provide modern particle image velocimetry measurements of these conical inertial wave excitations at low Ro, as well as introducing the physical picture of these waves in a particularly clear way. 13 1.3.4 Inertial Modes The previously discussed uid dynamics in the presence of rotation did not discuss realistic boundary conditions on the ow. Provided that Ro is very low and accelerations are small, the Taylor-Proudman theorem holds generally outside the boundary layers in the presence of walls. Wave beams that behave more or less as plane inertial waves may be excited in a bounded uid provided that they are generated in a short wavelength limit and dissipate before any closed trajectories are completed. However, contained rotating uids also exhibit normal modes of oscillation involving global coherent motions of the uid. These modes also obey Eq. 1.15, but must additionally satisfy the boundary conditions on the container. In cylindrical coordinates (s; ; z) in axisymmetric containers we may use an ansatz u = U(s; z) ei(m +!t) (1.19) where U(s; z) is a complex amplitude as a function of position. Solutions to Eq. 1.15 consistent with boundary conditions in a given container can be di cult to nd. However, analytical solutions in some geometries are known. Solutions are known in the interior of a sphere or spheroid subject to no-penetration boundary conditions (Un = 0, where n denotes the wall-normal direction). In the context of rotating spheres or spheroids, Eq. 1.15 is called the Poincar e equation, after Poincar e 1885 [7]. Bryan [8] found implicit solutions. Greenspan [6] and Kudlick [9] found explicit solutions for some modes as polynomials in s and z. Zhang et al. found explicit solutions for every mode in a sphere [10] and a spheroid [11]. Nonaxisymmetric inertial modes in a sphere are three-dimensional traveling 14 waves. As with plane inertial waves, the motion of uid parcels is everywhere sinu- soidal with dimensionless frequency 0 < ! < 2. However, the phase and amplitude pattern U(s; z)eim for a given mode is generally fairly complicated. One mode is depicted in Fig. 1.2. A series of papers by Kelley and collaborators [12{15] establish that inertial modes similar to those in the full sphere can be excited in spherical Couette ow. The non-slip boundary conditions and existence of the inner core in the ex- periment are signi cant departures from the full sphere, free-slip conditions of the theoretical mode calculation. Solutions for the spherical shell are not known in gen- eral. In a spherical shell with free-slip and non-slip boundary conditions, numerical calculations [16{18] show interesting behavior like wave attractors, ray focusing and internal shear layers. However, despite the presence of a core and non-slip bound- ary conditions of the experiments, the modes observed so far in spherical Couette ow show good agreement with the inviscid eigenmodes of Zhang et al.. In the special case where a full sphere mode has identically zero spherical radial veloc- ity, it will trivially satisfy no-penetration boundary conditions. Modes that t this criterion are among those observed [12], but this condition does not fully describe the observed modes. Inertial oscillations satisfying the boundary conditions of the spherical shell, their relation to those in the full sphere, and the weakly nonlinear behavior of inertial modes are a topic of considerable active study [16{21], as these modes are thought to be important in the low-viscosity uid cores and atmospheres of planets and wholly uid stars. It is important to establish to what extent modes like those of the full sphere 15 N = 1, ? = 0.3779, l = 6, m = 3 Fig. 1.2: Velocity (arrows) and spherical radial component of vorticity (color, see Appendix A) on a surface near the outer sphere boundary (0:95 ro) for a particular inertial mode. This mode, according to the classi cation of Zhang et al. [10], has subclass N = 1, azimuthal wavenumber m = 3, and dimensionless frequency ! = 0:3779. Greenspan [6], points out that on the surface of the sphere, a mode has the polar angular dependence of a single associated Legendre polynomial, Pml (cos ). This mode is also, then, uniquely deter- mined by (l;m; !). For this mode, l = 6. This mode is one observed by Kelley et al. [12] 16 play a role in spherical shells in general under a wide variety of forcing methods, as the existence of analytical solutions is a powerful theoretical tool. Zhang et al. have established, for example, the relevance of inertial modes to rapidly rotating convection in a spherical shell [22]. Some of the convective modes in an Earthlike geometry (ri = 0:35 ro) have nearly vanishing velocity near the inner core. Therefore, they should be weakly a ected by its presence. However, it does not seem from the experiments of Kelley et al. [12{15] that this is a necessary property either, at least for forcing by di erential rotation. Inertial modes are an orthogonal set of velocity modes in a rotating sphere, but it is not known if they form a complete set [23]. If the inertial modes are complete, then any velocity eld consisting of motions with frequencies less than 2 in the rotating frame could be written as a superposition of these modes. Liao and Zhang [23] point out that truncated approximations based on inertial modes may lead to a great computational savings in predicting the ow inside rotating spheres. It is also worth mentioning here another recent paper [24] by Liao and Zhang on a new set of orthogonal polynomials useful for expanding the geostrophic motions, that is, the axisymmetric steady (! = 0) motions. It is important to investigate the applicability of all of these results to rotating turbulence in a spherical annulus, as already a good deal of insight has been gained through knowledge of the analytical inertial eigenmodes in a sphere. 17 1.3.5 Rossby Waves In a 1939 paper [25], Carl-Gustaf Rossby deduced that waves could propagate on a zonal ow in a rapidly rotating thin atmosphere on a planet, starting from the principle of conservation of absolute vorticity3. The sum of the relative vorticity on the sphere surface, , and the component of planetary vorticity normal to the surface, 2 sin , must be constant. Here, is the latitude. Rossby used cartesian coordinates to model two-dimensional ow in a thin latitudinal band, with x^ to the East (prograde direction) and y^ to the North. The conservation of total vorticity normal to the planet surface reads: + 2 sin = 0 (1.20) Di erentiating with respect to time and introducing u and v, the relative velocity components in the x^ and y^ directions respectively, one arrives at an equation for the evolution of the vorticity in the rotating frame: @ @t = v: (1.21) In the above, = @xv @yu and = 2 cos =R, where R is the planetary radius. If ow in a narrow latitudinal band is of interest, the -dependence of can be neglected and taken to be a constant. This is the beta-plane approximation, and solutions to Eq. 1.21 are often called \ ow on the beta plane." Using Eq. 1.21, with u = U + u0 and v = v0, Rossby sought perturbations u0 and v0 that propagated without change of shape and constant velocity c. Perturbations of the form v0 = 3 This can be derived by applying Stokes? theorem to the the Kelvin circulation theorem. 18 sin k(x ct) are solutions, with the phase speed c given by c = U k2 : (1.22) The dispersion relation shows that long, low wavenumber waves with k < p =U propagate in the negative x^ (West, retrograde) direction, while waves with k > p =U propagate in the positive x^ (East, prograde) direction. Rossby waves are important to ow in Earth?s atmosphere, and indeed, this is the original motivation of Rossby?s work. The wavy form of the jet stream is an important example. These Rossby waves grow to large amplitude and break, leaving behind isolated high and low pressure systems (anticyclonic and cyclonic vorticity, respectively). Many large-scale weather patterns depend on Rossby wave propagation and breaking, and their generation by and pinning to surface topogra- phy. Some good reviews of the role of Rossby waves in the atmosphere and oceans are Dickinson 1978 [26] and Rhines 1979 [27]. A number of experiments study, in particular, turbulence on the beta-plane by use of a sloping bottom in an annular rotating channel. See, for example, Tian et al. [28], Baroud and collaborators [29{31], and Read et al. [32]. The -e ect in such experiments arises as the variation of height with the radial coordinate stretches or compresses regions of relative vorticity as they move inward or outward on a uni- form background vorticity. The sign of this geometrical -e ect and the resultant direction of propagation of Rossby waves depends on whether or not height increases or decreases with increasing radius. If the height H decreases with radius, as is the case in the interior of a sphere, the long wavelength Rossby waves should propagate 19 Eastward (prograde) and the short wavelength waves should propagate Westward (retrograde). Rossby waves in the interior of a sphere have been addressed numer- ically by Schae er and Cardin [33] in a sphere with di erentially rotating endcaps. They identify the turbulence in their quasi-geostrophic, depth-averaged model as Rossby wave turbulence. We expect that Rossby waves will be important to tur- bulence in a spherical annulus, but with an additional e ect. In this geometry, the height increases with radius for radii inside a cylinder tangent to the inner sphere, but decreases with radius outside the tangent cylinder. This important feature with regard to Rossby wave propagation is depicted in Fig 1.3 (a). There is a connection between Rossby waves in the interior of the sphere and inertial modes. According to Busse, Zhang, and Liao [34], certain slow inertial modes in the sphere, those that have the lowest frequency for their subclass and azimuthal wavenumber, are indistinguishable from Rossby waves. These slow inertial modes are nearly two-dimensional. The Rossby-like inertial modes discussed by Busse et al. propagate only in the prograde direction, while Rossby?s waves propagated in both. However, Rossby?s initial derivation included a prograde zonal ow. When U = 0, all waves would be retrograde. So it seems it is merely the zonal ow that allows prograde Rossby waves in the -plane. The propagation is always retrograde to the zonal ow. With the reversal of the sign of the -e ect in the sphere, in the absence of a zonal ow, we only expect prograde Rossby waves outside the tangent cylinder. Furthermore, since the inertial modes are not known for the spherical annulus, it is also not known what e ect the inner core will have in the sense of Fig. 1.3 (a). 20 Even if Rossby waves in the region outside the tangent cylinder are described by the analytical formulas of Zhang et al [10], those inside and their interactions with the outer region are not known. Further, it is not clear that experimentally ob- served waves in shear ow are well described by slow inertial modes. Schae er and Cardin [35] nd that the varying -e ect with radius results in a spiralling Rossby wave instability of the Stewartson layer. No single inertial mode has a spiral struc- ture, as all azimuthal variation in the known inertial modes are described by a single multiplicative eim factor. It is possible that a coherent superposition of modes could describe this, however. We must account for the role of zonal ow, because we generally drive dif- ferential rotation. Rossby?s original 1939 paper assumed a relative eastward wind velocity U . The inertial modes of the sphere propagate on a background of pure solid body rotation, but there are probably also modes that propagate on sheared pro les. A few exact nonlinear solutions are known for ow on the beta plane. A review by Barcilon and Drazin [36] collects a number of nonlinear solutions relevant for di erent zonal ow pro les. The implication of the nonlinear solutions is that the waves may exist at high amplitudes on background shear. This could make them robust features even in strong rotating turbulence. The existence of such non- linear solutions to the beta-plane equations and the analogy drawn between inertial modes and Rossby waves raises the question of nonlinear solutions to the Poincar e equation, perhaps as sums of inertial modes, inertial modes plus shear pro les, or single special inertial modes. 21 dH < 0 ds dH > 0 ds Ekman layers a) b) Pumping Fig. 1.3: (a) The direction of propagation of Rossby waves in a spherical an- nulus depends on the sign of dH=ds. Inside the tangent cylinder, long wavelength Rossby waves should propagate the opposite direc- tion of those outside the tangent cylinder. In (b), Ekman layers and Ekman pumping will always be present on sloping boundaries in this geometry whenever uid is rotating di erentially with respect to the walls. 22 1.3.6 Ekman Layers: Pumping and Stability In many real turbulent ows of interest in nature, and certainly for those that we can study in the laboratory, uid is contained within walls on which it must satisfy non-slip boundary conditions. Even in high Reynolds number ow where much of the bulk uid dynamics can be well described as inviscid, the tangential stresses at the boundaries will have a substantial e ect on the ow. If the ow is, as in the present experimental investigation, driven by relative motion of relatively smooth4 walls, then energy injection and momentum exchange take place entirely through the action of the boundary layers. This can be the case even without a solid boundary, as in the forcing of ocean currents by surface wind stresses. It is this problem in particular that spurred the initial work of V. Walfrid Ekman. Fridtjof Nansen, a Norwegian explorer and oceanographer, had observed that icebergs did not drift in the direction of the wind, but rather at an angle to it. In 1905, Ekman established [37] mathematically that a rotation of the ocean current direction with depth should be expected from a balance of shear stresses and the Coriolis force near the surface of the ocean. Ekman?s work assumed a uniform turbulent momentum di usion, as molecular viscous shear stresses in balance with the Coriolis force would lead to \the absurd result" of an oceanic boundary layer only a few tens of centimeters thick. However, Ekman?s name is also associated 4 If the walls are rough enough compared to the smallest scales of the turbulence, then pressure gradients across roughness elements can dominate the momentum exchange with the walls. In this situation, the viscous sublayer near the wall is irrelevant. Otherwise, viscous stresses are always important in a thin layer at the wall. 23 with a laminar layer in which viscous shear stress is balanced by the Coriolis force. In laboratory experiments on turbulent ow, the Ekman boundary layers are not necessarily turbulent. In the ocean, there can be sources of turbulent motion near the surface that are unrelated to that generated by an unstable laminar Ekman layer. For example, wave motion causes turbulent mixing in the near-surface ocean. This will probably be the case in many geophysical and experimental situations of interest, but we should also consider the stability of an Ekman layer resulting from the relative motion of uid near a wall. The Reynolds number threshold for instability and transition to turbulence in a laminar Ekman layer is typically given in terms of a Reynolds number based on the Ekman layer thickness = p = , Re = U : (1.23) Leibovich and Lele [38] establish that the critical Re for instability of the Ekman layer on the surface of a sphere is a complicated function of both the relative local direction of the velocity in the bulk and the latitude on the sphere. Typically, Re for initial instability to rolls in the Ekman layer is low, a few tens. In terms of the experiments presented here, this will generally be exceeded except in the case of regions in the ow that happen to be quiet because of the constraints of rotation. But even if the Ekman layer is everywhere unstable, the complicated spatial dependence of the instability could have consequences to the global ow on the interior of the sphere, as the boundary layers could be rather inhomogeneous. Matters may be simpli ed if the Ekman layers were everywhere fully turbu- 24 lent. However, it is not clear that this should always be the case in the experiments presented here. A number of papers, including Aelbrecht et al. [39] put the critical Re for sustained turbulence in the vicinity of 400-500. Noir et al. report a value greater than Re = 120 for sustained boundary turbulence near the equator of a librating5 sphere. Furthermore at large libration amplitudes, the latitudinal extent of the observed turbulence saturates, covering most but not all of the whole bound- ary up to Re = 400 or so. In the experiments here, where typical parameters are =2 1 Hz and U 1 m/s, we expect Re = 325. Whether or not to expect fully turbulent boundary layers everywhere on the surface is not clear, despite the enormous free stream Reynolds number. Furthermore, the experimental investigation by Aelbrecht et al. demonstrates a curious property of Ekman layer stability when the outer ow is oscillatory. For Re up to several hundred, the oscillatory Ekman layer always showed laminar ow near the maximum velocity portion of the cycle. At the velocity minima as the ow was reversing, however, the layer would become turbulent. The extent of the laminar and turbulent portions of the oscillation cycle was a function not only of Re but of the ratio of oscillation frequency to rotation frequency. This peculiar behavior has important consequences to the present experimental work. The question of damping of large scale coherent wave motions like Rossby waves and inertial modes hinges partially on whether or not the boundary layers are turbulent where the waves are strong. Zhang et al. [10] establish that the inertial modes all have zero internal dissipation in the case of free-slip boundaries. There is some internal dissipation due 5 Rotating with a modulated rotational speed. 25 to internal shear layers in the case of non-slip boundaries, but boundary dissipation is still the most important source of damping. This is addressed for laminar boundary layers in Liao et al. [40]. It would be expected that turbulent boundary layers would increase modal damping rates. However, the nding of Aelbrecht et al. for oscillatory layers raises the question of whether instability would happen for inertial modes? boundary layers. The behavior of the Ekman layers in this case has not, to our knowledge, been established. Perhaps there is a turbulent bursting phase, but at wave nodes instead of wave maxima. Additional Ekman layer e ects are relevant to inertial waves and modes in particular. At critical latitudes in the spherical shell, given in terms of the dimen- sionless frequency, !, by c = cos 1 ! 2 ; (1.24) the Ekman boundary layers erupt and radiate inertial wave shear layers into the ow [12]. These internal shear layers spawned from the Ekman layer are the viscous resolution of an inviscid singularity in the ow eld [16, 17, 41]. The energy carried to the boundary by the inertial modes at the critical latitudes is not re ected back to the ow, but instead stays in the boundary layer there, and this leads to a breakdown and eruption of the Ekman layer ow [41]. The shear layers driven by Ekman layer breakdown can be strong and the nonlinear self-interaction of the shear layers can transport angular momentum by generating zonal ows. [21, 41]. One nal property of the boundary layers in a rotating system needs to be addressed, one that depends on the centrifugal term of Eq. 1.7. When a boundary 26 is di erentially rotating with respect to the bulk uid, the uid inside the boundary layer will experience a greater or smaller centrifugal force. The pressure, however, cannot change signi cantly over the small thickness of the boundary layer. So the greater or lesser centrifugal force in the boundary layer will not be balanced by the gradient of the pressure, and uid near any horizontal or sloping surface will be accelerated in the direction of the net force resulting from this imbalance. This e ect is known as Ekman pumping and it will always be present on every boundary of the spherical annulus when the uid is not strictly rotating rigidly with the boundary. An example circulation pattern set up by Ekman pumping is depicted in Fig. 1.3 (b). The pattern depicted might be expected for ow driven by sub-rotating the inner sphere. The outer boundary rotates fastest of all, and uid is pumped out from the poles down toward the equator. The uid from the lower hemisphere meets that from the upper at the equator and is injected into the bulk. Near the inner sphere, the uid near the more slowly rotating inner sphere is pumped by the pres- sure gradient of the bulk uid from the equator of the inner sphere to the poles. Ekman pumping is also likely the most important sources of dissipation for the large scale vortices in rotating systems. In rapidly rotating turbulence, which we will dis- cuss next, energy tends to undergo a cascade to large scales. Boundary friction at these large scales resulting from Ekman pumping is a signi cant contributor to the total dissipation. In three-dimensional isotropic turbulence, the energy is cascaded to small scales and dissipated in intense shear events in the bulk. Dissipation in the boundary layers is a less important e ect there. 27 1.4 Rotating Turbulence 1.4.1 Rotating vs. 2D Turbulence The tendency toward two-dimensional ow introduced by the Taylor-Proudman constraint has important consequences in the turbulent ow of rotating uids. In a 1967 paper [42], Robert Kraichnan derived the expected scaling for the inertial range in strictly two-dimensional turbulence. In inviscid two-dimensional ow, where vor- tex stretching by uid strain is impossible, both energy and mean-squared vorticity (also called enstrophy) are conserved. The enstrophy is constrained by other physi- cal considerations to cascade toward smaller scales. Then, to conserve energy, there must be a net energy transfer toward larger scales. This \inverse cascade" of energy to the large scales of the ow is observed in exactly two-dimensional ows in nu- merics [43, 44], and in nearly two-dimensional experiments using, for example, soap lms or other thin uid layers [45, 46]. An up-scale transfer of energy is observed in rotating turbulence as well, expected due to its tendency toward two-dimensionality. However, it is important to keep a clear distinction between truly two-dimensional turbulence and rapidly rotating turbulence. Indeed, Kraichnan writes [42]: \The result is not directly applicable to meteorological ows because the constraints that render the latter two-dimensional break down at su ciently high [wavenumber] k." The Coriolis force acts weakly in proportion to advection and pressure forces at small scales, as discussed in Sec. 1.3.1, and this breaks the small-scale two-dimensionality in rotating turbulence. Furthermore, even in linear theory, any accelerations in 28 rotating ow must be associated with three-dimensional motion, in accordance with Eq. 1.15. This is in contrast to more strictly 2D ows, such as soap lms, in which velocities in the thin third direction can never become important. Even though the constraint of rotation is broken primarily for small scale motions, the di erences between two-dimensional turbulence and rotating turbulence persist to the large scales. One notable example is that of cyclone-anticyclone asymmetry in rotating turbulence. In experiment and simulation of strictly two-dimensional turbulence forced at intermediate or small scales, there are about as many vortices of positive vorticity as of negative [47, 48]. In experiments and numerics of rapidly rotating turbulence forced three-dimensionally at small scales, cyclones are preferentially excited [47, 49{51] and isolated cyclones in rotating tanks have di erent stability properties than anticyclones [52{54]. Phenomenologically, a tendency toward two-dimensionalization is noted in fairly low Rossby number rotating turbulence. Relatively recent theoretical work attempts to extend the Taylor-Proudman theorem to low Rossby number turbulent ow. Chen et al. [55] build on the work of Wale e [56] and others to establish the dynamics of the the \slow modes" in rotating periodic box turbulence. The normal modes of a periodic box can be described in Fourier space, each with wavevector k. In a weakly nonlinear system, modes with purely horizontal wavenumbers kz 0, decouple to rst order from the \fast modes" which have kz 6= 0. In this situation, the dynamics of the large scale purely two-dimensional modes obey Eq. 1.13 while the three-dimensional fast modes evolve independently according to Eq. 1.8. In the words of Chen et al. 29 \Notice that it is not implied by this result that the ow under rapid ro- tation will become two-dimensional, but it does mean that the dynamics will contain an independent two-dimensional subdynamics, to leading order." Chen et al. present wavenumber spectra conditioned on kz from a turbulence model forced with three-dimensional isotropic forcing at an intermediate single scale. The energy spectra show that the three-dimensional fast modes contain a good fraction of the total energy. Even the developed statistically steady ow is actually far from two-dimensional. This work may help explain the observation in experiment that rotating tur- bulence appears to weakly depart from columnar structure. This often explained simply by invoking the Taylor-Proudman theorem despite obvious accelerations in the rotating frame. The two-dimensional subdynamics will result in long range vertical correlations at fairly energetic and likely easy-to-observe scales. However, this does not mean that the ow is actually two-dimensional even for rotating tur- bulence that is fairly weak. Therefore, it is not always appropriate to take the Taylor-Proudman constraint too literally when the ow is turbulent. 1.4.2 Inertial Waves in Turbulence: Linear Propagation If we supply a three-dimensional perturbation to a rapidly rotating ow, Eq. 1.13 cannot hold. In the linear theory of rapidly rotating ow, we must look at Eq. 1.15, noting that a small three dimensonal perturbation u will lead to time varying vorticity (r u). If we restrict the problem to an in nite rotating space, 30 we may decompose a three-dimensional velocity perturbation into a sum of plane inertial waves, as they form an orthonormal basis [47]. The subsequent evolution of the perturbation may then be described as radiation of inertial waves. This pro- cess can be complicated when a spatially compact initial perturbation is considered, as such a perturbation necessarily contains a wide spectrum in both wavelength and direction. Radiation of energy away from the perturbation will proceed at the group velocity (Eq. 1.18). The group velocity cg k 1, so the energy in the long wavelength Fourier components will radiate away fastest. In a pair of papers, Davidson, Staplehurst, and Dalziel [49, 57] consider the role of linear inertial wave propagation in the evolution of small three-dimensional eddies in a rotating turbulent ow using both inviscid theory and experiment. Davidson et al. [57] note that in the inviscid limit, the axial component of the angular momentum, linear momentum, and vorticity of the original eddy must always remain inside a cylinder tangent to it and aligned with the rotation axis. Energy can radiate in all directions under the action of inertial waves. Davidson et al. show that a spherical vortex of diameter d evolving according to Eq. 1.15 will form a pair of increasingly elongated vortices with centers spaced a distance 2d t apart. This is consistent with the dispersive propagation of energy implied by Eq. 1.18. This process provides a way for a sea of small three-dimensional eddies to rapidly evolve toward an anisotropic, vertically elongated ow even in the absence of nonlinear interactions. The process is depicted in Fig. 1.4. Staplehurst et al. present empirical evidence that this linear propagation pro- cess is important to the vertical elongation of vortices in realistic rapidly rotating 31 ?,L ?,L d 2d?t d?t Fig. 1.4: An initially spherical isolated vortex evolves into a pair of vertically elongated vortices by inertial wave radiation, according to David- son et al. [57] and Staplehurst et al. [49]. The separation and vertical extent of these vortices are consistent with a spreading of energy at the inertial wave group velocity, given in Eq. 1.18. The vorticity ! and angular momentum L inside a cylinder tangent to the original blob must remain there in the inviscid limit of Eq. 1.15, but energy can radiate in all directions. 32 turbulence by studying two-point correlations in decaying grid turbulence in a rotat- ing tank. Davidson, Staplehurst, and Dalziel emphasize the role of phase coherence among the component inertial waves making up an initial three-dimensional per- turbation: in the linear dynamics, the energy spectrum of the original vortex blob is forever identical to that of the subsequently evolving ow, because the energy spectrum does not retain phase information. Vertical elongation is not a general feature of a random sea of linear, noninteracting inertial waves, but rather a speci c property of the particular phase-coherent superpositions needed to describe compact three-dimensional initial conditions. Kolvin et al. [58] establish the importance of inertial wave propagation to the transient evolution of rotating turbulence by forcing an initially quiescent rotating tank from the bottom using turbulent, three-dimensional source-sink ow. The instantaneous horizontal power spectrum in a sheet near the top of the tank is measured using particle image velocimetry. Despite the three-dimensional forcing with a broadband distribution of length and timescales, energy in long wavelengths arrives rst at the measurement plane, consistent with propagation at the group velocity of inertial waves, Eq. 1.18. The initial evolution from quiescent rotation to turbulence is well described by fronts of inertial waves arriving after a time H=cg, with H the height from the bottom forcing to the the measurement plane. The experiment is depicted schematically in Fig. 1.5 At later times, a buildup of energy at long scales is observed, consistent with a nonlinear inverse cascade of additional energy from small scales to large. An important observation of Kolvin et al. is that the slow group speed of the short wavelength (large k) inertial wavefronts results in 33 ? klong kshort H c (k)g t(k) = H Fig. 1.5: Schematic of the experiment of Kolvin et al. [58]. Three-dimensional turbulence is excited at the bottom of an initially quiescent rotating tank by source-sink ow. Velocity energy spectra are measured in a plane a height H from the bottom of the tank. It is observed that the longest wavelength disturbances arrive rst, followed by shorter and shorter wavelengths. The arrival time of motions with wavenumber k is consistent with the dispersive propagation of inertial waves at the group speed cg(k) / 2 k , as in Eq. 1.18. In the experiment, the shorter wavelength waves are observed to propagate through a turbulent background, suggesting that they are a relatively robust feature of moderate Ro turbulence. 34 their arrival at the measurement plane well after nonlinear e ects have started to cascade intermediate scale energy to large scales. The short wavelength inertial waves continue to arrive in in a manner well described by linear theory despite their propagation through established rotating turbulence. The implication is that inertial waves are robust to nonlinear advec- tion in the rotating frame, and are important to energy transport even through a turbulent background. As Kolvin et al. put it, \These waves can, therefore, not be regarded as small and slow perturba- tions to a static uid, but as a more general mode of energy transport." The fragility of inertial waves and inertial modes under the action of other turbulent motions is a matter of some contention, as the derivation of both assumes small am- plitude perturbations to solid body rotation. However, the \linear propagation" of inertial waves is, when viewed in the inertial frame, already a signi cantly nonlinear e ect. In the inertial frame, there is no Coriolis term, and it is the rapid advection by the fast azimuthal velocity (a high Reynolds number ow) that is responsible for the aforementioned e ects of rotation. When the azimuthal velocity of solid body rotation is large compared to the velocities in a frame rotating with it (when Ro is low), the mathematics in the rotating frame are signi cantly simpli ed by truly negligible additional nonlinearity. However, as Ro is increased, the exact role of the nonlinear term is always a matter of great di culty for both calculation and intuition. It should not be assumed a priori that its only role is to decohere and tear apart motions that are derived using linear analysis. The certainty of nonlinear 35 interaction of large amplitude inertial waves does not preclude observation of the \linear" e ects. 1.4.3 Inertial Waves in Turbulence: Nonlinear Interaction The observation of linear wave propagation in rotating turbulence does not minimize the importance of nonlinear interactions. However, inertial waves have a role to play here too. The nonlinear interactions of inertial waves are some of the rst nonlinear e ects to become relevant for three-dimensional forcing as Ro is increased. Smith and Wale e [59] discuss the importance of resonant wave triads in low Rossby number turbulence. Non-resonant triads are also rst-order nonlinear interactions among inertial waves, but the strength of the interaction of triads depends on time as ei(!1+!2+!3)t=Ro. When Ro is small, this phase factor will oscillate rapidly in time unless (!1 + !2 + !3) = 0. The time-averaged interaction is therefore heavily dominated by resonant triads. According to Smith and Wale e, the role of the inertial wave triads is that they transfer energy from strongly three-dimensional motions (those with large (z^ r)u) to more nearly two-dimensional ones. At low Ro, inertial waves with wavevectors and frequencies that add to zero tend to transfer energy to the wave with the more nearly horizontal wavevector, those with small values of kz= jkj. At rst order, uid motions with kz 0, which can be Rossby waves or other geostrophic motions, decouple from the motions with kz 6= 0 [55, 59]. Higher order interactions such as resonant quartets are needed transfer energy from the small kz inertial waves into the purely two-dimensional ow. The two-dimensional large scales in rotating turbulence are 36 energetic scales, and nonlinear interactions among them can also be important, in the manner of an inverse cascade discussed in Sec. 1.4.1. It appears, however, that these interactions can be of secondary importance in low Ro turbulence. Smith and Lee [47] investigate this by comparing simulations including iner- tial wave triad interactions to those with 3D interactions suppressed. The latter is equivalent to the 2D inverse cascade. In both cases, energy is injected at intermedi- ate scales around kf . If Ro is low enough, both the 3D and the 2D simulation result in the generation of energetic, nearly two-dimensional large scales. However, the purely two-dimensional nonlinear evolution results in a dramatically slower buildup of energy in the large scales than that a orded by the three-dimensional inertial wave interactions. Furthermore, the 2D inverse cascade process ends (at the termination of the simulation) with nearly symmetric distribution of cyclones and anticyclones, while the 3D cascade process results in a signi cant asymmetry in favor of cyclones. Inertial waves? role in the cascade of energy to large-scale, nearly two-dimensional motion is similar to the ideas of Davidson, Staplehurst and Dalziel [49, 57]. How- ever, the mechanism proposed by Smith and Lee is nonlinear where that of Davidson, Staplehurst and Dalziel is not. Davidson, Staplehurst and Dalziel suggest that the linear mechanism of Fig. 1.4 may be important where the forcing method results in a large number of small three-dimensional eddies, like von K arm an vortices shed behind an oscillating grid. It may be that both e ects generally co-exist. As Ro becomes larger, o -resonant triads and higher order nonlinear interactions will be- come more and more important. These two mechanisms are simply examples that elucidate di erent aspects of the evolution of rapidly rotating turbulence. 37 1.4.4 Inertial Modes and Rossby Waves in Turbulence The discussion so far of inertial waves? role in rotating turbulence has not made explicit mention of modes of containers or other system-scale waves like Rossby waves. Experimental work in rapidly rotating tanks [49, 58] could possibly (should probably?) result in excitation of these modes. The constraint placed on possi- ble large scale inertial wave motions by the boundary conditions is not necessarily particularly restrictive. However, the spectrum of allowable large scale modes in certain frequency ranges can have a signi cant role in shaping the overall ow. The experiments of Kelley et al. [12{14] nd that the power spectrum of induced linear magnetic eld uctuations near a turbulent spherical Couette experiment is dom- inated by a single inertial mode with a spectral power perhaps 104 times larger than the induction by any other uid motion. The external induced magnetic eld in such experiments and comparison with the external linear induction predicted for a pure inertial mode velocity eld are shown in Fig. 1.6 (a). The modes are excited by di erential rotation at Reynolds numbers exceeding 106. The physical mechanism of excitation of inertial modes in spherical Couette ow appears to be over-re ection [12, 14], wherein wave-induced ripples on a shear layer moving faster than the phase speed of the wave transfer energy to the wave. This dissertation will extend the results regarding inertial modes in spherical Couette ow somewhat by establishing some of the directly measured ow properties of the turbulent ow states associated with inertial mode excitation. In particular, we will present scaling of driving torque and ow uctuations. 38 U Sampled Region Towed Grid Inertial Mode ? a) b) l = 6, m = 1, ? = 0.47, experiment l = 6, m = 1, ? = 0.44, theory Fig. 1.6: Inertial modes in turbulent ows result in global ow correlations and inhomogeneity. At (a), a Mollweide projection of the external induced magnetic eld (top) in rapidly rotating turbulent spheri- cal Couette ow with weak axial applied eld is compared with the induction predicted for that expected from a single matching iner- tial mode (bottom). The experimental measurements of Kelley et al. [14] show a strong signature of the identi ed mode along with additional uctuating induction. At (b), a schematic depiction of the results of Bewley et al. [60]. Inertial modes excited by a towed grid in rotating, decaying turbulence result in oscillatory ows at large scales with respect to the measurement volume. Several iner- tial modes are observed as the turbulence decays. Excerpted images from Kelley et al., 2007 [14] reproduced here with permission from Taylor and Francis Group, Ltd. 39 Bewley et al. [60] observe the strong excitation of several modes of rotating channels (cylindrical and square) throughout the decay of rapidly rotating grid tur- bulence. The kinetic energy in a small measurement volume does not decay as a power law in time, but rather undergoes large uctuations at frequencies consistent with known inertial modes of the container. These inertial modes excited by grid turbulence are a strong source of inhomogeneity in the ow, as a few oscillatory system-scale motions dominate the kinetic energy in a given measurement volume, as depicted schematically in Fig 1.6 (b). A few experimental groups report magneto-Coriolis waves of system-wide scale in hydromagnetic systems. These waves are the combined result of restoration by Coriolis and Lorentz forces and their fast branch, with forces acting in concert on the uid6, shares features in common with Alfv en waves and inertial waves. Large scale wave motions identi ed with magneto-Coriolis waves in a cylinder are reported by Nornberg et al. [61]. Magneto-inertial waves are identi ed in magnetized spherical Couette ow by Schmitt et al. [62]. These latter two do not attempt to compare directly with predicted modes of the containers, although the addition of magnetic eld as a restoring force complicates that comparison. Observation of large scale motions in rotating turbulent ows speci cally as- cribed to inertial modes of the container are relatively scarce, though the results we are aware of suggest that modes of containers are robust components of the large scale motions if a mechanism exists to excite them, and perhaps they are not often identi ed. In experiments, this may be due partially to the di culty of global mea- 6 The slow branch has Coriolis and Lorentz forces in opposition. 40 surements that expose system-wide correlations like those available for the induced magnetic eld in the experiments of Kelley et al.. In numerics, it is probably due to the di culty of performing three-dimensional calculations at low enough Ekman numbers even in the absence of other turbulence. Quasi-geostrophic numerics such as those by Schae er and Cardin [33], while they can reach low Ekman number, are incapable of showing inertial modes, as they are limited to strictly two-dimensional motions. The role of Rossby waves in turbulence is more well established, especially in geophysical contexts. Rhines has written a good review on the role of Rossby waves in oceanic and atmospheric turbulence [27]. Rossby waves dominate the motion at long length scales in geophysical turbulence, and the wave \steepness" controls the nonlinearity at large scales. That is, long wavelength waves must grow to much larger amplitude than shorter wavelength Rossby waves for nonlinearity to be important. Rossby wave motions cause signi cant non-locality in the evolution of -plane turbulence: waves can radiate energy into previously quiet regions of uid. Turbulent straining and mixing coexists with propagating waves on the large scales of rotating turbulence. Rossby wave breaking (where a wave motion is wrapped up by its own vorticity) and interaction with zonal ows are phenomena of primary importance for planetary-scale transport in Earth?s stratosphere [63{65]. The quasi-geostrophic simulations of Schae er and Cardin and comparison with experiment [33, 35] demonstrate the importance of Rossby waves to extremely low Ro turbulence in a rapidly rotating sphere. Tian et al. [28] study ow transi- tions between Rossby wave ow states in a turbulent rotating annulus, and Baroud et 41 al. [30] measure the scaling properties of Rossby-type turbulence in a similar exper- imental geometry. Read et al. [32] demonstrate generation of banded zonal ows by rapidly rotating convection on the -plane. 1.5 Summary The preceding review of theory relevant to rotating turbulence and review of the literature on rotating ows is not comprehensive, but it touches on a number of phenomena that we may expect to be relevant. Rotating turbulent ows have a tendency toward two-dimensional correlations and vertical coherence. However, only those ows where uid accelerations are unimportant and where forcing is pre- dominantly two-dimensional will be described well by the theory of two-dimensional turbulence. Real rotating turbulence will be extremely anisotropic, but will still have a great deal of the energy contained in three-dimensional motions. Three-dimensional rotating turbulence is dominated by inertial waves. The linear propagation of inertial waves and Rossby waves quickly radiate energy into otherwise quiescent regions of rotating uid, but this non-local transport of energy is also robust to pre-existing turbulent ows and so is not limited to energy transport into quiet regions. Linear propagation of inertial waves and nonlinear interaction thereof both help to e ciently cascade energy to the two-dimensional large scales, which retain signi cant memory of the forcing method. In bounded ows, a few large scale inertial modes (or Rossby waves) can be an important component of the velocity eld, resulting in system-wide coherence and strongly inhomogeneous 42 turbulent ow. The boundary layers in rotating turbulent ow may be complicated, especially where inertial modes, Rossby waves, or other oscillatory ows interact with bound- aries in relatively quiet regions of inhomogenous rotating turbulent ows. The e ect of the boundary layer dynamics on rotating ows can be especially signi cant, as the tendency for an inverse cascade results in energy piling up in the otherwise nearly inviscid large scales. Boundary friction is likely to be the most important energy dissipation mechanism for the energy in many rapidly rotating ows. Tight non- local coupling of the bulk uid to the boundaries can arise from the \ uid elasticity" evidenced by Rossby waves, inertial waves, and a tendency toward two-dimensional ow. Instabilities and boundary inhomogeneity as the Ekman layers are modulated by inertial modes or Rossby waves could provide system-wide feedback on overall energy dissipation and transport. Boundary layer eruptions as the resolution of in- ertial mode singularities can result in strong zonal circulations and may therefore signi cantly a ect transport of angular momentum in the bulk. Rotating turbulence is anisotropic, inhomogeneous, wavy, and tightly coupled to boundaries. It is also commonplace in real physical systems, and as such, there is much to be gained from a solid predictive theory that incorporates all of these e ects in a realistic way. Moreover, we nd in rotating turbulence, as Lord Kelvin wrote [66] regarding the motions of a rotating column of uid: \Crowds of exceedingly interesting cases present themselves." 43 2. MOTIVATION AND PRIOR WORK 2.1 Earth?s Dynamo The magnetic elds of Earth and the Sun have signi cant in uence on hu- mankind. Dynamics of the magnetic eld of the sun are responsible for accelerating charged particles toward Earth, and the Earth?s eld acts as a shield for this en- ergetic charged particle radiation. The shield a orded by our magnetosphere is important for life on Earth and increasingly important as we expand technologically into near space. In 1919, Joseph Larmor hypothesized [67] that the magnetic eld of the sun was due to the motion of the electrically conducting uid thereof. Like the sun, the Earth has a magnetic eld because of the ow of conducting uid. In the case of the Earth, this ow is that of its liquid iron outer core, depicted in Fig. 2.1, along with the solid inner core and mantle. This dynamo process is a sort of continuous electrical generator resulting from a self organizing turbulent ow. The evolution of Earth?s eld is slow, and it is arguably of more immediate importance to have good predictions for the Sun?s magnetic eld dynamics. How- ever, we have access to long historical records of Earth?s eld dynamics in the form of magnetized lava ows and sediment deposition, and we can map Earth?s modern eld in great detail with satellite magnetometers. Understanding Earth?s eld is a key to unlocking more general predictions for planetary and stellar dynamos. Earth?s magnetic eld is dominated by a dipolar component tilted slightly with respect to Earth?s rotation axis. The higher-order multipole moments of the eld that we can measure are equal in strength when extrapolated to the core-mantle boundary. [68]. The dipole is a factor of ten stronger than the apparently white spectrum of higher multipoles. Earth?s eld is a dynamical quantity. Changes are observed on a wide range of time scales. Most notable, perhaps, are the erratic reversals evident in the paleomagnetic record. Deposited sediments and lava record the direction and strength of Earth?s magnetic eld in the distant past, and it is ob- served that the dipole component of the magnetic eld changes polarity on irregular intervals from a few tens of thousands of years to fty million years [69]. The dipole reversals are only one component of the variations of Earth?s eld. The smaller deviations, known as the secular variation, have been of interest to humankind for as long as we have used compasses, and for centuries we have recorded the local direction of the magnetic eld across much of the globe with respect to true North, in the interest of navigation [70]. The generation of magnetic eld by a velocity eld u is described by the induction equation, which can be arrived at from combining Maxwell?s equations in a moving conductor with Ohm?s law. In dimensionless form, the induction equation is @B @t = r (u B) + 1 Rm r2B (2.1) To fully describe the nonlinear dynamo process in a rotating planet, one must also 45 Inner Core Outer Core Mantle Crust* Atmosphere* ? ? Fig. 2.1: The dipole-dominated magnetic eld of the earth is generated by turbulent ow in the outer core. The inner core of the Earth is thought to be nearly pure solid iron with a radius 0.35 times that of the outer core. The outer core a liquid mix of iron and lighter metallic elements. The mantle is rock, very weakly conducting and slow owing. The thin rocky crust and atmosphere are drawn much thicker than to scale (*). Dynamics in the core are greatly in uenced Earth?s rotation, and many of the results discussed in Sec. 1.3 should be relevant there. 46 include a Lorentz force term in in the Navier-Stokes equations, Eq. 1.8. This feed- back of magnetic eld on the velocity eld is responsible for nonlinear saturation of dynamos. The magnetic Reynolds number of Eq. 2.1 expresses the strength of the di u- sion of magnetic eld by ohmic dissipation of currents. With length scale L, velocity U and magnetic di usivity m = ( ) 1 the magnetic Reynolds number is de ned as Rm = UL m (2.2) In any uid dynamo, the ow u must be su ciently vigorous to overcome the action of di usion. However, the critical value of Rm required for any particular ow eld to show a growing (@tB > 0) magnetic eld is a complicated matter of the ow geometry. Some analytical results for growing eld solutions of Eq. 2.1 for prescribed ows of homogeneous, incompressible uid were presented by Bullard and Gellman [71]. Magnetic eld generation by a velocity eld where Lorentz forces are negligible is called a kinematic dynamo process, and is important for the start-up phase of a self- excited magnetic dynamo. It is likely that subcritical bifurcations to dynamo action can exist in nature when a nearby object supplies su ciently strong eld. Some simple model ows are known to be subcritical dynamos [72]. However, to produce a eld initially from a dynamically insigni cant seed eld requires a kinematic dynamo phase. Inspired by particular mathematical ows known to be kinematic dynamos, a number of experiments have been performed, with only a few showing dynamo 47 action. The Riga, Latvia dynamo involves a helical down ow of liquid sodium and straight return ow in a large, rotating concentric tube apparatus [73]. This apparatus was inspired by the Ponomarenko dynamo [74], a theoretical kinematic dynamo where a cylinder rotates and translates in an in nite conducting medium. An experiment in Karlsruhe, Germany [75] pumped liquid sodium through a lattice of oppositely-signed helical ba es. Although they were successful dynamos, both of these experiments placed fairly rigid constraints on the geometry of the uid ow. The prescribed velocity elds chosen in much of the early work on kinematic dynamos were not consistent with solutions to the Navier-Stokes equations with any particular forcing or boundary conditions. This resulted in a need to build the right ow geometry out of stainless steel. A di culty in achieving self-organizing dynamos is that su ciently high mag- netic Reynolds number using liquid metals requires strongly turbulent ow. The magnetic Prandtl number, Pm = m = Rm Re (2.3) for liquid metals is on the order of 10 5 or 10 6, and the critical magnetic Reynolds number for dynamo ows is at least one, if not tens or hundreds. Therefore, there is no hope in a liquid metal experiment of generating a dynamo in a laminar ow, as the required hydrodynamic Reynolds number will be in the millions. The large velocity uctuations in a turbulent ow make it di cult to predict whether a particular apparatus will be a dynamo. High magnetic Reynolds number with su ciently self-organized ow appears 48 to be easily achieved in the liquid metal cores of planets and the plasma of stars. In experiment, turbulence greatly confounds analysis and experimental design, and the search for a turbulent laboratory free from the tightly constrained ows of the Riga and Karlsruhe experiments has proven di cult. A recent experiment in Von- K arm an shear ow (counter-rotating disks at either end of a cylinder) in Caderache, France exhibited a self-excited and dynamically interesting magnetic eld, replete with reversals [76, 77]. This is a turbulent dynamo in perhaps the most open ge- ometry used to date, allowing Lorentz forces to restructure the ow. It is a step toward a self-organizing turbulent ow free from strong boundary constraints on the ow. However, this experiment so far has only generated a eld when ferromagnetic impellers are used. This raises doubts to the validity of the results for planetary and stellar dynamos, which are too hot to support ferromagnetism. Numerical simulations have done better than laboratory dynamos in produc- ing earth-like dynamics. Glatzmaier and Roberts [78] were among the rst to nd a dipolar reversing solution in rapidly rotating convection in an earth-like geometry. Reversing dipoles are a common feature of modern geodynamo models [79], but they are not the only possibility. Christensen, Olson, and Glatzmaier [80] nd that in the range of numerically accessible parameters, many dynamo solutions can be found. A new review by Christensen et al. [81] collects a number of numerical dy- namo models that are \earth-like" in relative dipole strength and reversal frequency. Most geodynamo numerics focus on convectively driven ows, though it has been postulated that the precession of Earth?s axis could be an alternative candidate for forcing the core ow. Tilgner [82] has demonstrated that self-consistent ow driven 49 by precession of the rotation axis can result in a dynamo. However, despite the Earthlike behavior of these dynamo models, a mystery remains. None of the geodynamo numerics come close to matching the dimension- less parameters of planetary or stellar dynamos. The Ekman number is as much as nine orders of magnitude too large and the magnetic Prandtl number Pm in the nu- merical simulations is rarely lower than 0.1 or 0.01, with many simulations run near Pm = 1. Serious computational limitations exist in the ability to push toward more Earthlike parameters, and when attempts are made to do so, the long-time evolution of the resulting models can rarely be studied, as time evolution is traded for spatial resolution. Furthermore, Christensen et al. [81] note that the more the parame- ters are tuned toward those of the Earth, the less the resultant dynamo behavior agrees with the observations of Earth?s magnetic eld. The success of numerical models in capturing some of the salient features of the geodynamo must eventually be reconciled with the self-consistent turbulent ow at realistic parameters. While no experiment or numerics will reach Earthlike parameters in any of our lifetimes, a great deal could be learned about these unusual observations by comparing a self-excited laboratory dynamo with numerical models. It is suggested that perhaps turbulent momentum di usion resolves the vast discrepancy in the actual geodynamo Ekman number and the model Ekman number. This is similar to how the oceanic Ekman spiral of Section 1.3.6 is more consistent with a momentum di usion hundreds of times the molecular value. This seems to be a reasonable hy- pothesis, but to build predictive models of the geodynamo, we will need to establish a solid understanding of turbulent momentum transport in a rapidly rotating system 50 with similar geometry, and test any predictions in situ. The University of Maryland three meter experiment is the latest in a series of turbulent sodium experiments aimed at better understanding the geodynamo problem. As will be described more fully in Section 3.2, our experiment is designed to capture some of the important ingredients of the geodynamo. In particular, it will have high magnetic Reynolds number, turbulent ow, rapid rotation, and a geometry that matches the inner and outer core depicted in Fig 2.1, with the outer rotating tank representing the core-mantle boundary. The experiment still does not reach an Earthlike Ekman number1, but if it successfully produces a dynamo, it will come closer to Earthlike parameters and, possibly, Earthlike dynamics than any other to date. It will also present a testbed for numerical models that reproduces many of the di cult conditions that keep us from predicting the behavior of planetary dynamos in simulations. It is a rapid rotator, highly turbulent, and lled with low magnetic Prandtl number uid with a familiar geometry. If this device shows dynamo action, working to predict its dynamics with an eye toward realistically and consistently parameterizing the e ects of the rapidly rotating, magnetized turbulence will be fruitful in the more general problem of predicting planetary and stellar dynamos. 1 It is likely that this is literally impossible in a non-self-gravitating object without signi cant advances in materials science. 51 2.2 Atmospheric Turbulence As it turns out, no hydromagnetic2 data will be presented in this dissertation. However, the mechanically complete experiment was lled with water and the hy- drodynamic ow therein has yielded a good deal of interesting science, which forms the scienti c content of this work. These results are all relevant as a purely hydro- dynamic base case that can be compared with numerics in the interest of studying core ows. However, rapidly rotating purely hydrodynamic ow bears on other geophysical problems. The complicated turbulent circulation of the atmosphere is something of a post-hoc motivation for the experiments we have conducted. Few turbulent uid ows have more direct e ect on humankind3 than the motions of the Earth?s atmosphere. Intense weather events like hurricanes and tornado-spawning squall lines carve wide swaths of destruction. The general pattern of the transport of moisture is critical to the food and water supply for our populace. Our ability to predict the short-term weather is reasonably good in a practical sense. Good short-term prediction of intense storms and sophisticated real-time tracking of them helps to limit the loss of life and property by allowing people time to prepare and nd shelter well before a large storm arrives. However, the prediction horizon for atmospheric motions is not long enough to predict long-term trends in the weather. In recent decades, there has been a keen interest in climate prediction. We would like to understand how rising temperatures may change the familiar long-term averages of energy transport and material transport in Earth?s atmosphere and oceans. 2 We prefer hydromagnetic to magnetohydrodynamic or the acronym MHD. 3 With exceptions for the blood pumping through our veins [83] and the air in our lungs [84]. 52 Climate prediction is a thicket of interacting feedback loops of energy inputs and sinks, turbulent ows, chemical and phase changes, and biological responses. While an incompressible, unstrati ed experiment cannot capture most of these as- pects, it does appear to have some relevance to a speci c problem in climate predic- tion. The world?s structure of population centers, farmland, and uninhabited areas is largely contingent on the very long-term, persistent behavior of the turbulent ow of our atmosphere and oceans. The general circulation of the air mass on the Earth has profound consequences on the zones of habitability of our planet. A schematic cross section of the layers of the atmosphere and its general circulation are depicted in Fig. 2.2. Strongly three-dimensional turbulence tends to be localized in a thin Polar Cell Ferrel Cell Hadley Cell ITCZ Tropopause Polar Jet Subtropical Jet 90N 60N 30N Equator Boundary Layer Stratosphere Tropopause Troposphere Stratosphere Fig. 2.2: Schematic view of Earth?s general circulation and layered struc- ture. Convective upwellings in the inter-tropical convergence zone and the midlatitudes large-scale recirculations, which are de ected in and out of the page by the Coriolis force. Jet streams are strong azimuthal motions resulting from this process. Instabilities of the the jet streams feed back strongly on the convective energy release. This gure is adapted from a public domain image at http://www.srh.weather.gov/srh/jetstream/global/jet.htm, thanks to NOAA. boundary layer near the ground on the order of 1 km thick [85]. This layer is well 53 mixed and vigorously turbulent, with a Reynolds number approaching 1010, but is capped by stably strati ed uid above, which damps turbulent uctuations. The thickness of the well mixed boundary layer is a strong function of other factors. The troposphere is an often stably-strati ed region of the atmosphere that may be unstable to three-dimensional turbulence depending on the exact local conditions. The troposphere is strongly turbulent as well, but the turbulence there is typically closer to two-dimensional. The onset of vigorous moist convection results in thun- derstorms which can extend up to the depth of the troposphere, up to 8-15 km and occasionally into the stratosphere. The stratosphere, lying beyond the tropopause is tenuous and strongly strati ed as it is in radiative equilibrium. Typically, little mass is exchanged between the troposphere and stratosphere [86]. The stratosphere has only weak direct uid e ects on the troposphere, and so will not be discussed here. The wavy, slowly evolving character of the upper tropospheric turbulence seems to provoke atmospheric scientists into calling the three-dimensional boundary layer motions \turbulence" and the large-scale more nearly geostrophic motions a \mean ow" when studying the velocity uctuations near the surface [85]. Nev- ertheless, ow in the upper troposphere is generally nonlinear and aperiodic, and exerts an in uence on the details of the boundary layer turbulence [87]. So the tur- bulence in the troposphere could reasonably be described as strongly anisotropic, inhomogeneous turbulence. The general circulation of the earth is organized into several poloidal cells. The Hadley cells in Fig. 2.2 exist on on either side of the equator to around 30 N and 54 are essentially simple convective recirculations driven by a steady supply of vigorous thunderstorm activity in the inter-tropical convergence zone at low latitudes [88, 89]. All the circulation cells are signi cantly de ected by the Coriolis force resulting in the large-scale mean wind patterns near the surface in the troposphere4. The subtropical jet stream is a rapid region of meandering but largely azimuthal ow the upper troposphere at the down ow boundaries of the Hadley cells, caused largely by angular momentum conservation as uid is pumped from the equator to that region. The polar and Ferrel cells have a more complicated relationship to the convective processes that drive the atmospheric circulation. The polar jet stream is driven largely by turbulent eddy momentum transport in the midlatitudes as familiar weather systems like midlatitude cyclones form and mature [88, 89]. The two jet streams are fed by convective motions but exert a strong feedback on the convection. Rossby waves that form, grow, and break on the jet streams strongly in uence the development of all large-scale convective weather systems. The mid-latitude energy release and momentum transfer to the jet streams is especially sensitive to the meandering and the pressure systems associated with them. So the upper level, slow, tropospheric turbulence, is coupled in a rather tight feedback with the processes that sustains it. Furthermore, the upper level tropospheric winds are acted upon by a turbulent boundary layer drag and even at enormous Reynolds numbers, this turbulence does not appear to become free of the in uence of the mean shear and strong anisotropy aloft [3, 85, 87]. It rains a great deal near the equator and in the midlatitudes but it is rather 4 The trade winds, for example. 55 hot and dry near the down ow of the Ferrel and Hadley cells depicted in Fig. 2.2. This is an important factor in how we have structured our global population centers and farmland. However, the number of cells and jet streams is probably not pre- dictable solely by simple arguments regarding force and energy balances, especially since the polar jet heavily depends on rotationally organized turbulent transport of momentum. Detailed predictions of momentum transport in a turbulent atmosphere are likely needed for fully predictive climate models. It would be a vast oversimpli- cation to suggest that our experiment could serve as a climate model, as it is free from many of the detailed energy transports and feedback loops of energy and mois- ture by processes other than uid transport. However, we will present results that suggest that the \global circulation" in the experiment, the large-scale pattern of zonal ows and waves, is a rather sensitive function of the experimental parameters even in unstrati ed, non-convecting shear turbulence. Angular momentum trans- port away from the inner sphere and and large-scale ow patterns in the system are robust across broad ranges of the Rossby number de ned in Eq.1.9. In these ranges of Ro, the ow patterns remain the same, simply strengthening with increasing Ro. At critical values of Ro, however, there are abrupt transitions to other turbulent ow regimes. The ratio of measured azimuthal velocities to the outer sphere tangential velocity in the ranges Ro where multiple stability are experimentally observed are comparable to the ratio observed jet stream velocities (60 m/s) [88] to the tangen- tial velocity of the Earth. The Ekman number of the experiment, while two orders of magnitude higher than that of Earth, is still extremely low. In each case, the rotation should be of great importance. A comparison is shown in Table 2.1. 56 Tab. 2.1: Comparison of the dimensionless parameters of Earth?s atmosphere and those of the experiment. The Ekman number is based on the thickness of the uid layer in both cases, using Earth?s maximum troposphere depth of 15km [88]. A value of 1:5 10 5 m2=s is used for the kinematic viscosity of air [90]. In each case the Rossby number is de ned as U=( R). The Rossby number, in the atmospheric case is based on the maximum observed jet stream velocity of 60 m/s [88]. In the experimental case, it is de ned based on measured mean zonal velocity observed in one particular turbulent ow state out of multiple stable states observed in the experiment. Parameter Earth Experiment E 10 9 10 7 Rojet 0.2 0.2 This is not to suggest that we are close to a switch of the jet stream behaviors or global circulation patterns, but rather it seems that Ro is a critical parame- ter in rapidly rotating systems that can exhibit multiple stable turbulent states. Understanding the ow regime switches in a simple system free from the extra com- plications of the convecting, strati ed atmosphere could be instructive in some way, much as Rossby?s simpli ed dynamical model described in section 1.3.5 remains useful in our understanding of the jet streams? tendency to meander. 2.3 Prior work on Spherical Couette 2.3.1 Di erential Rotation vs. Convection Neither the rapidly rotating core of the Earth nor Earth?s atmosphere have strong boundary di erential rotation5. The forcing in Earth?s core comes from buoy- 5 It should be mentioned that spherical Couette ow can produce dynamo action. For example, see Guervilly and Cardin [91] 57 ancy driven convection or precessional torques as discussed in Sec. 2.1. Rotating shear ow, however, is in some ways not so di erent from rotating convectively driven ow. A strong analogy can be drawn between the turbulent shear ow be- tween between concentric cylinders and thermal convection [92, 93]. The Reynolds stresses6 associated with a wide variety of uid motions in rotating systems generate di erential rotation in the form of zonal ows (axisymmetric and vertically invariant azimuthal ows). Convection in rapidly rotating systems often causes di erential rotation in both theoretical studies [94{96] and experiment [32, 97]. The nonlinear interaction of inertial modes or other waves in a shell can drive strong di erential rotation [21, 98]. Such waves can be excited by precession or convection. Even generically forced homogenous turbulence in a rotating, sheared system exhibits Reynolds stresses that generate zonal motion [99]. Indeed, strong di erential rota- tion is a common feature of rapidly rotating turbulence. So, despite the seemingly more direct forcing of di erential rotation by the di erentially rotating boundaries, spherical Couette might not be such a bad model for vigorously nonlinear rapidly rotating ows. It will be shown later that the observed di erential rotation is not simply related to the strength of the \applied" di erential rotation based on the boundary angular velocities. The e ects of strong zonal ows on the energy and momentum transport may be more important than how they are set up in the rst place. 6 Advective forces resulting from persistent velocity correlations. 58 2.3.2 Non-Rotating Laminar States Curiously, much of the prior work in the shear ow between rotating spheres is not particularly relevant to the current work. There are few experiments or sim- ulations addressing the turbulent regime. The Chapter 1 review on inertial waves, inertial modes, Ekman layers, and other topics on rotating ow and turbulence is perhaps of more importance. With a few exceptions that will be noted at the end of this section, experiments in the spherical Couette geometry have focused largely on laminar mode transitions, up to the transition to chaos, using small devices. When turbulent ow is achieved, the transition threshold is noted, and the turbulence is not characterized further. Simulations have been largely focused on investigating the same laminar parameter regime as the experiments, and it is still not possi- ble to reach our experimental parameters using three-dimensional direct numerical simulations. It is useful to review a few instructive examples and clear, systematic studies to elucidate some of the basic laminar ow states from both experiment and simulation. A few results in similar geometries may be relevant, and will be discussed here, as well as the few turbulent spherical Couette results that have been published. In the ensuing discussion, it is useful to use some typical spherical-Couette- speci c de nitions of the Ekman and Rossby numbers. The Ekman number used from here on without other quali cation will be de ned using the gap width ? = ri ro and the outer sphere rotation rate o as E = o?2 (2.4) 59 and the Rossby number will be de ned using a dimensionless velocity scale based on the boundary di erential angular velocity and the length scale, U = ( i o)?. With these choices, the Rossby number of Eq. 1.9 becomes a dimensionless di erential angular speed: Ro = i o o (2.5) When the outer sphere rotation is zero, neither the Ekman nor the Rossby number is de ned, but the Reynolds number based on their quotient, Re = Ro E = ( i o)?2 ; (2.6) is always de ned and we will adopt this de nition of Re throughout the parameter plane. For completeness, we note that the radius ratio, = ri ro (2.7) is an important dimensionless parameter7 in spherical Couette ow. Many authors use the dimensionless gap width instead, usually denoted = ri ro ri : (2.8) For reference, in our experiment = 1:86. It is more typical to use the inner sphere tangential speed instead of the velocity ( i o)? in the de nition of Re, but the choices above appear to be most relevant to the nal experimental results, and the de nition of Re = Ro=E for every Ro and E simpli es the comparison of experiments where the outer boundary rotates and those where it does not. It is common in the literature to de ne E in terms of the outer sphere radius ro, and 7 Again, =0.35 for the three meter experiment. 60 sometimes to have an extra factor of two in the denominator. We note that some care should be used in comparing to other work and among other papers, as the choices for the external experimental parameters have various de nitions. It is easy to convert, provided explicit de nitions are provided. Other researchers have taken an interest in laminar spherical Couette ow to understand the routes to chaos therein. It turns out that an enormous variety of laminar ow patterns can be found as a function of the dimensionless parameters in Eqns. 2.4-2.7. The basic state of laminar spherical Couette ow with the outer sphere stationary for small enough radius ratio is a simple recirculation as shown in Fig. 2.3(a), though with a laminar and equatorially symmetric out ow jet at the equator. The jet as sketched is de ected and turbulent. Fluid is subject to Ekman pumping near the inner sphere boundary, and uid on the upper and lower halves gets pumped to the equator, emerging as a jet. For low enough Re, this jet is equatorially symmetric and axisymmetric, and a large- scale meridional circulation completes the uid circuit [100, 101, 104]. Along with this is an axisymmetric azimuthal velocity pro le throughout the uid, leading to a fully three-dimensional axisymmetric circulation. Despite the simple geometry and axisymmetric ow, the fully three-dimensional base state is more di cult to work with theoretically. According to Wulf et al. [101], spherical Couette ow is a prototype for systems where the possible dynamical instabilities vary continuously in space. Certainly, a great number of laminar states are possible as and Re are varied. Wulf et al. map the laminar bifurcations as a function of Re for two di erent radius ratios with the outer sphere stationary. 61 Jet Ekman Layer Ekman Layer ? ??+??0 Stewartson Layer a) b) Solid Body Fig. 2.3: The general circulation patterns for the special cases when the outer sphere is stationary (a) and rapidly rotating (b). At (a), when the inner sphere revolves faster than the outer, the Ekman layer on the inner sphere pumps uid toward the equator, where it erupts in a jet, with a global two cell poloidal recirculation. This jet is steady, planar, and axisymmetric in the laminar basic state ow, but de- velops wavy instabilities at higher Re [100, 101]. This jet persists and becomes turbulent at high enough Re. At (b), the basic state with rapid overall rotation (low E and nearly vanishing Ro) consists of motions con ned near and inside the cylinder tangent to the in- ner sphere. The Ekman layers on the inner and outer spheres are connected by a steady, nested free shear layer called the Stewartson layer after Keith Stewartson who derived its structure [102, 103]. The Stewartson layer resolves the velocity jump between the di er- entially rotating uid inside the layer and the uid rotating with the outer cylinder. 62 The rst bifurcations from the two-cell recirculation state in laminar spherical Couette ow, even leaving the outer sphere stationary, depend on the radius ratio and Reynolds number in a complex fashion. As Re is increased, the ow bifurcates from the axisymmetric two cell recirculation to a nonaxisymmetric rotating wave. Hollerbach et al. [100] systematically studied a wide range of radius ratios, from the narrow gap case of = 0:91 to a very wide gap with = 0:09, and found strong and unusual8 dependence on radius ratio. Hollerbach et al. establish that the the nonaxisymmetric wave is an instability of the equatorial out ow jet for all but the smallest radius ratios, where the instability is in the in ow near the poles. In the two cases considered that straddle our radius ratio, the instability is to a low azimuthal wavenumber (m = 2 or m = 3) slowly traveling wave. The low Re associated with these ow states is impossible to achieve in our experiment, but the idea of a wavy out ow jet may still have some relevance. 2.3.3 Turbulent Equatorial Jet If the inner sphere has an out ow jet in our experiment, that jet must be turbulent due to the large9 Reynolds numbers at which we operate. A turbulent out ow jet as depicted in Fig 2.3(b) is not a feature that seems to be established by a published direct observation in high Re spherical Couette ow, though it is estab- lished for related cases. Bowden and Lord [105] present a Schlieren visualization of a likely turbulent jet ejected from the equator of a magnetically levitating, rapidly 8 For example, a mode transition involving two di erent m = 6 waves 9 Up to 6 107. 63 rotating sphere in a large container. Hollerbach et al.[106] observe a turbulent jet from the equator of a periodically oscillating sphere starting at Re = 200. Kohama and Kobayashi [107] study the boundary layer on a rapidly rotating sphere in a large volume and report a turbulent jet and turbulent boundary layer below some critical latitude, which is about 40 at their highest Reynolds number10. A turbu- lent equatorial jet in spherical Couette ow has been visually established in a small apparatus here at Re 200; 000, but no satisfactory visual record has been worked out in the short time this was attempted. The out ow jet is a basic consequence of the imbalanced centrifugal forces near the sphere surface, and so should be robust at all Re. A boundary such as the inner sphere will always act as a centrifugal pump and drive an out ow when it is rotating more rapidly than the uid around it. This pumping and equatorial jet is therefore likely to be signi cant to many of our results. The jet almost certainly exists throughout a large region in parameter space, and may play a role in rotating turbulent states when the inner sphere is super-rotating. We might also consider a similar process at the outer sphere equator when the uid is rotating more slowly than the wall. This undoubtedly causes Ekman pumping down toward the equator as depicted in Fig. 1.3(b). This is the basic mechanism of Ekman spin up from rest [6]. However, to our knowledge, the possibility of a concentrated and turbulent radially inward jet at the outer sphere equator has not been addressed, as it does not correspond to any numerically or experimentally accessible ow. 10 Less than 2 105 based on the sphere radius and tangential velocity. 64 2.3.4 Stewartson Layer and Instabilities There is another laminar limiting case of interest of ow at low velocities in spherical Couette ow. The circulation at low Ro and low Ekman number, very near solid body rotation, is depicted schematically in Fig. 2.3(b). Stewartson [103] derived a steady solution to the ow between concentric spheres for rapid overall rotation and in nitesimal di erential rotation. In some sense, this solution is the resolution of the equatorial out ow consistent with the Taylor-Proudman theorem of Eq. 1.13, which must hold except in the boundary layers for vanishing Ro. The solution consists of nested shear layers near the cylindrical surface tangent to the inner sphere. The outer layer at largest radius scales in thickness11 as E1=4, and resolves the azimuthal velocity discontinuity. It also transports mass and angular momentum from the faster sphere to the slower, as per the arrows in Fig. 2.3(b). The outer layer at the smallest radius scales di erently, as E2=7 and is required to smoothly join the internal shear layer with the boundary layers. This layer has some reversed mass ux, the balance circulating in a less concentrated return ow in the region inside the tangent cylinder. The innermost layer, scaling as E1=3 is needed to resolve a 2nd derivative discontinuity that the other layers cannot. The Stewartson layer is a striking example of the e ects of rapid rotation, but as with the Taylor-Proudman theorem in general, the range of validity of the concept to rotating turbulence is limited. Stewartson stated the conditions for validity of the solution as RoE 1=3 1. Even if we interpret \much less than" generously and x 11 This layer would be a few centimeters thick in our experiment, could we reach su ciently low Ro 65 it at 1, the solution ceases to be valid at Ro = 0:003 at the lowest Ekman number we will present here. Our minimum achievable Ro is approximately seventeen times that, Romin = 0:05. Stewartson?s validity threshold is consistent with extrapo- lation from the experimental studies of Hide and Titman [108] and Schae er and Cardin [35] on instabilities of the Stewartson layer. Hollerbach et al. [109] studied supercritical mode transitions in comparison between theory and experiment in a spherical Couette apparatus, but again this work is much less strongly rotating, with E in the simulation four orders of magnitude higher than that in our experiment. At our Ekman number, using a quasigeostrophic model, Schae er and Cardin report Rossby wave turbulence at a Rossby number less than half of our lowest accessible Ro. So while the Stewartson layer and instabilities thereof are conceptually useful, the existence of such a layer consistent with Stewartson?s original solution should not be expected in any of our experiments. 2.3.5 Turbulent Sodium Spherical Couette As far as we know, there are three spherical Couette devices worldwide that operate in a strongly turbulent regime. The desire to perform hydromagnetic ex- periments in planetary geometries and the need for more vigorous forcing than that a orded by convection makes spherical Couette ow attractive, and any experiment that reaches an interesting value of Rm using liquid metals will be necessarily tur- bulent. Therefore, the other high Re devices are all liquid sodium experiments. The DTS12 experiment in Grenoble, France is a sodium ow with a permanent 12 Derviche Tourneur Sodium 66 magnet dipole inner core and therefore has a core region of highly magnetized uid [62, 110, 111]. It is a large device, about 42 cm in diameter, capable of Re up to probably 107, but the permanent magnet core confounds direct comparison of results in weakly magnetized ows. This device exhibits interesting hydromagnetic waves and future work to understand the similarities and di erences to turbulent spherical Couette without strong magnetic eld could prove fruitful. The other two highly turbulent devices are housed at our lab, and the author has assisted with experiments in both devices. The 30 cm device used by Sisan et al. [112] has a xed outer sphere, a rotating, conducting inner core, and strong ap- plied axial magnetic eld. Sisan et al. studied primarily hydromagnetic instabilities of the turbulent base state. The hydromagnetic instabilities were identi ed with the magnetorotational instability, a phenomenon that is a matter of some intense research in the astrophysical community. This device has a peak Re as de ned by Eq. 2.6 of about 4:4 106, but is incapable of outer sphere rotation. The 60 cm diameter device used in the inertial mode studies of Kelley et al. [12, 14] is capable of Re up to about 1:6 107 and the 3m device has reached 5:6 107 in the experiments presented here. The rapidly rotating 60 cm device and the newly constructed three meter device have a signi cant region of overlap in Rossby and Ekman and in this region they can complement each other. The three meter experiment has direct ow measurements in the rotating frame to allow directly access to turbulent uctuations and mean ows, while the 60 cm experiment allows the use of weak magnetic eld as a passive ow tracer to infer the global large-scale velocity eld using magnetic eld induction. The hydromagnetic three meter experiment with liquid sodium will 67 exceed the maximum Rm of the 60 cm device by about a factor of ten, the purpose for which it was originally constructed. It is also capable of much more rapid inner sphere super-rotation, achieving higher positive Ro for a given E. This region of parameter space is home to the most novel turbulent ow states presented herein. A few new results regarding the turbulent ows in the overlapping parameter space will also be reported here, with comparison of both direct ow and magnetic eld measurements. 68 3. EXPERIMENTAL APPARATUS 3.1 Introduction The design and construction of the experimental apparatus represents a signif- icant contribution to the experimental study of rotating hydrodynamic and, eventu- ally, hydromagnetic turbulence. The majority of the time spent1 on this dissertation work was in the preparation of the device, and the author?s contribution to the suc- cess of this massive experimental undertaking is signi cant. This description of the experiment?s engineering and construction will be detailed in a way that is not called for on purely scienti c grounds. The author leaves behind the usual lab note- books and repositories of drawings and design les. However, the complexity of the device warrants a more structured description of the author?s contribution to the mechanical, control, and instrumentation systems. Persistent and easily accessible documentation of various design elements is critical for future researchers in our lab, as well as possible international collaboration e orts. The detailed description in this chapter is intended to collect engineering information into something of a users? manual to help other researchers throughout the hopefully long life of the University of Maryland three-meter device. Those interested primarily in the scienti c content 1 A large amount, but worthwhile and enjoyable. of this dissertation may skip to the relatively terse overview of the geometry and capabilities of the nished device provided in Appendix B, and then resume with the scienti c results, starting with Chapter 4. Section 3.2 of this chapter will discuss the design goals for the experiment in the context of the original motivation of obtaining dynamo action in an earth- like laboratory experiment. This is of some importance, as the experiment was not designed for purely hydrodynamic experiments with water as a working uid. Some of the design choices may seem overly restrictive for the experiments presented in this dissertation. The remainder of the chapter will be structured to provide a complete overview of the engineering and design of the device, while dwelling in more detail on the contributions of the author. This device is a collaboration, and many hours were spent by our design team on all aspects of the experiment. Major subsystems designed largely by other members of the team will be discussed in the interest of completeness and continuity, but it will be made clear who was responsible for the largest portion of the engineering and/or construction thereof. Further detail on the contributions made by Santiago Andr es Triana throughout the course of the design process will be available in his forthcoming dissertation. 3.2 Design Motivation 3.2.1 Dimensionless Parameter Goals The original experimental goal of dynamo action in an Earthlike system led to design choices speci cally aimed toward safely conducting hydromagnetic ex- 70 periments at high magnetic Reynolds number and low Ekman number in a device geometrically similar to Earth?s core. Fig. 3.1 schematically depicts the basic ex- perimental apparatus, which consists of outer and inner stainless steel spheres with radius ratio = 0:35, making the uid volume the same shape as Earth?s uid outer core. The Earth?s inner core is thought to be nearly pure solid iron, so it is electri- cally conducting2. Several solid metallic inner sphere designs were considered but ultimately rejected as too costly, too heavy, or too far beyond the abilities of rea- sonable manufacturing processes to be feasible. We chose instead to approximate insulating electromagnetic boundary conditions by installing a thin, hollow stain- less steel shell. The inner shell is essentially transparent to uctuating magnetic elds at frequencies we might expect to encounter, becoming one electromagnetic skin depth ( = p 2= !) thick at a frequency of 7 kHz. The thicker outer shell is one skin depth thick at a frequency of about 280 Hz, which should be transparent to all except the induction at fast small scales. This boundary condition is fairly Earth-like, with the mantle and core mantle transition zone thought to be weakly conducting3. The goal of achieving high magnetic Reynolds number, Rm = UL m (3.1) demands a large apparatus capable of driving high velocity ows in a working uid with low magnetic di usivity. The magnetic di usivity, m = ( ) 1 is a material 2 But not ferromagnetic, as the core exceeds the Curie temperature for iron. 3 Though possibly heterogeneously so [113]. 71 1m Fig. 3.1: Brief schematic overview of the experiment. The spherical bound- aries are free to revolve independently. The ratio of their radii, ri=ro = 0:35, is the same as the radius ratio of Earth?s uid outer core to its solid inner core. 72 property of the working uid. Though the magnetic permeability, , of liquid sodium is close to that of free space, 0, it is one of the best uid electrical conductors, with an electrical conductivity only ve and a half times less than pure copper. Therefore, it is a good choice for high magnetic Reynolds number experiments. There are further advantages to sodium in large hydromagnetic experiments. It has a fairly low melting point, around 98 C. It is produced easily via the electrolysis of salt, so it is rather inexpensive. The density of liquid sodium is low, 0:93 kg m 3 , slightly less than that of water. Other groups have used mercury or gallium metal in smaller experiments, however, both of these metals are signi cantly more dense and expensive. As the three-meter experiment already contains 13,500 kg of working uid at the density of water, this is not an insigni cant issue. Furthermore, other liquid metals in common use are less electrically conductive than sodium, requiring larger experiments or higher speeds to produce the same magnetic Reynolds number. Sodium is, then, nearly the only choice for a large scale hydromagnetic experiment like this one. Once the magnetic di usivity of the working uid has been made as low as possible, further improvements in magnetic Reynolds number come from increasing the size and the ow velocities of the apparatus. The scaling of the power required to drive a turbulent shear ow apparatus is approximately P / U3L2: (3.2) Thus, the achievable ow velocity scales as the cube root of the input power. Power supply and cooling can be di cult for a large apparatus like this one. The installed 73 electrical infrastructure for the experiment as designed was one of the single most signi cant budget items. A major di culty in smaller apparatus is an issue of power density. Ultimately, all of the mechanical power supplied to the apparatus is dissipated as heat in the uid. The cooling requirements for a small apparatus can be extremely stringent, and every predecessor of the three-meter apparatus at the University of Maryland has ultimately had Rm limited by the impossibility of providing su cient cooling, rather than other design considerations. To achieve high enough Rm while avoiding power density problems, it is best to make the apparatus as large as possible. The major design constraint in this regard is the available laboratory space and access to it. The three-meter experiment is housed in a large high-bay laboratory with an overhead crane. Access from the outside to this laboratory space is by means of a large rolling door at one end, which is 10 feet (3.05 m) wide. The frame of this door is of heavy I-beam post and lintel construction, a load-bearing structure. The cost to enlarge this door was estimated early on to be a signi cant fraction of the total budget. The nominally three meter size of this experiment is simply the largest sphere that would t through the door. The nished vessel cleared the door?s width by approximately 16 mm (5/8"). The three-meter experiment is driven by a pair of 250 kW motors However, as the speed of the outer boundary is limited to a fraction of the maximum speed of the inner sphere, and, as the torques on the two spheres must balance, it is impossible for the outer motor to develop full rated power. The absolute upper bound on the mechanical power input for the three-meter apparatus is 320 kW based on the motor torque and the allowable boundary speed. The actual uid power demand, 74 however, is a property of the turbulent ow state as a function of inner and outer sphere rotation speeds. It is impossible to predict the power input required at a given Ro and E. Similarly, it is not possible to predict the actual ow velocities. We will discuss magnetic Reynolds number estimates based on measured ows in Chapter 7, where we give a brief prognosis of the prospects for dynamo action based on our hydrodynamic results. We may, however, estimate a maximum magnetic Reynolds number here. Us- ing a velocity scale that is an estimate of the maximum achievable tangential bound- ary velocity of the inner sphere and a length scale of the radius of the outer sphere, we have Rm = iriro m : (3.3) Based on a estimate of the limiting speed of the di erential boundary rotation taken from experiments conducted in water, the maximum possible Reynolds number in the three meter experiment is about 700. We are also interested in attempting model a dynamo such as that which arises in the Earth?s rapidly rotating core. The dynamo simulations discussed in Chapter 2 seem to rely on the ow organization a orded by the rapid rotation. Such a low Ekman number (Eq. 1.10) as that in Earth?s core, E 10 15, is impossible to achieve in experiments. However, we can study rotating turbulence at modest Rossby number at an Ekman number considerably lower than that which can be achieved in simulation. The Ekman number de ned using the maximum design 75 rotation rate and outer sphere radius ER = or2o (3.4) can be as low as 1:25 10 8. We denote this as ER here, as later chapters will use the uid gap width, not the outer sphere radius, as the relevant length scale, and we do not wish to cause confusion. The de nition using the core-mantle boundary radius (equivalent to our outer sphere radius) is common in the geophysical literature, so ER is more appropriate to the current discussion. Table 3.1 compares the dimensionless parameters achievable in the three meter experiment vs. what is estimated for Earth?s core. Tab. 3.1: Comparison of the relevant dimensionless parameters, Ekman and Magnetic Reynolds, of Earth?s core and the design goal parameters of the experiment. Parameter Earth Experiment ER 10 15 [114] 10 8 Rm 300-500 [115] 700 Sodium metal is a good choice as a working uid for rapidly rotating hydro- magnetic experiments. It has a low viscosity (7 10 7 m2=s, 70% that of water), and its low density compared to other commonly used liquid metals reduces the centrifu- gal load on the experimental device, allowing more rapid rotation. Ultimately, the lowest Ekman number that can be achieved is limited by the maximum allowable centrifugal stresses in the outer shell. This sets our highest rotational speed limit of 4 Hz, corresponding to our lowest Ekman number, ER 10 8. 76 3.2.2 Experiment Limitations While sodium metal is the clear choice to achieve the highest possible magnetic Reynolds number, and a good choice for reaching low Ekman number, it is not without its hazards, as it reacts violently with water and burns readily in air if it comes in contact with an ignition source4. The sodium oxide produced by burning or otherwise oxidizing sodium dissolves in water (damp skin, throat, eyes, mucous membranes) to form sodium hydroxide, which in low concentrations is an irritant and can cause severe caustic burns in high concentrations. Sodium res are di cult to extinguish. The best prevention of all possible sodium hazards is to attempt to eliminate the possibility of any large scale leaks. In a rapidly rotating experiment such as this one, the high centrifugal pressures developed at large cylindrical radius are of serious concern. Even small ori ces can leak signi cant mass in a short time. A three centimeter hole opening near the equator with the outer sphere rotating at maximum design speed could release a thousand kilograms or more of sodium in the few minutes before the sphere could be brought to a halt. Additionally, when the sphere is at rest, any opening far from the top of the experiment has the potential to leak a large amount of sodium slowly if the leak goes unnoticed for some time. In the interest of minimizing the possibility of large scale leaks of either sort, it was decided that the outer spherical vessel would have openings only at the top. A large cylindrical anged opening 1.5 m (5 feet) in diameter provides access to the interior of the outer vessel. This allows easy insertion and removal of a variety of internal 4 At the temperatures we intend to run the experiment, around 105 C, liquid sodium does not auto-ignite in air. However, if sprayed into air at signi cantly higher temperatures it will. 77 forcing systems, but signi cantly limits the amount of sodium that can leak out gravitationally. A centrifugal leak could still be signi cant, but the modest pressure at the lid rim (about 1.7 atmospheres at maximum speed) can be sealed e ectively without any special techniques. The prohibition of openings below the lid rim is not particularly limiting for the planned hydromagnetic experiments. As discussed in Sec. 3.2.1, the wall of the outer vessel is nearly transparent to magnetic elds, as long as they are uctuating more slowly than a few hundred Hertz. A great deal of information can be deduced from magnetic eld measurements external to the outer vessel. In regions of param- eter space where the experiment does not generate its own magnetic eld, applying a magnetic eld and observing the induced eld can still be instructive. It is sci- enti cally useful to have some direct ow measurements, pressure, velocity, and so forth to corroborate the magnetic eld measurements. However, measurements of the magnetic eld can serve as a primary diagnostic. For the purely hydrodynamic ows that will be presented in this dissertation, the limitation of access to the ow solely through the top lid (coupled with limited space for instrumentation) does prove to be something of a hindrance. Some seemingly rather basic quantities rel- evant to a rotating, turbulent shear ow, such as mean velocity pro les, may seem conspicuously absent. 78 3.3 Machinery Overview 3.3.1 Rotating Frame We will begin the description of the experimental device with a mechanical and power system overview describing the major mechanical and drive subsystems. We will leave instrumentation until Sec. 3.5. Figure 3.2 is an annotated cross section of the apparatus, and Fig 3.3 is an annotated photograph of the exterior. The outer sphere, Fig. 3.3 (a), has an inner boundary diameter of 2:92 0:005 m (115 in.)5 and 2.54 cm (1 in.) wall thickness. The entire rotating mass is supported by a load spreading base, Fig. 3.2 (A). The cylindrical section of the base is welded to a half inch thick stainless steel plate, and strong struts help spread the load. The plate, base, and struts, and lower portion of the frame, Fig. 3.3 (b), are a welded assembly that rests on the laboratory oor. The weight of the rotating mass (about 21 tons) is borne by a spherical roller thrust bearing, Fig. 3.2 (B), which ts into a seat in the cylindrical base. A part called the bottom head, Fig. 3.2 (C), ts into the bottom bearing inner race and bolts to the bottom sphere ange Fig. 3.2 (D). The original design called for an alignment pin in this location, but the manufacturer was unable to adequately align the hole in the vessel bottom ange with the vessel?s rotation axis. Therefore, spacer plates allow a range of radial adjustment for centering the bottom of the sphere on the rotation axis. In addition to its mechanical function, the bottom head conveys heat transfer oil from a concentric- ow rotary uid union, 5 Dimensions relevant to scienti c results will be given in metric and Imperial units. Those that are engineering details will only be given in Imperial, as was used in the design process. 79 Fig. 3.2 (E), to inlet and outlet manifolds, Fig. 3.2 (F), which distribute uid to a spiral wound half-pipe heat transfer jacket that covers much of the outer sphere surface. A 1.52 m (5 ft.) diameter anged cylindrical opening in the top of the outer sphere allows installation and extraction of the interior forcing package and allows access to the uid for making ow measurements. This opening is capped by a thick stainless steel lid, Fig. 3.2 (G), the lower surface of which completes the outer spherical boundary. The lid is located with respect to the sphere ange by two large location pins diametrically opposite on the 65 in. diameter bolt circle. A short shaft integral with the lid mates with the inner race of the outer sphere top bearing, Fig. 3.2 (H). The outer race of this bearing is held by a housing Fig. 3.2 (I) which bolts to the outer support frame seen in Fig. 3.3. The bearing (H) is free to move axially on the shaft of lid (G) to allow for thermal expansion of the vessel when it is brought to operating temperature for sodium experiments. The outer sphere is driven by a timing belt drive with a 25:3 reduction ratio. A large toothed pulley, Fig. 3.2 (J), bolts to the top lid and is centered on the rotation axis using a dial indicator. Four instrumentation ports, Fig. 3.2 (K), each 12.7cm (5 in.) in diameter allow direct measurement of ow quantities. These ports are located at 60 cm (23.625 in.) from the rotation axis. Removable port plugs approximate a ush surface at the outer sphere boundary, and are removable so that instrumentation may be installed in them easily. These port plugs will seal to the top lid with at graphite seals for the sodium experiments. For water, O-rings were used. A 16.8 cm (6.6 in.) shaft, Fig. 3.2 (L), concentric with the outer sphere axis 80 A B C H J G D E F F I K K L M N O P Q R S TT Fig. 3.2: A cross-sectional overview of the experiment mechanical design, excluding supporting structure. Cont?d next page. 81 Fig. 3.2: A) Load spreading base and bottom bearing outer race seat, integral with Fig. 3.3 b. B) Outer sphere bottom bearing, SKF#29344E. C) Bottom head. D) Outer vessel ange, bolts to B. E) Dual- ow rotary uid union, Deublin #H107. F) Outer vessel jacket manifolds. G) Top lid, bolts to outer vessel ange. H) Outer sphere top bearing, SKF #ECB 23956 CAC/W33. I) Outer top bearing housing, bolts to motor subframe, Fig. 3.4(e). J) Outer sphere pulley. K) Instrumentation port (4 total). L) Inner shaft. M) Inner shaft bearing, SKF#24030 CC/W33, #N030/#W030 lock- nut/washer, seal CR#64998 N) Inner sphere. O) Inner sphere bottom ange. P) Inner sphere top clamp. Q)Inner sphere bottom bearing, SKF#22216E, locknut #AN16, sealed by CR#36359. R) \Joe," bottom bearing bayonet mount. S) Hexagonal bottom bayonet, integral with outer vessel. T) 3/4"-10 studs (4). 82 supports and drives the inner sphere. The weight of the inner sphere and shaft is borne by the top lid, with the axial load carried by the inner sphere top bearing and locknut, Fig. 3.2 (M). This bearing is sealed against the uid by a pair of lip seals directly beneath. The inner sphere top bearing and the seals are nominally replaceable in situ without disassembling the experiment due to the design6 of the outer race housing, allowing prompt recovery from any small leak through the seals. The inner sphere, Fig. 3.2 (N), has 1:02 0:005 m (40.1 in.) outer diameter and a wall thickness of approximately 0.2 in. The inner sphere is hollow and liquid- tight. Axial location of the inner sphere with respect to the top bearing and radial location of the bottom of the sphere is provided by a ange integral to the inner shaft, Fig. 3.2 (O), which bolts into the bottom of the removable inner sphere unit. A clamping ring, Fig. 3.2 (P), radially locates the top of the inner sphere. The bottom ange and top clamping ring are su cient to transmit the torque. The bottom of the shaft is radially located by a bearing, Fig. 3.2 (Q), and a receptacle7, Fig. 3.2 (R), which mate with a hexagonal bayonet integral with the bottom of the outer sphere, Fig. 3.2 (S). The tip of this bayonet was adjusted to be concentric with the rotation axis of the bottom bearing (B). The concentricity with the rotation axis was measured by means of projecting a magni ed video image of the bottom bayonet onto a whiteboard to measure its runout as the outer sphere was rotated by hand. A hydraulic press was used to slightly slide the sphere ange (D) with respect to the bottom head (C). In this way, the axis of the vessel was aligned with 6 Detailed drawings in the dissertation of S.A. Triana. 7 We named it Joe, and don?t really know what to call it otherwise. 83 the bottom bearing to better than 0.4 mm (0.017 in.). Rounding out Fig. 3.2, four 3/4"-10 studs Fig. 3.2 (T) are welded to the bottom of the outer sphere concentric with the four ports in the lid above, in case any ba es or other sorts of forcing or boundary system were to be desired in the future. The great majority of the design of the vessel and forcing package described above was conducted by S.A. Triana, although the author was responsible for the initial thermal jacket design as part of designing the overall thermal system. In Figure 3.2, the author was responsible for the detailed design of components (A) through (C) as well as parts (I) and (J). S.A. Triana was responsible for the design of the rest of the rotating equipment. The large rotating parts were largely beyond our in-house fabrication capabilities, and were manufactured by outside companies. While it is not a strict demarcation, the author?s contribution to the design of the three meter machine was generally in the nonrotating infrastructure: the design of the frame, mechanical drive system, and thermal system. Therefore, the design of some of these systems will be discussed in detail here. For the more detailed design and mechanical drawings of the rotating equipment, the reader is referred to the dissertation of S.A. Triana. 3.3.2 Lab Frame The exterior of the experiment is depicted with annotations in Fig. 3.3, and a top schematic view of the frame and motor drive system is depicted in Fig. 3.4. A roughly cubical frame in four parts, Fig. 3.3 (b,c,d,e), is primarily a mechanical support for the top outer sphere bearing, Fig. 3.2 (H). The inner dimensions of the 84 frame are approximately 14 ft: 14 ft: 12 ft:; 8 in. (4:2 4:2 3:9 m). The frame is mostly of welded construction. Frame members are 4 in. square beams of AISI 304L non-magnetic stainless steel with 0.25 in wall thickness. The side walls of the cube are divided into two sections. The bottom section, Fig. 3.3 (b), comprises roughly the bottom third of the cube. This portion of the frame is designed as a catch basin with more than su cient volume to accept the entire sphere?s contents in the extremely unlikely event of a catastrophic complete leak. The oor of the bottom portion is 1/2 in. thick stainless steel to help spread the weight of the experiment across the concrete oor of the laboratory, and all seams of this oor are welded to the frame members around the bottom interior. The walls of the bottom section are 0.059 in. thick stainless steel sheet, welded to the frame members around the periphery. All seams of the frame are welded to create a nominally liquid-tight basin. Some areas were more di cult to weld than others, and the seams of this basin will be painted with silicone rubber sealant before sodium ll to help ensure that it is fully liquid-tight. The removable upper frame section, Fig. 3.3 (c), is bolted to the bottom frame section (b) in twelve locations, with a bolting plate at every frame upright. The upper frame section was installed after the sphere (a) was placed by a professional rigging team. The laboratory space has a crane on the ceiling su cient to lift 5 tons, but the sphere shell is approximately 7 tons. It was lifted into place by professionals with stronger equipment. Each frame section is designed to weigh safely less than 5 tons. The frame roof, Fig. 3.3 (d), depicted in top view in Fig. 3.4 (d), tops the frame with a framework capable of transferring large radial loads from the rotating 85 a b c h g d i e f Fig. 3.3: Annotated photograph of the experiment. a) Outer pressure vessel. b) Bottom frame section, secondary containment. c) Middle frame section, shield panels. d) Frame roof. e) Motor mount subframe. f) Inner motor mount. g) Outer motor mount. h) OSHA-compliant safety rail. i) The author, nowhere else referred to as \i." 86 sphere out to the cube walls. The roof is of the same 4 in. box beam construction, but is covered in expanded metal decking to provide a walking/working surface, and is surrounded by an OSHA-compliant handrail (h). The roof Fig. 3.3 (d) bolts to the upper sidewall section (c) in twelve places, and can be removed to allow insertion and removal of large equipment into the cube. The outer sphere ange protrudes through a square opening in the middle of the roof that is not much bigger than the sphere pulley. In this way, provided that the roof is installed, the outer sphere can not tip more than approximately 1 from the vertical. In ordinary disassembly, even if the roof is not to be removed, the outer sphere is guyed using four steel cables to the corners of the lower frame section (b) before the top bearing is removed. However, if the bearing were removed, the outer sphere pulley would come to rest against the inner surface of the roof opening. This e ects a secondary mechanical con nement for the rotating equipment in case of bearing failure. The motor mount subframe, Fig. 3.3 (e) and Fig. 3.4 (e) bolts to the lid roof and mates with the outer bearing housing Fig. 3.2 (I) to transfer loads from the outer sphere to the roof. The motor mount subframe is the only frame piece that must be removed to allow extraction of the lid Fig. 3.2 (G) and forcing package Fig. 3.2 (L-S) from the outer vessel. Two motors weighing about 1 ton apiece drive the inner and outer sphere respectively. The inner motor is held centered above the inner shaft by the inner motor frame Fig. 3.3 (f) and Fig. 3.4 (f). This frame is actually an integral part of the motor mount subframe, having been welded in place after initial positioning. The tall inner motor mount is sti ened by diagonal struts out to the edges of the motor mount subframe. The outer motor mount Fig. 3.3 87 g Y a d J K f e Z W U Z VX To Drives IG Fig. 3.4: Schematic top view of the experiment. a) Outer vessel. d) Frame roof. e) Motor mount subframe. f) Inner motor mount. g) Outer motor mount, movable in direction indicated. G) Vessel top lid. I) Outer bearing housing, bolts to (e). J) Outer sphere pulley. K) Instrumentation port. U) Motor power junction box. V) Motor ex conduits. W) Outer motor. X) Inner motor. Y) Outer drive belt, Gates 6160-14MGT-55. Z) Outer drive idler pulley. 88 and Fig. 3.4 (g) bolts to the motor mount subframe and can be freed to translate in the direction indicated by the arrow in Fig. 3.4 to allow adjustment of outer sphere drive belt tension. 3.3.3 Motor Drive System The rotation of the spherical boundaries is driven by a pair of 250 kW (350HP) induction motors. These 480 V three-phase motors are speed controlled to better than 0.2% by a pair of ABB ACH-550 variable frequency drives. Power from the drives arrives at the edge of the frame through a quartet of conduits terminating in a junction box, Fig. 3.4 (U). This junction box is supported on a beam that attaches only to the lowest frame section, so that most of the frame can be removed simply by disconnecting the cables in the exible conduits, Fig. 3.4 (V), using quick connects in the junction box. Generally, this is not necessary, but the ex conduits (V) must be disconnected to free the motor mount subframe for sphere access. The inner motor, Fig. 3.4 (X), directly drives the inner sphere shaft through a Lovejoy exible shaft coupling and a torque sensor to be described later. The outer motor, Fig. 3.4 (W) has a timing pulley mounted on its output shaft. A high horsepower Gates Powergrip timing belt Fig. 3.4 (Y) transmits power from this pulley to the large ring pulley on the outer sphere lid rim, Fig. 3.4 (J). The ratio of this pulley drive system is 25:3. An idler pulley, Fig. 3.4 (Z), ensures that the the belt only wraps around the large ring pulley over an arc of 180 , a design advised by an engineer at Gates. Static belt tension is adjustable by use of a large bolt that pulls and pushes the outer motor mount (g) in the direction of the arrow. 89 3.4 Mechanical Engineering Before moving on to the description of the scienti c instrumentation and data collection, some detailed design information will be presented regarding the author?s contribution to the engineering of the three meter device. The author was respon- sible for the design engineering and most of the construction of the experiment?s outer frame, motor mounts, and mechanical motor drive systems. This detailed de- scription has several motivations. Documentation of these design processes is of use in our upcoming preparation for an operational readiness review as we work toward sodium ll. These design tasks represent substantial contributions of the author to the successful completion of the device, and are worth documenting in that regard. Lastl, the author has received a more comprehensive than usual education in the design and construction of large experimental apparatus, and some exhibition of the mechanical engineering education received while enrolled at the University of Maryland seems appropriate for inclusion here. 3.4.1 Support Frame The outer cube frame supports and contains approximately 21 tons (19000kg) of rotating mass with over ve and a half megajoules of rotational kinetic energy at top design speed. This mass is well balanced and smooth running by itself but a bearing or other failure could result in very large forces applied to the frame as the sphere comes to a halt. Prediction of the strength of the frame and elimination of stress concentrating design elements is important to ensure a robust and long-lived 90 experiment and the safety of the experimental team. Furthermore, frame vibration is of substantial concern, a lesson learned from many prior rotating experiments in our lab. An initial goal of frame design was to make it as sti , light, and strong as possible. However, it is di cult to remove all mechanical resonances from the op- erating speed band for an experiment of this size and speed. Simple detuning by sti ening or adding mass is not useful with an experimental device that must oper- ate throughout a continuous range of speeds. Furthermore, while many resonances can be practically removed from the operating band, harmonics of the operating fre- quency can excite higher modes. For those troublesome vibrational modes that lie within the operating speed band, especially those at higher frequencies with more complex structures, prediction of the frame normal mode shapes is useful. This information can be used to place vibration measurement intelligently. It can also be used to decide likely locations for added damping to reduce the amplitude of vibration and subsequent vibrational stresses developed in the frame. To determine the strength of the frame and the vibrational mode shapes, we have used linear stress and mode shape analysis of nite element beam structures to model the cube frame. These analyses served as a guide toward rational assessment of proposed frame structures before we rendered one in stainless steel. Five or six major frame revisions were modeled by the author throughout the design process. It is worth discussing the methods used and the nal output of the predicted calculations. Our proposed frame designs were analyzed using linear stress and mode shape analysis, implemented using Algor, a commercially available professional nite el- 91 ement analysis software. Our frame is, with a few exceptions, well described by a rigidly connected network of beam elements. In the context of nite element analy- sis, a beam, depicted schematically in Fig. 3.5 is a linear elastic mechanical element that can support extension and compression, exure, and torsion when used in a nite element model. The stresses and vibrations of a network of beams can be e 1 e 2 e 3 k-node Flexure Torsion Extension/Compression Fig. 3.5: A beam is a mechanical element that can support extension, com- pression, exure and torsion. This is in contrast to a simpler truss element, which can only support extension and compression. As beams can be anisotropic with di erent sti ness in di erent direc- tions, analysis using beam elements requires knowledge of beam properties referenced to a local coordinate system in each beam. analyzed by calculating the displacements and rotations of all of the nodes in the network consistent with the elastic response of each element, which are determined for a given beam by the forces and twisting moments at its ends. In beam analysis, the calculation must take into account all three components of the force and the three components of the moment at every node. 92 The basic idea of the linear nite element methods used here can be under- stood through the direct sti ness formulation of nite element beam analysis. The equations at all the nodes can be written as one matrix equation, F = Kx (3.5) where F is a vector of forces and moments, x is a vector of translational and ro- tational displacements, and K is called the sti ness matrix, in which is encoded all the beam data: it represents the connectivity among all nodes and the elastic responses of the beams connecting them. The design process of a suitable support frame involves a certain optimization of this sti ness matrix by choosing beam prop- erties and connections intelligently. With applied forces and boundary conditions at certain nodes, Eq. 3.5 can be solved for the forces and displacements everywhere, and the stresses in the beam elements can be calculated afterward from their stress- strain relationships. The mode shapes and frequencies for small amplitude sinusoidal oscillations of a beam network are the solutions to the eigenvalue problem, (K !2M)x = 0; (3.6) where M is the nodal mass matrix expressing the mass loading of each node, either by explicit masses or those of the beams themselves The eigenvectors xn give the mode shapes and the eigenvalues !n are the corresponding vibrational frequencies. We must specify the connectivity of all the beams, their lengths, their material properties, and their cross sectional properties so that Algor can formulate the sti - ness matrix and solve the problems discussed above for a proposed frame design. 93 This is the heart of the design process. We iterate through di erent designs, in- specting the output for troublesome vibrational modes and areas of excessive stress, adding beams and modifying connections to move toward a better design. The beam material properties are homogenous throughout the frame. We use stainless steel, out of a desire to use nonmagnetic and relatively weakly conducting material in the vicinity of the experiment. A weight density of 0.289 lb/in:3 (8 g/cc), Poisson?s ratio of 0.3, and elastic modulus of 2:9 107 lb.in:2 (200GPa) are used throughout, as per values obtained from www.matweb.com for AISI 304L stainless steel8. To specify the cross sectional distribution of mass, which determines the sti ness of the beam, several properties of the cross-section must be speci ed. The beam cross section properties are speci ed with reference to the coordinate system of Fig. 3.5. Figure 3.6 depicts the two important beam sections required for the frame model. The square beam of Fig. 3.6 (a) is the basic frame member. The triple corner beam in Fig. 3.6 (b) is simply three square beams welded together. One of these triple composite beams forms each vertical corner of the cube. The area of the cross section is one required sectional property. The centroid of the section is another, as it de nes the location of the neutral axis of the beam. When the beam is bent, a plane through the neutral axis separates the metal that is in compression from that in tension. The centroid is the origin of the coordinate system in Figs. 3.5 and 3.6, labeled as \CM" in Fig. 3.6 (b). Higher moments of the area further de ne the distribution of mass around the centroid. The second moment of the area, called 8 These material properties vary only modestly across di erent types of steel. 94 e 3 e 2 CM e 2 e 3 a) b) y 3 Fig. 3.6: Cross sections of the two beam types needed in our frame model, a square beam (a) and a triple beam as we have in the corners of the frame (b). This schematic shows the 2 and 3 coordinate axes, the extremal distance from the three axis to the beam extremum, y3, and the centroid of the triple beam, labeled CM. The centroid is the origin of the beam local coordinate system. the moment of inertia 9 about the en axis, is de ned as In = Z x2dA (3.7) and has units of length to the fourth power. This integral is taken along the axis perpendicular to the nth axis: this is the moment of inertia for the cross sectional shape rotating about the nth axis. The section modulus Sn is de ned as Sn = In yn ; (3.8) where yn is the distance from the nth axis to the most extreme point on the section, as shown in Fig. 3.6. Finally, the torsional sti ness of the beam about the 1-axis, J1 9 The second moment of the mass per unit area would give the more familiar moment of inertia in kg m2. In beam bending, the moment of inertia is a property of the area. 95 must be known to specify the resistance of the beam to twisting. The expression for the torsional sti ness of a given beam section is too complicated to warrant space here, but formulas for sectional properties for many useful shapes can be found in Roark?s Formulas for Stress and Strain [116]. We tabulate the relevant section properties for the two beam sections of Fig. 3.6 in Table 3.2. Tab. 3.2: Beam sectional properties of the square and corner beams in Fig. 3.6 for input to Algor nite element modeling software. A is the area, I2 and I3 are the moments of inertia about the 2 and 3-axes, S2 and S3 are the section moduli about the same, and J1 is the torsional sti ness about the beam 1-axis. J1 for the corner beam is incorrect, but at worst that makes our calculations a lower bound on strength and sti ness. Property Square Beam Corner Beam A 3.75 in:2 11.25 in:2 I2 8.83 in:4 86.47 in:4 I3 8.83 in:4 63.14 in:4 S2 4.41 in:3 15.30 in:3 S3 4.41 in:3 12.65 in:3 J1 14.9 in:4 14.9 in:4 With beam properties collected, a model can be built. The Algor software provides a graphical user interface to draw a connected wireframe of the beam structure which is parsed into length and connectivity information for the sti ness matrix. Beam joint coordinates in the nished model are connected with beams divided into perhaps ve to ten segments between joints for better representation of the mode shapes and de ections. Fig. 3.7 shows the wireframe model in perspective view looking from an elevated vantage point West of the cube. The cylindrical base is modeled as a set of perfectly rigid constraints in space, 96 Fig. 3.7: Wireframe frame drawing from the nite element model, west per- spective view. Red circles and triangles are boundary condition con- straints. Green labels of LM are the locations of lumped masses representing the motor and distributed mass of the sphere. The small red points on the beam in the middle of the frame highlight the sphere model beam. The cube?s orientation in the lab has Y to the North and X to the East. The corner closest to the camera in Fig. 3.3 is the Southwest corner. 97 but the boundary constraints around the bottom edge of the cube are varied to investigate the solutions? dependence on edge boundary conditions. The cube is not bolted to the oor. This is intentional, as a sudden development of signi cant out of balance forces will result in much lower peak stresses on the cube if it can slide slightly with respect to the oor. This will dissipate a great deal of energy and lower the peak forces. However, it makes renders the boundary conditions around the bottom outer edge of the cube somewhat indeterminate. We assume that the center of base will not move under normal operation, so we always constrain it fully10, but we change the boundary conditions around the edge to compare cases. Fig. 3.8 shows the locations for constraints for four cases considered in the following text. Typically we prevent translation in the X and Y-directions at least, to model the 1/2in. plate welded inside the bottom frame section that adds a huge amount of sti ness at the bottom edges in the X,Y plane. However, we try di erent boundary conditions, some of them somewhat unphysical, simply to put bounds on what to expect. All bolted joints are modeled as rigid connections. For the purpose of accurate modal analysis, especially for modes that involve horizontal oscillations of the top of the frame, we must include a model of the sphere. We model the outer spherical vessel using a sti , nearly massless beam and three point masses. The two 6400 lb. masses are positioned away from the center of mass such that that the moment of inertia of the spherical shell about the X and Y-axes through the center of mass is correct. The balance of the mass of the uid is represented by a lumped mass at the center of mass of the sphere model. 10 In fact there is no base, per se, but just a set of eight constraints. 98 Base Base Base Base (a) ff2sph (b) ff2sph_e (c) ff2sph_r (d) ff2sph_u Fig. 3.8: The four constraint cases considered in the text and Fig. 3.13 are pictured here. The base is always modeled by rigid constraints on the inner ends of the support struts. We consider four cases, (a), all twelve edge nodes at the bottom of a vertical beam are constrained. At (b), only the nodes along one edge are constrained. At (c), a random selection with no symmetry is chosen. At (d), we use only the base as a constraint. This is not especially realistic, because the real cube has a 1/2" thick steel oor adding a web between all the struts. However, (d) can be considered a worst case. 99 6400lb (2900kg) 6400lb (2900kg) CM 33000lb (14960kg) Very Flexible Very Stiff Very stiff, nearly massless. R Model Top Bearing Anisotropic Compliance Tx, Ty, Tz, Rz Constrained Z Model Bottom Bearing Fig. 3.9: A trio of point masses on a large cross section, extremely light and sti beam is used to model the vessel. The 6400 lb. masses give the correct moment of inertia of the sphere about the X and Y-axes through the center of mass and the 33,000 lb. mass models the mass of the water, inner sphere, etc. R is the radius of the outer sphere. The bottom bearing is modeled by a spatial constraint that allows the joint to pivot about the X and Y-axes, but disallows translation. Rz, rotation about the Z-axis, is constrained to prevent spurious torsional oscillations of the sphere beam restored by the top bearing model. The top bearing model allows easy displacements with little stress in the Z-direction but is horizontally sti , modeling in a linear way the ability of the top shaft to slide in the bearing. 100 Unlike the shell, the uid moment of inertia is irrelevant for vibration analysis as the boundaries will just slip with respect to the uid. Pressure, however, will engage the mass of the water. The main bottom sphere bearing is a spherical roller bearing and is modeled here by a constraint that disallows translation in the the X,Y, and Z directions. It also disallows rotation about the Z-axis11, but allows the axis to pivot about the X and Y-directions, as is appropriate for a spherical roller bearing. The top bearing, which is axially free to slide on the shaft, is modeled as a set of four beams that will bend easily if Z-directed loads are applied at their midpoint but which are rigid in transmitting horizontal forces to the frame. Without this, the very rigid sphere shaft e ectively constrains the motor subframe?s distance from the sphere?s bottom constraint constraint. This exibility trick would not work if beam buckling were allowed, as such slender beams would crumple, but the linear analysis performed here can show no such e ects. Linear stress analysis does not re-calculate the beam lengths and nodal forces based on the new nodal distances. A more sophisticated model would use some sort of node that allowed local degrees of freedom to slip. 3.4.2 Static Stress Analysis Static stress analyses were performed on numerous iterations of frame models with loadings corresponding to di erent operational and accident scenarios. We will present only one example here. In this example, the frame is constrained around 11 This constraint is required to suppress spurious torsional modes of the model sphere that would arise if it were rotationally constrained by only the anisotropic \top bearing" beams. 101 the lower edge in twelve places, as depicted in Fig. 3.7 and Fig. 3.8 (a), with full translational constraints at the four corner locations and translational constraints in the wall-normal direction on the eight mid-span nodes. All nodes are free to rotate about all three axes. This constraint and loading is not of a sort that could be arranged precisely in the actual experiment, as the cube is not actually tied down. Even if it were, the concrete oor and tie anchors would not be well modeled by an in nitely rigid constraint. However, this stress model should be reasonably accurate for a radial di erential force applied between the experiment base and the bearing housing on the motor subframe. This is a load case we care about substantially, as it is similar to how the rotating sphere might apply radial forces to the frame. Figure 3.10 shows the result of the calculation. With 10,000lb. (44kN) applied force, the peak stress is just over 4100 psi12 (28 MPa), about 7 to 10 times lower than the yield stress for stainless steel. We use a working number of 30,000 psi for the yield stress of 304L stainless, but much of the steel used in the frame is rated to 40,000 psi as per the speci cations received from the supplier. 3.4.3 Vibrational Mode Analysis A few results for vibrational analysis of the frame are presented here. With su ciently constrained edges, the lowest vibrational mode of the frame is well above the operating speed range for the outer sphere, though it is still inside the allow- able speed band for the inner sphere. To situate us in the discussion of di erently constrained cases below, we will refer back to Fig 3.8. 12 lb./in:2 102 F ?S Fig. 3.10: Static stress distribution and 100x exaggerated de ection, with 10,000 lb. (44kN) force applied to the motor subframe near the point where the bearing housing bolts. The load is applied in the Y-direction, as indicated by the arrow. The peak stress is 4100 psi (28MPa), a factor of 7 to 10 lower than the yield stress of AISI 304L stainless steel. Constraints in the four bottom corners are (Tx; Ty; Tz) and in the eight mid-span edges they are (Tx) or (Ty), which prevents the edge from translating in the cube-wall-normal direction, mimicking the e ect of the plate. The label S indicates the location of a quantitative displacement inquiry at the top edge of the cube, which displaces about 0.011 in. (0.3 mm). We may use the force and displacement to calculate a spring constant for the cube lid in this loading con guration, 9 105 lb./in (1:6 108 N/m). Positive stresses here are extensional, negative are compressional. 103 The lowest vibrational mode with the fullest edge constraints, Fig. 3.8 (a), is depicted in Fig. 3.11 and has a frequency of 10.8 Hz. This mode involves primarily an X-direction13 tilting of the inner motor mount, Fig. 3.3 and Fig. 3.4 (f). This mode is likely troublesome and some additional struts with dampers could be useful in limiting its amplitude, or rigid struts might raise it su ciently to move it out of the experimental speed band. Due to time constraints, empirical measurement of the nalized frame?s vibration spectrum has not been performed. However, this will done as part of the upcoming transition to sodium. We consider another vibrational model with only one edge constrained, as in Fig. 3.8 (b). The lowest mode of this constraint case is depicted in Fig. 3.12 and has a frequency of about 5.5 Hz. This may be a slightly more realistic lowest mode, as there will be a tendency for one edge of the cube frame to be on the oor at all times. However, this model does not take into account the membrane elasticity of the 1/2 in. oor plate or the mass loading of the oor plate on the bottom of the lower frame section. The elasticity of this plate is probably more important, and it is likely that this resonant frequency underestimates the true frequency somewhat. A summary of the e ect of boundary conditions on the spectrum of mode frequencies is given in Fig. 3.13. The rst thirty lowest vibrational modes were calculated for comparison. The four constraint cases referred to in Fig. 3.13 are pictured in Fig. 3.8. The unconstrained, Fig. 3.8 (d), and heavily constrained, Fig. 3.8 (a) cases in Fig. 3.13 are probably reasonable upper and lower bounds for the frequencies 13 East-West in lab coordinates. 104 10.8438Hz f = 0 Fig. 3.11: De ected mode shape, lowest mode with full constraints, model 2sph. The mode resonant frequency f0 is 10.8 Hz. 105 = f 0 5.49 Hz Fig. 3.12: De ected mode shape, lowest mode with one edge constrained. The mode resonant frequency is probably underestimated as the plate on the bottom of the cube would be stretched in-plane by the mode pattern shown. The added sti ness would probably raise this fre- quency signi cantly. The real lowest frequency is probably some- where between this one and that of the mode shown in Fig. 3.11. 106 0 5 10 15 20 25 30 0102030405060708090 Mode Numbe r f (Hz ) Unconstrained Edges: ff2sph_ u Heavily Constrained: ff2sp h Random Edge Nodes: ff2sph_ r One Edge: ff2sph_ e Fig. 3.13: Frequency spectrum of the lowest thirty modes for four di erent edge constraint choices. Fig. 3.8 depicts the four constraint condi- tions presented here. These mode spectra show the general e ect of the edge constraints on the frequencies. Generally speaking, the lowest ve modes or so involve either the cube shearing, the cube and sphere tilting and various combinations of motor mount mo- tion. These modes, in most cases, are in the 10-20 Hz range. The exceptions to that for the one edge constrained and no edges con- strained case are unlikely to be particularly realistic, and are likely also to be damped by collisions with the oor and sliding. Modes 6-10 in the 20-30 Hz frequency band tend to be drumhead-type membrane modes of the walls, with various walls oscillating in and out of phase with the others. Higher modes than this tend to incor- porate the next-highest modes corresponding to the aforementioned motions. For example, two walls may de ect in a half-wavelength bulge while one wall has a full wavelength across it and de ects into a S-shaped curve. The wall modes in the 20 Hz and above range seem to strongly decouple from the motion of the model sphere. 107 of all the motions. So the wall-bulge modes will probably all fall into the 20- 30 Hz range, roughly, and motor mount modes in the 10-20 Hz range, no matter the real boundary conditions. Cube shearing/tilting modes need to be assessed carefully because it is unlikely that such modes can really be observed as linear vibrational modes. Excitation of such modes will cause boundary condition changes as di erent cube bottom nodes come in contact with the oor. Some of these motions may be of some concern with the outer sphere rotating rapidly. We have rotated the outer sphere up to o=2 2 Hz and the inner up to i=2 9 Hz with few vibrational problems. We intend, however, to nish vibrational measurements of the frame to assess the possibility of low-lying troublesome modes. We have done such measurements on portions of the frame as it was under construction and compared them to a partial frame model for model validation. In this process, we noticed that the actual vibrational spectrum tends toward broad peaks around frequencies corresponding to modes with every permutation and combination of bottom edge boundary conditions. Constraints of time have prevented empirical measurement of the complete frame. 3.4.4 Additional Contributions To round out the mechanical engineering section of this dissertation, brief mention will be made of some further signi cant tasks undertaken by the author in the design process of the hardware. The author rounded out the detailed design and produced mechanical drawings of the frame. Furthermore the author engaged in a great deal of heavy construction, assembling much of the frame, cutting out 108 beams with a bandsaw and plasma torch, and welding and bolting together the frame sections. The author had signi cant help from the rest of the team for the lower portions of the frame sections. However, by the time the roof and motor mount sections were constructed, they were completed essentially singlehandedly by the author and his trusty torch and welder. Working directly to turn drawings into solid steel structure was a satisfying experience and also an important part of an education in straightforward design of structures that can be easily built. Space will not be allocated in this dissertation for any mention of the thermal control system, as the great majority of the hydrodynamic experiments in this dis- sertation were run at su ciently low power input that passive cooling to the room and the large heat capacity kept the temperature of the uid nearly constant near room temperature throughout the experimental period. As it was not needed, no attempt was made to nalize the thermal system and get it on line for hydrodynamic experiments. However, much of this system was designed by the author to provide 120 kW of heating for melting the sodium. This system is also capable of cooling 500 kW, which is more than su cient for steady state operation at maximum power. Much of the thermal system had been installed by the team prior to experiments in water. Similarly, a ventilation system for the sodium experiments occupied a fair amount of time. Perhaps a year of the author?s time was spent on mechanical aspects of the drive system design. The drive requirements for the three meter experiment are rather complicated, as we desired a signi cant speed reduction on the outer sphere drive, but the unusual speed at which the large apparatus runs and the additional 109 constraints of the design made this di cult. A pinion and ring gear system was designed but the high tangential speed of the pulley rim resulted in unacceptable predicted wear characteristics. This was only revealed in the nal stages of design, after weeks working with professional gearing engineers. Ultimately, the nal drive design is simple, motors are mounted on the frame and the outer sphere is driven by a large timing belt reduction drive. A custom-manufactured pulley, depicted in Fig. 3.14 bolts to the experiment lid rim. The pulley has alternate large and small holes, so that the experiment lid may be bolted down with half of the bolts holding the pulley as well. The large holes in the ring pulley give ample clearance to the nuts on the lid bolts. In this way, the lid may be bolted solidly and the experiment can be rotated by hand to adjust the pulley?s concentricity with the rotation axis. After the pulley is aligned, it is bolted down tightly. The large ring pulley in Fig. 3.14 was designed by the author and fabricated by Motion Systems of Warren, MI. The pulley is made of 6061-T6 aluminum, hard-coat anodized to help protect it from any caustic residue and to give it long-lasting wear characteristics needed for a high speed belt drive. The other drive system pulleys, the motor pulley and idler, were stocked parts, though the outer motor pulley was modi ed by D.H. Martin to suit our motor shaft. The author also did some stress calculations regarding the strength of the in- ner sphere. The inner sphere design was worked out by S.A. Triana with estimated strength from analytical expressions. However, the nished fabrication of the inner sphere involved welding together two hemispheres supplied by a spun metal man- ufacturer. An attempt was made to weld the equator and polar regions with full 110 All dimensions in inches 14M Pitch/Gates Power Grip GT2 Teeth 400 Teeth Face Width:3 inches Material: 6061-T6 aluminum; hardcoat anodized Ring I.D. 62.0 48 thru holes equally spaced on a ? 65.00 bolt circle. Alternating holes ? 1.75 and ? 1.125 UMD Large Ring Sprocket Zimmerman 16 April 2007 Built by Motion Systems: http://www.motion-pulleys.com/ Fig. 3.14: Large ring pulley for outer sphere mechanical drive system. 111 penetration welds. However, there was no way to inspect the inside of the inner sphere to check if this was the case. The inner sphere maximum speed limit set by centrifugal stresses is barely a matter of personnel safety, as the inner sphere could explode with great force inside the outer spherical vessel and literally not make a dent. However, it is important to know the operational limitations of the device. So a nite element model of the inner sphere with centrifugal stresses was undertaken in Algor. An axisymmetric linear stress analysis with centrifugal loading was performed with several models of fully welded and \cracked" inner sphere, modeling the e ects of incomplete weld fusion. The incomplete weld fusion regions are, as expected, stress concentrators. The output of the stress analysis model is shown in Fig. 3.15. The model above was run at 1800 RPM (30 Hz) and at that speed, the stresses in the inside corner of any stress concentrating incomplete weld fusion regions near the poles are approaching the lower limit yield stress for AISI 304L. An upper speed limit of approximately 20 Hz gives a safety factor of two, and may be considered an upper safe speed. Experiments suggest, however, that this speed will be di cult to achieve due to the torque and power demand of the turbulent ow. It is likely that throughout much of parameter space the inner sphere maximum speed, at least without further speed reduction gearing, will be limited to little more than 12 Hz by the available torque of the motors. This is far from the maximum allowable upper speed. However, the inner sphere should not be run faster than i=2 =20 Hz if that should prove possible in some region of the ( o; i) parameter plane. 112 Fig. 3.15: Inner sphere stress analysis for centrifugal loading at a speed of 30 Hz. Inset shows the polar \cracked" region of concern. As expected, the region of highest stress is a small region near the tip of the crack. Based on this analysis, it is unlikely that the maximum speed of the inner sphere will be set by centrifugal stresses. The drive system will probably be torque limited before the maximum safe speed, an allowable value for which may be 20 Hz. This gives a safety factor of 2.25 based on this rather badly fused model. 113 3.5 Instrumentation and Control While it is ultimately scienti cally useful to study purely hydrodynamic states in a rapidly rotating, highly turbulent system like this one, the decision to do so was done late in the design and construction process. Ultimately, the initial goal of sodium data in this device as a dissertation project was too ambitious. Fill with water and mechanical testing had always been planned, as water has nearly the same hydrodynamic properties (density and viscosity) as liquid sodium, and as such provides a safe and easy way to do mechanical tness testing of the system. Some instrumentation was to be elded during this process, partially to do instrument debugging, but the results proved too interesting to pass up the chance to perform a more complete survey of the rapidly rotating hydrodynamics. Furthermore, the author, having spent many years designing and constructing, was excited to examine some scienti c results. However, little instrumentation design had been undertaken as the mechan- ical system went on line. An attempt was made to initially eld instrumentation that would perform measurements in water, but with an eye toward eventual use with sodium. As water data collection progressed, a few water-speci c sensors were added. Largely, though, instrument design and choices were made with the intent to minimize wasted work on the longer-ranging project. That is, we looked toward our sodium goal even as we instrumented the device for water. As discussed in Sec. 3.2.2, magnetic eld measurements are intended as the primary diagnostic in sodium. We needed direct ow measurements in water, but we largely limited our choices to 114 those useful in an opaque liquid metal, like ultrasound doppler velocimetry. An exception is made for the wall shear stress array discussed in Sec. 3.5.4. Although the three meter apparatus has a large rotating lid on which a great deal of instrumentation can be mounted, small vertical clearance, strong centrifu- gal and Coriolis forces, and a desire to re-use existing equipment required some special instrumentation design and constrained the commercial choices for certain types of instrumentation. Many common pieces of uid dynamics instrumentation are designed to be rack-mounted and occupy a much larger volume than is nec- essary for their function. Additionally, the budget for instrumentation, especially for equipment that would only be useful in water, was fairly limited. Therefore, much of the rotating instrumentation was designed and constructed in-house. Ad- ditional equipment was added to the rotating frame for the sake of remote control of various rotating systems. The time to spin up the massive volume of water is substantial, so being able to turn equipment on and o remotely and change the position of sensors without stopping the outer sphere is of great value. Some of the instrumentation outgrowth of the experimental campaign in water may be useful to other uid dynamics researchers, especially those working in tight spaces or with small budgets. Many measurements herein rely on simple equipment that uses inex- pensive, commonly available parts. Those pieces of equipment that seem especially useful will be discussed in some detail in the following sections. A general overview of the instrumentation and data collection will be provided for the sake of lasting documentation. 115 HDD w/ Diss. Data Win/DOS Data Program Time Sync Client Time Sync Server axl@:-$ 1000BaseT Ethernet, 192.168.1.x RS-232 Serial RS-485 Bus USB 802.11g Wi-Fi XBee Wireless Serial GigE Switch XP Network Bridge wave.umd.edu 192.168.1.10 RAID5 disk array: 0.75TB wave: archive/analysis, OSX foam.umd.edu 192.168.1.18 axl bigdisk (K:) foam: personal/control station: XP UVP Monitor Software erf-117.umd.edu 192.168.1.4 3motor: motor control, torque RX, Linux axl@:-$ torqsen10 axl@:-$ controltest InnerOuter Slow Variable Frequency Drives RS-232 <-> RS-485 Adapter XBee In shed On cube West wall highbay Dynamo lab Wireless AP 192.168.1.5 Web Config SSID: 3mwirelessAxis Webcam 192.168.1.11 Torque Sensor, Inner Shaft USB Wi-Fi 192.168.1.7 sphere DAQ computer: XP Matlab Acquistion + MCC Drivers Arduino Control via Hyperterminal ProStream (Camera) 192.168.1.6 labframe DAQ minibox: XP Matlab Acquistion + NI-DAQmx TwoEnc_v1.0.exe Encoder Acq. Met-Flow UVP-DUO UVP Monitor 192.168.1.17 Rotating Frame Lab Frame via bridge Arduino ?Controller (2x) US Rotator, Dye Injector XBee + Xbee Explorer USB XBee XBee Linux Data Program Remote Desktop Login 3mWireless WEP: ad95c56b07 Fig. 3.16: Block diagram of the instrumentation system with legend. 3.5.1 Data and Control Overview The data and control infrastructure for the three meter experiment is fairly complex. A number of coordinated devices perform control, communication, and data acquisition tasks. Fig. 3.16 is a block diagram of the installed data and com- munication infrastructure. The basic communications backbone for the system is a private, local TCP/IP network running mostly over wired ethernet. Communica- 116 tion between the outer sphere rotating frame and the lab frame is handled by an 802.11g wireless access point. This access point is con gured to be a largely trans- parent bridge to the rotating frame. Sustained data rates of about 20MBps are generally possible over this link, su cient for most tasks to date. A dual quad-core Mac Pro, Wave in Fig. 3.16, houses a large capacity RAID disk array and is used for many data analysis tasks. This machine is used a central repository for data access by all researchers in the lab. It also hosts a simple near-real-time status monitoring web page, http://wave.umd.edu/3m.php, which parses data out of the several data log les in the current day?s data directory and displays it on an automatically updating web page. A Debian Linux machine, 3motor, is used for control of the variable frequency drives, using a program written by S.A. Triana. It is used for speed control, can be used to perform automated speed ramps, and reads drive status data, logging it into a le (log le.dat) in the day?s data directory. This computer also acquires torque data from a wireless sensor on the inner shaft, using another program. Finally, 3motor provides a local time server for network synchronization of the acquisition computers, some of which are isolated from the internet14. The labframe DAQ computer at present acquires rotation speed data from a pair of optical encoders. The outer sphere has an absolute position encoder accurate to better than 0:02 . The inner sphere has a high resolution speed encoder. Both encoders? outputs are 14 We have found that the constant updating of the operating systems required for internet- connected computers can sometimes break our data acquisition computers in di cult to debug ways. 117 streamed to wave. The lab frame computer also acquires accelerometer data on frame vibration, and will eventually acquire magnetic eld data in the laboratory frame. In the rotating frame, the sphere DAQ computer runs a simple MATLAB continuous data acquisition routine, streaming data from sphere frame sensors to wave using an 802.11g USB adapter. The sphere frame computer runs Windows XP Professional, and is operated from the lab frame using Remote Desktop Protocol. There are two ethernet adapters on this computer. A gigabit adapter can connect to a camera in the rotating frame. The camera system is a separate network, running in the 10.0.0.x IP address block. A Win XP network bridge shares the USB WiFi connection over a 100BaseT ethernet port on the motherboard to allow the Met-Flow UVP Monitor to connect to the lab frame network. Software to control the UVP monitor runs on a lab frame machine. The UVP software is Windows-based, and the software seems to require a reasonably fast computer. Depicted in Fig. 3.16 is foam, which is the author?s computer, but the experimental physical control location need not be localized to any particular set of computers. Network connections to 3motor are best done from another Linux computer. We have notice a bug with terminal programs running on Windows that can result in a crash15. The nal piece of equipment to be mentioned in Fig. 3.16 is an Axis 2100 TCP/IP webcam, which can be viewed on a web browser on a computer connected to the internal network. This camera is mounted on the rear strut of the inner motor 15 Crashing this machine is inconvenient but not dangerous. The motor drives will continue to run at the last setting until 3motor is rebooted, or the drives can be manually shut down. 118 frame, aimed at the top of the experiment so that it can see the lid of the outer sphere and some of the inner shaft. This allows us to monitor for problems such as leaks or loose wires. 3.5.2 Rotating Instrumentation A detailed block diagram of the instrumentation in the sphere rotating frame for the water experiments is depicted in Fig. 3.17. The heart of the rotating frame ac- quisition is the sphere DAQ computer, whose internal layout is shown in Fig. 3.18. This computer is housed in a custom aluminum case, designed and built by the author16, which bolts solidly to the sphere lid. Acquisition is performed by a Mea- surement Computing PCI-DAS6402/16 acquisition card. This PCI card provides 64 channels of 16 bit single-ended acquisition. Breakout terminal blocks allow easy wiring. A number of channels are run to front panel BNC connectors on the alu- minum computer case. All instrumentation is designed to be single-ended, with one output terminal grounded to the large metal sphere lid and coaxial cables. The com- puter has a solid state disk to avoid problems with centrifugal and Coriolis forces on a rapidly spinning mechanical hard drive. It has a power supply designed for wide range 9-15 V input and runs directly from four 12 V, 35 Ah batteries that power all of the rotating instrumentation. Batteries were chosen instead of slip ring power delivery for cost and noise rea- sons. Battery power and a large ground plane in the form of the lid help to ensure minimal pickup of noise from the variable frequency drives running the experiment?s 16 And possibly the only computer on the planet with functional hoodscoops for ventilation. 119 192.168.1.7 sphere DAQ computer: XP Matlab Acquistion + MCC Drivers Arduino Control via Hyperterminal ProStream (Camera) Met-Flow UVP-DUO UVP Monitor 192.168.1.17 MOSFET Shield Axlprivate network via 3mwireless 802.11g US transducers, 500kHz, 4MHz Arduino w/ firmware US_rotator_v_1_3.pde MOSFET ?shield? motor driver board Unipolar stepper motor with gears PCI Bus Measurement Computing PCI-DAS6402-16 Acq. Card RB 41-44 (4x) Constant temperature anemometer box, 4 Ch. Omega F2020-100-B-100 Platinum RTD (1/Ch.) for wall shear stress FP 0-2 Kistler 211B5 pressure pransducer (1/Ch.) Pressure box: 3Ch. Lo-cap probe cable Rsense 500?? 0.01? Battery negative terminals Parallel port cable Omega 4-20mA thermocouple transducer +V Thermocouple (K type) FP 6 Parallel port relay box Current shunt INT 17 DC-coupled polar pressure transducer Phase reference photodiode 10:1 voltage divider +Vbatt Battery voltage monitor FP 4 INT 16 FP 5 Fig. 3.17: Block diagram of the rotating frame instrumentation. 120 motors. These drives are the only practical choice for high power mechanical drive, but they emit signi cant radio frequency energy and great care must be taken to avoid unintentional reception of this noise. Isolating the electrical circuits in the lab frame from those in the rotating frame is useful to minimize noise pickup. Instru- mentation requiring voltages other than 12 V is powered from well- ltered DC to DC converters also housed in the computer case. There is a terminal strip inside the case, and another bolted to the sphere lid to distribute power to the instruments. The computer section is isolated from the data acquisition equipment by a shielding wall, to minimize digital noise ingress into the analog system17. Two channels of acquisition, labeled INT16 and INT17 in Fig. 3.17 are con- nected internally to the computer to measure battery voltage and system current draw. Three channels on the front panel, FP0-FP2, acquire pressure signals from a trio of Kistler 211B5 AC-coupled pressure transducers, which are powered and am- pli ed by a homemade conditioning box, designed by S.A. Triana and built by the author. The pressure conditioning box provides a constant current source of 5 mA to power each transducer, and has an overall gain of 70 on each channel. A low pass lter on the output with a -3 dB cuto of about 160 Hz rolls o the response to help prevent aliasing, especially important as the high frequency machine noise can be substantial in the microphonic pressure sensors. For all of the experiments herein, a sampling rate of 512 Hz was used, and each pressure channel was attenuated ap- proximately 10dB at the Nyquist frequency of 256 Hz. The overall sensitivity of 17 The MCC DAQ card is, of course, connected to the motherboard, but these devices are designed to isolate the sensitive analog circuitry from the PCI connection. 121 each pressure channel at the ampli er output is about 0.58kPa/V. These sensors can resolve uctuations faster than about 0.05 Hz at full sensitivity, with a rollo at lower frequencies. Channel FP4 acquires a DC-coupled pressure transducer installed in a bore that reaches to near the north pole on the outer sphere. This transducer allows us to monitor the net static pressure relative to atmospheric inside the sphere. This is important for safety reasons and for preservation of the machine, as overpressure will lead to leaks through the lip seals. This pressure transducer is not primarily intended to be a scienti cally interesting measurement, though it has a reasonable frequency response and can be used as a scienti c diagnostic. The DC response, provided that the sphere is sealed, can be used as a surrogate measure for the inten- sity of the global uid circulation the experiment, which sometimes proves useful. Channel FP5 in Fig. 3.17 acquires the output of a photodiode, which is illuminated by a LED in the lab frame once per rotation. This gives an absolute position refer- ence to the outer sphere which is useful in some circumstances, as well as providing a second measurement of the sphere speed that is synchronized with the data ac- quisition. Channel FP6 acquires the output of an Omega thermocouple transducer. The thermocouple is immersed a short distance into the uid. Temperature is an important safety diagnostic in sodium, and is used to provide some temperature compensation of the calibration of the wall shear stress sensors. It is also used to correct the relative viscosity in determining the nal Reynolds number at which ex- periments are run. A note about the thermocouple; it picks up some high frequency noise which does not a ect the mean over a few seconds needed for its primary pur- 122 pose in temperature control, but if a time series of temperature is used to condition any data (like the wall shear stress), it should be numerically low pass ltered rst. Four channels, RB41-44 acquire signals from constant-temperature ush-mount wall shear stress sensors of the author?s design, to be discussed separately later. Also depicted in Fig. 3.17 are a parallel port relay box that allows remote switching of four devices. This was used in the water experiments for LED lighting, camera power, and a pressure relief valve. The Met-Flow UVP-DUO ultrasound velocimetry device in the rotating frame has, like the main computer, had its hard drive replaced by a solid state disk to ruggedize it for rapid rotation. It is powered by a commercial 120 V inverter (not shown). The inverter feeds a power strip that could be used for ordinary mains-powered instruments of other types. The UVP Monitor is typically connected to two or three ultrasound transducers mounted in instrumentation ports in various con gurations. Software allows selection of these transducers remotely using built-in switching. One 500kHz transducer is mounted intrusively into the ow on a rotatable mount driven by a remotely-controlled stepper motor and worm gear drive. The mount holds the transducer 45 degrees from the vertical, pointing down into the sphere, and the mount can be rotated about a vertical axis. The mount allows 290 degrees of rotation. This allows some degree of ow mapping, and allows tuning of the transducer direction to better pick up uid oscillations or to turn the transducer out of a mean ow that is too strong. Generally, the trandsucer is rotated so that it is facing upstream, to minimize measurements in its own wake. The ultrasound rotator is controlled by an Arduino microcontroller board with 123 analog ground bus internal shield wall BNC Connectors Connected to Breakout Cards +24V DC-DC +/-15V, +5V DC-DC Power Distribution Solid State Disk GigE NIC PCI PCI DAS6402-16 Mini-ITX Mainboard 9-12V ATX Power Breakout Card Breakout Card Rear BNC, Wall Shear Fig. 3.18: Internal layout of the sphere frame computer. Custom case, com- plete with air cooling hoodscoops, was designed and constructed by the author to accommodate the large acquisition card and power conditioning circuitry. The main board is an Intel D201GLY2 low power consumption Mini-ITX form factor board. The computer is powered by a boost converter that will run down to 9 V. A solid state disk is used, as Coriolis and centrifugal forces can damage a mechanical hard disk. The main data acquisition is done by a Measurement Computing PCI-DAS6402/16 64 channel, 16 bit PCI card. 124 switching MOSFET motor drivers, and accepts direction commands using Hypert- erminal and a USB serial connection. Not described here but included in Appendix E is a device that may be of considerable interest to researchers with smaller devices. Before the UVP monitor was installed in the rotating frame, an older prototype unit with no network connec- tion was used in the lab frame, with a resonant RF transformer coupling the 4 MHz signal into the rotating frame. This transformer is a simple pair of copper loops with adjustable capacitors, and works well on small devices. This method on the three meter device was made obsolete by the purchase of the networked UVP-DUO, but those without su cient space on a rotating table may nd the non-contact resonant coupling method useful. A few nal remarks on the ultrasound measurements should be made. While ultrasound Doppler velocimetry can be used to measure the velocity far from the ultrasound transducer, there are practical limitations on the maximum achievable depth due to the high velocities typical in the experiments herein. The 4 MHz measurements may extend 10 cm into the ow, and the 500 kHz perhaps 40 cm when measuring a signi cant component of the mean ow. When the 500kHz transducer is rotated to be nearly orthogonal to the mean ow and the experiment is operated at the lower end of the speed range, the depth may be occasionally extended to 1 m or more. In addition to the depth limitations, the large particles required to scatter 500 kHz ultrasound are subject to some unusual transport dynamics in some of the ow states described here. We have noticed that some observed turbulent ow states are poor mixers for the fairly large inertial particles we must use, and it 125 is therefore not always possible to acquire good velocity data. 3.5.3 Speed Encoders Accurate speed measurement is provided by a pair of pulse encoders attached to the inner and outer sphere motors. These encoders are capable of quadrature operation and output 2500 pulses per shaft revolution. At motor speeds which may rise as high as 33 Hz for the outer motor at maximum speed, the resulting pulse rate is high (83.3 kHz). Simpler encoding schemes in the past have used analog input to measure pulse rates, and high rates can be divided down with extra circuitry. However, the National Instruments USB-6210 on the lab frame computer provides a set of high speed digital inputs that can be used to perform frequency counting tasks. These inputs are used to acquire encoder data. A connection diagram is shown in Fig. 3.19. The outer encoder is set up as an absolute position encoder. The two quadrature channels are used, and the counter may then count up or down depending on the direction of rotation. The \Z" input of the quadrature counter provides a once-per-revolution reset of the outer sphere position. The encoder has such an output, but since it is mounted on the motor side of the reduction drive, it could not be used. Instead, a photointerrupter is provided that pulses once per revolution, interrupted by a small metal ag approximately 0:07 wide. A lack of inputs restricts acquisition of the inner encoder to a frequency count of a single pulse channel. This has proved adequate for our needs, though it is less ac- curate if the motor may vibrate torsionally more than 1/2500 of a revolution, as this 126 +5V +5V +5V +5V 192.168.1.6 labframe DAQ minibox: XP Matlab Acquistion + NI-DAQmx TwoEnc_v1.1.exe Encoder Acq. to private data network National Instruments USB-6210 DAQ Outer encoder 20833.33 PPR Position of OS Inner encoder 2500PPR Speed of IS A_outer = PFI0 B_outer = PFI1 Inverter (1/6 Hex) Outer sphere flag interrupter Z_outer = PFI2 A_inner = PFI3 GND = D GND A, B, Z lines have 2200? pullup resistors to +5V. Analog inputs will be used for accelerometer acquisition. Fig. 3.19: Encoder acquisition block diagram in the lab frame. Both pulse encoders rotate at the motor speeds. The outer motor encoder runs at 400=48 2500 pulses per revolution because of the speed reduction drive. The inner encoder is directly driven at the motor speed and outputs 2500 pulses per revolution of the inner. These high pulse rates are counted by timer/counter circuitry in the NI USB-6210 card. The outer sphere encoder is an absolute position encoder, with a single photo-interrupter to reset the outer counter once per revolution of the outer sphere. A DOS executable writes outer sphere position and inner sphere pulse count to a le on wave. 127 causes extra counts. A DOS program written by the author, TwoEnc v1.1.exe18 A samples the counter synchronously with an analog input running in MATLAB on the lab frame computer and writes those samples to a le on wave. Source code for this program is available in Appendix E. The outer sphere position is written in degrees, and the inner sphere count is recorded as an integer from 0-2500, with the output resetting once per revolution. An analog input dummy routine (no in- put or output) running at 500 Hz is currently used to provide a sample clock to the counter acquisition program. When the sodium experiments go on line, mag- netic eld sensors in the lab frame will be acquired along with accelerometers for safety interlocking. For both scienti c and vibration analysis purposes, a sample- synchronous, high resolution measurement of the outer sphere position is a useful auxiliary measurement. It will also allow for post-processing phase synchronization of magnetic data in the rotating frame with that taken in the lab frame, by aligning the phase reference photodiode signal with the outer sphere position measurement. We may, however, devise a wireless trigger for synchronizing these measurements using hardware triggers. Then measurements can be made phase synchronous with the outer sphere position acquisition. 3.5.4 Wall Shear Stress Array Wall shear stress measurements provide a more reliable and more broadband measurement of mean ow uctuations than the ultrasound velocimetry. In some 18 Data taken prior to this dissertation used TwoEnc v1.0.exe, which mistakenly reports the outer sphere speed 0.5% too high. 128 turbulent ow regimes, the mixing of the particles needed to scatter 500 kHz ultra- sound is poor. Furthermore, the sampling rate for ultrasound velocimetry is limited by the round trip time to the furthest measurement location. Wall shear stress can be sampled faster and does not su er from any issues of inertial particle transport, so it is an attractive ow diagnostic. For some of the experiments presented in this dissertation, a Dantec model 55R46 wall shear stress sensor powered by a TSI model 1750 constant temperature anemometer (CTA) circuit was used to make single-point measurements. This sensor failed partway through the experimental campaign. It was desirable to replace it with multiple sensors to allow cross-correlations, but com- mercially available units rated for use in water cost along with the CTA circuitry needed cost approximately $5,000 per channel in a form factor that can be rotated with the outer sphere. The commercial sensors are small and can therefore resolve short length scales in turbulence. They are also faster than the current probes. However, neither of these characteristics was mandatory for these experiments. The failed wall shear stress sensor and constant temperature anemometer were replaced instead by a system designed and built by the author. A ush-mount wall shear stress sensor is typically a thin platinum or nickel lm deposited on a ceramic substrate, covered with a thin coating of thermally conductive electrical insulator. This lm forms a very stable temperature dependent resistor, usually with a value of 5 15 . The high engineering cost and low sales volume of these devices undoubtedly contributes to their high price. The constant temperature anemometer circuit is, in principle, a simple device that applies a variable amount of electrical power to heat the sensor. It uses a fast feedback loop that keeps the electrical resistance, and 129 therefore the temperature of the sensor, constant. The power required by the sensor to maintain constant resistance is a function of the uid ow past it (and therefore the heat ux away from it). This variable power constitutes the output signal of a constant temperature anemometer. Inspection of commercially available constant temperature anemometry circuits shows them to be fairly complex devices, possibly to aid circuit reproducibility, to increase circuit bandwidth, or improve other circuit properties. However, a simpler circuit su ces for lower-accuracy measurement of relatively slow turbulent uctuations. The system described here uses very low cost platinum resistive temperature detectors as ush mount wall shear stress sensors. Four sensors were inlaid into a machined plastic substrate in a square array 10.2 cm on a diagonal, depicted in Fig. 3.20 (b). At the location of the ports, the two probes on a line of latitude are separated by approximately 9:7 in azimuth. Those on a line of longitude are separated by about 3:9 in latitude. The RTD device used is an Omega F-202019, which measures approximately 2 mm square. Like a hot lm ush mount sensor, it consists of a platinum lm covered with an insulating coating. The resistance of these sensors is well controlled and the temperature coe cient thereof is reasonably large and extremely stable, consistent with their intended function as temperature- variable resistors. They have some disadvantages. The lowest resistance available for o -the-shelf RTDs is 100 . This requires the use of a higher voltage CTA circuit, possibly ruling out the use of commercial circuits. Furthermore, as shown 19 Available from www.omega.com at the time of this writing for $1 per sensor in a quantity of 100, Omega part number F-2020-100-B-100. 130 Omega F-2020 RTD, 4x Unknown (glass?) Pt Film 0.5mm Ceramic, Sensing Surface (a) 10.2cm (b) Fig. 3.20: A schematic of an Omega F-2020 sensor layered structure, (a). The at white thermally conductive ceramic surface is used as the sens- ing surface, while the unknown blue side faces away from the uid. Four such sensors are arranged in a square array (b), with a 10.2 cm diagonal. The sensors and their connecting wires are sealed into shallow trenches in the array substrate plastic surface using silicone sealant. This holds them in place and provides electrical insulation from the water, which is critical. 131 in Fig. 3.20 (a), the platinum lm is deposited on a 0.5 mm thick ceramic substrate that is used in this application as the sensing surface, as it is at and likely made of highly thermally conducting ceramic. The blue coating, likely some sort of glass, is thinner but it is not at and cannot be ush mounted with a surface. Measurement through a thick ceramic layer will cause undesirable thermal lag and probably lead to phase shifts between the sensor variations and uid uctuations when they are fast. This was not assessed in detail, as the sensors performed adequately for the task at hand. Resistive temperature sensors are available in many physical con gurations, some of which may be more suited for use in CTA service. The reader is encouraged to investigate other options, as some could be signi cantly superior while retaining very low cost and drop-in compatibility with the circuit described here. As mentioned before, the high resistance of the lowest resistance resistive tem- perature detectors on the market requires the use of high excitation voltage to get signi cant self-heating. Not all commercial CTA bridges will work properly with a sensor of this resistance. Furthermore, the four channels required for the four sensors of Fig. 3.20 (b) are by themselves cost prohibitive for casual needs. A nearly minimal CTA circuit was designed, the schematic for which is shown in Fig. 3.21. Resistors R1, R2, Roh20, and the F-2020 sensor form a Wheatstone bridge. The resistors in the left half of the bridge, R2 and Roh, nominally have a value ten times their counterparts in the left half of the bridge. In the nal circuit, R1 is nomi- nally 27 and R2 is 270 . The overheat resistor is nominally adjusted to a value ten times the resistance of the RTD sensor at the desired overheat, which can be 20 This is a xed resistor in series with a variable one for ner adjustment. 132 ? + In Out Gnd RTD, Omega F-2020 +24V Regulator FZT491A 1/2 AD8662 3x 0.1 ?F 10?F10?F PWR ? + 100k?100k? 330? +15V 1/2 AD8662 Zero-Flow Offset 3x 0.1 ?F BNC Out PWR 8 4 +15V +15V One Per Channel 2R 1R Roh U1 U2 Q1 Fig. 3.21: Schematic of simple constant temperature anemometer circuit. The sensor is an inexpensive resistive temperature detector. R1, R2, Roh and the sensor form a Wheatstone bridge that is balanced when the sensor is at a particular temperature, adjustable by the overheat adjustment, Roh, which is a xed resistor in series with a poten- tiometer for better resolution. As uid ows past the sensor, the AD8662 op amp and transistor vary the power to the bridge to keep the sensor temperature constant. Because this voltage may typi- cally exceed the input range of the acquisition card, it is inverted and a zero- ow o set is applied to put the zero- ow voltage at the top of the allowable acquisition input range. As the ow increases, this voltage decreases. In this way, the voltage always remains within the -10 V to 10 V range of the data acquisition card. The output voltage for the experiments here is 11:75 Vbridge. The zero- ow bridge voltage is around 3 volts. 133 calculated using the temperature coe cient of resistance. This is standardized at 0.00385 = C for most RTD devices. When the RTD sensor is cooler than the overheat setpoint, the bridge is unbalanced, and the output of U1 op amp rises. This increases the current through bipolar transistor Q1 until it is su cient to keep the sensor at the setpoint. If the sensor is hotter, less power is delivered and it cools toward the setpoint. The speed of this feedback loop is extremely high, essentially keeping the sensor at constant temperature at all times. In such a circuit it is possible for the loop gain to be too high, and then the circuit may oscillate. It is typical in a CTA circuit to intentionally roll o high frequencies by including a capacitive feedback from the output to the negative input of op amp U1. This was tried initially but the circuit remains stable without any rollo , so capacitance was omitted here. It is also typical to provide an small inductor in series with the sensor connections to tune out cable capacitance from a long coaxial cable. In this application, the CTA circuits are a short distance from the sensors themselves, connected with twisted pair cable. No inductive compensation was needed, but may be desired if long coaxial cables are necessary. The output of many CTA circuits is simply taken from the top of the bridge, at the junction of R1 and R2. To help ensure that the output was always within the range of the 10V maximum of the acquisition card, this voltage is fed to a unity gain inverting ampli er with o set based around U2, the other half of the AD8662 dual op amp. A well-bypassed potentiometer allows adjustment of the zero ow voltage to put it near the top of the data acquisition range, maximizing the range 134 in which it can vary. Increasing wall shear stress leads to a decreasing voltage. An updated version of this circuit will use a non-inverting stage instead, but the design is presented here as implemented. Because of the extremely high gain of the open loop op amp, board layout needs to be neat with short leads. Furthermore, it is critical that the sensor leads be completely insulated from contact with water. The high gain feedback circuit misbehaves spectacularly when water leaks into contact with the leads. A printed circuit board template for this circuit and parts placement diagram is reproduced in Appendix E for the reader interested in reproducing this circuit directly, though a re-worked layout could result in more channels on the same board space. The wall shear stress system was not exhaustively tested in a known ow to independently assess it for frequency and phase response. Little can be said about the suitability for turbulent ow applications in general, especially those focused on ow spectra. Calibration was done in situ against the torque on the inner sphere. Fluctuations out to the Nyquist frequency of 256 Hz were easily observed with su ciently high mean ows, and uctuations out to several tens of Hz at least were resolvable in a zero- ow situation. Response to the mean ow and uctuations was more than adequate for our purposes. It is likely that smaller surface-mount type RTD sensors used with this circuit would be considerably faster. Characterization and improvement of this circuit for more sophisticated measurements is a matter for future work. As with all simple hot-element sensor circuits, the temperature of the uid bears directly on the power delivered to the sensor if a xed \hot" resistance is used. 135 An updated circuit could include simple temperature compensation in the form of a temperature-sensitive overheat setpoint resistor, Roh. Excellent absolute tempera- ture control can eliminate the di culties caused by the xed overheat resistor in the circuit of Fig. 3.21. However, it is expected that the sorts of experiments in which extremely low cost, modest performance wall shear stress arrays may be most useful will be in small, simple apparatus. In these devices, precise temperature control is not usually practical, with the apparatus being run at whatever room temperature is on a given day. Simple temperature compensation could extend the utility of this sensor to make absolute measurements, and residual temperature dependence could be calibrated out. 3.5.5 Torque Sensor The torque on the boundaries of a turbulent shear ow is an integrative mea- surement of the turbulent azimuthal wall shear stress. It quanti es the global energy input and the ux of the axial component of angular momentum from boundary to boundary. Therefore, torque is a quantity of fundamental interest in turbulent shear ow experiments like this one. Some examples of prior work with torque as a pri- mary diagnostic are discussed in Sec. 5.2. The variable frequency drives on the inner and outer motors calculate an estimate of the torque based on the electrical power supplied to the motors and the speed. At low motor speeds, though, this power dissipation is dominated by motor winding losses and the accuracy of the reported torque su ers. Since it has no speed reduction, the inner sphere is more seriously a ected by this, and we decided to add a more sensitive torque measurement system 136 to the inner sphere. Fig. 3.22 is an annotated photograph of the inner sphere torque sensor arrangement. Torque from the motor is transmitted by a modi ed Lovejoy CS-285 exible shaft coupling, Fig. 3.22 (a) to a Futek TFF-600 reaction ange torque sensor. The sensor, Fig. 3.22 (b), is a strain-gauge based device with a maximum measurement rating21 of 1130 N m, su cient for the maximum torque output of the motor. The sensor bolts between the lower half of the exible coupling and an anodized 6061 aluminum adapter, Fig. 3.22 (c), designed and machined by the author, which trans- mits the torque to the inner sphere shaft, Fig. 3.22 (d). Both the bottom surface of the modi ed coupler and the top surface of the adapter housing have a circular radial location pin and keyslots with square keys that mate with the sensor anges. The adapter housing has a pair of diametrically opposing at surfaces with ten 3/8"-16 tapped holes for solid mounting of electronic equipment to the sides of the housing. The torque sensor signal is acquired by a high resolution wireless acquisi- tion system, Fig. 3.22 (e), which is powered by a 10 Ah lithium-ion battery pack and power supply, Fig. 3.22 (f). The battery box contains a CUI PK-25-D5-D12 DC to DC converter, which accepts the variable 6-8.3V voltage from the battery pack and supplies steady 12 V outputs at 850 mA. Battery power is su cient for approximately 12 hours of continuous acquisition. A schematic of the battery box, including low voltage cuto circutry to protect the battery from excessive discharge, is available in Appendix E. The battery box and wireless torque sensor are mounted diametrically opposite to help preserve shaft balance. The internal layout of these 21 The sensor will withstand 150% of this safely. 137 a b c d e f Fig. 3.22: Photo of the torque sensor arrangement. (a) Flexible shaft coupler. (b) Futek TFF-600 torque sensor. (c) Adapter coupling. (d) Inner sphere shaft. (e) Wireless torque sensor, Fig. 3.23. (f) Inner shaft battery box. 138 devices was carefully considered with regard to large centrifugal accelerations. The devices have been tested to approximately 100 g with the inner sphere rotating at 10 Hz. It is likely that mounting of the battery will need to be improved to reach much higher speeds, although as discussed in Sec. 3.4.4, the maximum operating speed of the inner sphere and shaft is likely to be torque limited. A block diagram22 of the custom wireless strain gauge transmitter, designed, built, and programmed by the author, is depicted in Fig. 3.23. The Futek TFF-600 is excited with a DC voltage, and the millivolt output of the bridge is ampli ed by a low noise instrumentation ampli er with a gain of approximately 100. The output of this ampli er is sampled by a high resolution 22 bit analog to digital converter running at a sampling rate of approximately 30 Hz. A large dynamic range is desirable to measure small uctuations around the large range of mean torques the sensor is subjected to. The 22 bit converter is not strictly necessary in light of the mechanical confounding errors of seal and bearing drag, mechanical noise, and hysteresis/repeatability limits on the sensor itself; a 16 bit converter would likely su ce. However, the circuit as built has very low inherent noise and an improved mechanical torque measurement system with less confounding drag and an enhanced dynamic range sensor could fruitfully take advantage of this bit depth. In fact, rather minute uctuations can be resolved simply by lowering the confounding drag. While a strain gauge type sensor may su er from repeatability and hysteresis issues that limit its absolute accuracy, it may still be useful to measure tiny relative changes 22 Full schematic is provided in Appendix E 139 Futek TFF600 Amp ADC Microcontroller SPI Bus RS-232 TTL XBee 802.15 2.4GHz XBee 802.15 RS-232 TTL Acquisition Computer Fig. 3.23: Block diagram of the wireless torque sensor on the inner sphere shaft. Torque from the motor is transmitted through a Futek model TFF-600 reaction ange strain gauge torque sensor. The output of this sensor is ampli ed, sampled at about 30 Hz using a 22 bit ana- log to digital converter. This converter is controlled by an Arduino microcontroller board which transmits the data through a 2.4 GHz wireless serial link to a computer, 3motor, in the lab frame. A pro- gram called torqsen7 on this computer writes incoming data to a le. 140 provided that medium-term drift can be quanti ed or controlled. The sensor and acquisition as built can detect faint nger pressure on the rim of the sensor. It is estimated that the system as built approximately 18 bits of useful dynamic range in terms of the ratio of minimum resolvable relative uctuations to the maximum measurable torque. Machinery improvements may allow more of this range to be used. Conversions are clocked and controlled by an Arduino microcontroller board. This is an inexpensive microcontroller system based around an AVR ATMEGA168 microcontroller. It is supplied with a pre-programmed bootloader that allows it to be programmed through a USB to serial converter included on the board itself. No external programmer is needed. A free and open source development environment with access to many intuitive function libraries speeds the code development process. The microcontroller communicates with the analog to digital converter over SPI, a synchronous serial protocol. The binary 2?s complement output of the ADC is converted to a an ASCII encoded signed integer. The output integer 1801990 corresponds to 1130 N m based on calibration of the ampli er and ADC against a known millivolt source and the factory calibration of the Futek sensor, 1:68 mV/V full scale23. Data is transmitted to the lab frame using Digi XBee wireless serial devices. The Arduino communicates using a TTL RS-232 type serial connection with one XBee housed in a radome on the torque transducer box. The XBee in the lab frame 23 Bridge output per volt of bridge excitation. 141 is connected to 3motor using a USB to TTL serial converter24, and the torqsen7 program receives data and writes it to an ASCII le. The incoming data is times- tamped by 3motor as it is received. This timestamp is roughly synchronized to the rotating acquisition with network time synchronization. The XBee serial link is occasionally interfered with, probably by other 2.4 GHz wireless networking. This results in occasional dropped samples, and occasionally a few seconds of interrup- tion. However, since the torque dynamics to be measured are slow, this is not a serious problem. This issue may be addressed if magnetic sensors and good syn- chronization are desired in the future. 3.6 Sixty Centimeter Apparatus A smaller predecessor of the three meter device has yielded interesting scienti c results [12{15] in rapidly rotating spherical Couette ow of sodium, as discussed in Sections 1.4.4 and 2.3.5, with some results depicted in Fig. 1.6. During the design and construction phase of the three meter system, the author assisted from time to time with instrumentation development and scienti c investigations with this device. Some data from the 60 cm apparatus will be presented herein to illuminate some of the turbulent ow states discovered in the three meter device. As discussed in Sec. 2.3.5, the 60 cm device partially overlaps the parameter regime of the three meter device. However, the primary diagnostic in the 60 cm sodium device is the induced magnetic eld outside the outer sphere. With an applied eld su ciently 24 An XBee Explorer board from www.sparkfun.com 142 weak to be dynamically insigni cant, the induced magnetic eld can be used to infer global information about the uid ow, consistent with Eq. 2.1. This global ow information is a good complement to the local direct ow measurements and global torque measurements we have available in the three meter device. In those states where data is available from both experiments, we will include some 60 cm data, and so we describe the salient aspects the 60 cm brie y. This device is more fully described in the dissertation [117] of D.H. Kelley. The experiment is depicted schematically in Fig. 3.24. In most respects, it is simply a 1/5 scale version of the three meter device. The two cylindrical shafts labeled in Fig. 3.24 (a) are the hydrodynamically relevant25 exception. These cylin- drical shafts are approximately half the diameter of the inner sphere and rotate with the outer sphere. The inner sphere?s drive shaft is mounted on a pair of bearings near the tips of the large shafts. This is in contrast with the three meter apparatus, where the inner sphere shaft extends all the way to the poles of the outer sphere and is only 16% of the inner sphere diameter. This di erence in the geometry has a slight noticeable e ect on the observed ow states, in particular, the bounding values of Ro for particular states seem to be shifted. Some of the induced magnetic eld data from the 60 cm apparatus presented in this dissertation is recently collected data intended to more fully explore states discovered in the three meter apparatus. Other data has been pulled from relevant portions of the 60 cm data archives. The older magnetic eld data was collected 25 There is an additional hydromagnetic di erence: the inner sphere of the 60 cm apparatus is solid copper, not hollow stainless steel. 143 M21 M1 (a) (b) Shafts B0 Bind Fig. 3.24: Schematics of the 60 cm sodium spherical Couette apparatus, adapted from the dissertation of D.H. Kelley [117]. This experi- ment is proportional to the three meter apparatus, and ve times smaller, with the exception of the large shafts labeled in (a). These shafts rotate with the outer sphere, and slightly modify the geome- try of the experiment with respect to the three meter device. Small red circles denote the location of Hall e ect sensors for two di erent arrays. At (a), an array constructed by the author, has 21 probes spaced equally on a meridian. At (b), a more sophisticated array built later by D.H. Kelley allows decomposition of the induced eld into spherical harmonics up to l = 4;m = 4. In both cases, a weak, approximately uniform and axial eld, B0, is applied as in (a). The Hall sensors are aligned such that they are perpendicular to the applied eld and respond to the cylindrical radial component of induced magnetic eld, Bind. 144 with an array of 21 Honeywell SS94A1F Hall e ect sensors that was constructed by the author. This array, depicted in Fig. 3.24 (a), is xed in the lab frame and lies on a single meridian spaced a short distance from the outer sphere surface. Probe M11 is at the equator. The newer data is collected with a three-dimensional array constructed by D.H. Kelley [117] of substantially similar design to that used by D.R. Sisan in experiments in highly magnetized spherical Couette ow [112, 118]. This array, depicted in Fig. 3.24 (b), can be used to decompose the cylindrical radial component of the induced eld, Bind, into spherical harmonic components of degree l and order m, up to l = 4;m = 4. It is also xed in the lab frame. It will be made clear which array was used when magnetic eld data is presented in the coming chapters. 145 4. RESULTS I: TURBULENT FLOW STATES 4.1 Introduction Now that we have laid the groundwork to help us understand rotating, tur- bulent shear ow, motivated our experiment from a geophysical standpoint, and described the experiment and its measurement capabilities, we are ready to dis- cuss experimental results. First we will situate ourselves again in parameter space with a brief reminder of the de nition of the relevant dimensionless parameters in terms of the dimensional experimental parameters. The experimentally adjustable parameters are the outer and inner boundary rotation rates, o and i. The xed experimental parameters are the viscosity of water, taken to be = 1 10 6 m2=s, the radius ratio = ri=ro = 0:35, and a length scale close1 to the gap width ? = 1 m. One adjustable dimensionless control parameter is the Ekman number, de ned using the gap width ? = ri ro and the outer sphere rotation rate o as E = o?2 : (4.1) The Ekman number, in the context of the experimental results, characterizes the importance of the outer sphere rotation. Generally speaking, we consider this a 1 The gap is 0.95 m. However, 1 m is convenient and we probably have more than 5% error in our knowledge of the viscosity. proxy for the overall rotation, but speci cally it is the relative importance of viscous forces to Coriolis forces for uid rotating at o. As we will see throughout the following chapters, the outer boundary rotation is important even at exceptionally high di erential rotation. The other adjustable dimensionless parameter is the Rossby number, a dimen- sionless di erential angular speed, Ro = i o o (4.2) For convenience of comparison with the outer sphere rotating and without, we will always de ne the Reynolds number this way: Re = Ro E = ( i o)?2 (4.3) Although Re is not an additional independent dimensionless parameter to Ro and E, and is not needed when both spheres rotate, we will nd that it can be useful to use Ro and Re in interpreting the dependence of the data on the experimental parameters. It is not the usual convention to use the de nition of Eq. 4.3 when only the inner sphere revolves, but rather to use a velocity scale equal to the tangential velocity of the inner sphere; however, to de ne Re in terms of Ro and E without any extra factors of two, we choose the de nition in Eq. 4.3. Approximate boundaries of the range of Re, Ro, and E explored in the experiments to be described are given in Table 4.1. In this chapter, we will give a broad overview of the observed turbulent ow states, dividing the results into three categories based on the Rossby number and observed behavior. First, we will present results with the outer sphere stationary, 147 Tab. 4.1: The approximate bounds on the dimensionless parameters considered in the experimental results presented here. The values of Ro and E are exclusive of the outer sphere stationary cases, which have Ro =1 and E =1. Constraints on time prevented investigation of high negative values of Ro. Parameter Minimum Maximum E 9 10 8 5 10 6 Ro -5 100 Re 2:5 105 6 107 Ro = 1 and E = 1, with Re ranging approximately over the values in Table 4.1. There is a single turbulent ow state when the outer sphere is stationary, with a scaling for the torque on the inner sphere that is similar to that of the turbulent drag observed and predicted for other shear ows, like pipe ow and ow over a at plate. However, also observe a system-wide coherence, consisting of a slowly rotating wave motion consistent with an m = 1 azimuthal wavenumber. We also observe this wave in the 60 cm apparatus, and magnetic eld data is included to further characterize the state. We then turn our attention to Ro < 0, where Kelley et al. [12, 14, 15] observe inertial modes in the spherical Couette geometry. Indeed, we observe these modes as well, holding E constant and varying Ro in the range 5 < Ro < 0, and we compare to the prior published results. We note some new behaviors, including that one observed mode is apparently part of a resonant triad. We also look at the torque on the inner sphere as a function of Ro and E. Several ow transitions exist in the Ro < 0 regime corresponding to di erent inertial modes, but these can just barely be detected in the torque. 148 Finally, we will conclude this chapter with results for 0 < Ro < 100. A number of turbulent ow transitions are observed here as well, but unlike those for Ro < 0, the torque required to drive the inner sphere has a more complicated dependence on Ro. There are several distinct transitions as Ro is varied with xed E, with bistable ranges of Ro where spontaneous transitions involving the two adjacent states are noted. Each di erent ow state is characterized by its frequency content, much like the inertial mode states for Ro < 0. However, the Ro > 0 states may also be characterized by di erent dynamical uctuations of the torque. When Ro > 0, it is often the case that the uctuations in the uid drag on the inner sphere lie at low enough frequencies and are su ciently correlated over the surface of the sphere that we may detect them. 4.2 Ro =1, Outer Sphere Stationary 4.2.1 Flow Properties There is not much information available on the uid ow in turbulent spher- ical Couette systems, and even less is known when both spheres revolve. When only the inner sphere rotates, as is considered in this section, there are some good measurements of time series and ow spectra for weakly turbulent states at di erent radius ratios. The turbulent regime reported by Wulf et al. [101] for both = 0:752 ( = 0:33) and = 0:667 ( = 0:5) is characterized by chaos in the amplitude of successive velocity extrema. A common feature of both radius ratios investigated, however, is the persistence of \aperiodic coherent structures" as well as a frequency 149 spectrum that exhibits signi cant peaks. The weakly turbulent regime for = 0:667 exhibits a broad spectral peak at about 0:25 i and a comb of incommensurate fre- quencies that lie higher. This lowest frequency is about a quarter of the frequency observed at the laminar onset of traveling waves. For = 0:75, the lowest peak in the turbulent spectrum is about half that observed in the rst laminar rotating wave state. To our knowledge, no one has ever published a thorough characterization of the turbulent dynamics of Ro =1 spherical Couette ow for = 0:35 or nearby radius ratios. The only previously reported very low frequency drifting state reported in strongly turbulent, purely hydrodynamic spherical Couette ow seems to be that in the dissertation of D.R. Sisan [118]. Sisan noted a weak, low frequency wave state below onset of hydromagnetic instabilities. In this case the \precession below onset" reported by Sisan was not always observed, and it was ascribed to experimental imperfections, but the data presented here suggests that this may not be the case, at least at our Reynolds numbers. Based on the work of Wulf et al.[101], we may expect a slowly drifting wave in the weakly supercritical turbulent state, and we might expect it to have a lower azimuthal wavenumber and signi cantly lower frequency than that of the rst lam- inar nonaxisymmetric state. Hollerbach et al. [100] predict that the saturated state arising from the rst nonaxisymmetric laminar transition for geometries near our radius ratio involves either a m = 2 or m = 3 drifting wave. The laminar oscillation frequency for our radius ratio, which is in between those given by Hollerbach et 150 1200 1220 1240 1260 1280 1300 ?2.5 ?2 ?1.5 ?1 ?0.5 0 0.5 1 1.5 x 10 ?3 ?it 2? P ?U2i Fig. 4.1: Time series of two pressure sensors installed in adjacent instrumenta- tion ports (Fig. 3.2 (K)), 90 degrees apart. Ro =1, Re = 4:4 107. A large-scale pressure oscillation consistent with an m = 1 azimuthal pattern is observed. The red curve leads the black, consistent with a pattern propagating in the prograde direction with respect to the inner sphere rotation. Other turbulent uctuations are present, but the strong wave dominates. 151 al. [100] may be estimated2to be 0:128 i if the rst laminar wave is m = 2 and 0:192 i if it is m = 3. If the same trend seen in Wulf et al. [101] holds at this radius ratio, we might expect a frequency considerably lower than these laminar frequen- cies, at least just above the onset of chaos. Our Reynolds number is many orders of magnitude above the likely critical value3, but we do observe a slowly drifting turbulent wave motion. Figure 4.1 shows a time series of the wall pressure4 at 23:5 colatitude for two sensors spaced 90 in azimuth. The Reynolds number (Eq. 4.3) is 4:4 107. The observed correlation in the pressure sensors is consistent with a prograde drifting pattern with m = 1, and it has frequency near 0:04 i. This is possibly in qualitative agreement with the prior studies discussed above, though it is perhaps initially surprising that wave motion survives to such high Re. Figure 4.2 depicts a power spectrum of wall shear stress from the three meter experiment at Re = 4:4 107 and power spectra of cylindrical radial induced mag- netic eld from the 60 cm experiment5 at Re = 3:5 106. The sensor, M10, closer to the equator of the 60 cm experiment responds to the weaker, higher frequency peak seen in the three meter data at 0:066 i. A midlatitude magnetic sensor, M8, reproduces the three meter peak at != i = 0:040. The spectra have been normal- ized so that the 0:04 i peaks are equal in strength. Despite an order of magnitude di erence in Re, the agreement for the large-scale wave motions is excellent. It should be noted that the \wave" frequency peak is rather broad. In this case, the 2 From quadratic interpolation of the four nonlinear drift speed results of Hollerbach et al. [100]. 3 Rec where turbulence sets in is not known for this , but it is probably at most a few thousand. 4 We have checked these oscillations to rule out free-surface dynamics. 5 Using the array of Fig. 3.24 (a). 152 PSD raehS llaW m3 8M mc06 01M mc06 ?/? i 01 4? 01 3? 01 2? 01 1? 01 0 01 1 01 2 01 3 01 4 01 4? 01 3? 01 2? 01 1? 01 0 01 1 01 2 01 3 f -3/5 f -11/3 f -5/3 Fig. 4.2: Power spectrum of wall shear stress (black) from the three meter experiment and of the induced magnetic eld from the 60 cm exper- iment for two di erent sensors from the array of Fig. 3.24 (a), one closer to the equator (M10) and one at midlatitude (M8). Frequency is made dimensionless by the inner sphere rotation rate. For the 60 cm spectra (red, blue), Re = 3:5 106. For the three meter spec- trum (black), Re = 4:4 107. The strongest peak has != i = 0:04. The magnetic eld spectrum shows an overall decrease in power with frequency not seen in the shear data. This is discussed in the text, and appears to be a characteristic of the induced magnetic eld, not of the turbulence in the 60 cm system. In the low frequency region, where the wall shear stress spectrum is nearly at, the slope of the magnetic data depends on frequency roughly like f 3=5. 153 half-power fractional bandwidth is around 15%. The wave motion, though it has a well de ned center frequency, is deeply modulated with low frequency uctuations. This is similar to the data of Wulf et al. for the weakly turbulent state. They nd no signi cant correlation between the amplitudes of successive maxima of a ltered time series, but temporal correlations are clear in the frequency spectra. The ob- served waves in the three meter and 60 cm experiments also seem to be consistent with the frequency of the \precession below onset" reported by Sisan. Sisan does not state the frequency, but the weak- eld end of the \O1"6 spectrogram, Fig. 4.13 (a) of the dissertation [118], suggests a weak peak near the right frequency. Aside from the strong modulated waves, the overall shape of the uctuation spectrum for wall shear and magnetic eld data is di erent, likely due to magnetic ltering of the uctuations. The ltering e ect likely has two causes. The induced eld in a multipole expansion representation depends on polar wavenumber l like r (l+1). Furthermore, the linear magnetic induction, Eq. 2.1 will be in uenced by stronger di usion and weakened advection of elds at shorter spatial scales. These small scales map to higher frequencies as they are swept past the array by the mean ow. It is also the case that very high frequencies will be ltered by the nite conductivity of the titanium alloy outer sphere. However, the shell of 60 cm is about one skin depth thick at 28 i. Since the magnetic eld uctuations are already deep below the noise oor by this frequency, the ltering due to electromagnetic skin e ect plays no role. A few reference slopes are provided in Fig. 4.2. The f 11=3 scaling is a 6 Odd equatorial symmetry, m = 1. 154 600 610 620 630 640 650 660 670 680 690 M2 M4 M6 M8 M10 M12 M14 M16 M18 M20 ? i t 2? Probe Fig. 4.3: Spacetime diagram of the radial induced eld on the array of Fig. 3.24 (a), for Ro = 1, Re = 3:5 106. The vertical axis maps to latitude, with sensor M11 at the equator. Time is mea- sured in units of inner sphere revolutions. This data is temporally bandpass ltered with passband between the -3dB points satisfying 0:02 < != i < 0:12 to highlight the large-scale wave. As can be seen from Fig. 4.2, the overall induced magnetic eld spectrum is broadband. 155 weak- eld prediction by Pe ey et al. [119] for di usively dominated magnetic eld uctuations generated by E(k) k 5=3 turbulence. Pe ey et al. make a pre- diction for mixing-dominated induction that should apply at longer wavelengths, EB(k) k E(k)=!. The dispersion relation relating k and ! is taken to be Tay- lor?s frozen turbulence hypothesis, ! = U k, there predicting EB(k) k 5=3 for the external eld uctuations. However, no k 5=3 spectra are noted7 even in direct ow uctuations. If there is a k 5=3 spatial spectrum of velocity uctuations, it seems that Taylor?s hypothesis fails. Given that some of the most energetic motions in the ow are large-scale and oscillatory, we might expect this. Weak turbulence advected rapidly past the sensor by a uniform free stream ow is probably not the right picture in this case. The large-scale structure of the observed slow turbulent wave motion is made clearer by the global magnetic eld measurements from the 60 cm apparatus. Fig- ure 4.3 is a space-time diagram of the twenty-one sensors pictured in Fig. 3.24 (a). These sensors are spaced out along one meridian xed in the lab frame. The signal from each sensor was ltered with a phase-preserving lter with passband extend- ing from 0:02 i to 0:12 i, to help distinguish the \wave" region of the frequency spectrum from the rest of the turbulent uctuations. Approximately 3.5 periods at 0:04 i are shown in Fig. 4.3 as the wave sweeps by the Hall sensor array. The sym- metry across the equator is similar to the \shift-and-re ect" symmetry of laminar wide gap wave states reported by Hollerbach et al. [100]. The motion in the lagging hemisphere in Fig. 4.3 lags most of the time. As with all other quantities associated 7 Perhaps the nearly k 11=3 region in the magnetic spectrum is simply a coincidence. 156 051- 001- 05- 0 05 001 051 01 4- 01 3- 01 2- ? 8M -? 41M )seergeD( Pr(??) Fig. 4.4: The probability distribution of the lag angle between the Northern and Southern hemisphere wave maxima of Fig. 4.3. A running cross- correlation of the ltered data of Fig. 4.3 is taken with a four period window (with T = 2 =0:04 i) and histogram taken of the resulting lag angles for a long data set. The most probable lags are between 30 and 45 . with this state, the lag angle between hemispheres is a statistical property of the turbulent ow state and therefore is strongly uctuating. A probability distribution of the lag angle is depicted in Fig. 4.4. Typically, one hemisphere lags the other by 30 to 45 . There is substantial short-term variation in the phases of the two hemispheres, possibly due to other strong uctuations in this frequency band, but the picture of Fig. 4.3 may be considered typical, as it has a lag close to the mean value in Fig. 4.4. 157 The wall shear power spectrum8 is indicative of strong turbulent uctuations over a wide range of length and time scales, even if it is not consistent with ap- pearance of an inertial range of K41 turbulence. The wall shear spectrum shows measurable broadband uctuations out to the Nyquist frequency of 37 i and a nearly at spectrum below about 0:4 i with the exception of the strong waves. Even the \coherent" waves have large temporal and spatial variation. We may ex- pose this further by looking at the probability distributions of measured uctuations compared to previously studied turbulent ows. As an example, Fig. 4.5 depicts a probability distribution of wall shear stress at Re = 4:4 107, normalized by the mean shear stress. The distribution is nearly log-normal, with high shear stress events much more likely than low. The same was observed in high Re turbulent Taylor-Couette ow by Lathrop et al. [120] and Lewis and Swinney [121]. Interest- ingly, there is little about Fig. 4.5 to suggest a strong wave. It may be inappropriate to think of the wave as separate from the turbulence. We hypothesize that the large-scale wave motion for Ro =1 spherical Couette ow is a slow modulation of the inner sphere?s turbulent equatorial jet, such that the jet is not (statistically) axisymmetric, at, and of uniform strength. The resulting recirculation pattern may be complex in this case. The jet itself should be a robust feature over a wide range of Re, because it relies only on a centrifugal pumping e ect from the gradient of the pressure in the uid being out of balance with the centrifugal force near the inner sphere. This will be the case whenever the inner sphere is rotating faster than the bulk uid in its vicinity. This jet should persist even 8 And any other quantity we can measure. 158 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 10 ?5 10 ?4 10 ?3 10 ?2 10 ?1 10 0 10 1 ? w /? ? w ? Pr( ? w /?? w ?) Fig. 4.5: The probability distribution of wall shear stress at Re = 4:4 107 is approximately log-normal, similar to the ndings of Lathrop et al. [120] and Lewis and Swinney [121] in Taylor-Couette ow. Here the wall shear stress has been normalized by its mean. 159 with a fully turbulent Ekman layer near the inner sphere, as turbulent momentum transfer spins up uid near the sphere surface. The boundary layer near the inner sphere in the three meter experiments is probably completely turbulent. It is possible that this is not always so in the 60 cm experiments. Kohama and Kobayashi [107] study the transition to a turbulent boundary layer on a rotating sphere in air up to Reynolds numbers Rei = ir 2 i = (4.4) of approximately 5 105. We use Rei to distinguish this inner-radius based Reynolds number, which is approximately one quarter of Re as we have de ned it in Eq. 2.6. At the highest Re in their study, the turbulent boundary layer extends up to 40 latitude. Above that latitude laminar spiral vortices and a steady laminar layer are observed. A rough extrapolation of their results might require a fully turbulent boundary layer to have Rei of a few million to make the turbulent transition latitude reach the pole. As Rei for the three meter data exceeds 10 million, we expect a fully turbulent Ekman layer on the inner sphere. However, Rei for the 60 cm data shown in Fig. 4.2 is just under 900,000. If the boundary layer still exhibits a laminar portion anywhere in the Reynolds number range 9 105 < Rei < 1 107, it must not have much e ect, given the good agreement of the observed wave states between the two experiments. It should be noted that the experiments of Kohama and Kobayashi entrain quiescent uid by Ekman layer suction. The spherical Couette case will be in uenced by the turbulent uctuations already present in the entrained uid. We cannot so simply 160 predict the behavior of the ow in spherical Couette ow at = 0:35 from data as ! 0. Indeed, the results of Kohama and Kobayashi [107] and Bowden and Lord [105] note no wavy modulation of the equatorial jet position or strength as uid is en- trained from and pumped into quiescent surroundings in either the laminar or the turbulent case. It is clear that wavy behavior requires something more. Con nement of the resultant jet the azimuthal and meridional recirculation of uid in the gap is probably a necessary ingredient. Hollerbach et al. [100] come to this conclusion regarding the laminar states in comparison to the results for an isolated rotating sphere, and note that it takes a long time to establish the steady state circulation in wide gaps. This is consistent with our observations in the three meter experiment. The jet from the inner sphere begins immediately; a vigorous equatorial out ow can be observed visually as polystyrene particles settled on the inner sphere surface scatter when it is started. The slow wave state, however, takes several thousand inner sphere rotations to set in. In fact it takes several hundred rotations of the inner sphere before the wall shear stress array on the outer sphere registers a shear signal. Even with turbulent momentum mixing, it takes a long time to build up angular momentum in the gap. At early times no slow wave is observed, even if turbulence is already established. We may propose a simple mechanism for out ow jet de ection and modula- tion of strength in high Reynolds number spherical Couette ow that satis es the constraints of the observations. The mechanism should perhaps not depend too 161 L H Strong Outflow Weak Outflow Fig. 4.6: Schematic of large vortex feedback on the boundary layer on the inner sphere. The pattern as drawn would appear as an m = 2 pat- tern on the probes, not m = 1 as we observe. However, as discussed in Sec. 4.4, we observe several signi cantly di erent turbulent ow states when the outer sphere revolves, and di erent bulk ows may be selected in that case. 162 much on the exact details of the inner sphere boundary layer, applying equally to a turbulent or partially laminar boundary layer. Since the out ow jet is driven by centrifugal forces that are not balanced by the pressure gradient in the uid, the pressure distribution over the surface of the inner sphere will a ect this out ow. In a large-scale-axisymmetric turbulent basic state, the pressure distribution would be axisymmetric, on average. However, we propose that regions of high and low pres- sure near the inner sphere associated with large-scale circulations could modulate the strength of the turbulent out ow jet into a large-scale pattern. The modulated out ow and resultant recirculation may then have the right spatial dependence to sustain the large-scale vortex patterns, closing the feedback loop. A schematic of this process is shown in Fig. 4.6. De ection of the jet from the midplane could proceed similarly, provided that the large-scale motions on the lower hemisphere were not completely symmetrical with those on the upper. Then, the impinging boundary layer ows at the equator would be unequal in strength, the extra vertical momentum of the stronger ow would de ect the jet away from the high pressure side. It is worth mentioning that large-scale, low-frequency uctuations similar to those observed here are an area of interest in large industrial mixers. So-called \macro-instabilities" in these devices involve large-scale, precessing, turbulent vor- tices. These ows can contain a substantial fraction9 of the total uid kinetic en- ergy [122{125]. Roy et al. [123] report slow variations in the large-scale circulation patterns in a dished-bottom, impeller driven mixer. The \macro-instablity" in this 9 Values of 10%-30% are found in the literature, depending on mixer geometry. 163 case is identi ed with the in uence of the large-scale circulation on the trailing edge vortices and axial out ow from the impeller. The large departures from axisym- metric ow pattern have important consequences for turbulent transport and the structural soundness of large mixing tanks. The three meter device is a particularly simple geometry for a turbulent mixer, and some mixers have a free surface, but there are no other signi cant di erences10, when the outer sphere is stationary. Much of the work surveyed on the topic of these low frequency mixer dynamics makes the point that these ows cannot be characterized without long observation times. Among other things, this makes simulations exceedingly di cult. Lavezzo et al. [125] expended signi cant e ort in a modern computation to increase their sim- ulated time from three impeller revolutions to twenty-one. The time series of wall shear stress used to make Fig. 4.2 was nearly sixty thousand inner sphere rotations long. Welch?s method of power spectral density estimation was used, using a sliding window of 4000 inner sphere revolutions. Motions varying on long timescales com- prise a great deal of the uctuation energy. The successful prediction of turbulent ow and turbulent transport requires that we understand when and why such ener- getic but slowly evolving dynamics may occur. Several thousand rotations? worth of simulated time would be needed just to see the beginning of the dynamics presented here if the initial condition were not well chosen. 10 Up to and including the manufacturer of the outer vessel. 164 4.2.2 Torque The torque demanded by the ow at constant boundary speeds has been well characterized in high Reynolds number Taylor-Couette ow with stationary outer cylinder [120, 121], and some data exists with rotating outer cylinder [126, 127]. The torque required to drive the ow between two spheres has received fairly little attention. In the turbulent case, Sisan [118] reported the hydrodynamic torque as part of an investigation on the e ects of strong magnetic elds in this geometry. The scaling of the torque required to drive the inner sphere where the outer sphere is stationary is similar to the case with the outer cylinder stationary in the Taylor- Couette system. Following Lathrop et al. 1992 [128], we de ne the dimensionless torque G on the inner sphere in terms of the uid density , viscosity and inner sphere radius ri as G = T 2ri : (4.5) The argument can be made on dimensional grounds, as in Lathrop et al., that the dimensionless torque G should vary as G / Re2 in the asymptotic range of Reynolds number. The measurements of Lathrop et al. , further re ned by Lewis and Swinney[121], demonstrate that the torque in Taylor-Couette ow has no single scaling exponent even at high Re. Whether we should expect this in the spherical geometry is not obvious, but our confounding errors and limited range of Re mean that it is not worthwhile to investigate this. We use a simpler model and t a power law for the torque with the outer sphere stationary. The power law t is shown in Fig. 4.7 for a range of Re accessible in this experiment, exhibiting a Re1:89 165 dependence. At lower rotation rates, the measurement is signi cantly confounded by the bearing and seal torque. A constant drag torque, Gd = 3:3 1010, is subtracted from the data before the t is carried out. This drag is based on measurements with the sphere emptied of uid11, and is fairly constant with rotation rate. The error bars on Fig. 4.7 are constant error bounds of approximately 1:1 1010 based on the maximum observed range of seal uctuation in several repeated trials with the spheres empty. Typically, the medium-term uctuations over a day?s run, especially for a single continuous ramp, are signi cantly smaller than this, but systematic day- to-day o sets of this magnitude are, unfortunately, common. Curiously, the typical change from day to day is fairly well described by a discrete change by this amount, possibly due to varying engagement by one of the two lip seals. Therefore, we choose between 3:3 1010 and. 2:2 1010 as the constant drag removed. To be concrete, we model the e ect of the confounding drag on the dimensionless torque as G = Gmeas 8 >>< >>: 2:2 1010 3:3 1010 (4.6) with the choice made for better collapse. There are other minor uctuations which cause a good deal of scatter in the data at low speeds, but cannot t a constant to deduce the drag, as we do not know the scaling exponent independently. However, we choose to correct for the major component of the drag, and use the restricted model of Eq. 4.6 based on repeated observations that the mean drag is close to one of the values listed, uctuating only slightly about one or the other. 11 The air torque is assumed to be negligible. 166 10 6 10 7 10 810 9 10 10 10 11 10 12 10 13 G Re = ? i? 2 Fig. 4.7: Dimensionless torque, G vs. Reynolds number Re. Based on measurements with the sphere empty, a constant drag torque of Gd = 3:3 1010 has been subtracted from the data. Error bars are 1:1 1010 based on the worst case variation implied by Eq. 4.6. The model t is G = 0:003Re1:89. We will use this model to ex- press the Re-dependence of the torque in the rotating states, and so we de ne G1(Re) = 0:003Re1:89. This is the predicted torque as a function of Reynolds number when o = 0, in other words, when Ro =1. 167 The torque t in Fig. 4.7 will be used later to normalize the torque mea- sured with both spheres rotating. In some subsequent discussion when both spheres revolve, the Reynolds number is calculated with Eq. 4.3 and the predicted torque, G1 = 0:003Re 1:89; (4.7) is used to normalize the measured torque. The quantity G=G112 seems to be dom- inated by the Ro-dependence of the torque, and provides a useful summary of the torque results. This will be discussed in detail in Chapter 6. 4.2.3 Summary The torque as a function of Re of Fig. 4.7 is ultimately similar to other turbu- lent shear ow experiments. In terms of the torque and the wall shear stress uctu- ations of Fig. 4.5, turbulent spherical Couette ow when only the inner sphere re- volves is not much di erent, to our measurement capabilities, than turbulent Taylor- Couette ow reported by Lathrop et al. [120] and Lewis et al. [121]. However, the turbulence contains large-scale, unusually slow but energetic wave motions. These are reasonably coherent in the sense that they exhibit peaks in temporal auto- correlation (frequency spectra) and spatial cross correlation, but nevertheless are signi cantly uctuating and should be considered part of the overall inhomogeneous turbulence. 12 For experiments that do not use multiple uids to extend the range of Re, this is the ratio of the dimensional torques. 168 4.3 Ro < 0, Outer Sphere Rotating 4.3.1 Flow Properties This section will be somewhat brief compared to Section 4.2, as the inertial mode states presented here are more well understood, reported earlier by Kelley and collaborators [12, 14, 15]. However, the broader conclusions of Chapter 6 will discuss this, so it is worth mentioning some properties of the turbulent ow that we observe when Ro < 0. A broad view of the observations for 4:2 < Ro < 0 is provided in the wall pressure spectrogram of Fig. 4.8. Each vertical line is a power spectrum, with frequency normalized by the outer sphere rotation speed. The logarithmic color scale represents the magnitude of the power spectral density. Signi cant strong peaks are noted over ranges of Ro and many can be identi ed as inertial modes. The identi cation in Fig. 4.8 of the annotated peaks (b), (c), (d), (e) and (f) with particular inertial modes similar to those in the full sphere is based partially on observation of modes with these frequencies in Kelley et al., 2007 [14]. The nature and role of the low, broad energy peak around the frequency of (a) for large ranges of Ro are not known. We will not provide a thorough comparison of these observations with the observations and predictions of Kelley et al.. More solid identi cation of the observed modes by azimuthal wavenumber should be possible using the wall shear array of Sec. 3.5.4. However, detail regarding the mechanism of excitation of these modes and the veri cation that they are identical to those seen before is outside the focus of this dissertation. More information on inertial modes, including 169 ?0.1 ?0.5 ?1.3 ?1.7 ?2.1 ?2.5 ?2.9 ?3.3 ?3.7 ?4.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ro a b c d e e ? o ? f X l=4, m=1, = 0.612/?o? Mode (f) Magnetic Field Fig. 4.8: Spectrogram from a wall pressure sensor at 23:5 colatitude as a function of Ro for Ro < 0, E = 1:6 10 7, showing inertial mode states with annotations. A series of inertial mode states are excited. The frequencies are as follows. The broad, persistent peak, (a), is at 0:083 o. At (b), a small region of 0:333 o is probably (6, 5, 0.333). At (c), initial frequency of 0:38 o is likely (6, 3, 0.378). Peak (d), starting at 0:51 o is probably (4, 3, 0.500). Peaks (e) are an in- teresting case. The middle-frequency peak is probably (6, 1, 0.44). However, the peaks at 0:27 o and 0:75 o are previously unre- ported, and the three frequencies form an approximately resonant triad. Peaks (f), at 0:61 o and 1:22 o are probably (4, 1, 0.612) and its second harmonic, which could excite the mode (4, 2, 1.23). The discontinuity in the spectrum, especially evident at higher fre- quencies and marked with an \X," is due to a gap of inaccessible Ro. This covers approximately 1:2 < Ro < 0:7 where the in- ner sphere motor cannot revolve slowly enough. The inset shows a Mollweide projection of the induced magnetic eld from the 60 cm experiment for the (4, 1, 0.612) mode, (f), showing how much the simple spatial pattern dominates the induced eld. 170 mode identi cation, can be found in the dissertation of S.A. Triana. Here we are concerned more with the properties of the turbulence associated with these states with strong inertial modes. As can be seen from Fig. 4.8, signi cant broadband uctuations are observed. The equivalent spectrogram of the magnetic induction presented by Kelley et al. [14] shows a banded structure of the broadband uctuations, coexisting with the strong inertial mode. However, the broadband part of the uctuation spectra here is simple (see also Fig. 4.9). It may be that the banding observed by Kelley et al. is not a property of the turbulence itself. Instead, it may be a characteristic of the magnetic induction by this particular velocity eld. Two-dimensional motions do not cause any induced eld with a uniform, axial applied eld, so low-frequency inertial modes and other nearly two- dimensional motions will exhibit weak induction. The existence of broad bands higher than 2 o in the spectra of Kelley et al. are due to frequency shifts, !lab = ! + m o of the azimuthal wavenumber spectrum because the magnetic sensors are in the lab frame. It seems likely that much of the structured appearance of the broadband magnetic eld uctuations reported previously is because of particulars of the induction and the measurement frame of reference. Nevertheless, here we present point measurements of what is undoubtedly extremely inhomogeneous turbulence. It may be that banded spectra are associated with the smaller scales of turbulent ow nearer the equator, for example, far from our sensors? location. We can see the broadband turbulent power spectrum more clearly in Fig. 4.9, a power spectrum of the wall shear stress for Ro = 1:8. The strong peaks include the mode (4, 1, 0.612) and its second harmonic, as in Fig. 4.8 (f). It also appears 171 10 ?2 10 ?1 10 0 10 1 10 210 ?10 10 ?8 10 ?6 10 ?4 10 ?2 ?/? o PSD ?/? o = 2 Fig. 4.9: Wall shear stress power spectrum for Ro = 1:8, E = 1:6 10 7, with strong inertial mode l = 4,m = 1,!= o = 0:612. The measured frequency is slightly higher, 0:618 o, as is typical as Ro increases above onset. Unlike the turbulent wave of Fig. 4.2, the fractional bandwidth of the inertial mode peak is extremely narrow. The in- ertial mode power is approximately 400 times higher than the tur- bulent background. It is strong, but signi cantly less so than the relative power of the modes in the magnetic observations of Kelley and collaborators [12, 14, 15]. It is not thought that this is a de - ciency in the power of the wave, but rather that the ltering of the turbulent uctuations by the magnetic eld emphasizes the large- scale inertial mode in the magnetic measurements. For reference, we include the maximum possible inertial mode frequency , 2 o, as a vertical dashed line. The spectrum is nearly at below this frequency aside from the few strong peaks. 172 that the third harmonic is present in the wall shear stress data. The inertial mode peak is strong, approximately four hundred times the background turbulence level. This is not as strong as the relative strength reported by Kelley et al. who put the (4, 1, 0.612) mode nearly six thousand times stronger than the noise. However, this is likely due to the reduced strength of higher frequency magnetic uctuations similar to Fig. 4.2. We therefore interpret this discrepancy not as a de ciency in the strength of the inertial mode, but rather that our wall shear stress sensor is signi cantly more sensitive to the turbulence than the induced eld measurements are. The turbulent uctuations at the measurement location have considerable total power with respect to the mode. While the mode itself stands out signi cantly, it is narrow-band, and the range of energetic uctuations is wide. The spectrum is almost at below the inertial mode upper limit of 2 o. Figure 4.10 shows the probability distribution of wall shear stress uctuations normalized by the mean wall shear. The low shear and high shear tails are similar to those of Fig. 4.5, but the weak bimodality is the signature of the wave. A time series of the wall shear stress for Ro = 1:8 is shown in Fig. 4.11. These results, Fig. 4.8 through Fig. 4.11, establish that the inertial modes coexist with even stronger turbulence than could be deduced from the measurements of Kelley et al.. Furthermore, we note signs of inertial mode nonlinearity, for example, in the second harmonic of Fig. 4.8 (f) and the resonant triad of Fig. 4.8 (e). The turbulence is probably extremely inhomogeneous, and without measurements throughout the bulk uid, we cannot quantify the total energy in the turbulence vs. that in the 173 0 0.5 1 1.5 2 2.5 3 10 ?4 10 ?3 10 ?2 10 ?1 10 0 ? w /?? w ? Pr( ? w /?? w ?) Fig. 4.10: Wall shear stress probability distribution for Ro = 1:8, E = 1:6 10 7 normalized by its mean value. Compare with Fig. 4.5. In this case, the strong, narrow inertial mode leaves its signature with a bimodal distribution of wall shear stress. The upper and lower tails of the distribution seem to have the same log-normal character as that in Fig. 4.5. 174 250 255 260 265 270 275 280 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 ? w /?? w ? ?ot 2? Fig. 4.11: Time series of the wall shear stress at Ro = 1:8,E = 1:6 10 7, normalized by its mean value. The strong oscillation is an inertial mode, l = 4,m = 1,!= o = 0:612. Turbulent uctuations are more evident here than in the magnetic measurements of Kelley et al. [14]. 175 inertial mode, nor can we rule out that the wave lives mostly in a quiet region of the ow. Nevertheless, we establish that there is an even more turbulent background than is apparent from the magnetic data of Kelley et al., suggesting that inertial modes are fairly robust to the presence of other turbulence. 4.3.2 Torque We conclude this section with a brief mention of the torque, which we will revisit as part of the broad Ro dependence in Chapter 6. Figure 4.12 shows the dimensionless torque on the inner sphere as a function of Rossby number. A constant seal drag has been subtracted. In this case the constant drag subtracted is less than that of Fig. 4.7, at Gd = 2:2 1010, as discussed in Sec. 4.2. The gap around Ro = 1 is due to the fact that the inner sphere motor can not be stably rotated slowly in the range of about 1:2 < Ro < 0:8, where Ro = 1 corresponds to i = 0. This gap is not explicitly shown in Fig. 4.8, but it is the reason for the spectral discontinuity marked with an \X" there. As Ro is changed for xed E, as in this plot, both Ro and Re vary. The resultant scaling is dominated by the Re dependence, and is similar to that for the outer sphere stationary, close to Re2 = (Ro=E)2. Residual dependence on Ro is pictured in the inset of Fig. 4.12, where the torque on the inner sphere is divided by the predicted value at the same Re, if the outer sphere were stationary13. This compensated torque plot makes it easier to see the small e ect of the inertial mode transitions. The transition from the triad Fig. 4.8 (e) to the mode Fig 4.8(f) can 13 G1, see Eq.4.7 176 be seen at Ro 1:5 and the cessation of the mode Fig. 4.8(f) can just barely be detected in G=G1 at Ro 2:4. The compensated inset of Fig. 4.12 shows that the Re-dependence of Eq. 4.7 overpredicts the torque at small negative Ro, rising slowly toward agreement as the magnitude of Ro is increased. It is not known if the large change in the apparent value of G=G1 across Ro = 1 is a uid e ect or an e ect of directional seal wear leading to direction-dependent drag. The much quieter broadband spectrum for low negative Ro in Fig. 4.8 does suggest the turbulent uctuations are much weaker overall, so perhaps the drop is due to a real ow transition. The inertial mode excitation smoothly varies across Ro = 1, but it is possible that other properties of the turbulence and angular momentum transport do not. 177 ?4?3.5?3?2.5?2?1.5?1?0.66 10 9 10 10 10 11 10 12 Ro G ?4.5?4?3.5?3?2.5?2?1.5?1?0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Ro ? G G Fig. 4.12: Dimensionless torque vs Rossby number, E = 1:6 10 7. Despite the inertial mode ow transitions evident in Fig. 4.8, there is a small e ect on the torque, which varies roughly quadratically with Ro. As Ro is changed, Re = Ro=E also changes. The inset shows G divided by the value it would have at the same Reynolds number with the outer sphere stationary. This compensated torque plot exposes the residual dependence of G on Ro more clearly, eliminating the large Re dependence typical of a turbulent shear ow. The arrows denote two inertial mode state changes. It appears that a slight drop in the compensated torque can be seen for two mode transitions. 178 4.4 Ro > 0, Outer Sphere Rotating 4.4.1 Narrow Range, Ro < 5 The ows in this section, those when the inner sphere super-rotates, have been the main focus of the experimental campaign in water. Like the case when Ro < 0, a number of di erent ow states may be identi ed in ranges of Ro, with boundaries that depend weakly on the Reynolds number14 if at all. These ow states are also marked by strong wave motions. Unlike the negative Rossby states, however, each state transition involves an general re-organization of the turbulent ow, and some of the wave motions are much more like that of Sec. 4.2, deeply modulated by low frequency uctuations. The spectrogram of Fig. 4.13 provides an overview of the uctuation spectra for Ro < 4:5. Turbulent ow transitions are evident near Ro = 0:5, Ro = 1, Ro = 1:8 and Ro = 3:3. In some regions of Ro, like the region discussed in detail in Chapter 5, the ow exhibits bistable behavior, transitioning between each of the turbulent states observed adjacent to the bistable range and spending a long time in each. The mean torque and the size and character of temporal torque uctuations are very di erent in each identi ed ow state, unlike the case when Ro < 0. Large, low frequency temporal uctuations in the torque are common for the states iden- ti ed for positive Rossby number. In the bi-stable ranges the torque demanded by one turbulent state is considerably higher than the torque demanded by the other. Fig. 4.14 shows the dependence of the torque on Rossby number for xed Ekman 14 Or, equivalently, the Ekman number E. 179 5.00 5.0 1 5.1 2 oR H L LL ? ?o 1.0 2.01.5 2.5 3.5 4.03.0 7-6-5-4-3-2-1-01 log(PSD) Fig. 4.13: Pressure spectrogram, E = 1:6 10 7. Several turbulent transitions are noted. It is clear that the turbulence changes character at each transition, and that the wave states here are mostly broadband compared to those of Fig. 4.8. This spectrogram will be discussed in more detail in Chapter 5. 180 number. Like Fig. 4.12, the inset shows the compensated torque, G=G1. The ranges of Ro where this relation is not single-valued are those where the ow is bistable. Long enough observation times here expose ow transitions between the states that appear on either side of the overlapping range. The torque in these regions exhibits a bimodal probability distribution15, and when this is the case, the torque is reported as the two modes of the distribution instead of the mean. Because of the transitions, neither the torque nor the mean power dissipation16 are a monotonically increasing function as Ro is increased. Throughout the rest of the dissertation, we adopt labels for the states on either side of the most pronounced torque transitions. The high torque state, H, exists alone from Ro = 1 to Ro = 1:8. The low torque state, L, becomes possible around Ro = 1:8 and exists alone until approximately Ro = 3, where a subsequently lower torque state becomes possible, the LL state. Unlike the Ro < 0 inertial mode state transitions, the factor G=G1 varies a great deal between the two adjacent states. The H state torque for a given Re is 1.4-1.7 times that of the L state torque in the transition region. The boundaries of onset are suggested by Figs. 4.13 and 4.14, but precise determination of boundaries requires acquisition of more data than time allowed17. A manual division of the torque into state based on qualitative identi ca- tion of the states results in a somewhat di erent set of state boundaries. Figure 5.2 in the next chapter is perhaps more suggestive of the points of bifurcation where 15 Skip ahead to the next chapter, Fig. 5.4, for an example. 16 The torque on the inner sphere and the speeds of the spheres are su cient to know the power input. 17 A much faster experiment at the same dimensionless parameters would make this possible. 181 0.5 0.9 1.3 1.7 2.1 2.5 2.9 3.3 3.7 10 10 10 11 Ro G 0 1 2 3 4 0 0.5 1 1.5 2 H L LL ? G G Ro H L LL Fig. 4.14: Torque as a function of Rossby, E = 1:6 10 7, with compensated torque inset. Where bimodal torque distributions are noted, the torque is split into two branches corresponding to the two modes. Overlapping regions are associated with intermittent transitions be- tween the two turbulent ow states adjacent to the switching region. new states come in, though it is not as objective. More thorough characterization of the H and L states are the topic of Chap- ter 5, but we will state an additional conclusion here to frame the broader picture of Ro > 0 state transitions. Fig. 4.15 shows time series of the dimensionless torque in the H/L transition regime, at Ro = 2:13, as well as the measured zonal18 ve- 18 Azimuthal motions that are invariant in the vertical coordinate, which is a strong assumption. Often used in the literature on rotating ows. 182 locity, normalized by the outer sphere tangential velocity19 and low pass ltered with a cuto frequency comparable to that used to lter mechanical noise out of the torque. The bistablity and slow uctuations in the zonal velocity are clearly anti-correlated with the torque on the inner sphere. As we will see later in this sec- tion, this is typical of all of the turbulent states for Ro > 0 that correlate with the ow uctuations at all. This anti-correlation seems to involve zonal- ow induced transport barriers to angular momentum. Because of this, the velocities (and wall shear stress) at the measurement location increase in the higher shear, lower torque state, even as the torque20 decreases. In Chapter 5 we discuss and interpret these zonal ow observations more fully. 4.4.2 Wide View, 0 < Ro < 62 We have performed some experiments to explore the turbulent ow states out to high positive Rossby number. In doing so, we have identi ed three more states on the basis of ow spectra and torque uctuations. The distinctions drawn here are more qualitative. Some transitions are extremely subtle, and the transitions Ro higher than approximately21 Ro = 5 do not show long residence times in identi able adjacent states. It is only the H/L and L/LL transitions that seem to be clearly qualitatively bistable, and the transitions for higher Ro are more smeared out and gradual. However, up to Ro = 15, the system still exhibits torque that uctuates in 19 Meaning that it can be interpreted as the local Rossby number of the mean zonal ow, as mentioned in Table 2.1. 20 And probably the total power dissipation. 21 The end of the LL state range. 183 0 0.05 0.1 0.15 0.2 0.25 0 500 1000 1500 2000 G u? ?oro ?ot 2? 5.2 ? 01 01 0.2 ? 01 01 5.1 ? 01 01 Fig. 4.15: Torque and low pass ltered velocity time series, Ro = 2:13, E = 5:3 10 7, 10 cm from the wall at 60 cm cylindrical radius. The onset of the high torque state involves a rapid crash of the mean zonal velocity with a corresponding abrupt rise of the torque. The return to the low torque involves an exponential decay of the inner sphere torque associated with spin-up of the mean zonal velocity. The torque uctuations are anti-correlated with the slow velocity uctuations. 184 time in a way that is correlated strongly with the ow measurements. Wall shear stress frequency spectra over the range 1:8 < Ro < 62 are depicted in Fig. 4.16, with six states labeled. In addition to the H, L, and LL states, we identify three more, B, Q, and E, and we will brie y discuss their characteristics below. The L to LL state transition is fairly subtle in the spectra, with the upper large frequency peak increasing in frequency and broadening. It is easier to see in Fig. 4.13. This transition exhibits torque bistability like the H to L transition does, with the LL state exhibiting about 10% lower torque than the high. Fig. 4.17 (a) shows the correlated torque and slow wall shear stress dynamics over 4500 outer sphere rotations. Like the H to L transition, the lower torque state is associated with higher mean wall shear stress at the measurement location, and the slow uctuations of the wall shear stress and the torque are clearly anti-correlated. As in the H/L transition, the new state has lower torque, but higher wall shear stress at the measurement location22. The LL to B (\bursty" state) transition is fairly indistinct, and may probably be considered a modulation on the LL state. In Fig. 4.16, the spectrum for Ro = 5:3 shows the growth of a broad peak at a frequency of 0:34 o, in addition to the peaks at 0:18 o and 1:5 o shared23 with the LL state spectrum at Ro = 4:6. The torque in of the B state exhibits positive-going torque bursts, shown for Ro = 6:3 22 As a reminder, this is 60 cm cylindrical radius and even for the velocities, never more than 15 cm from the wall. 23 The 0:18 o peak is a feature of most of the states for Ro > 0. It is likely to be a prograde inertial mode, (3, 1, -0.1766). 185 PSD 10 ?1 10 0 10 1 10 2 10 3 10 0 10 10 10 20 10 30 Ro 1.8 2.1 2.5 3.0 3.5 4.0 4.6 5.3 6.1 6.9 7.9 9.0 11.6 14.9 19.0 24.2 30.8 38.8 49.1 61.7 ? ?o H L LL B Q E 10 ?2 Fig. 4.16: Wall shear stress spectra vs. Ro, wide positive Rossby ramp. For clarity, spectra have been shifted vertically by successive multipli- cation by 50. 186 in Fig. 4.17 (b). At this Ro, the bursts come fairly rapidly, with a burst every 125 rotations of the outer sphere. The bursts are associated with peak torques 5%- 10% higher than the lowest values. As with the lower states, low frequency wall shear stress uctuations are anti-correlated with the torque. Higher torque implies lower wall shear stress at the measurement location24. The faster oscillations on the wall shear stress are the wave at 0:18 o, which at this outer sphere speed falls inside the low pass lter. The B state exists over a large range of Rossby number, approximately 5 < Ro < 15. The Q state starting25 around Ro = 14:9 is associated with a disappearance of the waves in the spectra of Fig. 4.16. The Q (\quiet") state spectra are rather featureless, with the exception of a narrow peak that appears at Ro = 14:9 with a frequency of 4:5 o and moves up in frequency as Ro increases. In addition to the nearly featureless, broadband wall shear stress spectra, the temporal uctuations in torque greatly diminish26 and are not seen to correlate with uid motion anymore. By Ro = 25, a weak buildup of energy is seen in the 2 3 o frequency range, at Ro = 61:2 it is centered around 3 o, which is consistent with the Ro =1 wave doppler shifted by the outer sphere speed. A reasonable picture of the temporal dynamics of the E state is that the ow uctuations involve the same wave as that for Ro =1, but the sensors rotate slowly around that turbulent ow, sampling the dynamics from their slowly moving reference frame. However, though the frequency 24 The reader is assured that the wall shear stress plots are not upside down. 25 The data in this region is widely spaced, so this is not a good estimate of where the waves die out. 26 To the point where the resolvable uctuations are dominated by the seals, most likely. 187 0.9 1 1.1 0 1000 2000 3000 4000 0.9 1 1.1 1.2 0.95 1 1.05 1.1 300 400 500 600 700 800 900 1000 0.8 0.9 1 1.1 ? ?? ? G ?G ? ? ?? ? G ?G ? ?o 2? (b) (a) t Fig. 4.17: Torque and wall shear time series, corroborating the anti- correlation of these two quantities. Time is measured in outer sphere revolutions. Part (a) is a time series of several L/LL state transitions at Ro = 3:2. The torque (top) and wall shear stress (bottom) normalized by their mean value. The same quantities are shown in (b) for the B state at Ro = 6:3. The wall shear stress has been low pass ltered the same as the torque. 188 is right, the wave is extremely weak compared to that in Fig. 4.2. Furthermore, as we will see in the next section, the torque in the E (\Ekman") state is not yet the same as that in the outer sphere stationary state. In other words, G=G1 has not yet returned to unity. We will present evidence in Chapter 6 that G=G1 may take on two di erent values, at least as a transient, in the E region. We attribute the weak wave and the lower relative torque in this region to the existence of an Ekman boundary layer on the outer sphere when it revolves that is not present when it is stationary. At the slowest outer sphere rotations we can achieve, approximately one revolution per 30 s, the Ekman number is still extremely low on the outer boundary, E = 5 10 6. This is considerably di erent from the situation with a nonrotating27 outer boundary. It may be that Ekman suction and pumping on the outer boundary is responsible for the di erence in the two states seen in the E region. It is not the case that Ro! 100 can be identi ed with Ro!1. 4.4.3 Wide Ro > 0 Torque and Shear We will conclude this section with the overall view of the torque for large positive Ro. Figure 4.18 shows the compensated torque G=G1 for 1:8 < Ro < 62. The six states discussed in the last section are labeled for reference. The general trend in the torque is that a global minimum in G=G1 is found between Ro = 7 and Ro = 8. The Rayleigh line, where the angular momenta of the uid on the two 27 Actually, the outer boundary rotates once per day with the Earth, with an Ekman number of about 10 2, crossed with our axis according to our latitude. If this mattered, however, the 60 cm data, with an Ekman number an order of magnitude higher, would probably be di erent. 189 boundaries at the equator28 are equal, RoR = 1 2 1 = 7:16 (4.8) is shown as a vertical line. The global minimum of G=G1 is found near RoR. Above the Rayleigh line, inviscid, axisymmetric, axially invariant ows are unstable to axisymmetric overturning motions. It is not known to what extent the Rayleigh line is important in spherical Couette ow, as the uid angular momentum varies along the boundaries. The relationship between the angular momentum distribution on the outer boundary and inner boundary does not lead to obvious predictions as it does for axisymmetric ows in axially invariant geometry, like the ow between two in nite rotating cylinders. This line is not particularly special in delineating state transitions. The ow on the Rayleigh line shows typical B-state bursting dynamics, which persist much higher and lower in Ro. It is possible that local centrifugal instability plays some role in the di erence between states at low positive Ro and high positive Ro, but we have scant evidence for this. It may simply be an interesting coincidence that the global minimum of G=G1 is near the Rayleigh line. The behavior below Ro = 4 is consistent with Fig. 4.14. As the LL state transitions in to the B state as Ro is further increased beyond Ro = 4, the compen- sated torque G=G1 falls until it reaches RoR, where there is a shallow, nearly at minimum that shows B state dynamics. With further increase in Ro, G=G1 starts to rise and the Q state, free of waves, sets in. From the B/Q transition on, G=G1 rises toward unity. 28 Or any equal latitude. 190 Ro G? G H L LL B Q E RoR 2 5 10 20 50 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 Fig. 4.18: Compensated torque vs. Ro for 1:8 < Ro < 62 and 9:2 10 8 < E < 2 10 6. The Ro-axis is logarithmic. At Ro = 61:2, the outer sphere revolves once every 13 s. Note, however, that G=G1 has not yet reached unity. At Ro = 61:2 the ow still feels the outer boundary rotation. RoR denotes the Rayleigh line, which is where the angular momenta of the uid at a given latitude on the boundaries are equal, and is close to the global minimum of G=G1 for Ro > 0. The rough area in which the six states identi ed in Fig. 4.16 exist are labeled here. 191 A broad view of the wall shear stress as a function of Ro is consistent with the observations of shear-torque anti-correlation discussed in Sec. 4.4.2. Figure 4.19 shows what we might call the \shear stress concentration factor." The wall shear stress sensors on the outer wall are calibrated in situ against the torque on the inner sphere, under the assumption of a particular29 constant factor relating the local wall shear stress on the outer sphere to the torque on the inner, 1 = CTinner: (4.9) The quantity = 1 concisely expresses how changing Ro a ects the spatial distri- bution of the wall shear stress on the outer sphere. Ultimately, the torques on the two spheres must balance, and so the integrated torque contributed by the shear stress on the outer sphere must equal the torque on the inner sphere. However, at di erent Ro, the spatial distribution of the stress that integrates to a given total torque is not the same. Figure 4.19 depicts = 1 for the wide range of positive Ro. This quantity peaks near RoR, where G=G1 has its minimum. Therefore, we see that the anti-correlation between wall shear stress and torque persists over the whole range of positive Ro, both in the temporal dynamics at a xed Ro and also as Ro is varied. The lowest mean value of G=G1 coincides with the largest mean = 1. A possible general mechanism for the Ro > 0 states we have observed may involve angular momentum throttling by zonal30- ow induced transport barriers. The results here are consistent with larger and larger zonal circulations at the core 29 Equation 5.7, which assumes wall shear stress falls o linearly with cylindrical radius. 30 Or if not zonal, fast azimuthal circulations with a 3D dependence. 192 0 10 20 30 40 50 60 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Ro ? ? ?w Fig. 4.19: Mean wall shear stress made dimensionless by the value it would have if the outer sphere were stationary, based on calibration against the inner sphere torque described in Sec. 5.3.1. 193 of the experiment as G=G1 approaches its minimum from either side. It may be that each distinct torque state is associated with a di erent zonal circulation pat- tern, with the inner-most ow31 highest at the peak of Fig. 4.19 and the minimum of Fig. 4.18. If there is really stronger zonal ow with reduced angular momen- tum transport heading toward RoR, it is clear that we can not make any kind of maximization argument for energy dissipation or angular momentum transport to explain these ows. The large zonal circulations that arise seem to reduce angular momentum transport. They seem to trap that angular momentum which is shed from the inner sphere at small cylindrical radius. Each state also has associated wave states, which may feed o the shear or modulate the boundary layer like the hypothesized state in Sec. 4.2. As the Ekman pumping on the inner sphere becomes stronger and stronger, we eventually expect a jet like Fig. 2.3 (b), and we may expect more 3D motion to become important. That jet, however, represents a large gradient of vertical velocity, which by Eq. 1.15 gives a source term for vorticity. Eq. 1.15 implies a large variety of wave states, Rossby waves and inertial modes, and the long wavelength versions of these all consist of large-scale vorticity patterns. We suggested in Section 4.2 that even in the Ro = 1 case, non-trivial large- scale recirculation patterns in the bulk can interact with the boundary layer on the inner sphere in a self-reinforcing way. Perhaps this is the case for all of the states when Ro > 0. large-scale, nonlinear waves of vorticity, whether they are Rossby waves or other strong inertial modes, could modulate the boundary layer 31 That at smallest cylindrical radius 194 on the inner sphere and the strength of the inevitable out ow from its surface, or they could grow on the (presumably) zonal shears observed. We should work to understand whether the wave patterns are important to the observed zonal ows; it seems unlikely that the changes in the waves would change simultaneously with the observed mean ows if they were not. 195 5. RESULTS II: H/L BISTABILITY IN DETAIL 5.1 Introduction The preceding parameter space overview established the existence of multiple turbulent ow states with transitions well de ned at di erent Ro. This chapter will focus on one of them, the high/low (H/L) state transition discussed brie y in Section 4.4. The material in Section 5.2 through Section 5.5 is nearly a verbatim excerpt from a paper recently submitted to Physics of Fluids. The author of this dissertation is rst author of the paper, and was responsible for all data collection and data analysis, generation of gures, and drafting of the text of this paper. The co-authors of the submitted paper, S.A. Triana and especially D.P. Lathrop made some intellectual and editing contributions to this paper, these being roughly com- parable in scope to those in the rest of this dissertation. As is the case with this entire dissertation, the experimental team worked together to improve the experi- ment. However, this work is predominantly the author?s work regarding the analysis of the H/L transition in detail. Section 5.6 is not part of the Physics of Fluids sub- mission. It is new data taken with the help of Matthew Adams, who is currently working with the 60 cm apparatus toward his Ph.D.. 5.2 Torque-Derived Measurements 5.2.1 Torque Revisited We will begin the discussion of the H/L state transition with a reminder of the torque results for Ro > 0 and how they di er from what we might consider the base-state case with Ro =1. Fig. 5.1 is a reproduction of Fig. 4.7, and shows the torque on the inner sphere as a function of Reynolds number when only the inner sphere revolves. In the context of this chapter, we note mainly from Fig. 5.1 that G is mono- tonically increasing in Re. At rst glance1, this might be the general expectation for turbulent torque scaling in a shear ow experiment. The higher the di erential rotation between the boundaries, the higher the torque. However, we established in Chapter 4 that this is not necessarily the case when we impart global rotation. When we hold the outer sphere speed constant and super-rotate the inner sphere as in Fig. 5.2, we nd that the mean torque on the inner sphere is non-monotonic as the dimensionless di erential speed, Ro, is increased. In ranges of Ro in Fig. 5.2 where the torque data are presented in overlapping branches, the ow exhibits bistable behavior with associated abrupt changes in inner sphere torque. A representative time series of the torque in the rst bistable regime is shown in Fig. 5.3, with E = 2:1 10 7 and Ro = 2:13. The onset of the high torque state is abrupt, taking on the order of 10 rotations of the outer sphere. The torque overshoots the high state mean value at high torque onset by 10-15%. The end of 1 If you hadn?t read the preceding chapters... 197 10 6 10 7 10 810 9 10 10 10 11 10 12 10 13 G Re = ? i? 2 Fig. 5.1: The dimensionless torque G vs. the Reynolds number Re for the case o = 0, stationary outer sphere. This is the same as Fig. 4.7, reproduced here to keep this gure closer at hand and and to keep the ow of this section as written. We recall that the torque is monotonic with Re and the dashed line is the t G = 0:003Re1:89. 198 0 0.5 1 1.5 2 2.5 3 3.5 G Ro 0 14 x10 10 12 x10 10 10 x10 10 8x10 10 6x10 10 4x10 10 2x10 10 H L LL Fig. 5.2: Mean value of the dimensionless torque G vs. Ro at E = 2:1 10 7. Ro is varied by increasing the inner sphere speed in steps of Ro = 0:067, waiting about 450 rotations per step. Three curves are di erentiated by circles, triangles, and squares. In the ranges of Ro where the symbols overlap, ow exhibits bistable behavior. Several other ow transitions exist below Ro = 1:5 but do not involve signi cant measurable torque variation. H and L denote the torque curves of the the \high torque" and \low torque" states discussed from here on. There is a second bistable regime starting around Ro = 2:75, the lower torque state of which we label LL. 199 the high torque state exhibits a slow exponential decay of the torque to the low torque value with a mean characteristic time of 40 rotations. In addition to the full transitions between the two torque levels, there are \excursions" where the torque decays toward the low mean value or rises toward the high mean value without fully reaching the other state. The qualitative bistability in Fig. 5.3 manifests itself quantitatively in the bimodal probability distribution of the torque shown in Fig. 5.4. The division of data into high torque and low torque states was done manually so as to exclude the transition regions. The resulting individual distributions of G for the high and low states are also shown in Fig. 5.4. There is a small region of overlap in the \high" and \low" state individual distributions due to the di erence drawn between \transitions" and \excursions." The same manual division in states is used throughout this chapter to condition other data on state. With the division as shown, the high state mean torque is 1.4 times that in the low state. The torque uctuations in the high torque state are much higher than in the low; the standard deviation of the high state torque is 1.8 times that in the low state. These low frequency uctuations are clear in the velocity and wall shear, and this will be further discussed in Sections 5.3.1 and 5.3.2. In both Fig. 5.3 and Fig. 5.4 the torque data has been numerically low pass ltered. In Fig. 5.4, the lter cuto frequency was chosen where the apparent uid torque power spectrum crosses the mechanical vibration noise oor, thereby retaining the hydrodynamically relevant uctuations. The interval between transitions is somewhat irregular, and it is di cult to 200 1000 2000 3000 4000 5000 6000 ?ot/2? G 01 01x 01 01x 01 01x 01 01x 01 01x 01 01x 01 01x6 7 8 9 10 11 12 H H H H H H HL L L L L L L Fig. 5.3: Time series of G at xed Ro = 2:13 and E = 2:1 10 7, with time made dimensionless by the outer sphere rotation period. The raw torque signal has been numerically low pass ltered (fc = 0:05 Hz, 15 rotations of the outer sphere.). 201 10 ?13 10 ?12 10 ?11 10 ?10 G Pr(G) 6 8 10 1210 10 x10 10 x 10 10 x 10 10 x H L Fig. 5.4: Probability density of the dimensionless torque at Ro = 2:13; E = 2:1 10 7. The full distribution is denoted by small points. Solid circles denote conditioning on low torque state, and open circles on the high, with Gaussian solid and dashed curves for the low and the high respectively. The mean and standard deviation of the low torque state data are hGi = 6:82 1010 and G = 2:74 109. In the high torque state they are hGi = 9:53 1010 and G = 5:02 109. The data were low pass ltered at fc = 0:5 Hz. 202 get good statistics on the state switching due to the slow dynamics, which allow observation of only 10-20 transitions per experimental run at a xed pair of system parameters. However, over 45 transitions at the parameters in Fig. 5.3, we observe the statistics shown in Table 5.1. Tab. 5.1: Statistics of the interval between high torque onsets. t0H is the time interval between two subsequent high torque onsets made dimensionless by o=2 , so the time interval is measured in outer sphere rotations. Ro = 2:13, E = 2:1 10 7. These are the mean, standard deviation, maximum and minimum of the interval between high-going transitions. h t0Hi t0H Max( t 0 H) Min( t 0 H) 717 313 1917 390 We also observe that the probability the system is in one state or the other is a function of Ro in the bistable range. Above a threshold value of Ro, we begin to observe state transitions to the low state, and as Ro increases, the high torque state becomes less likely. Fig. 5.5 shows the probability, based on measurements over 4000 rotations, that the system is in the high or low torque state over the rst range of Ro where state switching is observed. Outside of this range, transitions were not observed for more than 4000 rotations of the outer sphere, and those points have been assigned a probability of one or zero accordingly. The probability that the system was in the low state was t to Pr(L) = 8 >>< >>: 0 : Ro < Roc 1 exp( (Ro RocRoc )) : Ro > Roc (5.1) with = 8:25 and Roc = 1:80. The high torque probability is given by Pr(H) = 1- Pr(L). The physical implication inherent in the exponential form is that there is no 203 1.6 1.8 2 2.2 2.4 2.6 0 0.2 0.4 0.6 0.8 1 Ro Pr(H,L) L H Fig. 5.5: Probability that the ow was in the high torque (open circles) or low torque (closed circles) state as a function of Ro over the rst bistable range with xed E = 2:1 10 7. The dashed and solid lines are ts to the exponential form of Eq. (5.1). upper threshold where the high torque state becomes impossible but only becomes less likely as Ro is increased. However, the lower threshold for state transitions is well de ned at Roc. 5.2.2 Net Angular Momentum The torque measurements presented so far have focused only on the inner sphere. To get a complete picture of the angular momentum transport processes in the system, we must examine the torque on both boundaries. To do so, we record the torque reported by the variable frequency drives. Therefore, we are able to detect 204 the uctuating uid torque on the outer sphere, though with less sensitivity than our inner sphere measurements. Perhaps the simplest expectation for the torques would be that the inner sphere torque level would be promptly mirrored by the outer sphere torque. That would indicate that the angular momentum ux across the gap switches between two statistically steady values depending on the internal ow state, and furthermore that the net angular momentum of the uid should be constant when averaged over the turbulent uctuation timescales. This must be true with su ciently long averages. The inner and outer boundaries are rotating at constant speeds of 2.35 Hz and 0.75 Hz. So, on averages over arbitrary long times the uid can neither be accelerating or decelerating. However, the net torque on the system presented in Fig. 5.6 reveals long periods of angular acceleration and deceleration interspersed with plateaus where the net angular momentum of the uid remains nearly constant. The net torque is shown in Fig. 5.6 along with the separate inner and outer torques with the steady outer sphere bearing and aerodynamic drag subtracted. Up to a constant of integration, we can calculate the net angular momentum L(t) about the rotation axis from the net torque, L(t) = Z t 0 Tnet(t 0)dt0: (5.2) Due to measurement limitations, we cannot integrate the torques from the start of the boundary rotation where we know the initial angular momentum is zero. Instead, we set the initial value of L to zero at an arbitrary time and make the resulting quantity dimensionless by dividing it by the angular momentum the uid 205 would have if it were in solid body rotation with the outer sphere, L0 = L I uid o : (5.3) The moment of inertia of the uid lling the gap is I uid = 8 15 (r5o r 5 i ); (5.4) which has a value of (1:14 0:02) 104 kg m2. At the onset of the high torque state, as at Fig. 5.6(a), there is a prompt response of the outer sphere torque, indicating a certain amount of increased angular momentum transport. However, the increase in the torque on the outer sphere is insu cient to fully oppose the increased inner sphere torque. At this point, the net torque becomes steadily positive, and the uid accelerates. As this happens, the uid torque Go on the outer sphere tends to slowly decrease in magnitude, though with large uctuations. Eventually the inner sphere torque G starts the slower transition to the low torque state. At a point during the high to low transition, as at Fig. 5.6(b), the net torque becomes negative and the the net angular momentum starts to decrease. The torque on the outer sphere continues a slow decay toward a value opposite and equal to that on the inner sphere, sometimes reaching a net torque uctuating about zero as in times after Fig. 5.6(c). Overall, the torque measurements in the rotating system show a behavior that contrasts sharply with the case with the stationary outer sphere. When both spheres rotate the angular momentum ux in the range of Ro considered here may be partially a slow \store and release" process where long lasting imbalances in the torques on the inner and outer spheres can lead to uid spin up and spin down. It 206 L? ?0.9 0 0.9 1.8 x10-2 x10-2 x10-2 ?ot/2? 6000 6500 7000 7500 8000 0 10 x1010 ?10 x1010 G Gnet Go a b c Fig. 5.6: The angular momentum, inner torque, outer torque, and net torque. Ro = 2:13, E = 2:1 10 7. The upper plot shows the dimension- less angular momentum L0 as de ned in Eq. (5.3). The lower plot shows the inner torque G, the outer torque Go, and their sum Gnet. The bearing and aerodynamic drag on the outer sphere have been subtracted o , and the torques have been low pass ltered as in Fig. 5.3. 207 is worth noting that the average magnitude of torque is similar between the outer stationary and the outer rotating cases for similar di erential rotation across the range of Fig. 5.2. In both cases the mean torque on the inner sphere is between G = 1010 and G = 1011. Although the dynamics of the angular momentum transport seem quite di erent, the magnitude of the transport has not changed drastically. 5.3 Direct Flow Measurements 5.3.1 Mean Flow Measurements All measured ow quantities undergo transition in a qualitatively bistable way with the inner torque. The time averaged wall shear and measured velocity near an instrumentation port decrease sharply the onset of the high torque state, as shown in Fig. 5.7. The velocity is measured in a shallow range near the outer sphere surface at 23:5 colatitude, 60 cm cylindrical radius from the axis of rotation. This location is about 9 cm outside the cylinder tangent to the inner sphere equator. We calculate the time averaged azimuthal velocity at this location, hu it = humeasit sin(23:5 ) ; (5.5) and make that dimensionless by the outer sphere equatorial tangential velocity, hu0it = hu it oro : (5.6) Ultrasound doppler velocimetry only measures the component of the velocity parallel to the beam axis. In this experiment, the beam from the transducer near the outer 208 sphere is inclined 23:5 from pointing straight down, and lies in the plane normal to cylindrical radius at the transducer location. This orientation of the transducer axis makes it sensitive to the cylindrical radial, azimuthal, and vertical velocity components, us, u , and uz. Near the transducer, us has very little projection onto the beam direction. We expect little time averaged response to uz since z- independence and the experiment?s rotational symmetry make persistent vertical velocities at the probe location unlikely. So we believe that Eq. 5.5 is justi ed. The distance along the beam y0 has been made dimensionless by the gap width ?. The wall shear stress sensor has been calibrated against the torque measure- ments from the data in Fig. 5.1 with the outer sphere stationary. The mean wall shear w on the outer sphere at the probe cylindrical radial location so was assumed to be w = T 4 r2oso : (5.7) This assumes that the wall shear falls o linearly with cylindrical radius. This is motivated by the nearly constant angular momentum pro le seen in turbulent Taylor-Couette [121]. Under these assumptions, we t the constant temperature bridge voltage V and mean wall shear calculated from the measured torque by Eq. (5.7) to V 2 = A 2=3w +B 1=3 w + C (5.8) as in [128]. We then use the calibration coe cients A,B and C to calculate the wall shear from the measured bridge voltage for subsequent runs with the outer sphere rotating. 209 2000 3000 4000 5000 G u? ?0.1 0 0.1 0.2 ?ot/2? 0.03 0.06 0.09 ?? w y? 11 01 01x 9 01 01x 7 01 01x 10 14 18 01 01x 1 01x 0 1 01x 0 < >t Fig. 5.7: Velocity, wall shear stress, and torque. The top plot is the mea- sured velocity u0, de ned in Eq. 5.6 as a function of distance from the transducer and time. It is time averaged over 30 rotations of the outer sphere. This measurement should be dominated by the azimuthal velocity u . The velocity u0 is made dimensionless by the outer sphere tangential velocity, and so can be interpreted as a lo- cally measured Rossby number. The middle plot is dimensionless wall shear stress 0w = 4 r 3 o w=( 2ri), and the lower plot is the di- mensionless torque G. The wall shear stress and torque have been low pass ltered with fc = 0:05Hz as before, which is comparable to the time averaging of the velocimetry. 210 In contrast with observed behavior with outer sphere stationary, the wall shear at the measurement location on the outer sphere does not always increase with the inner sphere torque. At xed parameters in the torque switching regimes, the time averaged wall shear behavior is anti-correlated with the inner sphere torque. The non-monotonic behavior of the wall shear measured at the sensor location cannot be representative of the integrated shear stress over the whole outer sphere. Fig. 5.6 shows that the torque the uid exerts on the whole outer sphere is indeed greater in the high torque state. However, the mean shear stress at the measurement location in the high torque state is only 65% of that in the low torque state (see Fig. 5.9). This indicates a di erent latitudinal pattern of shear stress on the outer sphere. This observation is consistent with the mean velocity measured near the probe location. In the high torque state, the measured velocity hu0it for y0 > 0:05 averaged over 30 rotations drops to 41% of its low torque state value. Using dye injections and a camera mounted in the rotating frame, we have also observed visually that the azimuthal zonal circulation essentially stops during the short duration torque overshoot. Despite the increase in angular momentum input into the system from the inner sphere, and the increase in angular momentum ux from boundary to boundary, the time averaged uid velocity and transfer of momentum to the wall at the sensor locations show a sharp decrease at the onset of the high torque state. It is clear from the shear and velocity measurements that the low torque state is associated with something like a fast central zonal ow. This central circulation drops suddenly at the high torque onset, leading to greatly increased torque on the fast moving inner sphere. When the uid around, above, and below the inner sphere 211 is circulating faster, there is less torque on the inner sphere. The unusual aspect of this is that it takes less torque, and therefore less power input, to maintain the faster circulation. This suggests formation of a transport barrier to energy and angular momentum in the low torque state. This change in transport is also important for the observation that the total angular momentum is decreasing (See Fig. 5.6, line (b)), while the low torque state fast circulation is growing. 5.3.2 Flow Fluctuations The turbulent uctuations in the system show a signi cant change through the transition. The measured velocity uctuations in the high torque state are larger than those in the low torque state, although with a lower mean. Fig. 5.8 shows the probability density of the dimensionless velocity conditioned on the low and high torque states, with Gaussian curves for comparison. The mean velocity measured in the low torque state is 2.45 times that seen in the high torque state. However, the uctuations in the high torque state are 1.5 times the low torque state uctuations. Based on this measurement, the high torque state has a turbulence intensity u0=hu0i, of 57% while that for the low torque state is 16%. Some of the di erence in uctuation intensity could potentially be attributed to di erent velocity anisotropy between the two states. The assumption of insensitivity to vertical veloc- ities discussed in Sec. 5.3.1 is only reasonable for time averages, not for uctuating velocities. The wall shear uctuations are similar in magnitude between the two states. Wall shear distributions are shown in Fig. 5.9. The low torque state mean wall shear 212 ?0.1 0 0.1 0.2 0.3 0.4 10 ?2 10 0 u? Pr( )u? LH Fig. 5.8: The probability density of the dimensionless velocity conditioned on the torque state, with solid circles denoting the low torque state and open circles denoting the high. The dashed and solid lines are Gaussian curves with the same mean and standard deviation as the data set. The standard deviation in the high torque state is u0 = 0:043, and in the low, u0 = 0:029. The high state mean hu0i = 0:076 and the low state mean hu0i = 0:186. These data are for a distance from the transducer y0 = 0:03, with the transducer beam 45 from the vertical in the same plane as in Fig. 5.7. The velocity is divided by sin(45 ) under the assumption of azimuthal dominance discussed above. Velocity sampling rate was 100 Hz 213 5 10 15 20 25 10 ?6 10 ?4 10 ?2 Pr(? ? w ) ? ? w x10 10 x10 10 x10 10 x10 10 x10 10 LH Fig. 5.9: The probability density of the dimensionless wall shear conditioned on torque. Solid circles again denote the low torque state, and open circles the high, with Gaussian solid and dashed curves. The mean and standard deviation in the low torque state are h 0wi = 1:63 10 11 and 0w = 1:81 10 10. In the high torque state they are h 0wi = 1:06 1011 and 0w = 1:71 10 10. 214 10 ?1 10 0 10 1 10 2 10 ?8 10 ?6 10 ?4 10 ?2 PSD L H ? ?o Fig. 5.10: Power spectra of wall shear stress at Ro = 2:13 and E = 2:1 10 7. Frequency has been made dimensionless by the outer sphere rota- tion rate. The black curve is the spectrum from the low torque state, and the gray curve is that in high torque state. The low torque spectrum has prominent peaks at != o = 0:18 and at != o = 0:71 and its harmonics. In the high torque state there are slight peaks at != o = 0:40 and 0:53 and a broader buildup around != o = 0:22 is about 1.5 times that in the high torque state, and the standard deviation between the two states is within 5%. In the low torque state, 0w=h 0 wi is 11% compared to 16% for the high torque state. The high torque state wall shear distribution is signi cantly skewed, with a skewness of about 0.6. Gaussian curves with the same mean and standard deviation as the measured distributions are included for comparison. High wall shear uctuations are signi cantly more likely in the high torque state. 215 The ow uctuations are broadband overall. Kelley et al. [14] made the case for strong inertial waves on a somewhat turbulent background ow with comparable absolute magnitude of Ro. However, without ow measurements little could be said about the nature or strength of the turbulence in the ow. The turbulence is clear here from the direct measurements of Fig. 5.8-5.10. However, power spectra of the ow quantities, shown for the wall shear in Fig. 5.10, reveal prominent narrow frequency peaks. These may be inertial or Rossby waves or coherent superpositions thereof. The high torque state is characterized by stronger broadband low frequency uctuations. The slow uctuations in wall shear and velocity in the high torque state are very similar to the torque uctuations in that state. The weak peak around 0:40 o in the high torque state spectrum appears to be associated with the same wave as that responsible for the 0:71 o peak in the low torque spectrum. This wave increases in frequency and strength through the high to low torque transition until the frequency remains steady at 0:71 o. Again, the high to low transition involves a steadily increasing azimuthal circulation at the measurement location. The increasing frequency of this wave suggests that it is being advected by the azimuthal ow. That the frequency of this wave increases with increasing Ro is more evidence of this advection. This can be seen in the spectrogram in Fig. 5.11. If the 0:40 o peak in the high torque state and the 0:71 o peak in the low torque state are the same wave, we have some extra information about the relationship among the wave frequencies observed in the system. The frequency of this wave observed in the frame rotating with the outer sphere, !obs, can be written in terms of a natural oscillation frequency !w and azimuthal wavenumber m in a 216 frame rotating faster than the outer sphere by an amount 0, !obs = !w +m 0: (5.9) Assuming that the ratio of wave advection angular speeds 0L= 0 H is that of the means of the velocity distribution in Fig. 5.8 and noting the measured frequencies at the wave onset and in the steady low torque state, we can write a system of equations for wave?s frequency !w in the faster rotating frame: 0:71 o = !w +m 0 L (5.10a) 0:40 o = !w +m 0 H (5.10b) 0L = 2:45 0 H: (5.10c) For any m, the solution for the wave frequency in the frame rotating with angular speed 0 is !w = 0:186 o, near that of the other, steady frequency observed wave. There is a sign ambiguity due to the fact that we do not know if the waves are prograde or retrograde. So not only is the 0:71 o wave in the low torque state advected by the fast inner circulation, but it seems to be the result of nonlinear interaction between the other observed steady frequency wave and the azimuthal circulation. Fluctuation spectra as a function of Ro over a broader range allow us to delin- eate the observed ow states more clearly. Fig. 5.11 shows a spectrogram of the wall pressure at one sensor location as a function of Ro for xed E = 1:3 10 7. The logarithmic color scale denotes the power in each frequency bin, and the vertical axis is frequency made dimensionless by the outer sphere rotation speed. Below 217 5.00 5.0 1 5.1 2 oR H L LL ? ?o 1.0 2.01.5 2.5 3.5 4.03.0 7-6-5-4-3-2-1-01 log(PSD) Fig. 5.11: A spectrogram of wall pressure at 23:5 colatitude shows evi- dence of several ow transitions as Ro is varied. E = 1:3 10 7 and 0:1 Ro 4:4, waiting 430 rotations per step with steps of Ro = 0:1. Instead of averaging the power spectra over an entire step of Ro, there are 30 spectra per step in Ro, so some tempo- ral evolution is visible at a given Ro. This gure is identical to Fig. 4.13, reproduced here for the sake of the detailed discussion. 218 Ro = 1 there are a number of di erent wave states, some of which are visible here. It is possible that at extremely low Ro there are transitions similar to those seen in simulations by Hollerbach et al. [109], however at these low Ekman numbers, the Stewartson layer instability onset is probably much lower Ro than the slightest di erential rotation we can achieve [35]. At similar E with Ro somewhat lower than the lowest here, Schae er and Cardin [33] report Rossby wave turbulence in numer- ical models. It may be that we have observed such turbulence, and investigation of those model predictions may be fruitful in the future. As Ro is increased, we would not expect agreement; the quasigeostrophic model used to achieve this low E at modest Ro uses depth averaged equations and therefore cannot exhibit three dimensional e ects. The waves visible at lowest Ro in Fig. 5.11, are probably over- re ected inertial modes [14]. These are already three dimensional, and we might guess that the higher Ro behavior involves further 3D motion. We do not focus on lower Ro here, but we note that a variety of di erent turbulent ows are likely present in a small region of parameter space. Between Ro = 0:5 and about Ro = 1:0 there are a pair of waves with frequency increasing with Ro. A transition to the high torque state ow occurs around Ro = 1. This ow is more turbulent but still shows some evidence of the waves in the prior state. It is unclear whether this transition can be reliably detected in the inner sphere torque, although there seems to be some e ect. The high torque state ow starting near Ro = 1.0 is observed alone until the Rossby number exceeds Roc = 1:8. Between Ro = 1:8 and Ro = 2:4 in Fig. 5.11 several transitions between the high and low torque states are noted. Above the rst observed torque switching regime, 219 the waves dominating the low torque spectrum in Fig. 5.10 remain. The frequency of the wave at 0:18 o varies only a small amount with Ro over the measured range, while the higher wave frequency increases with Ro. Again, this is consistent with advection of the higher frequency wave by increasing azimuthal velocities. The nature of this higher frequency wave is still unknown. One possibility is a Rossby wave of one particular wavenumber; Rossby waves are advected by the mean ow. The multipole vortices studied by Van Heijst, Kloosterziel, and Williams [52, 129] are another possible candidate. These consist of a central cyclonic vortex with weaker anticyclonic satellites. The whole multiple vortex structure slowly rotates about the central cyclone axis. Our observations would be consistent with what Van Heijst et al. call a tripole, two satellites on a central vortex. This structure appears to be nonlinearly stable even at similar E to our experiment and Ro > 1 based on the relative velocity of the stirred vortex. The lower frequency wave may be an inertial mode of the sphere. The inertial modes observed in spherical Couette ow at lower Ro oscillate at a frequency near the natural frequency of the corresponding full sphere mode over wide ranges of Ro [14]. An inertial mode of the full sphere exists near the observed frequency. That mode has N=1, m=1, ! = 0:1766 in the notation of Zhang et al. [10], or (3,1,-0.1766) in the (l;m; !)notation of Greenspan [6] and Kelley et al. [14]. The spectrum of the ow associated with the \LL" curve in Fig. 5.2 is clearly present in this spectrogram beyond Ro = 3.3. With longer residence times at each step of Ro, intermittent transitions between the \L" and \LL" states can be observed at lower Ro consistent with Fig. 5.2. This transition from the ?L" state to the \LL" 220 seems to involve an increase in the wave speed and intermittency of the higher frequency \L" state wave, resulting in a more turbulent, broadened frequency peak. The lower frequency wave remains relatively una ected. As in the \H" to \L" transition on which we?ve focused, the \L" to \LL" transition is also accompanied by an increased mean azimuthal circulation and an increase in the local wall shear at the measurement location. The ow spectra and other uctuation measurements provide a clear distinc- tion among the various turbulent ow states present in rapidly rotating, strongly sheared spherical Couette ow. We expect that further measurements will help illu- minate the precise nature of the observed waves and allow us to understand whether they have a prominent, primary role in the overall dynamics observed here or are present as a peculiar secondary feature of the turbulent ow. 5.4 Discussion We hypothesize that the state transitions discussed here involve formation and destruction of a fast zonal circulation at small cylindrical radius. The fast mean circulation and strong local shear in the low torque state, along with the falling total angular momentum of Fig. 5.6 mean that there must be a change in the shape of the angular mean angular velocity pro le (s) from that of the high torque state. The ow must change to favor, on average, fast uid at the center of the experiment and slower uid at larger radius. A simple possibility would be a central zonal circulation bounded by a sharp shear layer with waves, shown schematically in 221 (a) L (b) H Fig. 5.12: A sketch of the possible mean ow states. The low torque state is labeled L and the high labeled H. The low torque state at (a) is characterized by fast zonal circulation near the core of the ex- periment and attendant waves. The waves may exist throughout the sphere. In the high torque state, the velocity pro le must be atter. The zonal circulation has been destroyed by angular mo- mentum mixing, so the uid near the inner sphere is slower and the torque higher. Fig. 5.12(a). The abrupt transition to the high torque state may involve a collapse of this fast central zonal ow. The very short timescale of the azimuthal ow crash, 10 rotations of the outer sphere, suggests that rapid mixing is responsible for the crash; it would be di cult for other processes to act quickly enough. So, we might expect a fast relaxation to a less steeply varying angular velocity pro le, as in Fig. 5.12(b). The bistable regime we have focused on here is in a part of parameter space that is far from the known basic states discussed in Section 2.3 for spherical Couette 222 with and without overall rotation. We are also far from previously studied turbulent regimes. So, it is not obvious what ows to expect when Ro is moderate and E is very low. The basic state spherical Couette ow for low E and in nitesimal Ro shows a free shear layer on the cylinder tangent to the inner sphere equator, the Stewartson layer [102]. Outside the Stewartson layer the uid is locked in solid body rotation with the outer sphere [103, 130, 131]. No angular momentum is transported across the Stewartson layer. Instead, high angular momentum uid from the Ekman boundary layer on the inner sphere is pumped vertically to the outer sphere inside the Stewartson layer. This layer also resolves the angular velocity jump across the tangent cylinder. In quasigeostrophic simulations by Schae er and Cardin, at E comparable to that in our experiment, the Stewartson layer becomes unstable at very low Ro, Ro 10 3, [35, 108]. These simulations nd Rossby wave turbulence lower than any Ro lower than any presented here [33]. The angular velocity pro le (s) in the Rossby turbulence regime predicted by Schae er and Cardin still shows evidence of a concentrated shear like the Stewartson layer. The inertial wave overre ection presented by Kelley et al. [14] demonstrates that three dimensional uid motions become important at low E and moderate Ro. However, the ow with strong inertial waves is still very di erent from that with the outer sphere at rest. When the outer sphere is stationary, an equatorial jet erupts radially out- ward from the inner sphere equator, carrying angular momentum toward the outer sphere [101, 104, 132, 133]. The boundary layer on the outer sphere pumps uid poleward. This forms a large scale poloidal circulation throughout the volume even at low Re. At higher Re, a series of nonaxisymmetric instabilities and transition 223 to turbulence are observed in wide gap ow [101]. The equatorial jet persists in the turbulent ow [105]. The mean angular velocity pro le in highly turbulent ow with outer sphere stationary is near (s) s 3=2 [112, 118]. We hypothesize that we could approach this state, plus an overall constant, at high enough Ro that the di erential rotation completely dominates the dynamics. The shape of the mean angular velocity pro le, not just its overall magnitude, is very di erent between the previously studied turbulent states. We might expect that each turbulent ow transition could involve a signi cant change in the the mean angular velocity pro le. Complex patterns of zonal ow (along with waves and localized vortices) are a prominent feature of rapidly rotating turbulent ows. The schematic ows of Fig. 5.12 are presented as two of the in nite number of possibilities consistent with the observations we present. In the low torque state schematic, Fig. 5.12(a), the azimuthal ow outside the central zonal circulation is slow, but not locked to the outer sphere. The torque measurements of Fig. 5.6 suggest it spends a long time slowing. The Ekman suction on the outer boundary drains angular momentum from this exterior uid region faster than the ux across the zonal ow boundary can replenish it until an equi- librium is reached. The boundary of the zonal ow supports wave motion; again, these waves could be Rossby waves or other vortex structures. The waves could extend throughout the uid volume or be fairly localized. The abrupt onset of the high torque state is associated with a destructive instability of the zonal ow. The ensuing angular momentum mixing provides a mechanism for the sudden reduction in the observed azimuthal circulation. When this happens, the inner sphere, now 224 surrounded by much slower mixed uid, demands more torque from the motor. The angular momentum ux from the inner sphere now exceeds the ux out of the sys- tem from Ekman pumping on the outer sphere. Overall, the net angular momentum begins to increase. However, since the fast zonal circulation near the experiment core is gone, the measured velocities at small radius remain slow. The increased angular momentum ux from inner sphere to the bulk uid persists throughout the high torque state, and the uid angular momentum rises. At the transition from the high torque to low torque state, the slow exponential decay of the inner torque and rise of the observed uid velocity and wall shear means that the angular momentum transport barrier bounding the central zonal ow has re-formed, but the inner sphere must still provide angular momentum and energy to spin up the fast central ow. At this point, since the exterior uid is now weakly coupled to the fast inner sphere, it starts to spin down. There is one direct piece of supporting evidence for Ekman circulation related to spin down of uid outside a central zonal ow in the existing velocimetry. Beyond y0 = 0:05 in Fig. 5.7, the observed velocity is high and positive, consistent with a fast prograde circulation. However, the the negative going jet for y0 < 0:05 in Fig. 5.7 is correlated with the angular momentum, and is probably a manifestation of the Ekman circulation on the outer sphere. The data in Fig. 5.13 and that in Fig. 5.7 is a ected by unusual probe mount- ing geometry. In this data, the port insert with a ush mounted ultrasound probe is turned 90 degrees from the orientation that makes the plug insert ush with the inner sphere surface. This forms a wedge shaped cavity. So this measurement is 225 ?0.1 0 0.1 3000 3500 4000 4500 5000 5500 6000 6500 ?0.04 ?0.02 0 0.02 0.04 ? o t/2? t L? Fig. 5.13: Jet velocity, time averaged and depth averaged from y0 = 0:02 to y0 = 0:025 and angular momentum, Ro = 2:13, E = 2:1 10 7. The velocity in the observed jet in Fig. 5.7 peaks at the same time the angular momentum starts to decrease. It is likely that this jet is related to the Ekman circulation on the outer sphere, though it is not likely to be a direct observation of ow in the Ekman layer, but rather its interaction with a wall cavity formed by the particular transducer mounting used for this run. 226 not necessarily a direct measurement of the Ekman layer on the outer sphere. If the Ekman layer is stable, it would be much too thin to see with our instrumentation. However, the correlation between the near-wall jet seen in Fig. 5.7 and this mea- surement is clear, and perhaps it is caused by the Ekman layer ow over and into the cavity. In subsequent experiments with less intrusive transducer mounting, we do not observe this near wall jet clearly. 5.5 Dynamical Behavior Geophysical and astrophysical systems exhibit bistability. The dynamo gener- ated magnetic elds of the Earth and Sun reverse polarity. Ocean currents, namely the Kuroshio current in the North Paci c near Japan and the Gulf Stream both exhibit bistability in meander patterns [134]. Multi-stability and spontaneous tran- sitions among turbulent ow states have been noted in a number of turbulent lab- oratory ows as well. The mean circulation in highly turbulent thermal convection cells is observed to switch direction abruptly [135]. Hysteresis in the large scale ow states is seen in surface waves excited by highly turbulent swirling ows in a Taylor Couette geometry with a free surface [136]. Von Karman ow in a cylinder between two independently rotating impellers has been shown to exhibit multi-stability and hysteresis of the mean ow despite very high uctuation levels at high Reynolds number [137, 138]. Magnetohydrodynamic experiments in the Von Karman geom- etry have succeeded in producing dynamos that show reversals of the generated magnetic eld [76, 77] from a very turbulent ow. 227 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 0 0.05 0.1 0 0.02 0.04 0.06 0.08 u?(t-? t)/Ro u?(t)/Ro u?(t)/Ro u?(t)/Ro (a) Ro = 1.80 (b) Ro = 2.13 (c) Ro = 2.40 L H H L Fig. 5.14: Time delay embeddings of the slow velocity uctuations at y0 = 0:02 for three values of Ro below, in, and above the bistable range. Arrows in (b) show the direction taken by the transitions. The time delay t is 4.5 rotations of the outer sphere in all three cases. The dimensionless velocity as de ned previously has been scaled by Ro, roughly collapsing the mean velocity and uctuation levels of each state. These data have been low pass ltered with fc = 0:1 Hz, or a period of 7.5 rotations of the outer sphere. 228 That we can see bistable behavior requires a special sort of separation of spatial structures or time scales. We must be able to recognize the bistable behavior beyond the turbulent uctuations and their high dimensional dynamics. If a few ow states well separated in phase space play a major role in de ning the dynamics, ltering and projection may provide some insight. In many ows, however, we rely on observation of changes in mean ow or low frequency dynamics to provide information on what sort of ltering and projection to use. A priori knowledge of relevant states analogous to those relevant in pipe ow and plane Couette ow [139{ 142] near turbulent transition may allow suitable projection of the full dynamics onto a lower dimensional subspace of possible uid motions. There are many uid degrees of freedom that we recognize as irrelevant in the de nition of the observed multiple \ ow states." The actual ow in each is associated with a trajectory through a very high dimensional phase space, and the division of that space into a small number of regions that correspond to di erent \states" can be non-obvious. In this sense, the novelty of bistability in experiments may be that we notice it at all. We investigate phase space trajectories using low dimensional time delay em- bedding of low pass ltered data. Figure 5.14 is a 2D embedding of a low pass ltered velocity time series. The lter is a 4th order Butterworth with a cuto frequency fc = 0:1 Hz. This corresponds to a period of 7.5 rotations of the outer sphere. This velocity signal was plotted against the same signal 4.5 rotations prior. Data are shown for three di erent values of Ro at E = 2:1 10 7. At (a), Ro = 1:8, which at the low threshold of the rst bistable range, in (b), Ro = 2:13, the same as in Fig. 5.3, where the bistable switching is evident, and in (c), Ro = 2:4, above 229 the range where spontaneous transitions are observed. The velocity is rescaled here by the outer sphere tangential speed to make it dimensionless. It is also scaled by Ro, the expected dimensionless velocity scale relative to the outer sphere. We expect that within each state, the mean measured velocity increases with Ro. Scaling by Ro seems to equalize the mean velocity observed in the pure high torque state in Fig. 5.14(a) and that observed in the pure low torque state in Fig. 5.14(c) with the mean velocities of the corresponding states in Fig. 5.14(b). Figure 5.14(b) seems to represent a heteroclinic connection between two turbulent attractors, with connections between the high torque ow state in (a) and the low torque ow state in (c). The arrows in Fig 5.14(b) show the direction of ow in phase space as the system undergoes several transitions. Below the critical Rossby number in Eq. 5.1, we expect all connections to be broken between the two di erent attractors. Above the bistable range, we recall the probability of state as a function of Ro of Eq. 5.1 and Fig. 5.5 and expect that connections between attractors do not break. Rather, they become extremely unlikely. 5.6 Magnetic Field Measurements We have performed some experiments in the 60 cm experiment in an attempt to reproduce the H/L transition switching state, and we have evidence that it is reproducible there. These magnetic eld measurements are not included in the Physics of Fluids submission but are added to this chapter because they t best with the detailed discussion of the H/L state transitions herein. We do not know 230 to what extent magnetic eld measurements of the bistable torque regime will be performed in the future2, and we wish to include these observations here even if a few discrepancies could bear more investigation. We will propose resolutions of these discrepancies, but these should be tested. Figure 5.15 shows the torque normalized by its low state value and one coef- cient3 of the spherical harmonic expansion of the cylindrical radial induced eld outside of the 60 cm experiment. This data was taken using the three dimensional array of Figure 3.24 (b). The Gauss coe cient plotted in Fig. 5.15 is that for the spherical harmonic with l = 3 and m = 1. This coe cient is not a signature of the (3,1,-0.1766) inertial mode, but rather of the higher frequency wave that, in the rest of this discussion, lies at 0:71 o when Ro = 2:13. The g31 Gauss coe cient is clearly correlated with the torque transitions. The torque behavior is roughly consistent with the H/L transition in magnitude and dynamics. The g31 coe cient is not identi ed with the l = 3, m = 1 dominated velocity eld of the inertial mode since the induced magnetic eld arising from the linear induction of an l = 3 velocity eld acting on the axial, axisymmetric applied mag- netic eld will be comprised only of even polar wavenumbers. This is established by the velocity and magnetic eld selection rules laid out for the induction equa- tion by Bullard and Gellman [71]. The induced eld should be dominated by polar 2 We will try in the three meter sodium experiments, but it may be impossible to reproduce the same hydrodynamic states exactly if we nd dynamo action! 3 Known as a Gauss coe cient. 231 1 1.1 1.2 1.3 1.4 1.5 G rel 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 1 1.5 2 2.5 ? o t/2? g31 Fig. 5.15: Dimensionless torque (top) and magnitude of the l = 3, m = 1 coe cient in a spherical harmonic decomposition of the induced cylindrical radial magnetic eld outside the 60 cm experiment. This coe cient of the spherical harmonic expansion of the magnetic eld is also known as a Gauss coe cient. The torque data is consistent with the observations in the H/L transitions, with the H state about 1.3 times higher than the L state. The torque is poorly temporally resolved because of the low sampling rate of the motor drives on the 60 cm experiment, but the rst L to H transition appears to show overshoot, and the H state decays seem slower than the onsets. The g31 Gauss coe cient is unmistakably present only in the low torque state, and one brief excursion around 1750 outer sphere rotations is noted. The source of the g31 coe cient is discussed in the text. For this data, Ro = 2:7, E = 3:2 10 7, which is higher Ro than what we would predict based on the three meter data. This is discussed in the text. 232 10 -2 10 0 PSD g21 H L 0 1 2 3 4 5 6 7 10 -2 10 0 ?/?o PSD g31 H L Fig. 5.16: Induced magnetic eld spectra from the 60 cm experiment at Ro = 2:7, E = 3:2 10 7. The largest peak in the g21 spectrum is at a frequency near 1:18 o, which is what should be seen for the (3, 1, 0.1766) wave doppler shifted into the lab frame. Interestingly, the main e ect of the state transition is that the peak is sharper in the L state. The g31 coe cient is clearly the one that turns on and o with the state transitions. Its frequency is 1:87 i, consistent with the higher frequency L wave peak at this Rossby number. wavenumber l = 2, arising from the interaction of the l = 3 velocity eld with the l = 1 dipole component of the applied eld, and the magnetic eld induced the axisymmetric (m = 0) applied eld will preserve the azimuthal wavenumber of the underlying velocity eld. Therefore, if the inertial mode (3, 1, -0.1766) is present, its magnetic signature will be in the g21 Gauss coe cients, which have l = 2, m = 1. Figure 5.16 shows power spectra for the g21 and g31 Gauss coe cients condi- 233 tioned on the torque state. The g21 coe cient shows a frequency consistent with the (3, 1, -0.1766) inertial mode. The negative frequency inertial modes are prograde4, and a m = 1 prograde wave will be doppler shifted5 upward by o, leading to the observed frequency near 1:18 o. There is not much change in the observed wave in the torque state conditioning, but we note from Fig. 5.10 that the direct ow mea- surements show uctuations that may obscure the in uence of the weaker 0:18 o wave at their location. We recall from Fig. 4.16 that the peak near 0:18 o persisted over much of the range of positive Ro. The fairly steady g21 peak in Fig. 5.16 might imply that this wave is present during the high torque state, but that local turbulent uctuations in the three meter data obscure it at the measurement location at 23:5 colatitude. Identi cation of the g31 coe cient with the higher frequency wave is not so straightforward and requires some discussion. First of all, this is the coe cent that changes most strongly with the torque, consistent with the three-meter observations. The dominant frequency in the g31 L state spectrum is consistent with what we would expect from the L state wave. The magnetic induction arising from this motion is m = 1, and assuming it is also prograde, the frequency 1:87 o in the lab frame corresponds to 0:87 o in the rotating frame. We can see from Fig. 5.11 that this is close to what we would expect for the higher frequency L state wave at this Rossby number. There are two di culties with the identi cation of the g31 coe cient with the 4 Due to the choice of ei!t instead of e i!t in the inertial mode ansatz. 5 !meas = ! +m o. 234 higher L state peak. The pressure measurements in the sphere frame imply an even azimuthal wavenumber pattern for the upper peak in the L state. Furthermore, the g31 coe cient is substantially weaker than the g21 coe cient, even though the upper L state wave is signi cantly stronger. These discrepancies would be explained if the higher frequency L state peak is associated with a Rossby type wave near the core of the experiment. Two dimensional motions acting on an axial, uniform eld cause no induction. The actual applied eld is not perfectly uniform and axial, nor are any motions precisely two-dimensional. However, the eld is probably close to a uniform and axial idealization near the axis of the magnet coils, and Rossby type waves will be very close to two dimensional. Therefore the induction by a 2D wave near the experiment core would be nom- inally invisible provided that the only applied eld were that which is intentionally applied. However, Earth also applies a transverse eld to the experiment. The transverse component of this eld is dominated by azimuthal wavenumber m = 1, and by the selection rules of Bullard and Gellman, a m = 2 velocity eld acting on a m = 1 applied eld may give m = 1 magnetic induction. The transverse eld applied by Earth, while weak, may interact with the very strong two-dimensional motion. This would explain the lacking strength and the azimuthal wavenumber discrepancy. We would expect weak induction with l odd (broadband in l) and m = 1 from the action of an m = 2 Rossby type wave interacting with Earth?s magnetic eld. The discrepancy in the onset Rossby number, Ro 2:5 instead of Ro = 1:8, for this state must be addressed as well. We identify this bistable regime with the 235 H/L transition, rst because it is the rst ow transition that has a signi cant e ect on the torque. Second, the relative change in the torque is about 30%, and as we noted in Chapter 4, the H/L transition has the biggest di erence in the mean torque between the two states, with the other torque dynamics not exceeding about 10%. Third, the frequencies seen in the magnetic induction are consistent with those of the L state at Ro = 2:7 when doppler shifted according to the observed magnetic azimuthal wavenumber. In contrast, the upper peak of the LL wave state measured in the rotating frame is always higher than 1:0 o. The 0:87 o peak observed here is therefore more consistent with the L state. The Rossby number discrepancy has a plausible explanation. We recall from Fig. 3.24 that the geometry in the 60 cm experiment is slightly di erent from that in the three meter experiment, with the inclusion of large shafts that rotate with the outer sphere. It may be that the delayed onset in Rossby is simply due to the need to spin the inner sphere a bit faster to achieve maintenance of the fast zonal ow and its associated transport barrier, given that it must interact with a large, slower shaft. Enough of the state characteristics match to make a strong case that this is the same physical process with the same waves, but with a shift in Ro related in a intuitively simple way to the geometrical di erence. If this ow transition is essen- tially identical to that observed in the three meter experiment, the apparently small e ect of the perturbed geometry is interesting. We observe a measurable di erence in the dynamics, but we also show some robustness of the physical processes to changes in geometry. The evidence presented here also corroborates the hypothesis 236 that the (3, 1, -0.1766) inertial mode is an important feature of the states where torque switching is observed, and it suggests that the other, higher frequency wave is more nearly two dimensional, like a Rossby wave. Indeed, a Rossby type wave propagating on the zonal ow responsible for the angular momentum transport bar- rier is consistent with the discussion in Sec. 5.3.2 and Sec. 5.4, and would explain the increasing wave speed as the zonal ow speed increases with increasing Ro. 237 6. TURBULENT SCALING 6.1 Torque Scaling 6.1.1 Torque Collapses In Chapters 4 and 5 we have established that a number of di erent turbulent ows exist in certain regions in the (Ro;Re), or equivalently, (Ro;E) parameter plane for high Reynolds number, rapidly rotating spherical Couette ow. Much of the prior discussion focused on the dynamics of angular momentum, with large changes in the driving torque required to maintain states at a given point in the parameter plane. We propose that at high enough Re, the dimensionless torque G can perhaps be expressed as G = f(Ro) g(Re): (6.1) We predict that the relation of Eq. 6.1 should hold approximately for all turbulent swirling ows when overall rotation is added. In general, G = h(Ro;Re), but we propose that at least as Re ! 1, this relation will factorize into Eq. 6.1, and we will present evidence that this is nearly the case in the three meter data. The function g(Re) in Eq. 6.1 should be similar to that for turbulent drag in all geometries. While we suggest that g(Re) may share similarities with other experiments, we may not go so far as to suggest that it should be universal. Even for well studied ows, this part of the relation 6.1 remains incompletely understood, and will depend on the geometry and the niteness of the Reynolds number, as well as the details of the boundary layers. The variation of G with Re can be complicated for nite Re, as seen in Taylor-Couette ow by Lathrop et al. [120], where a turbulent ow transition is noted as Re is increased. However, in a generic geometry, we expect that the function g(Re) expressing the Reynolds number dependence of the turbulent drag will be a monotonic function bounded from above by Re2 power law scaling, as is the case in other turbulent shear ows like Taylor and plane Couette ow, pipe ow, and others. Also, while we suggest a single function g(Re) is appropriate to capture the dependence on the overall velocity scale, with Re = UL= , our data suggests that this might not be true in spherical Couette ow until g(Re) has reached Re2, as might be the case in rough-wall shear ows. We propose that the relation f(Ro) of Eq. 6.1 completely expresses the in- uence of the di erent turbulent ow states discussed in previous chapters on the torque. It is a non-universal, geometry-dependent relationship. We have established that a wide variety of ow states may arise when Ro is varied, even in Re 107 turbulent ow. We observe drops in torque as Ro is increased and multi-stability, meaning that f(Ro) is not necessarily monotonic or single-valued1. We see that 1 Although we condition our data on ow state, the long term averages are single-valued. We do not, however, rule out parameter space path-dependence and hysteresis in turbulent swirling ows a priori. See, for example, Ravelet et al. [137] and Mujica and Lathrop [136]. 239 the e ects of rotation extend to rather high Ro. The observed dynamics, even at Ro = 100, are never exactly consistent with those at Ro = 1, so that the need to know f(Ro) to predict G is not at all limited to weak di erential rotation. It does seem, however, that f(Ro) is simpler at high Rossby number. The form of Eq. 6.1 implies that f(Ro) plays a similar role as the ow ge- ometry in other turbulent drag laws. The relation f(Ro) encapsulates both the geometry dependence and the particular transport properties of the distinct ow states. In other turbulent ows, the drag force is often represented as cf (Re)Re2, with the friction coe cient cf (Re) expressing the e ects of the ow geometry, in- cluding (implicitly) the behavior of the large scales and the e ects of wall roughness, while showing a residual Re-dependence based on viscous e ects in the boundary layers. At high enough Re, it is anticipated that cf (Re) asymptotes to a constant for a given geometry. However, when both boundaries of a turbulent swirling ow revolve, many extremely di erent ows are evidently possible in the same geometry. Indeed, we notice more or less intense zonal circulation and di erent drifting large- scale wave motions in di erent ranges of Ro, which will remain the same if Ro is held constant and only Re is varied. So identifying the e ect of changing Rossby number as a Ro-dependent geometry of the ow may be a reasonable physical picture. We have used the quantity G=G1 in Chapter 4 to compensate for the Reynolds number dependence of the torque as Ro is changed, and we propose that it can be identi ed as the Ro-dependent prefactor, f(Ro) = G G1 ; (6.2) 240 at least as Re ! 1, though G=G1 will be shown to not be completely Re- independent for nite but large Re. The compensated torque G=G1 for rotating data over the small range of Re that we can access here shows an approximate collapse, as depicted in Fig. 6.1 for the narrower range of the Ro > 0 data and in Fig. 6.2 for the wide-range positive Rossby data. We remind the reader that these data all have a constant drag removed, consistent with G = Gmeas 8 >>< >>: 2:2 1010 3:3 1010 ; (6.3) with the choice made for better collapse. Without this drag removal, there is a great deal of splaying of the low Ro tails, as Re will drop many orders of magnitude between Ro = 1 and Ro = 0 while the drag torque stays constant. Beyond the subtracted drag, residual failure of the collapse seems to have two components. One is a random long-term and medium-term variation in the confounding torque of the seals. It is clear from Fig. 6.1 that this dominates the failure of collapse of G=G1 for low Rossby number, again, small dimensional seal drag here is signi cant. As Ro is increased, this random drag variation is a decreasing percentage of the total dimensional torque, and the scatter is improved. There may also be a systematic variation, which we leave for a moment. Fig. 6.1 and Fig. 6.2 suggest the common Ro-dependence of all the observed states, just what would be needed for the relation f(Ro). The compensated torque 241 Ro ? G G 1.3 ? 01 -7 6.2 ? 01 -7 2.2 ? 01 -7 8.1 ? 01 -7 6.1 ? 01 -7 3.1 ? 01 -7 Ekman 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.8 1 1.2 1.4 1.6 1.8 2 Err. H L LL Fig. 6.1: G=G1 for the narrow positive Rossby range, showing the H, L, and LL approximate collapse. Collapse is better as Ro and Re are in- creased, due to reduced seal torque as a percentage of total torque. Some selected error bars are plotted based on the maximum observed systematic di erence in the seal torque, Gdrag = 1:1 1010, and error bars are given for the highest Ekman number (slowest overall rotation) and the lowest Ekman number (fastest). These error bars are both o -scale at the lowest Ro, but a systematic shift of this level is not ever observed within any given run, and the drag model of Eq. 6.3 seems justi ed. This e ect is thought to be the result of varying seal engagement of one of the seals from day to day. In any given constant-E run, the drag torque is reasonably described by choice of the better of Gdrag = 3:3 1010 or Gdrag = 2:2 1010, and for whatever reason, the drag does not switch values mid-run. We can see that the di erence between these values is not a good estimate of the random errors within a given run. Furthermore, we can see that the lower E, higher Re, faster rotation runs have un- ambiguous separation between states even in terms of the maximum seal error. The equations for the solid line ts are given in Eq. 6.4. 242 G=G1 for the H, L and LL states can be modeled by: G G1 = 8 >>>>>>< >>>>>>: 0:10Ro+ 1:8 for 0:5 < Ro < 2:4 : H 0:03Ro+ 0:9 for 1:8 < Ro < 3:5 : L 0:05Ro+ 1:1 for 3:3 < Ro < 4:6 : LL ; (6.4) with some possibility that the borders in Ro will extend beyond the stated range if one waits long enough at a xed Ro. We establish the Ro-boundaries fairly well with the aggregate data here, al- though there is still some uncertainty stemming from the fact that state transitions become extremely rare in the border regions2. In fact, we imply in Sec. 5.2 that the H state does not ever become impossible. In practice, we do not always observe a transition at a given Ro as we step through parameter space, so the overlap ranges of Ro will exceed the overlap implied in Eq. 6.4. We prefer to do continuous ramps to limit the seal variation within each Ekman number presented, and as it stands, each Ekman number in Fig. 6.1 represents a ten to twelve hour continuous run, about the limit of the batteries on the rotating frame and torque sensor. Lengthening the time at each torque step would have split each Ekman number over two days, which is undesirable from a practical point of view. The two wide Ro ramps of Fig. 6.2 provide some evidence that the approxi- mate collapse exists over the full range of Ro, but we note the discrepancy at the high Rossby number end, where the two constant i wide range ramps depart sig- ni cantly. When the outer sphere is turned on, G=G1 rises signi cantly. After a 2 See Fig. 5.5 243 Ro Ro R H L LL B Q E ? G G 1.3 ? 01 -7 6.2 ? 01 -7 2.2 ? 01 -7 8.1 ? 01 -7 6.1 ? 01 -7 3.1 ? 01 -7 Ekman ? i 4.7 Hz 3 Hz 10 0 10 1 10 2 0.8 1 1.2 1.4 1.6 1.8 Fig. 6.2: G=G1 for the wide positive Rossby range, showing the entire stud- ied range of Ro > 0 with state labels corresponding to Fig. 4.16. Symbols are the same as Fig. 6.1 with black circles and cyan tri- angles representing two wide-range ramps with i=2 = 4:725 Hz and i=2 = 3 Hz respectively. The three data points at large Ro with G=Ginfty > 1 in the state range labeled E appear to be a real physical phenomena. In the wide-range 4.725 Hz data, the torque on the inner sphere persisted a short time and then dropped to a value consistent with the lower branch curve. However, in the 3 Hz data, this drop occurred on the third ramp step. We hypothesize that this drop has something to do with the formation of an Ekman layer on the outer sphere, and it is possible that the spin-up required to get the torque drop at 3 Hz took longer than the rst couple of ramp steps. The large di erence between the outer sphere stationary and the outer sphere rotating at 0:030 Hz seems strange at rst. How- ever, the Ekman number in the outer boundary layer drops from E =1 to 5:3 10 6 when this happens. Caution is advised, then, in identifying seemingly high Ro with asymptotically high Rossby number. The solid line t is G=G1 = 0:17 log10Ro + 0:67, which is not a reasonable t form for an asymptotic approach to unity as Ro!1, but it does not seem that we should expect that anyway. 244 few ramp steps3, this higher branch dropped to the lower in the middle of a step. It is not know if this might be a bistable or hysteretic state, or if it was just a long- time transient. There were other ow changes associated with the torque change, with increased mean velocities. In this case, the transducer was oriented to pick up nearly meridional circulations, which increased near the transition. Furthermore, the north polar pressure was observed to drop along with the torque increase. When the vessel is sealed, the polar pressure givesn another corroboration of the conclusions regarding fast zonal ow. The polar pressure registers lower when the zonal ow is fast, and higher when it is not. So the conclusion regarding the three points of the 3 Hz data in Fig. 6.2 is the same as is reached in the rest of the states: lower torque is concurrent with faster zonal circulation. It may be that exceptionally long waiting times may permit observation of a bistable state in this region. Furthermore, this state may be hysteretic4. However, the extremely long times it may remain in either state make investigation of this phenomenon in the three meter experiment impossible. Once low enough Ro is reached, the 3 Hz wide-range Ro data seems to agree well with that at 4.725 Hz. We t G=G1 vs. log10Ro for Ro > RoR (Ro > 7:16) and obtain G=G1 = 0:17 log10Ro+ 0:67: (6.5) This is not a reasonable model if we expect an asymptotic approach to unity. How- 3 1.5 hours! 4 Though no other state transitions seem to be so, excessively fast slewing of parameters can make transitions appear hysteretic. 245 ever, the observed multiple states at high Ro and the di erence between the high Ro wave state and that with the outer sphere stationary may imply that there is not really a physical reason to expect an asymptotic approach from Ro = 100 to Ro =1 without further transtions. The state is simply di erent. Though we do not have multiple data sets to collapse, we include ts to our measurement of G=G1 when 4:5 < Ro < 1 for the sake of completeness. These ts quantify the small contribution of the inertial modes. Figure 6.3 shows ts to G=G1 for the three ranges. The compensated torque for Ro < 0 can be modeled as G G1 = 8 >>>>>>< >>>>>>: 0:135Ro+ 0:30 for Ro > 1:3 : (a) 0:135Ro+ 0:27 for 2 < Ro < 1:5 : (b) 0:135Ro+ 0:24 for 4:5 < Ro < 2:4 : (c) : (6.6) The strong inertial modes simply seem to add a small amount to G=G1. The percentage change is about 6% for the (4, 1, 0.612) inertial mode that is active in range Fig. 6.3 (b). In Eq. 6.6, we forced the slope for the two points in Fig. 6.3(a) to be the same as the others. 246 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 0.5 0.6 0.7 0.8 G/G? Ro a b c Fig. 6.3: G=G1 for the negative Rossby range. The in uence of the inertial modes active in (b) and (a) seems to be additive to the background contribution. Fits are given in the text. 247 6.1.2 Constant Rossby Number We mentioned before the possibility of residual Re-dependence in the quantity G=G1. The data of Fig. 6.4, G=G1 for a few ramps at constant Ro supports this idea. Figure. 6.4 includes data for Ro = 2:15, where the inertial mode (4; 1; 0:612) is excited, and see that it ts into the overall picture, but with a signi cantly lower torque for a given Reynolds number than the other states. The rotating states scale a little more steeply with Re than the data with the outer sphere stationary, even when G=G1 is rather low, as with the Ro = 2:15 inertial mode state. In Fig. 6.5, we divide the dimensionless torque by an alternative factor, CRe2, with C chosen to make G=G1 the low torque state torque be close to unity, and note that the rotating states seem to be closer to the asymptotic scaling. We believe that the higher velocities and velocity uctuations that are present in the rotating states cause this. Even when the torque is relatively smaller than that demanded at the same Re by the outer, the fast velocities and more vigorous velocity uctuations caused by spatial concentration of the momentum transport mean that the boundary layers may be thinner. The stresses at the walls are then dominated by the pressure drops across the small roughness elements of the boundary surfaces, and viscous e ects are negligible. This could happen even though the integrated stress is lower, because the local Re in the boundary layer is higher. The normalizing factor we chose, G1, does not yet scale this steeply. So, we cannot yet identify G=G1 with a completely Re-independent Rossby dependence, and we cannot factorize the torque into a Rossby dependent friction factor and a 248 10 7 10 8 0.5 1 1.5 Re Ro = +2.15, H Ro = +2.15, L Ro = -2.15 Ro = ? G G ? Fig. 6.4: G=G1 for selected data at constant Ro. The rotating states appar- ently scale a little more steeply than the nonrotating. Each value of Ro is done in a single continuous ramp, starting at high Ro and going downward, in the hopes of \breaking in" the seals for the day and stabilizing the medium-term drift. 249 single scaling with Reynolds. However, if we were able to reach the asymptotic G Re2 scaling in the outer sphere stationary state, G=G1 should become completely independent of Re. In other systems, it is possible that G=G1 does not have a systematic Re dependence because of the particulars of the system. We cannot rule out that G=G1 = f(Ro) for other geometries at large but nite Re, because other systems may have the same residual e ects of viscosity in all ow states. The prediction of Eq. 6.1 is one of the signi cant testable predictions of this dissertation, and can be tested approximately by investigating G=G1 or G=Re2. The complex Rossby dependence in spherical Couette ow, especially at low Ro, will be di erent in other geometries, as it appears to involve inertial modes of the sphere here. The slight changes in the switching boundaries seen at the end of Chap- ter 5 in the perturbed geometry of the 60 cm experiment make it clear that even small changes in geometry have an e ect. A slightly perturbed geometry results in a f(Ro) with minor shifts in state switch boundaries. However, a di erent geometry, like Taylor-Couette ow, will exhibit a much di erent f(Ro). Furthermore, it is an- ticipated that geometrical details that signi cantly change the engagement with one rotating boundary or another would result in signi cant changes in f(Ro). Examples of apparatus changes expected to greatly in uence f(Ro) might be roughening of the inner sphere in spherical Couette or changing of the end-cap speeds in a Taylor- Couette apparatus. The latter is of interest in Taylor-Couette experiments aimed at tuning the bulk velocity pro le to study the magnetorotational instability[143]. Usually, the ends of a Taylor-Couette apparatus revolve with the outer cylinder, which is a closed tank. Split end rings or end rings coupled to the inner should 250 10 7 10 0.6 0.8 1 1.2 1.4 1.6 Re G 2C Re Ro = +2.15, H Ro = +2.15, L Ro = -2.15 Ro = ? Fig. 6.5: G normalized by an alternative scaling G Re2 with a normalization with respect to the low torque state. The scaling in the rotating states, even when the relative torque is low with respect to outer stationary, seems to be somewhat closer to Re2. It is possible that this is due to the shear concentration of Fig. 4.19, resulting in higher velocities and closer-to-asymptotic turbulent boundary layers even when the total angular momentum transport is throttled by zonal ow transport barriers. 251 signi cantly modify f(Ro). We note a recent result by Ravelet et al. [127], where turbulent ow in a wide gap Taylor-Couette apparatus shows a bifurcation to a secondary ow state as a quantity similar to the Rossby number5 is varied. The torque on the inner cylinder seems to give results consistent with ours at high enough Re. The rotation number of Ravelet et al. selects di erent turbulent ow patterns and the torque for di erent ows approaches the same scaling when Re exceeds 104. The highest Re reached is about 2 105. Data soon to be submitted for publication by Paoletti et al. on high precision torque measurements in independently rotating Taylor-Couette ow reaches the same conclusion as well. The compensated torque G=G1 is a function primarily of Ro [144]. It is, however, nothing like the dependence noted in spherical Couette ow, consistent with the prediction that geometry plays a critical role. 6.2 Fluctuation Scaling We conclude this chapter with some analysis of the scaling of the turbulent uctuations and waves in the system. We investigate the collapse of wall shear stress spectra with Re for two di erent constant values of Ro. The frequencies of all of the motion at a given value of Ro should scale with the overall system timescale. Whenever the outer sphere revolves, we have normalized the frequencies by the outer sphere rotation frequency. We have not explicitly discussed this as an expected scaling, but we will see below that it it appropriately collapses the 5 They use a di erent parameter, which they call the rotation number (1 )(Rei+Reo)=(Rei Reo), and they also use the abbreviation Ro. 252 frequency content of the turbulent motions in the system. We de ne !0 = ! o ; (6.7) the dimensionless frequency of the turbulent motions. We will investigate power spectra of the wall shear stress. The wall shear stress is linearized using the procedure described in Sec. 5.3.1, and has units of Pa. We recall the expected stress scale from the equations of motion Eq. 1.8, LU . We chose this as the dimensionless pressure scale, and we adopt it here as a dimensionless wall shear stress scale as well. With L = ? = ri ro, U = Ro oL, and = o, we may construct a quantity, 2 3oRo 2?4 that has the units of the power spectral density, Pa2=Hz. We use that quantity to make the power spectra dimensionless as S 0(!= o) = S(!= o) 2 3oRo2?4 : (6.8) We will see that this collapses the broadband portions of the spectra fairly well for the low torque state at Ro = +2:15. However, when Ro = 2:15, it is the power in the strong inertial mode (4, 1, 0.612) that collapses best. The broadband background collapses upon further multiplication by Re2=3. The power in the frequency band identi ed with the inertial mode in both states scales di erently than the broadband background. 6.2.1 Ro = +2:15, Bistable L State We choose to look at uctuation scaling using the frequency spectra6 due to the wide variety of wave motions we observe. Figure 6.6 depicts the dimensionless 6 Rather than, say, the standard deviation of uctuating quantities. 253 power spectra S 0(!0) as de ned in Eq. 6.8. The low-frequency uctuations in the frequency range labeled LF, and the turbulent 0:71 o wave in the frequency range labeled TW both show good collapse in their dimensionless power spectra. However, we notice that the inertial mode (3, 1, -0.1766) in the frequency range marked W is stronger in some of the spectra. This wave grows in relative strength as Re is increased. Furthermore, the high frequency end of the spectrum does not show a good collapse here, with the power in the high frequency uctuations falling with Re. We can investigate this more quantitatively by calculating an approximation of the total dimensionless power in each of the marked frequency bands, P!0 = X !0 S 0(!0) !0: (6.9) In Fig. 6.7, we plot the Reynolds number dependence of P!0 for four of the bands marked in Fig. 6.6. The t for the LF region has a slight Reynolds number dependence but is close to at. The inertial mode peak, Fig. 6.6 W, grows one power of Re faster than the broadband low-frequency turbulence. This seems to be at the expense of the high frequency uctuations, as there is a steep decrease in the bands around 25 o and 90 o, scaling like Re 2:1 and Re 2:6 respectively. Because Re = Ro=E, and Ro is held constant, we see that the inertial mode power in the band W is growing in importance as rotation is increased. The frac- tional power in the inertial mode rises from 6% of the total spectral power at the lowest rotation rate to 25% of the total spectral power at the fastest rotation rate. 254 10 ?3 10 ?2 10 ?1 10 0 10 1 10 2 10 ?15 10 ?14 10 ?13 10 ?12 10 ?11 10 ?10 10 ?9 10 ?8 10 ?7 S ?(?/? o ) ?/? o 10 3 LF W M DTW Fig. 6.6: Wall shear stress dimensionless power spectra S0(!0) conditioned on the low torque state Ro = +2:15, 5 106 < Re < 5 107. Dark blue is the lowest Re and deep red is the highest. The expected scaling for the power spectral density based on the expected scaling of uid stresses collapses the low-frequency broadband region well. The the turbulent L state wave in the region TW collapses as well. However, we see that the region marked W, which we identify as the inertial mode (3, 1, 0.1766), scales a little more steeply than the rest of the turbulence, and the high frequency region of the spectrum scales less steeply. We label the M and D frequency ranges for reference in Fig. 6.7. 255 Re P LF ~Re0.2 P M ~Re-2.1 P D ~Re-2.6 P W ~Re1.2 0.13 0 low torque state, the higher frequency of which cannot yet be identi ed with an inertial mode. However, the lower frequency wave in many of the Ro > 0 states appears to be the inertial mode (3, 1, -0.1766), which persists through several ow transitions to extremely high Ro. It is deeply mod- ulated like the mode discussed above, in fact, it is more so. The wide range over which it exists makes it somewhat di cult understand the role it plays. The same can be said for the low-frequency, as of yet unidenti ed10 wave for Ro < 0 visible in Fig. 4.8 at 0:083 o, which is only completely absent when the (4, 1, 0.612) mode is strong. What we do know about both of these low-frequency waves is they are relatively una ected by the Rossby number. The (3, 1, -0.1766) inertial mode has much of its kinetic energy located at large cylindrical radius, near 10 The lowest wavenumber possibility for a retrograde mode near this frequency is (23, 20, 0.0896). 266 10 -2 10 -1 10 0 10 1 10 3 10 4 10 5 10 6 ? ?o S(? ) ? ? ?o = 2 -4/3 Fig. 6.12: Velocity power spectrum, Ro = 2:15, E = 1:1 10 7, the same as inset of Fig. 6.11, showing a knee at 2 o. This suggests that inertial waves or modes are important to the low-frequency, probably large to medium scale portion of the broadband turbulence. the equator of the outer sphere, perhaps making it understandable that it would be less modi ed by the ow nearer the inner sphere. But what is it doing there? Perhaps it plays a very general role in angular momentum transport in much the same way that a few simple inertial modes may arise to transport heat in rapidly rotating spherical convection [22]. It may be that the (3, 1, -0.1766) mode plus one or two other Rossby-type modes associated with the other peaks in Fig. 4.16 are associated with the bulk of the angular momentum transport for 0 < Ro < 15. We should mention again a point that we touched on brie y in Sec. 4.3. Gen- 267 erally speaking, in addition to the strong single inertial modes excited as part of the ow when both spheres revolve, the broadband portion of the spectra of rotating states tends to be rather at at frequencies less than twice the outer sphere rotation rate. This is true for Fig. 4.9, Fig. 6.6, Fig. 6.8, and the velocity power spectrum inset of Fig. 6.11 which we reproduce here in Fig. 6.12, as it is a nice example. Below twice the outer rotation rate, the spectrum is somewhat at, and a knee is present around 2 o with higher frequencies falling o like ! 4=3. The persistence of a knee near 2 o in many states when the outer sphere revolves suggests that inertial waves and modes play a role in the broadband turbulent background, as they cannot exist above this frequency. The at spectrum by itself is not much evidence, but the fact that rollo occurs in the vicinity of 2 o is suggestive of turbulence consisting of a sea of inertial modes. 268 7. SUMMARY, SPECULATION, AND FUTURE WORK 7.1 Summary We have described a series of hydrodynamic experiments in turbulent, rapidly rotating shear ow of water between concentric rotating spheres. The world-unique experiment is the result of a long design and construction process in which the author played a key role. In most of these experiments, we have reached unprece- dented dimensionless parameters for this geometry. We drive strong turbulence with Reynolds number up to 56 million and study the e ects of rapid rotation, lowering the Ekman number down to 10 7. In most regions of parameter space that we have studied, the ow is characterized not only by intense turbulent uctuations, but by large-scale energetic wave or vortex motions. For Ro =1 with the outer sphere stationary, we observe low-frequency oscil- lations consistent with slowly precessing large-scale vortical motions in the system with m = 1 and broken equatorial re ectional symmetry. We hypothesize that they result from a pressure-driven modulation of the equatorial out ow from the turbulent Ekman boundary layer near the rapidly rotating inner sphere. These mo- tions are con rmed over at least an order of magnitude in Re by comparison of data from the sixty centimeter experiment with that from the three-meter experi- ment. These low-frequency oscillations contain a great deal of energy and imply that some energetic non-universal large scales in high Re turbulence may go unnoticed if experimental observations or simulation runs are not conducted over long enough times. When Ro < 0, we con rm the excitation of inertial modes similar to those of the full sphere1, as reported previously in this geometry at similar parameters [12{ 15]. We establish that these inertial mode motions appear to be robust to turbulence in the system, and show signs of nonlinear evolution. Furthermore, as Re is increased (or E decreased, equivalently), the inertial modes appear to further dominate the turbulence. We hypothesize that non-linear processes involving inertial modes may play a role in non-local angular momentum transport by zonal ow generation, reducing that which must be transported by the turbulent uctuations. When Ro > 0, we observe that transitions between the turbulent ow states are associated with more dramatic changes, including the torque on the inner sphere, measured mean velocities and uctuations. The temporal uctuations of torque on the inner sphere are stronger for turbulent states in the range 0 < Ro < 15. The mean torque at a given Reynolds number is signi cantly di erent for each of the observed turbulent ow states for all Ro, but the mean di erence in G=G1 between adjacent states is dramatic when 0 < Ro < 5. We nd that the wall shear stress and mean azimuthal velocity at our measurement location at 23:5 co-latitude is anti-correlated with changes in the relative torque. We analyze one turbulent ow transition in detail and conclude that di erent angular momentum transport barriers 1 Modes indistinguishable from the full sphere modes by our measurements, at least. 270 due to di erent zonal ows are responsible for the turbulent ow transitions. The wall shear stress and azimuthal velocities at our measurement location are higher in the lower relative torque states because of the stronger zonal ows, despite a lower total angular momentum ux away from the inner sphere. The regions of parameter space where particular turbulent ow states ex- ist appear to be bounded simply by constant Rossby number, with the overall strength of the motion, turbulent uctuations, and angular momentum transport scaling with Re in a similar way to other turbulent ows. We hypothesize that the torque in a turbulent swirling shear ow with overall rotation should take the form G = f(Ro) g(Re) at least as Re!1 with g(Re)! Re2. We nd that the quantity G=G1, the dimensionless torque divided by the t to torque measured at the same Re when Ro = 1, is a nearly Re-independent quantity. The compensated torque G=G1 summarizes the e ect of the very di erent turbulent ows occurring at dif- ferent Rossby number on the torque. When Re is high enough, we hypothesize that G=G1 should become a relation that only depends on Ro. At the Reynolds numbers we investigate G=G1 = h(Ro;Re) may be interpreted as a friction factor that ex- presses, in part, the e ect of the self-organizing ow geometry on the turbulent drag forces on the walls. The quantity G=G1 should be a useful quantity to express the e ects of the di erent possible turbulent ows on the angular momentum transport in any turbulent rotating shear ow. The experiments presented here show the rich behavior possible in inhomoge- neous, anisotropic turbulent ow with and without outer boundary rotation. We make a few concrete predictions that should extend to other turbulent shear ows. 271 We nd that inertial modes appear to be important as part of the large-scale en- ergetic ow in all rotating states. Furthermore, we note a knee in power spectra at 2 o when both spheres revolve suggesting that the turbulence for uctuation frequencies 0 < ! < 2 o may reasonably described as a sea of interacting inertial modes or inertial waves. However, the inhomogeneity and waves in high Reynolds number shear ow between concentric spheres do not require outer boundary rota- tion, as we see from the large-scale wave when Ro =1. In all cases, the turbulence is inhomogeneous, anisotropic, wavy, and in uenced by the boundary layers to form non-trivial large-scale ows that contain much of the energy. These energetic non- universal, large-scale ows are not annoyances to be eliminated from a turbulence experiment, but important parts of a realistic high Reynolds number ow that must be understood fully in order to make good predictions about the turbulent transport properties in real systems. 7.2 On Extreme Inhomogeneity We might o er some further speculation on the interesting dynamics of the large scales presented here with regard to what to expect in inhomogeneous systems. When we have presented long-time dynamics throughout this dissertation we have made heavy use of numerical low pass ltering and band-pass ltering to reveal some of the uid behavior. We might re ect a bit on why this was necessary. The underlying issue is that fast turbulent uctuations are so large that they generally obscure the mean shifts and slow uctuations. The slow uctuations in low pass 272 ltered data are clearly and unambiguously correlated with other dynamics, as in the torque and azimuthal velocity time series of Fig. 4.15. The torque uctuations are low-pass ltered naturally2 by mechanical damping and integration of stress uctuations over a large area. The velocity uctuations are not. The velocity uctuations in the H and L states, for example, overlap signif- icantly in the tails of the distributions. This is clear from Fig. 5.8. Typically, we have large levels of velocity uctuations, with RMS velocities 16% of the mean in the low torque state and approaching 60% in the high torque state. The large high frequency uctuations can obscure the low-frequency uctuations which are well correlated with other measurements like the torque. Evidently, there are a wide range of length, time and velocity scales for the velocity uctuations that do not present su cient perturbation via advective non- linearity to promptly change the large-scale ows. Undoubtedly, these uctuations matter to the overall transport when the Reynolds stresses from long lasting veloc- ity correlations act over long times, but in the instantaneous measurements, they appear as noise. We might think of a simpler system that shows the same behav- ior. Figure 7.1 depicts two turbulent mixers connected by pipes such that they will generally pump uid into the other. Let?s imagine that the typical ow rates in the pipes when one mixer blade is stationary may correspond to one-tenth of the azimuthal velocities typical in each mixer. Each mixer has its own turbulent dynamics. If the experiment is large enough and powerful enough, the ow in the apparatus could be turbulent throughout for 2 Though mechanical noise is still be ltered out with a low pass lter. 273 ?2?1 U = ? L u = U/10 Fig. 7.1: Two identical turbulent mixers are cross connected by pipes, allow- ing them to pump uid into each other with weak ow in the pipes. We may expect a wide variety of collective dynamics as the ratio 1= 2 is adjusted. 274 any speed ratio, including when one blade is stationary. But the turbulence would be extremely inhomogeneous, with relatively weak ow and small uctuations in the the inoperational tank. It is likely that the collective turbulent motion when both impellers revolve would be much di erent than when only one mixer was operational. With both impellers rotating in the same direction, the mass ux and uctuation levels in the pipes would typically be much higher than with only one rotating. And we might expect that the collective dynamics would be a sensitive function of 1= 2. In this thought experiment3, it is not irrelevant that each system is turbulent. The large-scale turbulent pressure and velocity uctuation pro le and infrequent, intense uctuations at the inlet and outlet ports will greatly a ect the mass ow and the level of the uctuations advected through the pipes. But at the same time, the relative in uence of the two systems on each other will be fairly weak on short timescales. Large vortices shed from the second im- peller cannot directly advect the ow from the rst. The collective dynamics would probably be, in general, slow, with some time needed for the mass ux to change, and for swirling uid in the pipes to spin up and down. The in uence of the other system on the dynamics of the rst would be undeniable, but it would take time to act. The fast velocity uctuations in an active mixer of the coupled system might not show much clear di erence from a single running mixer until a large change occurred in the ow between mixers. Many of the measured uctuations would be irrelevant to the behavior of the other system except in their average e ects on the long-time collective dynamics. The physically imposed inhomogeneity makes it hard 3 Which may easily be turned into a real experiment. 275 to imagine a situation where the ow in system as a whole could approach a tur- bulent state that had any properties that only depended on the total power input. It would seem very unusual if the large-scale ow state, especially the circulation through the pipes, was found to decouple from the ratio of the impeller speeds in this hypothetical system. However, if point measurements were made in tank num- ber one, it might be di cult to appreciate the di erence between the situation when impeller two was rotating and when it was not, unless time series were long and the fast, smaller scale uctuations in the measurements were ltered out. We might imagine that a similar process is responsible for the states we observe in the three-meter apparatus. Instead of two mixers weakly coupled with pipes, we might imagine that regions of the experiment act as lightly coupled turbulent systems. Zonal ow transport barriers and regions of relatively quiet uid with weak turbulence could partially isolate parts of the uid from each other. Motions near the outer sphere may always be expected to locked more to the outer sphere by the e ects of the rotation, while those inside the tangent cylinder and near the inner sphere will always feel more in uence of the inner sphere. These regions of the uid are not uncoupled, and it is not irrelevant that each sub-region of the uid is fully turbulent. But they may communicate mostly via wave-driven momentum uxes and slowly acting Reynolds stresses due to persistent correlations of faster uctuations. The internal dynamics of each uid sub-region could be somewhat robust, and the communication between the regions potentially somewhat weak. Increasing the overall strength of the velocity uctuations does not imply that the nonlinear advection between systems will destroy the dynamics of the inde- 276 pendent systems. At a given speed ratio in either case, the individual systems? \internal" dynamics increase in strength in proportion to the cross coupling. There will always be a need for the second dimensionless parameter, Ro = i= o 1 in spherical Couette and, similarly, the speed ratio 1= 2 in the case of the mixing tanks. The same may be said for any other dual-driven system. Taylor-Couette ow with independently rotating cylinders should always have Ro as an important parameter, and von K arm an ow with independently rotating impellers will depend sensitively on 1= 2, even as Re ! 1. In such inhomogeneous systems, at any point the strong uctuations from the \far" impeller cannot be expected to wholly overwhelm the uctuations from the \near" impeller just because the overall vigor of the ow is increased, and that is really what would be needed to homogenize an inhomogeneous system. Indeed, von K arm an ow is a system known to exhibit multi-stability and even hysteresis in highly turbulent ow [137]. 7.3 Potential Impacts 7.3.1 Atmospheric Dynamics We may make some speculative predictions as to the applicability of this work in other ows that show some of the same characteristics. In Chapter 2 we mentioned that the mean circulation in the three-meter experiment lled with water and the jet streams in the atmosphere of the Earth are reasonably close to the same Rossby number, and the Ekman number in the atmosphere for motions on the scale of the troposphere thickness is only two orders of magnitude lower than that in the 277 experiment. We might brie y revisit this idea now that we have discussed the wide variety of motions that are possible in the experiment. We hypothesize that zonal ow transport barriers and/or interactions of large vortical motions with the turbulent boundary layers in the experiment are relevant to the development of the di erent turbulent ow states, and we establish that the important control parameter selecting di erent ows is Ro. In the case of the Earth, Ro and Re are slaved together for a given ow pattern. However, if our observation applies two orders of magnitude lower in E, as it might if all the boundary layers are fully turbulent and we have reached an asymptotic regime, there may be some applicability to climate prediction. The large-scale ows herein may be considered the global general circulation of the experiment. We have observed complex, multi-stable behavior in a similar parameter regime to our atmosphere that seems to rely only on inhomogeneity of the turbulence (not the uid) and a dimensionless ratio of two advective timescales, Ro. It is probable that the in uence of the large-scale ows on the net transfer of momentum and energy out of the boundary layers is of some importance in all of the states observed here. However, each state is present over a fairly wide range of Ro, making it clear that modest changes of energy input and uctuations do not change the general pattern except at critical values of Ro. If the turbulence in Earth?s atmosphere behaves in a similar way, perhaps it would allow some aspects of possible changes to global circulation patterns to be modeled with simpler models for the energy inputs than those commonly used, pro- vided that the turbulent momentum uxes were parameterized appropriately. It 278 may be that the properties of the turbulent momentum transport could dominate over subtle issues of the energy budget. Working to understand the multi-stability in systems free of the additional complications of moisture, strati cation, dust trans- port, and radiative processes could prove fruitful. 7.3.2 Dynamo Prognosis The three-meter experiment, as we discussed in Chapters 2 and 3, is intended to model the owing molten iron in the outer core of the Earth. We will soon ll it with liquid sodium metal and perform these experiments, but we may ask what we have learned from the purely hydrodynamic experiments we have completed. One of the critical ingredients in a successful turbulent dynamo is some measure of self-organization. Other hydromagnetic experiments [76, 151, 152] suggest that large ow uctu- ations are important in raising the critical magnetic Reynolds number of turbulent liquid metal dynamos above a value that the experiment was designed to achieve. Some experiments are designed using water models with ow visualization, and the time-averaged velocity eld tuned toward a pattern that would act as a kinematic dy- namo at an achievable magnetic Reynolds number. These experiments have shown that this condition is insu cient, because the ow is rarely close to an instanta- neous snapshot of the mean ow [152]. Large scale uctuations frequently spoil the suitability of the actual velocity eld as a low critical Rm dynamo. Consequently, these apparatus do not self-generate even though their mean ow would if it were steady. 279 The induction measurements of Volk et al. [152] show that the uctuations of magnetic induction and ow eld in turbulent Von K arm an ow at the desired rotation rate ration have largely at, featureless spectra, indicating that there is no preferential time or spatial scale for uctuations4. However, we nd that no state in spherical Couette ow is free from strong waves, some of which are extremely coherent in terms of frequency and spatial correlation. And though we discussed the uctuating nature of these waves, those in the rotating states, especially for Ro < 0 inertial mode excitation, are easily recognizable in time snapshots of the magnetic induction. While they are uctuating, they remain similar to an instantaneous snapshot of an inertial mode. Furthermore, inertial modes seem to be strengthening with respect to other uctuations as the rotation rate is increased. We might hope that these self-organizing turbulent ows could be useful for dynamo action, and we may look in particular at inertial modes because they are in a sense the \most coherent" motions in the system. It is known that one important ingredient in some dynamo scenarios is ow he- licity [153]. The helicity is the volume integral of the helicity density h = u (r u). Helicity is important for the so-called -e ect in dynamo action, wherein elds are twisted and stretched by helical motion, such that the eld lines resemble the Greek letter . The the -e ect may result in a dynamo when combined with the di er- ential rotation. Such a dynamo is referred to as an !-dynamo [153]. It seems that an inertial mode may generally provide some -e ect. Figure 7.2 shows a Mollweide 4 Volk et al. do nd a low-frequency modulation at 1/100 the disks? frequency involving their midplane shear layer, but lter it out. 280 h = U ? (4, 1, 0.612) 0 2? 0 ? Fig. 7.2: Mollweide projection of the helicity density h = U r U for the (4, 1, 0.612) inertial mode of the full sphere. The vorticity = r U given in Appendix A. While the net helicity of every inertial mode seems to be zero, the modes possess large-scale patches of helicity, which is thought to be a useful ingredient for dynamo action. projection of the helicity density of the inertial mode (4, 1, 0.612) calculated from the velocities given by Zhang et al. [10] and the curl of those expressions, provided in Appendix A. This m = 1 inertial mode has an m = 2 pattern of large-scale patches of helicity density. The volume integral of the helicity density is zero, which can be anticipated by symmetry arguments. But the large-scale patches of alternating helicity density may be bene cial for a dynamo. We may estimate the maximum magnetic Reynolds number for this inertial mode motion from our measured velocities. Figure. 6.11 shows that the mean am- plitude of the (4, 1, 0.612) inertial mode is U=( oro) = 0:014. Furthermore, this mode would only show velocity uctuations of about half the global maximum am- 281 plitude because of the measurement location. So we may expect U=( oro) 0:028 peak velocity for this mode. The magnetic Reynolds number for this motion at this dimensionless velocity scale and the outer sphere rotating at 4 Hz, using a typical length scale5 of 2 m is Rm = 0:028 2 1:5m 4Hz 2m 0:083m2=s = 25: (7.1) The typical velocity of the di erential rotation in this state might be U=( oro) = 0:2, giving Rm = 180 for the zonal ow. So, at maximum design rotation rate, we may expect a strong self-consistent ow that includes useful ingredients of helicity and di erential rotation and has Rm based on actual ow measurements greatly exceeding unity. Furthermore, while we nd that the inertial mode is uctuating, its motion typically rather strong with respect to other turbulent uctuations. The water measurements seem to support the idea that we could achieve high Rm large- scale-coherent ow organized by rapid rotation, possibly one that is useful as an !-type dynamo. 7.4 Future Work Drawing to a close here, at the culmination of a long and interesting program of study toward a Ph.D., one must wonder what the future holds. In this case, the most immediate future is known. The author has devoted the best part of a decade toward this experimental apparatus, and would prefer to be among the rst authors 5 One quarter of the circumference at 30 latitude, as typical of where the helicity patches are strong. 282 on the rst publications if it is indeed a liquid sodium dynamo. If all goes smoothly in the next year or so, the intention is to ll with sodium and begin hydromagnetic experiments in a brief post-doctoral stint. The large and di cult task of elding the experiment means that this appa- ratus may not have been been wrung dry of results6 even in terms of what can be learned from the hydrodynamic ows. However, the author did reach a point of diminishing returns at the end of the experimental campaign. In fact, one might say that a point of collapsed returns was reached, where new ideas for measurements that would give more insight yielded zero new results for a couple of weeks, and that spelled the end of the experimental campaign. It is just simply too di cult to make the hydrodynamic measurements necessary to better characterize the ow in this device. Therefore, a similar device, smaller and faster to reach similar dimensionless parameters, but with optical access could nish the scienti c story we have presented here. We have generated a long list of hypotheses regarding the inhomogeneity, the mean zonal ows, and the location and nature of the observed wave states. It would be gratifying to have these hypotheses con rmed and possibly even more interesting to falsify them. The author has used a great deal of imagination in this dissertation work. What we know about the time-averaged zonal ows in this apparatus might be best described as educated imagination. This has been, at times, frustrating, to have a rather small peephole into such an interesting ow. It would be something of a relief to shine a few watts of laser sheet into a seeded ow of a 1 m spherical 6 Though it will be thoroughly wrung dry of water before the sodium ll. 283 Couette apparatus rotating some 10 Hz or so. Although this is a massive experiment by many standards, it seems like a small task now. An interesting question in more general terms is the relationship between the ux of some quantity through a turbulent ow and storage of the same quantity by the uid. We recall the angular momentum uctuations of Figure 5.6 in Chapter 5. In the author?s opinion, this was one of the most surprising ndings to come out of this experimental campaign. The z-component of angular momentum enters and leaves via the boundaries, but the question of how much of it is present at any time in the uid between the boundaries is not something that occurred to the author before these results were found. A turbulent shear ow apparatus with very sensitive strain-gauge torque measurements on both the inner and outer boundaries could allow much more information about the angular momentum transport and storage, and would allow absolute integration from uid start-up. Corroborating these results with direct measurements of the relative angular momentum in the rotating frame using velocimetry could be quite fruitful. In terms of the medium term goals of the scienti c program in the University of Maryland three-meter experiment and suggestions for the author?s successors on this project, it is important to develop as many ow diagnostics as is possible. Per- manent magnet potential probes may stand in for wall shear stress sensors [154]. Ultrasound velocimetry is useable but some additional design is necessary to im- prove the transducers used by Sisan et al. [118] for deeper measurements in sodium. A magnetic eld array like that used by Sisan [118] and Kelley [117] can be elded quickly, but longer-term plans call for a higher resolution magnetic eld array with 284 three-dimensional sensors. Because of the large size of the experiment, it is possible that instrumenting the inner sphere could be fruitful as well. A more sophisti- cated version of the wireless acquisition device used for the torque, but with many channels of 16 bit acquisition should now be possible (see Appendix E), and could acquire torque as well as data from an array of eld and ow sensors on the inner sphere. Ultrasound transducers could be installed in the inner shaft or on the inner sphere, and this would be considerably safer than installing them at low latitudes on the outer sphere. It is important to have good ow measurements to corrob- orate magnetic measurements and to participate in the process of understanding appropriate parameterizations for the e ects of turbulence or reduced models based on intelligent truncations of the velocity eld into inertial modes and geostrophic components. Work of this nature will be critical for prediction of high Reynolds number hydromagnetic ows for many decades to come. 285 APPENDIX A. VORTICITY OF INERTIAL MODES IN A SPHERE The net helicity H = Z U dV (A.1) and the helicity density h = U (A.2) for inertial modes in a sphere may be interesting quantities relevant to magneto- hydrodynamics in spherical Couette ow, given the observed strong excitation of such modes in that geometry [12, 14] and the importance of helicity to the dynamo problem. The vorticity = r U; (A.3) of the inertial modes of a full sphere can be calculated by taking the curl of the velocities given by Zhang et al. [10]. The expressions are recorded here for conve- nience in future work. The analytical form for the net helicity is not immediately applicable to this dissertation so no attempt has been made to calculate the volume integral given in Eq. A.1. A numerical estimate for the net helicity of several modes has been made by calculating the helicity density on a grid of points and summing, and as expected by symmetry considerations, it seems to be zero. However, most in- ertial modes exhibit large regions of alternating-sign helicity density. A few notable exceptions will be discussed at the end of this appendix. The velocity of each inertial mode is given by Zhang et al., 2001 in the form U = hUs; U ; Uzi ei(m +2 t). In cylindrical coordinates, the components of = h s; ; zi ei(m +2 t) are, for equatorially symmetric modes, (Eqns. 4.1 in Zhang et al.) s = NX i=0 N iX j=0 CijmN 2i 1 (1 2)j 1 (2i)(m+m + 2 j) s2j+m 1 z2i 1 = i NX i=0 N iX j=0 CijmN 2i 1 (1 2)j 1 (2i)(m+m + 2j) s2j+m 1 z2i 1 z = NX i=0 N iX j=0 CijmN 2i (1 2)j 1 (4j)(j +m) s2j+m 2 z2i (A.4) Note that the above summations all start at i = 0. The rst term in the i sum will be zero for the radial and azimuthal vorticities, but it is written in to avoid index confusion. For the equatorially antisymmetric modes, with velocities given by Eqns. 4.6 of Zhang et al., s = NX i=0 N iX j=0 CijmN 2i 1 (1 2)j 1 (2i+ 1)(m+m + 2 j) s2j+m 1 z2i = i NX i=0 N iX j=0 CijmN 2i 1 (1 2)j 1 (2i+ 1)(m+m + 2j) s2j+m 1 z2i z = NX i=0 N iX j=0 CijmN 2i (1 2)j 1 (4j)(j +m) s2j+m 2 z2i+1 (A.5) The coe cients CijmN for the equatorially symmetric and antisymmetric modes can be found in Zhang et al.. 288 Some special inertial modes of subclass N = 0 with l m = 1 have identically zero helicity density U . The most obvious example is that of the spinover mode, (N; l;m; !) = (0; 2; 1; 1). This mode consists of solid body rotation about an axis orthogonal to the rotation axis. In this case, it is clear that the velocity and vorticity will be everywhere perpendicular. However, it appears that this is the case for every inertial mode with (N; l;m; !) = (0; l; l 1; !). These modes are also those with zero spherical radial velocity, and are therefore modes of the spherical shell [17]. These modes are among those observed in spherical Couette ow [14]. It is unknown whether this property of mutually perpendicular velocity and vorticity everywhere is of any interest in real physical systems, but it seemed su - ciently notable for inclusion here. 289 B. SHORT FORM EXPERIMENTAL OVERVIEW For those not interested in the detailed description of Chapter 3, a short-form experimental description su cient to situate the reader for the scienti c results of Chapter 4 and forward is reproduced here: The three meter spherical Couette apparatus allows independent rotation of the nominally 3 m diameter outer shell and 1 m diameter inner sphere. Instru- mentation in the rotating frame allows measurements of velocity, wall shear and pressure on the turbulent ow of water, as well as the torques required to maintain the boundary speeds. The apparatus is depicted in Fig. B.1. The stainless steel outer vessel has a 2:92 0:005 m inner diameter and is 2.54 cm thick. It is mounted on a pair of spherical roller bearings that are con- strained by a stainless steel framework. The vessel top lid, which is inserted in a 1.5 m diameter cylindrical anged opening, has a hemispherical bottom surface that completes the outer sphere. The top lid has four 13 cm diameter instrumentation ports centered on a 60 cm radius. The ports have inserts that hold measurement probes nearly ush with the inner surface of the outer sphere. The 1:02 0:005 m diameter inner sphere is supported on a 16.8 cm diameter shaft held coaxial with the outer shell by bearings at the bottom of the outer sphere and in the top lid. The outer sphere is driven by a 250 kW induction motor mounted to the support 1m Fig. B.1: A schematic of the apparatus showing the inner and outer sphere and locations of measurement ports in the vessel top lid at 60 cm radius (1:18 ri). Data from sensors in the ports are acquired by instrumentation, including an acquisition computer, bolted to the rotating lid and wirelessly transferred to the lab frame. Also shown is the wireless torque sensor on the inner shaft. 291 frame. A timing belt couples the motor to a toothed pulley on the lid rim, reducing the shaft speed by a factor of 8.33. The inner sphere is directly driven from a sec- ond induction motor through an elastomeric coupling and a calibrated Futek model TFF600 torsional load cell. A rotating computer acquires data from sensors that include a Dantec model 55R46 ush mount shear stress sensor driven by a TSI model 1750 constant tem- perature anemometer and three Kistler model 211B5 pressure transducers. These transducers are AC coupled with a lower -3 dB cuto frequency of 0.05 Hz, allowing sensitive measurements of pressure uctuations on top of the high steady centrifugal background pressure encountered in rapid rotation. A thermocouple is used to mon- itor uid temperature and a DC-coupled pressure transducer in a port near the pole of the outer sphere allows monitoring of the overall system pressure. The sampling rate for these sensors is 512 Hz. The rotating acquisition computer is networked to the lab frame using wireless ethernet, allowing storage of the data on a lab frame computer and remote control of the acquisition and various auxiliary equipment. A Met-Flow UVP-DUO pulsed Doppler ultrasound velocimeter is mounted in the rotating frame and paired with Signal Processing transducers. Communications with this instrument are handled over the wireless connection. Some velocity data in this paper was acquired with a Met-Flow UVP-X1-PS in the lab frame using a resonant transformer arrangement to couple the signal to and from the transducer in the rotating frame. The ow is seeded with 150 m 250 m polystyrene particles with nominal density of 1050 kg=m3. An emission frequency of 4 MHz was used for the measurements presented here. Limitations on the product of maximum 292 measurable velocity and measurement depth constrain the velocity measurements in this paper to be very close to the wall, typically in the range of 30-100 mm deep. Three transducers are installed in the experiment, and can be con gured to measure the velocity along the spherical radial direction, a direction parallel to the rotation axis or in a direction contained in the plane normal to the cylindrical radial direction, inclined at an angle 45 or 23:5 with the vertical. Power for instruments on the outer sphere rotating frame is supplied by a bank of four parallel 35 Ah 12 V sealed lead acid batteries. DC-DC converters supply needed voltages for instrumentation and an inverter provides 120 VAC for the UVP-DUO. The millivolt signal from the Futek torque sensor mounted on the inner sphere shaft is ampli ed and digitized, and the data transmitted to the lab frame using a wireless serial connection. The torque is sampled approximately 32 times per second. Power for the inner torque sensor is provided by lithium polymer batteries. The induction motors? speeds are controlled by variable frequency drives under computer command. Optical sensors monitor the inner and outer sphere speeds which are controlled to better than 0.2%. The drives estimate the motor torque from electrical measurements and the torque measurements agree well with the calibrated torque sensor at motor speeds above 2 Hz. This allows measurement of the torque required to drive the outer sphere over much of the operational range of the experiment. There is a confounding e ect of the inner shaft?s bearings and seal, and a confounding e ect of aerodynamic and bearing drag on the outer sphere. The average outer sphere confounding torque can be subtracted o by measuring 293 the torque demanded with no di erential rotation. 294 C. MECHANICAL DRAWINGS This appendix collects some relevant mechanical drawings produced in the course of this dissertation work. AA ? 14.250?0.005 A ?14.9612?0.0006 3/8-16 drilled and tapped 1.00 deep 12 places equally spaced on a ? 16.00 BCD A.005 ? 17.00?0.010 A.010 ? 2.00 through hole 4 places eq. spaced on a 19.500 BCD 1.055?0.050 2.945?0.001 A.001 ~4.00 o?all top view housing section AA housing ? 22.0?0.10 retaining ring (drawing 3/4 scale as above!) ? 14.25?0.010 ? 17.00?0.010 through hole for 3/8 hardware 12 places eq. sp. on a ?16.00 BCD top view side view 0.50?0.010 OS Bearing Housing / Retainer Zimmerman 25 May 2007 max. radius 0.075 radius 1/4? bevel 45? 1/16? deep material: 304 SS break all sharp edges material: 304 SS break all sharp edges all dimensions in inches 82.00 121.00 74.00 36.00 nom. 37.00 shimmable -0.5/+arb. 19.00 95.00 nom 31.60 > 30.88!! probably no shims needed nom 30.88 adjustable +1.6/-3.0 nom 30.88 nom. 41.00 adj. ?2.67 43.00 43.00 91.00 in 8 places: 8.00 x 8.00 x 0.625 0.625 triangle bracing ?1.25 holes 4.00 apart 1 - 8 welded studs on frame 8.00x4.00x0.625 3 places; hole and fastener same as 8x8 locator pin hardened 0.750 loc. pin Motor Mount - Top View Zimmerman 23 May 2007 Lip Seal CR #64998 ?/??? GapCR seal # HDL-3135-V 12.125 0.625 shims 0.500 nom 1.875 74.00 26.125 74.57 raw 20.2? floor 156.625 183.70 187.625 31.00 47.00 72.00 38.375 25.00 8.625 Motor Mount - Side View Zimmerman 23 May 2007 47.62 raw 33.1? 44.0040.00 38.00 o?all 30.00 22.00 18.00 25.00 30.00 2.00 2.00 42.00 7.00 nom? 5.50 min! 0.625 2.0021.00 26.00 22.00 0.500 30.00 8.00 22.00 26.00 8.00 4.00 6.50 1.00 Outer Motor Sliding Mount Detail Zimmerman 23 May 2007 B o t t o m H e a d O v e r v i e w Z i m m e r m a n 1 6 J u l y 2 0 0 3 1 o f 5 P . 2 - B o t t o m V i e w P . 3 - C r o s s S e c t i o n P . 4 - T o p V i e w P . 5 - F r o n t V i e w b e a r i n g s u r f a c e A b o t t o m f r o n t b o t t o m v i e w A ? 8 . 6 5 8 9 ? 0 . 0 0 0 6 ? 7 . 8 7 5 ? 0 . 0 1 0 . 0 2 0 A ? 4 . 7 2 9 ? 0 . 0 0 1 . 0 1 0 A ? 3 . 9 3 7 5 ? 0 . 0 1 0 b o r e 7 . 0 6 ? . 0 1 0 d e e p . 0 2 0 A 1 / 4 - 2 0 x 0 . 2 5 d e e p d r i l l e d a n d t a p p e d . 0 1 0 A 1 / 2 - 1 3 x 1 . 0 0 d e e p d r i l l e d a n d t a p p e d 6 p l a c e s e q u a l l y s p a c e d o n a 6 . 3 0 ? 0 . 0 1 0 b o l t c i r c l e d i a m e t e r A A A A B o t t o m H e a d B o t t o m V i e w Z i m m e r m a n 1 5 J u l y 2 0 0 3 . 0 2 0 A 2 o f 5 . 0 1 0 A ? 1 2 . 8 5 ? 0 . 0 5 3 . 5 6 ? 0 . 0 1 0 c r o s s s e c t i o n A A 0 . 5 8 4 ? 0 . 0 5 0 1 . 5 ? 0 . 0 1 0 B o t t o m H e a d C r o s s S e c t i o n Z i m m e r m a n 1 5 J u l y 2 0 0 3 3 o f 5 4 5 ? c h a m f e r . 0 6 2 5 d e e p 9 0 R a . 0 0 1 A b l o w u p o f g a s k e t s e a t a r e a g a s k e t s e a t - s e e i n s e t r e l i e f R . 1 2 5 1 2 ? t y p i c a l ? 1 . 9 9 5 + 0 . 0 0 0 / - 0 . 0 0 2 B o t t o m H e a d T o p V i e w Z i m m e r m a n 1 5 J u l y 2 0 0 3 . 0 0 5 A t o p v i e w ? 1 . 6 2 5 + 0 . 0 0 5 / - 0 . 0 0 0 t h r u h o l e 2 8 p l a c e s o n a 3 7 . 0 0 ? 0 . 0 6 2 5 b o l t c i r c l e d i a m e t e r . 0 6 2 5 A 1 2 ? 1 2 ? 4 o f 5 ? 4 0 . 0 0 ? 0 . 2 5 0 B o t t o m H e a d F r o n t V i e w Z i m m e r m a n 1 5 J u l y 2 0 0 3 4 . 0 0 ? 0 . 1 0 0 A . 0 1 0 A . 0 2 0 A 0 . 8 7 5 ? 0 . 0 3 0 ? 2 . 0 0 t h r u h o l e d r i l l e d a n d 2 " N P T f e m a l e t a p p e d a t e a c h e n d 1 . 1 " d e e p s t a n d a r d t a p e r 2 . 0 0 ? 0 . 0 5 0 2 0 . 0 0 ? 0 . 0 5 0 f r o n t v i e w . 0 1 0 A. 0 2 0 A 5 o f 5 e d g e r a d i u s . 1 2 5 . 0 0 1 A . 1 0 0 A . 1 2 5 ? 0 . 0 5 0 B a s e O v e r v i e w Z i m m e r m a n 1 1 A u g 2 0 0 3 1 o f 4 B a s e T o p V i e w Z i m m e r m a n 1 1 A u g 2 0 0 3 2 o f 4 ? 2 7 . 0 0 ? 0 . 1 0 ? 1 4 . 1 7 5 6 + 0 . 0 0 2 3 / - 0 . 0 0 0 0 ? 1 2 . 3 4 1 ? . 1 0 0 t o p v i e w A A A A B a s e C r o s s S e c t i o n Z i m m e r m a n 1 1 A u g 2 0 0 3 3 o f 4 c r o s s s e c t i o n A A 4 . 0 0 ? . 1 2 5 1 9 . 5 0 ? . 1 2 5 1 . 6 3 4 ? 0 . 0 2 0 1 . 5 0 ? . 0 5 0 A . 0 0 1 A . 0 5 0 A . 1 2 5 A . 1 2 5 A D. MODAL AND STRESS ANALYSIS DIRECTORY This appendix lists information regarding the nal modal and static frame analysis les. Tab. D.1: This is a le list of the nal run Algor models as discussed in this dissertation, both static and vibrational models. These les can be accessed with user access to wave.umd.edu, and are located in the di- rectory /data/3m/axl/Dissertation/algor frame/. PDF output of mode shapes are available in: /data/3m/axl/Dissertation/ gex/algor/modes/ and PDF outputs of static analyses are available in /data/3m/axl/Dissertation/ gex/algor/static/. Filename Type Constraints 2sph Modal All Edges 2sph s Static All Edges 2sph u Modal Only Base 2sph r Modal Randomly Selected 2sph e Modal One Edge E. INSTRUMENTATION INFORMATION This appendix collects schematics and other instrumentation information for instruments developed by the author that may be useful to other researchers, both in the Lathrop Lab and elsewhere. E.1 Torque Transducer Schematics The schematics for the wireless torque transmitter and its power supply are shown in Figs. E.1, E.2, and E.3. The home-made circuitry was constructed using point-to-point wiring on perf-board in a con guration that could plug in to the Arduino microcontroller board. Future versions of this device will be laid out on a printed circuit board, probably with multiple channels of acquisition addressed by the SPI bus protocol. Newer multi-channel ADC chips like the Analog Devices AD7606 may provide a good alternative to multiple one channel SPI-addressed converters. Futek TF600, 1.68mV/V FSO Vcc , Clean Analog, Nom. 6.5V, Actual ?V 12.740V +Vcc -Vcc SPI_ADC_Xbee _v_2_0.pde Arduino U3 NC RESET 3v3 GND VinAI0AI1AI2AI3AI4AI5 D0 RXD1 TX D11 D12D2 D3 D4 D5 D6 D7 D8 D9 D10 D13 GND AREF GND 2.4GHzXBee Vcc Dout Din DO8 RESET RSSI PWM1 NC DTR GN D AD4CTSON VREF Associate RTSAD3AD2AD1AD0 (20 Pin) U4 22 bit ADCMCP3553 U2 1 Vref 2 In+ 3 In- 4 Vs s SCK5 SDO6 CS7 Vdd 8 (MSOP8) LT1167 U1 1 Rg 2 3 4 -V s Ref 5 6 +Vs 7Rg 8 ? + 500 ? , G = 99.9 A +Vcc A A A Analog Ground D D D GRN D YE L In Out Gnd D Digital Ground Arduino Status A 2.5V Precision Cb Cb C b 0.1 ?F Chip Cap 0.1 ?F 0.1 ?F D +5V Digi +3v3 Digi J1, DB9 1 2 3 4 -Vcc Voltage Regulation on Separate Page Gain Programming Resistor Wireless Torque Transducer Dan Zimmerma n 9/13/10 Arduino Microcontroller XBee Wireless Transmitter Resistive Strain Gauge Full Scale, 1130N*m = 1801990 Output is an integer: according to mV Calibration 10/16/08 Surface Mount Daughterboard D Instrumentation Amplifier Analog-Digital Conv. Fig . E.1 : Sc hemati c of th e wireles s torqu e transduce r b ox , exclu din g voltag e regulation/ p ow er conditionin g circuitr y. Th e strai n gaug e senso r is ampli e d by a Linea r T ec hnolog y LT116 7 instrume ntatio n ampli e r of gai n 10 0 an d digi - tize d wit h a Micr oc hi p MCP355 3 22 bi t ADC . Th e co nv ersion s ar e co ntro lle d by an A rduin o micr oco ntrolle r b oar d runnin g r m w ar e writte n by th e author , SP I AD C X b ee v 2 0. p de , publishe d b el ow an d av ailabl e fo r do wnloa d at h ttp:// w av e.umd.edu/axlc o de/SP I AD C X b ee v 2 0. p d e Thi s co de read s sample s an d transmit s the m in ASC II signe d in tege r fo rm at vi a a Dig i XBe e 2.4GH z wireles s seria l connectio n to a compute r in th e la b fr am e. 310 In Out Gnd A A S1A S1B +12V -12VGNDRED BLK GRN A GRN A 15 ?F-6.5V (-Vcc) In Out Gnd A A A 15 ?F +6.5V (+Vcc) RED LM7805 LM7905 A D 100 ?H 100 ?H 15 ?F 0.1 ?F 0.1 ?F 15 ?F Clean +12V Analog Side In Out Gnd LD33V D 15 ?F 0.1 ?F+3v3, Digita l In Out Gnd A +5V, Digital Torque Box Power Conditioning Dan Zimmerman 9/13/10 Conditioning Filter LM7805 This circuit is i nside the torque box, drawn separatel y for clarity. +/- 12V and Ground from Battery / Converter Box, PowerPole Connectors Analog Digital Main Power Switch Fig . E.2 : T orqu e b ox p ow er conditioning . 780 5 an d 790 5 linea r regulator s wit h voltag e divider s (se e data sh eet ) pr ovid e nominall y 6: 5V fo r th e instrume ntatio n ampli e r an d strai n gaug e excitation . A balanced-p i ne tw or k lte r separate s analo g an d digita l p ow er an d ground , all owin g D C to pas s bu t helpin g to kee p digita l pulse s ou t of th e se ns iti ve milli volt-le ve l analo g circuitr y. A 780 5 regulato r pr ovide s + 5 V fo r th e Arduin o an d an LD33 V regulato r pr ovide s +3. 3 V fo r th e XBe e p ow er suppl y. Th e Arduin o ha s an on- b oar d 3. 3 V regulator , bu t it is insu cie nt to p ow er th e XBee . 311 In Out Gnd ? + +Vin -Vin CNT +Vout COM -Vout Trim LD33V LM311 270k? 10k? 1k? 10k? 3.7:1 Divider LM311 Power +Vin / -Vin + + 3.7V LiPoly 3.7V LiPoly Internal batteries fused and switched for charging (not shown). External battery permanently in series. Hysteretic comparator shuts off DC-DC converter @ about 6V Shield Wall +12V, 850mA 0.1?F 15?F 0.1?F 15?F LC Feedthrough CUI PK25-D5-D12 DC-DC Converter, Isolated +12V, 850mA RED GREEN BLACK Inner Shaft Battery Box Dan Zimmerman 9/13/10 Ground tied to box. Battery Side Floating w.r.t. Box Fig. E.3: DC-DC converter to accept the 6 V to 8.3 V output of the lithium polymer battery pack and output 12 V to power the torque sensor box. The comparator circuit is used to disable the DC-DC converter when the battery voltage drops below 6 V. Lithium polymer batteries do not survive deep discharge, and this is a simple protective circuit. Typically, the battery voltage rebounds a bit when load is removed, and the hysteresis built into this circuit is insu cient to keep the battery box o when the battery is just barely discharged to 6 V under load. However, large gaps and uctuations appear in the acquired torque signal, signaling that the end of useful battery life has been reached, and the battery will not discharge below 6 V loaded in any case. 312 E.2 Resonant UVP Transformer A resonant coupling ring arrangement was used early in the experiments for transferring the narrowband radio frequency pulses used in the ultrasound doppler velocimetry technique. A schematic and annotated photograph of the device are pic- tured in Fig. E.4. This system was adapted from a technique for coupling parallel- wire transmission line to a rotary antenna array found in an old American Radio Relay League antenna handbook [155]. This resonant coupling technique is currently enjoying something of a renaissance for \wireless power" delivery to small electronic devices [156, 157]. A signi cant amount of power may be transferred between elec- tromagnetic resonators even with spacings of many resonator diameters. As the resonators are moved apart, however, more and more power is dissipated in the losses associated with large amounts of stored eld energy. In the devices tested in our lab, we have always used close-coupling for best transfer e ciency, which should exceed 99% according to a LLNL NEC-2 nite element electromagnetic model. Rea- sonable results may be achieved with signi cantly wider ring spacing, however, if this is deemed necessary to accommodate the mechanical arrangement of a small experiment. Ring size and geometry are not particularly critical, but will a ect the ca- pacitance required to resonate. We have implemented the with rings of 20 cm as well as the 45 cm diameter rings shown here. Both systems showed good results. The capacitors are selected by measuring or calculating the inductance of a single loop of the desired size and choosing C = 1=((2 fo)2L), with fo the emission fre- 313 quency and L the loop inductance. Nearby metal reduces the inductance of the rings and a ects tuning, but if loop inductance is measured in situ, this is not a se- rious problem. Because of the relatively loose coupling of the rings, this transformer provides a resistive impedance step up somewhere between 2:1 to 3:1 depending on the geometry. However, we have found that the AC impedance1 of a typical ultrasound trans- ducer immersed2 is not 50 + j0, but typically somewhat lower and capacitatively reactive, 10 j10 for one example. So the impedance step-up is not a problem. For better impedance match, the ratio of the diameter of the rings could also be adjusted. The adjustable capacitors can be used to tune out residual reactance. A low cost MFJ Enterprises 259B impedance bridge is used for tuning the system to near 50 , as the resonant circuits are very sharp and tuning is critical. However, it should be possible to tune without an impedance bridge by tuning best echo sig- nal if the capacitors are known to be approximately the right values. This circuit should be relatively noise immune with strong elds dropping o quickly, but devices and wiring that produce interfering energy in the range of the ultrasound emission frequency should be kept at least a few ring diameters away. The system can be scaled down e ectively for use in smaller experiments, where it might nd maximum utility. However, some care should be used in capacitor 1 The letter "j" is the engineering convention for p 1. 2 A signi cant portion of the real part of the transducer electrical impedance is due to radiation. Impedance matching should be undertaken with the transducer immersed in the working uid, or at least similar uid. 314 a b d c a b c d ~50? To UVP Transducer Fig. E.4: Annotated photograph and schematic of the resonant inductive cou- pling arrangement used to transfer 4 MHz signals into the rotating frame. A parallel (nearly) resonant circuit is connected to the ul- trasound velocimetry unit via coaxial cable and a series (nearly) resonant circuit is connected to the ultrasound transducer. The photograph shows the physical layout of the system implemented on the three meter apparatus, with the lab-frame ring (a) and tun- ing capacitor (c) comprising the parallel resonant circuit connected to the UVP monitor and the rotating ring (b) and capacitor (d) connecting to the transducer in the rotating frame. 315 selection as the inductive reactance XL = 2 foL of the rings at the frequency of interest becomes small. Large capacitance will be needed, and dielectric losses might start to become an issue. Good radio-frequency capacitors should be chosen in any case, but the resistance of the capacitors will only cause signi cant losses when their reactance must be very large. For smaller devices (say, less than 10 cm for 4 MHz), heavy copper multi-turn inductors may be advantageous in this regard. When low- reactance rings are used, some parallel-connected xed capacitors will probably be required, as variable capacitors of su cient capacitance will not be available. The device used on the three-meter experiment uses a combination of xed polystyrene and variable mica trimmer capacitors in parallel to provide su cient capacitance with some tuning range. E.3 CTA Circuit Board and Layout The four-channel CTA circuit described in Section 3.5.4 was built with circuit boards made in house. Some circuits were made with double-sided board with a ground plane and others with single-sided. Neither construction seemed to have any signi cant advantage. Single-sided is therefore probably the recommended construc- tion, as they are easier to etch, with no masking needed on the back. Boards were made using a toner-transfer process. Reversed board layout patterns are printed onto glossy paper, in this case a magazine cover, using a laser printer. The laser toner is then ironed on to a clean board. The board with paper is soaked in water, and the paper can be rubbed o leaving the toner behind. The laser toner is a 316 good resist for ferric chloride copper etchant, which is then used to etch the boards. The board foil pattern suitably reversed for toner transfer and a parts placement diagram are included below. 317 AxlCTA v3.0 AxlCTA v3.0 U1, U2 AD8662 or similar ~10?F Tantalum 100k Output GainRoh Overheat, pot 1k? R2 Zero Offset, 1k? R1 FZT491A Power 0.1?F Chip Caps CTA Rev 3 Parts Placement Guide Copper Side SMT Parts Dielectric Side, Leaded Parts Sensor Connection This view has resistors as built, R1 = 3x82?, R2 = 270? 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