ABSTRACT
Title of dissertation: Accretion Physics Through
the Lens of the Observer:
Connecting Fundamental Theory
with Variability from Black Holes
James Andrew Hogg, Doctor of Philosophy, 2018
Dissertation directed by: Professor Christopher Reynolds
Department of Astronomy
Variability is a generic feature of accretion onto black holes. In both X-ray
binaries and active galactic nuclei, variability is observed on nearly all accessible
timescales and across the entire electromagnetic spectrum. On different timescales
and at different wavelengths it has unique signatures that can be used to character-
ize the accretion processes generating the emission and probe the accretion disks,
which would otherwise be impossible. Despite having been observed for over fifty
years, interpreting this variability is difficult. Simple phenomenological models have
been used to explain the behaviors and geometries of the observed accretion disk,
but they have yet to be rigorously tested in a full magnetohydrodynamic frame-
work. In this dissertation we use high-resolution numerical models to investigate:
(1) “propagating fluctuations” in mass accretion rate that give rise to the nonlin-
ear signatures of accretion on viscous timescales, (2) the dynamics of truncated
accretion disks which are invoked to explain the spectral variation of outbursting
X-ray binaries and the bifurcation of AGN accretion states, and (3) the large-scale
magnetic dynamo behavior in thick and thin accretion disks.
We find that the structured variability readily seen in the light curves from
accreting black holes (i.e. log-normal flux distributions, linear relations between
the RMS and the flux, and radial coherence) quickly and naturally grows from
the MRI-driven turbulence and that these properties translate into photometric
variability. For the first time, we identify the large-scale magnetic dynamo as the
source of the low-frequency modulations of the disk stress that cause this structure.
We introduce a bistable cooling law into hydrodynamic and magnetohydrodynamic
simulations to study the manifestation of a truncated accretion disk in each regime.
We find that rather than a truncation edge, the transition is better described by a
“truncation zone” when the angular momentum transport and heating is governed
by MRI-driven turbulence instead of a true viscosity. Additionally, we find that
the hot gas in the simulation buoyantly rises in a gentle outflow and eventually fills
the entire volume, instead of simply being confined to the innermost region. The
outflow interacts with the disk body and enhances the magnetic stresses, which could
produce stronger quasiperiodic variability. Finally, we conduct an investigation of
the large-scale magnetic dynamo using a suite of four global magnetohydrodynamic
disk simulations with scaleheight ratios of h/r = {0.05, 0.1, 0.2, 0.4}. Most notably,
the organization that is prevalent in accretion disk simulations and described as a
“butterfly pattern” does not occur when h/r ≥ 0.2, despite the dynamo action still
operating efficiently.
Accretion Physics Through the Lens of the Observer:
Connecting Fundamental Theory with Variability from Black Holes
by
James Andrew Hogg
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2018
Advisory Committee:
Professor Christopher Reynolds, Chair/Advisor
Professor Coleman Miller
Professor Richard Mushotzky
Dr. Lisa Winter, External Examiner
Professor William Dorland, Dean’s Representative
©c Copyright by
James Andrew Hogg
2018

Preface
The research presented in Chapters 2-5 of this dissertation has been published
in The Astrophysical Journal and is presented here with minimal modification. Each
chapter corresponds to the following papers:
• Chapter 2 - “Testing the Propagating Fluctuations Model with a Long, Global
Accretion Disk Simulation,” The Astrophysical Journal, Volume 826, Issue 1,
Article ID. 40, 20 pp. (2016)
• Chapter 3 - “The Dynamics of Truncated Black Hole Accretion Disks. I.
Viscous Hydrodynamic Case,” The Astrophysical Journal, Volume 843, Issue
2, Article ID. 80, 15 pp. (2017)
• Chapter 4 - “The Dynamics of Truncated Black Hole Accretion Disks. II.
Magnetohydrodynamic Case ,” The Astrophysical Journal, Volume 854, Issue
1, Article ID. 6, 17 pp. (2018)
• Chapter 5 - “The Influence of Accretion Disk Thickness on the Large-scale
Magnetic Dynamo,” The Astrophysical Journal, , Volume 861, Issue 1, Article
ID. 24, 15 pp. (2017)
To me, the contents of this dissertation form a coherent story. In the first act,
we investigate propagating fluctuations in mass accretion rate and in the second act
we explore the dynamics of truncated accretion disks. In the third act, we focus on a
reoccurring character who always seems to be tangled up in the action as we search
for the driver of variability in our simulations - the large-scale magnetic dynamo.
ii
Many of the same topics are revisited in each chapter and the story develops as new
pieces of evidence come to light. We see how each of the many characters driving the
disk behavior builds off, depends on, and, ultimately, affects the others. At times,
it may seem like the story was written with the dynamo as the main character from
the outset. This could not be further from the truth as the dynamo was only meant
to be a minor player. We originally wrote it into the story at each juncture for its
novelty, but ended up discovering it serves the role of the mysterious figure who
lurks in the background and connects all the otherwise unexplainable occurrences in
our story. At the end we try to uncover its identity, and do a little, but it manages
to just remain in the shadows.
I’m not sure where the story goes from here or how it ends, but it leaves me
with a few questions. What makes the dynamo tick? Are there other important
characters who have yet to make their debut? Maybe there is an accomplice or two?
It seems like the series is just beginning and the answers to these questions must
come in a sequel or two....
iii
Dedication
To those who open it and learn something new.
iv
Acknowledgments
More often than not, my life as a graduate student mirrored the accretion
disks I studied. There was an underlying bulk flow taking me in the right direction,
but I often felt like I was chaotically getting knocked about by hiccups like codes
that mysteriously crash or paper introductions that I just couldn’t write. Often
these frustrations were quickly followed by successes that forced me in the opposite
emotional direction, like telescope proposals being accepted despite unlikely odds or
great conversations at conferences.
To properly thank everyone who has helped me along my way to a Ph.D.
and aided in dampening the intrinsic instabilities of graduate school would require a
document as long as this dissertation. Nevertheless, I would like to thank everyone in
the Astronomy department at UMD, including the faculty, staff, post docs, students
I taught as a T.A., and of course my fellow graduate students. These people made it
a great place to work for six years and I feel incredibly lucky to have been surrounded
by such great folks.
I will forever be grateful for the mentorship of Chris Reynolds and it is hard to
encapsulate all that it has meant. His foresight and expertise enabled these projects,
but even more than that, his supervision shaped who I became as a scientist. I
always felt like an equal in our partnership and he always encouraged me to pursue
my own ideas. Consequently, many of the rabbit holes I went down were dead ends,
but they helped build intuition for the problem, and it taught me that failure is an
unavoidable part of creating new knowledge. A few were not dead ends and turned
v
into results or accomplishments that I am incredibly proud of, so I am thankful for
the opportunities.
While this dissertation focuses on work done as a graduate student at the
University of Maryland, my Ph.D. journey and scientific training actually began
1507 miles away at the University of Colorado. In Boulder I gained my first real
research experience under the supervision of Lisa Winter. That work sparked my
interest in accretion physics which led to the work presented here. I am certain I
would not have been as successful in graduate school (or even gotten in) if I had not
had her guidance. Her selflessness, knowledge, and enthusiasm mean the world to
me, and I am thrilled she has agreed to serve as a member of my advisory committee.
Earning a Ph.D. is taxing on those around you. I am thankful for the support
and love from my family and girlfriend over the last six years. Any time I needed
reassurance or a confidence boost I came to them and, while they willingly encour-
aged me and offered support, I know it passed on a lot of undue stress. This degree
is as much yours as it is mine.
I would be remiss without acknowledging the companionship of my cat, Stella,
throughout my grad school adventure. Oddly enough she was born on August 29,
2012, which my first day as a University of Maryland student, so she has been with
me since the beginning. Despite her best efforts to contribute to my papers by walk-
ing across my keyboard, her text was never (knowingly) included in the submitted
work, so she has never received authorship credit. If there are any egregious typos
in this work, please attribute them to her.
vi
Table of Contents
Preface ii
Dedication iv
Acknowledgements v
List of Tables x
List of Figures xi
List of Abbreviations xiii
1 Introduction 1
1.1 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The Accretion Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 The Magnetorotational Instability . . . . . . . . . . . . . . . . . . . . 16
1.4 Techniques for Simulating Accretion Disks . . . . . . . . . . . . . . . 22
1.5 The Dissertation in Context . . . . . . . . . . . . . . . . . . . . . . . 33
2 Propagating Fluctuations In Mass Accretion Rate 37
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Numerical Model and Convergence . . . . . . . . . . . . . . . . . . . 41
2.2.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3 Turbulence and the Effective α Description of the Model Disk . . . . 54
2.3.1 Fluctuating Effective α . . . . . . . . . . . . . . . . . . . . . . 54
2.3.2 Characterizing Velocity and Length Scales of the Turbulence . 57
2.3.2.1 Measuring Turbulent Velocity . . . . . . . . . . . . . 57
2.3.2.2 Measuring Turbulent Scales . . . . . . . . . . . . . . 58
2.4 Disk Dynamo and Intermediate Timescale α Variability . . . . . . . . 62
2.4.1 General Dynamo Behavior . . . . . . . . . . . . . . . . . . . . 65
2.4.2 Modulation of Stress By Dynamo . . . . . . . . . . . . . . . . 68
2.5 Propagating Fluctuations in the Model Disk . . . . . . . . . . . . . . 75
vii
2.5.1 Overview of Global Behavior . . . . . . . . . . . . . . . . . . . 75
2.5.2 Log-normal Ṁ Distribution . . . . . . . . . . . . . . . . . . . 78
2.5.3 Correlations Between α, Ṁ , & Σ . . . . . . . . . . . . . . . . 79
2.5.4 Radial Coherence . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.5.5 RMS-Flux Relationship . . . . . . . . . . . . . . . . . . . . . 84
2.6 Proxy Light Curve Analysis . . . . . . . . . . . . . . . . . . . . . . . 86
2.6.1 Signatures of Propagating Fluctuations . . . . . . . . . . . . . 89
2.6.2 Timing Properties . . . . . . . . . . . . . . . . . . . . . . . . 92
2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3 Truncated Accretion Disks In Hydrodynamics 102
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.2.1 Simulation Code . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.2.2 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.2.3 Cooling Function . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.3.1 The Truncated Disk . . . . . . . . . . . . . . . . . . . . . . . 116
3.3.2 Disk Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.3.3 Mass Accretion and Angular Momentum Transport . . . . . . 127
3.3.4 Disk Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.3.4.1 Energy Terms . . . . . . . . . . . . . . . . . . . . . . 136
3.3.4.2 Bound Outflow . . . . . . . . . . . . . . . . . . . . . 139
3.3.4.3 Synthetic Light Curves . . . . . . . . . . . . . . . . . 141
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4 Truncated Accretion Disks In Magnetohydrodynamics 148
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.2.1 Simulation Code . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.2.2 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.2.3 Test Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4.2.4 Cooling Function . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.2.5 Resolution Diagnostics . . . . . . . . . . . . . . . . . . . . . . 159
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.3.1 The Truncated Disk . . . . . . . . . . . . . . . . . . . . . . . 162
4.3.2 Disk Dynamics and Variability . . . . . . . . . . . . . . . . . . 167
4.3.3 Mass Accretion and Angular Momentum Transport . . . . . . 173
4.3.4 Disk Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.3.5 Disk Dynamo . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
viii
5 Towards A More Complete Understanding Of The Large-scale Magnetic Dy-
namo 196
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.2.1 Simulation Code . . . . . . . . . . . . . . . . . . . . . . . . . 200
5.2.2 Simulation Setups . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.2.3 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.2.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.3.1 The Global Dynamo . . . . . . . . . . . . . . . . . . . . . . . 208
5.3.2 Measuring αd . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
5.3.3 Magnetic Helicity . . . . . . . . . . . . . . . . . . . . . . . . . 216
5.3.4 Mass Accretion Rates . . . . . . . . . . . . . . . . . . . . . . . 220
5.3.5 Observational Signatures . . . . . . . . . . . . . . . . . . . . . 221
5.4 Extended Domain Tests . . . . . . . . . . . . . . . . . . . . . . . . . 225
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
6 Conclusions 237
Bibliography 253
ix
List of Tables
2.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2 Resolution Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3 Ellipsoid Fit Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5.2 Resolution Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.3 Upper Hemisphere Dynamo Coefficient Fits . . . . . . . . . . . . . . 216
5.4 Lower Hemisphere Dynamo Coefficient Fits . . . . . . . . . . . . . . 217
x
List of Figures
1.1 Conceptual drawing of MRI growth . . . . . . . . . . . . . . . . . . . 17
1.2 Schematic of Riemann problem in cell discretization with first-order
spatial reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.3 Schematic of Riemann problem with second-order spatial reconstruction 30
2.1 Snapshots of density and |B2| in Zeus simulation . . . . . . . . . . . 44
2.2 Average disk thickness in Zeus simulation . . . . . . . . . . . . . . . 46
2.3 Quality factors in Zeus simulation . . . . . . . . . . . . . . . . . . . . 49
2.4 Magnetic tilt angle in Zeus simulation . . . . . . . . . . . . . . . . . 51
2.5 α-parameter time dependence . . . . . . . . . . . . . . . . . . . . . . 55
2.6 α-parameter histograms . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.7 Time variability in turbulence properties . . . . . . . . . . . . . . . . 59
2.8 Turbulence autocorrelations . . . . . . . . . . . . . . . . . . . . . . . 63
2.9 Snapshot of Bφ in Zeus simulation . . . . . . . . . . . . . . . . . . . 64
2.10 Height dependent Bφ PSDs . . . . . . . . . . . . . . . . . . . . . . . 66
2.11 Bφ and BRBφ spacetime diagrams in Zeus simulation . . . . . . . . . 69
2.12 PSDs of Bφ and BRBφ . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.13 Spacetime diagrams of Σ, Ṁ , and emission proxy . . . . . . . . . . . 72
2.14 Ṁ time variability and PSD . . . . . . . . . . . . . . . . . . . . . . . 73
2.15 Ṁ histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.16 Ṁ , Σ, and α correlations . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.17 Ṁ coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.18 Ṁ timelag behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.19 Ṁ RMS - flux relation . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.20 Emission proxy variability and radial profile . . . . . . . . . . . . . . 87
2.21 Light curve properties . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.22 Light curve PSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.1 Gas density during heating transient . . . . . . . . . . . . . . . . . . 116
3.2 HD truncated disk density and temperature . . . . . . . . . . . . . . 118
3.3 Radial structure of hot/cold gas fractions in HD truncated disk . . . 120
3.4 Maps of 〈vφ〉/vk and 〈vr〉/vk. in HD truncated disk . . . . . . . . . . 122
3.5 HD outflow launching . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
xi
3.6 Time variability of hot/cold gas fractions in HD truncated disk . . . . 126
3.7 Hot/cold mass accretion rates in HD truncated disk . . . . . . . . . . 127
3.8 Stress maps in HD truncated disk . . . . . . . . . . . . . . . . . . . . 129
3.9 Maps of momentum equation terms in HD truncated disk . . . . . . . 131
3.10 Time variability of angular momentum equation terms in HD trun-
cated disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.11 Radial profiles of energy equation terms in the HD truncated disk . . 134
3.12 Time variability of energy equation terms in the HD truncated disk . 135
3.13 Outflow properties in HD truncated disk . . . . . . . . . . . . . . . . 140
3.14 Light curves from HD truncated disk . . . . . . . . . . . . . . . . . . 144
4.1 Disk thickness profile in MHD truncated disk . . . . . . . . . . . . . 161
4.2 MHD truncated disk density and temperature . . . . . . . . . . . . . 163
4.3 Radial structure of hot/cold gas fractions in MHD truncated disk . . 166
4.4 Map of average mass flux in MHD truncated disk . . . . . . . . . . . 167
4.5 Outflow structure in MHD truncated disk . . . . . . . . . . . . . . . 169
4.6 Radial profiles of gas properties in MHD outflow . . . . . . . . . . . . 171
4.7 Radial profile of turbulent intensity . . . . . . . . . . . . . . . . . . . 172
4.8 Time variability of hot/cold gas fractions in MHD truncated disk . . 173
4.9 Hot/cold mass accretion rates in MHD truncated disk . . . . . . . . . 174
4.10 Profiles of momentum equation terms in MHD truncated disk . . . . 175
4.11 Stress maps in MHD truncated disk . . . . . . . . . . . . . . . . . . . 177
4.12 Radial profiles of energy equation terms in the MHD truncated disk . 180
4.13 Light curves from MHD truncated disk . . . . . . . . . . . . . . . . . 181
4.14 Spacetime diagrams of Bφ in MHD truncated disk . . . . . . . . . . . 183
4.15 Plasma β maps and α-parameter profile . . . . . . . . . . . . . . . . 187
4.16 PSD of BRBφ in truncation zone . . . . . . . . . . . . . . . . . . . . 188
5.1 Confirmation of scaleheight ratio stability . . . . . . . . . . . . . . . 203
5.2 Spacetime diagrams of Bφ as a function of disk thickness . . . . . . . 208
5.3 PSDs of Bφ as a function of disk thickness . . . . . . . . . . . . . . . 210
5.4 Dynamo coefficient fits in disk upper atmospheres . . . . . . . . . . . 214
5.5 Dynamo coefficient fits in disk lower atmospheres . . . . . . . . . . . 215
5.6 Mean Bφ and magnetic helicity . . . . . . . . . . . . . . . . . . . . . 218
5.7 Histograms of mass accretion rates in different thickness disks . . . . 222
5.8 Synthetic light curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
5.9 Extended azimuthal domain Bφ spacetime diagrams . . . . . . . . . . 226
5.10 Extended azimuthal domain Bφ PSDs . . . . . . . . . . . . . . . . . . 227
6.1 Truncated disk schematics . . . . . . . . . . . . . . . . . . . . . . . . 240
xii
List of Abbreviations
ADAF Advection Dominated Accretion Flow
AGN Active Galactic Nucleus
Be Bernoulli Parameter
BHB Black Hole Binary
CAR Continuous Time, Auto-Regressive
CDAF Convection Dominated Accretion Flow
CV Cataclysmic Variable
D.o.F Degrees of Freedom
EMF Electromotive Force
GBHB Galactic Black Hole Binary
GRRMHD General Relativity, Radiation Magnetohydrodynamics
HD Hydrodynamics
IMBH Intermediate Mass Black Hole
ISCO Innermost Stable Circular Orbit
LLAGN Low-Luminosity Active Galactic Nucleus
LSST Large Synoptic Survey Telescope
MCMC Markov Chain Monte Carlo
MHD Magnetohydrodynamics
MRI Magnetorotational Instability
NICER Neutron star Interior Composition Explorer
OU Ornstein-Uhlenbeck
PDF Probability Density Function
PSD Power Spectral Density
QPO Quasi-Periodic Oscillation
RHD Radiation Hydrodynamics
RMS Root-Mean Square
SED Spectral Energy Distribution
xiii
SMBH Supermassive Black Hole
TESS Transiting Exoplanet Survey Satellite
UMD University of Maryland
UV Ultraviolet
YSO Young Stellar Object
xiv
Chapter 1: Introduction
Gas accretion onto compact objects remains one of the great unsolved prob-
lems in astrophysics. The accretion process, broadly, is the process of a massive body
acquiring mass by drawing in material from its surrounding environment courtesy of
its gravitational attraction. From the smallest scales of pebble-sized chondrule ac-
cumulation from the solar nebula that forms comets and asteroids up to the largest
possible scales of large galaxy clusters pulling in gas and smaller galactic halos
from the cosmic web, we see many different forms of accretion. While they are all
fascinating, this dissertation will focus on the accretion processes occurring in gas
disks around black holes. In particular, we model black hole accretion flows with
the overarching goal of connecting accretion physics with spectral and photomet-
ric variability that can be observed with telescopes and used to probe black hole
systems. We explore the physics driving “propagating fluctuations in mass accre-
tion rate” in Chapter 2 and the dynamics of truncated accretion disks in Chapter 3
and Chapter 4. Throughout this dissertation work the large-scale magnetic dynamo
consistently appears as a key player in the disk behavior so we dedicate Chapter 5
to characterizing its properties as the disk scale-height ratio is changed.
We begin here by reviewing the basics of black holes, accretion theory, and
1
accretion disk modeling before diving into the main thesis where the individual
studies are discussed. This is meant to serve as a general overview of accretion
and introductions to each of the specific topics studied are kept to the relevant
chapters. This chapter closes with a brief discussion of how this dissertation fits
into the advancement of accretion physics at large, and the dissertation closes with
a summary of the main results and what lies on the horizon. The essence of this
thesis and what we hope to convey is that more physically grounded variability
models are desperately needed to fully leverage the next generation of time-domain
surveys, we are in the midst of an exciting period of rapid advancement, and there
are many fundamental aspects of accretion still to be uncovered.
1.1 Black Holes
Black hole systems fall into two classes. The first is occupied by stellar mass
black holes which form as the final remnants of massive stars. A star of mass greater
than M∗ ≥ 30 M will spend several million years on the main sequence fusing
hydrogen into helium. As the fuel is exhausted, heavier elements are consumed
and the star evolves off the main sequence. Following several periods of burning
successively heavier and heavier elements, the star is left with an iron core. At this
point the heating mechanism is removed since no more energy can be extracted.
Lacking thermal support against its own self-gravity, the massive star is doomed to
collapse upon itself in a sudden, catastrophic event. This event typically leads to one
of the shortest lived and most violent events in the cosmos - a supernova explosion.
2
In less than a second, a black hole of a few to tens of solar masses is formed and
the remaining material is ejected. Alternatively, there is mounting evidence some
of the higher mass stellar mass black holes may be formed from direct collapse, i.e.
with no explosion in a “failed” supernova (Adams et al., 2017a,b; Gerke et al., 2015;
Heger et al., 2003).
The majority of these massive stars are found in binary systems. If the binary
survives the supernova, the black hole from the progenitor massive star and the
companion star will be in close proximity when the secondary evolves off the main
sequence itself. When the envelope of the secondary star’s atmosphere expands past
the Roche lobe it feels a greater pull to the black hole. It will then be accreted and
can be seen as an X-ray binary.
The second class of black holes, supermassive black holes (SMBHs), are roughly
a million times more massive than stellar mass black holes and believed to reside in
the center of all galaxies. Observationally, ≈ 10% of all SMBHs are active galactic
nuclei (AGNs) because they are actively accreting at high rates. Dynamical pro-
cesses in the host galaxy or gravitational interactions with other galaxies can drive
dust and gas from the interstellar medium to the center and feed the SMBH. At
lower levels, accretion can occur when outflows from massive stars in the center
of the galaxy provide a background of ambient gas. In more exotic cases, stars
that wander too close to a quiescent black hole can get ripped apart and trigger an
accretion episode in a tidal disruption event.
Even though they are common to most massive galaxies, the origin of SMBHs
remains a mystery and is vigorously debated in the literature. At high redshift,
3
i.e. early cosmic time, massive quasars have been discovered. Current theories
hypothesize they must have formed initially at high masses by the direct collapse of
massive gas clouds, through runaway collisions of stars or black holes in clusters, or
from the first stars like stellar mass black holes. These scenarios all have challenges
they must overcome before being fully accepted.
The important influence of SMBHs on galaxy formation and evolution is not
debated, however. Through feedback processes, i.e. radiative and mechanical energy
injection, the energy released through accretion can regulate the growth and devel-
opment of galaxies and star formation. The quintessential example is the M − σ
relationship where the mass of the central supermassive black hole is seen to scale
with the velocity dispersion of the host galaxy bulge. This relation is typically inter-
preted as an example of mechanical feedback from outflows launched by the accretion
disk around the SMBH acting on its host galaxy, or at least evidence of a mutual
evolution between the comparatively small scales of the accretion environment of
the black hole and the larger galaxy in which it lives.
Despite the discrepancy between masses, the diversity in environments, and
the differences in initial feeding, the two classes of black hole systems seem to be
quite similar. This similarity has led stellar mass black hole systems to be referred to
as “microquasars” as they essentially look like scaled down AGNs in their behaviors
and properties. For our theoretical exploration of the accretion physics powering
these objects, it suggests the same processes are operating in both types of systems
and motivates a general approach for understanding their properties. In fact, much
of what will be studied has application in most, if not all, ionized accretion disks,
4
including those around accreting white dwarfs (CV systems) and neutron stars.
Before proceeding, this introduction would be remiss by failing to mention a
third class that has been hypothesized to exist with masses of ≈ 1000 M between
the two standard black hole populations, aptly named “intermediate mass” black
holes (IMBHs). Several candidates of black holes belonging to this elusive species
have been found, but they come with ambiguity. If they exist, these black holes may
have primordial origins so positively identifying an IMBH would bolster the direct
collapse models of massive black hole formation. However, much of the difficulty
in finding IMBHs is that their mass determinations are typically based on their
luminosity, which is assumed to obey the Eddington limit, discussed below. A
number of IMBH candidates have recently been show to be super-Eddington pulsars,
but it is not known if black holes can produce super-Eddington luminosities.
1.2 The Accretion Problem
Before diving into the advanced numerical modeling of this dissertation, it is
prudent to frame the context for this work by revisiting the analytics that under-
pin accretion theory in order to build an intuitive understanding of the governing
processes.
The simplest accretion imaginable is a central massive object embedded within
a uniform medium of zero angular momentum. This scenario is referred to as Bondi
accretion after the seminal work of Bondi (1952) and results in a gas inflow that will
be steady and spherically symmetric. This problem has two characteristic length
5
scales. The first is called the Bondi radius,
GM
rB =
c2s,∞
where G is the gravitational constant, M is the mass of the central object, and c2s,∞
is the sound speed of the ambient medium with the sound speed being set by the
gas pressure P and gas density, ρ, √
γP
cs = . (1.1)
ρ
This is the farthest radius where the gravitational pull of the object is strong enough
to dominate the gas motions and ρ first begins to increase above the ambient value.
This is often called the “accretion radius”, racc. The second length scale is the sonic
radius, rs
GM
rs = .
2c2s
Two fundamental equations govern the gas dynamics of the inflow and the location
where the gas must become transonic. These are the (stationary and spherically
symmetric) continuity equation
dρ ρ d(vr2)
= (1.2)
dr vr2 dr
and the Euler equation describing the gas momentum
dv 1 dP GM
v + + = 0. (1.3)
dr ρ dr r
Combining and rearra(nging giv)es an equation o[f the(form, )]
1 c2− s d(v
2) −GM − 2c
2
1 = 1 s
r
. (1.4)
2 v2 dr r2 GM
6
This equation has six distinct families, but accretion from a static distant reservoir
is only allowed when the inward flow becomes transonic at rs as the radial velocity of
the flow accelerates from subsonic to supersonic. In a Bondi flow the mass accretion
rate is determined by the properties of the ambient medium as well as the mass of
the accretor,
G2M2ρ∞
Ṁ = 4πqs , (1.5)
c3s,∞
where, ( )(5−3γ)/(2γ−2)
1 2γ
qs(γ) = , (1.6)
4 5− 3γ
and is a function of the adiabatic index, γ. For an ideal gas, γ = 5/3, qs = 1/4.
In this idealized case, Ṁ can be arbitrarily high. However, associated with
the infall is the release of gravitational potential energy,
GMm
∆Eacc = (1.7)
R∗
where m is the mass of an individual particle and R∗ is the surface of the accretor,
or in the extreme case of a (non-spinning) black hole, the Schwarschild radius,
2GM
Rs = . (1.8)
c2
It is reasonable to assume that much of this energy is released in the form of elec-
tromagnetic radiation. Radiation fields from a central point exert a pressure
L
Prad = (1.9)
4πR2c
that is proportional to the source luminosity (L) and inversely to the square of the
local radius (R). This pushes back against the gas with a force
Frad = Pradκm (1.10)
7
and impedes the inflow. For our application we are primarily concerned with ionized
flows where
κ = σT/mp (1.11)
and σT is the cross-section provided by Thomson scattering of the photons on the
electrons which drag the protons, i.e. the dominant mass, mp. Setting this equal to
the gravitational force on a proton, a critical upper limit on the luminosity is found,
4πGMcmp
LEdd = . (1.12)
σT
If the luminosity exceeds this value (the Eddington luminosity), presumably due to
an exceptionally high accretion rate, particles will on average feel a net outward
push that slows or halts accretion. Below this limit, the luminosity will be
GMṀ
Lacc = (1.13)
R∗
if all of the kinetic energy of the gas is given up to radiation.
The simplicity of the spherical Bondi accretion flow also breaks down if the gas
has any angular momentum when it is fed from large distances as a nonzero specific
angular momentum prevents it from accreting directly. Instead, shocks and other
dissipative processes circularize the gas and cause it to settle into the lowest energy
orbit around the central object corresponding to its specific angular momentum.
This orbit will be circular and have a radius of
l2
Rc = (1.14)
GM
where l = Rvφ is the specific angular momentum. In the absence of angular mo-
mentum transport, the gas will orbit indefinitely. However, if there is a source of
8
viscosity, ν, and there is a radial shear, then neighboring annuli can exchange an-
gular momentum. For Keplerian orbits in a Newtonian gravitational potential the
angular velocity is, ( )1/2
GM
ΩK(R) = , (1.15)
R3
and the shear arising from this differential rotation is,
dΩ 3
R = Ω. (1.16)
dR 2
By assuming the gas radiates its thermal energy away efficiently, we can make the
“thin disk” approximation, where we treat the disk in two dimensions by integrating
vertically through a height, h, to get a surface density,
Σ = ρh. (1.17)
The torque that a specific annulus feels from the one directly exterior that it is in
contact with is,
dΩ
Y (R) = 2πνR3Σ . (1.18)
dR
Additionally, the annulus we are considering is also the source of torque for its inte-
rior neighbor, therefore, the net torque this one annulus feels is equal to ∂Y/∂RdR.
Looking at Equation 1.18, we see that the torque will always be negative for
a Keplerian disk so angular momentum is being transported outward. As a parcel
of gas loses angular momentum it sinks deeper into the gravitational potential. The
mass accretion rate at an annulus in the disk is thus given by the mass within an
annuli and the velocity at which it flows inwards
Ṁ = −2πRΣvR. (1.19)
9
We expect the inwards velocity to be quite low and since it will be negative, we
multiply by −1 so the mass accretion rate is positive for the sake of convention.
We can now write two conservation laws that describe the mass accretion and
the angular momentum transport. The mass conservation is given by
∂Σ ∂
R + (RΣvr) = 0, (1.20)
∂t ∂R
and the conservation of angular momentum equation becomes,
∂ 2 ∂ 1 ∂YR (ΣR Ω) + (RΣv 2rR Ω) = . (1.21)
∂t ∂R 2π ∂R
These two conservation laws can be combined to yield the disk evolution equation
for a Keplerian disk, [ ]
∂Σ 3 ∂ 1 ∂ 1
= R 2 (νΣR 2 ) . (1.22)
∂t R ∂R ∂R
For illustration, we for now discuss a steady thin disk and take ν to be constant,
but will discuss the evolution of an accretion disk with a time variable effective
viscosity and the dramatic affect it has on the disk behavior in Chapter 2. By
removing the time dependence, we arr[ive at(, )1/2]
Ṁ − R∗νΣ = 1 . (1.23)
3π R
With this expression we can examine some scalings. Taking the limit that R∗ << R,
we find that Σ ≈ Ṁ/(3πν) and vr ≈ −3ν/(2R). For a fixed accretion rate, this a
basically a statement about the flow continuity in that the surface density decreases
as you increase the viscosity and the radial velocity increases. Furthermore, the
radial velocity decreases with increasing radius, as would be expected since the
10
overall disk evolution is slower further ou√t as the dynamical timescale,
R3
tdyn = , (1.24)
GM
increases. Finally, for a given viscosity, the surface density increases with increased
accretion rate. Basically, this is the statement that denser disks accrete at a higher
rate than less dense disks for a given angular momentum transport efficiency.
Through the mutual torquing of annuli work is done and the gas heats. To
maintain the thin disk, this energy is radiated away in the electromagnetic radiation
that we observe. The viscous dissipation rate per unit plane surface area is given
by, ( )2
Y (R) dΩ 1 dΩ
D(R) = = νΣ R , (1.25)
4πR dR 2 dR
which becomes [ ( )1/2]
3GMṀ R∗
D(R) = 1− . (1.26)
8πR3 R
This basic observable is completely independent of the nature of the viscosity pro-
vided it regulates both the mass accretion and dissipation.
So far, we have made much headway in understanding accretion without spec-
ifying exactly what gives rise to the disk’s viscosity. In fact, understanding the
source of the effective viscosity in the disk and its behavior remains a central focus
for all of accretion theory. The densities around compact objects are generally too
low to have a significant molecular viscosity from the interaction of individual par-
ticles. Estimates of the Reynolds number, or the ratio of the fluid inertia to viscous
dissipation, for a luminous accreting black hole place Re > 1014 (Frank et al., 2002)
which is much too high to allow for any appreciable angular momentum transfer.
11
However, hydrodynamic flows with high Reynolds numbers have a unique char-
acteristic: turbulence develops easily. This was recognized by Shakura & Sunyaev
(1973) as a viable way to drive accretion and serves as the foundation of the “α-
prescription” which is still commonly used to describe disk accretion. Their key
insight was that turbulence can introduce a stress,
dΩ
Trφ = νρR . (1.27)
dR
The effective turbulent kinematic viscosity, ν, with units of length2/time, can be
linked to the length scale of the turbulence eddies and their velocities,
ν = vturblturb. (1.28)
Turbulent eddies are expected be some fraction of the disk height and they will be
less than the sound speed since any supersonic gas will quickly shock and force the
gas to be subsonic. These assumptions can be subsumed into the α-parameter such
that, ( )2
h
ν = αcsh = α Ω, (1.29)
r
where 0 < α < 1 and is a dimensionless scaling parameter. They also proposed that
with an α-parameter the local vertically averaged stress can be related directly to
the local vertically averaged total pressure as
Trφ = αP. (1.30)
It is worth nothing that strictly this α-parameter is 2/3 the previous α parameter
associated with the turbulent properties. This term is often neglected since it is of
order unity, but it can be the source of ambiguity.
12
Introducing the α-parameter allows for an estimate of the viscous time,
tdyn
tvisc ∼ . (1.31)
α(h/r)2
Unsurprisingly, a larger accretion efficiency (higher α) leads to shorter viscous times
and faster gas inspiral. Similarly, thinner disks have longer accretion times for a
given α, since the turbulent eddies span a smaller range of radii and the sound
speed is lower because the disk is cooler.
Parameterizing the viscosity this way allows the disk structure to be deter-
mined. Shakura & Sunyaev (1973) found three distinct regions that can form, de-
pending on the disk temperature. The first is a radiation dominated regime where
the pressure from radiation is greater than the thermal pressure and the dominant
opacity comes from electron scattering. Importantly, the disk scale height is inde-
pendent of radius and depends only on the mass accretion rate
3κṀ
hrad = , (1.32)
8πc
with a fixed opacity, κ. Radiation pressure dominated disks are expected for M =
10 M black holes when the luminosity exceeds L ≥ 3 × 1037 ergs s−1 and for
M = 107 M SMBHs when L > 5× 1043 ergs s−1. The second disk region occurs if
thermal gas pressure dominates, but electron scattering is still the dominant source
of opacity. The final possible region is a gas pressure dominated region where free-
free emission dominates.
The work in this dissertation will focus formally on the gas pressure domi-
nated systems, but it is worth a slight digression to mention the curious behavior
13
of radiation pressure dominated accretion disks. The linear dependence of the vis-
cous stress in the α-formalism on the total pressure introduces two instabilities that
were recognized shortly after the α-prescription was proposed (e.g. Lightman, 1974;
Lightman & Eardley, 1974; Pringle et al., 1973; Shakura & Sunyaev, 1976), but have
yet to be fully understood.
Firstly, there is a local thermal instability. The heating rate of an α-disk has
the form, ∣∣∣∣∣dΩ ∣
∣∣
Q+ ∼ 2hrTrφ ∣∣. (1.33)dR
In radiation pressure dominated regions the viscous stress heating the disk is given
by,
αaT 4
Trφ = , (1.34)
3
where a is the radiation density constant. Taking the radiation pressure as the
main support, the disk thickness in hydrostatic equilibrium is expressed in terms of
temperature (rather than Ṁ as before)
2aT 4
h = . (1.35)
3ΣΩ
Using Equation 1.34 and Equation 1.35 in Equation 1.33, means that locally, for a
constant α and Σ, the heating rate will scale as Q+ ∝ T 8. The cooling in this disk,
on the other hand, is given by
− ∼ 4caT
4
Q . (1.36)
3κΣ
and therefore scales as Q− ∝ T 4. With this steep temperature scaling of the disk
heating coupled to the less responsive cooling, a small perturbation away from equi-
librium will cause the disk to catastrophically heat with a positive perturbation
14
or catastrophically cool with a negative perturbation. Piran (1978) identified the
following inequality to describe this sc∣enario ∣
∂ lnQ+ ∣∣∣∣ ∂ lnQ− ∣∣∣> . (1.37)∂ lnT ∂ lnT ∣
Σ Σ
Initially, simulations investigating the stability of accretion disks found that
the disks were stable because of time delays between the stress that injects energy
and the response of the pressure once the energy is dissipated (Hirose et al., 2009;
Turner et al., 2002). However, those simulations were diffusive and the simulation
domain was small. The same initial disk conditions are unstable to thermal runaway
when evolved under different algorithms (Jiang et al., 2013) and other independent
simulations also find thermal instability (Fragile et al., 2018). Recent work finds
that there is not a clear delineation between a disk being stable or unstable, but
rather there is a probabilistic component as fluctuations can delay the instability or
stave it off completely in some realizations of the same conditions (Janiuk & Misra,
2012; Ross et al., 2017). The work of Jiang et al. (2016) showed that changes in the
iron opacity for hot AGN disks can provide a stabilizing effect that prevents thermal
instability on one side of the iron “bump.”
The second instability arises from a counterintuitive property of the angular
momentum transportation when radiation provides the dominant pressure. The
vertically integrated viscous stress is given by,
W = 2αhP, (1.38)
for which we can use equation 1.35 and see that all else being constant, W ∝ Σ−1.
This is because the temperature and pressure are independent of the gas density,
15
unlike the gas pressure dominated disk regions. Consequently, rings form in the
disk as lower Σ annuli have higher integrated stress so material is driven into the
denser regions of lower stress. These rings are expected to have widths of order h.
The numerical evidence of this instability developing is shaky because of resolution
concerns, but simulations find evidence that rings of material can form on timescales
consistent with this viscous instability (Mishra et al., 2016).
1.3 The Magnetorotational Instability
The α-disk model of Shakura & Sunyaev (1973) provided a reasonable angular
momentum transport mechanism in accretion disks, but no instability was identified
that could drive the turbulence. For an unmagnetized accretion disk whose gas fol-
lows Keplerian orbits, the Rayleigh criterion for hydrodynamic stability is satisfied
because the angular momentum profile increases with radius so no turbulence is ex-
pected. To reconcile this discrepancy and the significant challenge it posed, Shakura
& Sunyaev invoked magnetic fields to explain the anomalous behavior. Accretion
disks around black holes and other compact objects are hot and ionized, therefore it
is reasonable to expect one of the many magnetohydrodynamic (MHD) instabilities
is at play. In a groundbreaking paper seventeen years later, Balbus & Hawley (1991)
proved this intuition correct with the rediscovery of the magnetorotational instabil-
ity (MRI). This instability had previously been discovered by Velikhov (1959) and
Chandrasekhar (1960) to explain the stability of Couette flow in the presence of a
magnetic field, but its applicability to accretion disks was not appreciated for over
16
Figure 1.1: Illustration of the basic principle behind the angular momen-
tum transport between two initially neighboring particles that separate
and the growth of the MRI. The dashed line represents the spring-like
magnetic field tethering the two particles. The arrows’ direction shows
how the force is felt and the size corresponds to the amplitude.
three decades.
Conceptually, the MRI is relatively simple and arises by slightly perturbing
gas on circular orbits and stretching two elements of gas connected with a magnetic
field. Independently, each one of these actions would introduce restorative forces
that lead to stable, oscillatory behavior. If gas on a circular orbit is nudged radially
inwards, it has an excess of angular momentum for its new radius and it will want
to move to larger radii. Similarly, if it is pushed outwards, the dearth of angular
17
momentum will make it fall inward(s. The epicyc)lic frequency,1/2
2Ω d(R2Ω)
κ = , (1.39)
R dR
is the natural frequency of the oscillation. Separately, if two gas elements threaded
with a magnetic field are pulled apart, they feel a tension force proportional to
(B · ∇)δB for a small induced field,
δB = (B · ∇)ξ = ikBξ (1.40)
where k is the wavenumber, ξ is a small spatial displacement and B is the total
magnetic field strength (a scalar). Using the Alfvén velocity,
B2
v2 zAz = , (1.41)4πρ
the system takes the form of a simple harmonic oscillator and effectively acts like a
system with a spring constant equal to (k · vA)2 in its linear growth phase.
Coupling the effects of these two normally stabilizing effects leads to the dra-
matic instability that powers accretion. In an idealized sense, the local, differentially
rotating disk can be thought of as two neighboring annuli where the inner element
orbits more quickly. If the two annuli are tethered by magnetic fields, the field will
be stretched as the inner annulus pulls ahead of the outer annulus. The field resists
this stretching and tries to enforce rigid rotation. This tugging applies a negative
torque on the inner annulus and a positive torque on the outer annulus, thus caus-
ing the inner annulus to lose angular momentum and sink to smaller radii while the
opposite happens to the outer radii. This further stretches the field, enhances the
18
tension force, and exacerbates the infall, leading to instability. An illustration of
this scenario is shown in Figure 1.1.
The simplest manifestation of the MRI involves an initial field that is purely
poloidal. Geometrically, the problem is treated in cylindrical coordinates (R, θ, φ)
with axisymmetric perturbations with the spacetime dependence ei(kRR+kzz−ωt). The
background flow is taken to be constant on cylinders, the magnetic field is weak so
that its initial state has a negligible influence, and BR = 0. Finally, the Boussinesq
approximation of incompressibility (∇ · v = 0) is used so magnetoacoustic modes
can be ignored. In terms of the vertical wavenumber, kz, the dispersion relation
takes the form: [ ( )2]
k2
ω̃4 − 2 kRκ + N −N 2 2 2 2z R ω̃ − 4Ω kzvAz = 0. (1.42)k2z kz
In this equation k2 = k2z + k
2
R ,
ω̃2 = ω2 − k2 2zvAz, (1.43)
and N2z and N
2
R are the vertical and radial elements of the Brunt-Väisälä frequency,
3
N2 = N2R +N
2
z = − (∇P ) · (∇ lnPρ−5/3) (1.44)5ρ
of buoyant oscillations in the atmosphere.
Stability demands ω2 > 0,(giving the foll)owing stability criterion,
2 2
k4v4 2 2 2
dΩ 2 dΩ
z Az + kzvAz N + +Nz > 0. (1.45)d lnR d lnR
For a buoyantly stable disk with NR, Nz > 0, the stability criterion is satisfied if,
dΩ2
> 0. (1.46)
dR
19
This will always lead to instability for astrophysical Keplerian-like accretion disks
with Ω ∝ R−3/2 no matter how weak the magnetic field since some unstable wavenum-
ber in the perturbation will be accessible. T∣he gro∣wth rate of the instability is fast,
| | 1∣∣∣ dΩ ∣∣ωMRI = ∣, (1.47)
2∣d lnR ∣
or ω = 0.75Ω for a Keplerian accretion dis√k, and corresponds to a wavelength of,
16 vAz
λMRI = 2π . (1.48)
15 Ω
It should be noted that there is an upper limit to the strength of the magnetic field
that can seed the MRI. If the field is so strong that the ratio of the gas pressure to
pressure from the magnetic field is less than a value of 3, the critical wavelength of
the MRI exceeds the disk height and thus cannot drive the instability.
Assuming the necessary conditions are met, the instability will grow, be-
come non-linear, and ultimately saturate in a turbulent state. The early growth
of the MRI resembles a vertical, interchange instability. At the end of the linear
growth phase the channel solution becomes unstable to parasitic secondary Kelvin-
Helmholtz type instabilities (Goodman & Xu, 1994), evolves nonlinearly, and even-
tually saturates in a state of subequipartition between the magnetic pressure and
thermal pressure (a ratio of ∼ 1 : 10) with the field dominated by its toroidal
component. The saturation of the MRI driven turbulence is a crucial component
in understanding how angular momentum is transported in accretion disks, but is
still poorly understood. Dynamical equations have been derived to describe the
local kinematic and magnetic stresses in the disk which ultimately associate sat-
uration with the dissipation of turbulent the kinetic and magnetic energies (Kato
20
& Yoshizawa, 1995; Ogilvie, 2003). The helicity (winding) of the turbulent eddies
and magnetic field has also been invoked as a way to mediate the saturation as the
magnetic helicity can back react on the kinetic helicity and restrict further growth
(Blackman & Field, 2002). It has also been proposed that the MRI may saturate
when the growth rate of the secondary parasitic modes is comparable to its own
(Pessah & Goodman, 2009).
As a final comment on the MRI, it is important to mention that MRI driven
turbulence is much more nuanced than that envisioned in the simple α-disk and in
some respects is fundamentally different. The details of the differences are thor-
oughly presented in Pessah et al. (2008), but essentially the two key assumptions
that frame the α-disk model are: 1) that the disk stress can be treated as a verti-
cally averaged shear viscosity and 2) that the stress depends linearly on the shear
rate. Real astrophysical disks, however, are much more complex than a simple ver-
tically averaged thin disk model can properly capture since vertical gradients and
correlations within the flow have to be fully considered, which can violate the first
assumption. This is a less serious indiscretion than the second assumption, however.
Analytical models predict, and numerical models confirm, that there is indeed a non-
linear scaling of the disk stress with shear parameter. Luckily, this has little effect
on our work and we will frequently refer to the α-model despite this discrepancy.
The MHD simulations used in the studies of this dissertation use either Newtonian
or pseudo-Newtonian gravitational potentials so the shear rate is nearly constant for
the bulk of the disks used in the analyses. Nevertheless, it is crucial to be mindful
of this sensitivity because in other contexts, like a star accreting from a disk where
21
the gas has to decelerate to match the stellar rotation, there can be strong changes
in the shear which can break the assumptions of the α-disk.
1.4 Techniques for Simulating Accretion Disks
Analytic theory is essential for understanding the accretion phenomenon, but
only numerical models can capture the global and nonlinear behaviors of the tur-
bulence. To date, a large and rich body of simulation work has been dedicated to
studying how MRI driven turbulence behaves and transports angular momentum
including studies using unstratified shearing boxes (e.g. Hawley & Balbus, 1991;
Hawley et al., 1995), stratified shearing boxes (e.g. Brandenburg et al., 1995; Stone
et al., 1996), unstratified global models (e.g. Armitage, 1998; Armitage & Reynolds,
2003; Armitage et al., 2001; Hawley, 2001; Sorathia et al., 2012), stratified global
disk models (e.g. Hawley, 2000; Hawley & Krolik, 2001; Reynolds & Miller, 2009;
Sorathia et al., 2010), relativistic global models (e.g. De Villiers et al., 2003; Gam-
mie et al., 2003; McKinney, 2006), and global models with radiative transfer (e.g.
Jiang et al., 2014; McKinney et al., 2014; Sa̧dowski et al., 2015). Here, we present
a general overview of the fundamentals of accretion disk simulation, some points of
caution, and the two algorithms applied in this dissertation.
Like all modeling efforts, simulating accretion disks is a game of approximation.
Fully capturing the physics of an accreting system would require simulating the
motions of each and every particle, and the forces acting upon them. While this
is a noble goal, it is impossible. Luckily, a few simplifying assumptions make the
22
problem tractable and allow us to create models that seem to resemble physical
systems. The starting point for most accretion disk simulations, and where we will
begin, is to consider the gas as either a magnetized or viscous fluid, depending on
the aims of the study. The codes used in this dissertation are grid-based which
means the physical domain is divided into cells defined by a numerical mesh and the
calculations are made based on the values of the discrete cells and their neighbors.
This approach offers many benefits because the fluid equations are known well and
the calculations can be made in coordinates that naturally lend themselves to the
problem. In our case we will use spherical coordinates which is a more natural
way to follow the geometry and scaling in the calculation, rather than a rectilinear
geometry, for instance. Additionally, the full volume can easily be divided into
sub-domains that can largely evolve independently so a single CPU core does not
need to know the state of the entire simulated disk, and instead only the values of
cells around the perimeter of the sub-domain volume. At each time step, each core
performs the calculation for the sub-volume of the disk it is assigned and only needs
to communicate with the cores performing calculations on the other nearby regions
in the simulation. This allows the simulations to be parallelized efficiently so the
calculation can be distributed across thousands of cores on a supercomputer.
Grid-based simulations bring their own technical challenges, though. The
timestep needed to stably integrate the evolution for a cell in a given direction
x is set by the shortest crossing time scaled by the Courant-Friedrichs-Lewy (C0)
number. In hydrodynamics the shortest crossing time is set by the summation of
the fluid velocity and sound speed while in MHD the crossing time is set by the
23
velocity plus the speed of the fast magnetosonic wave,
C0∆x
∆tx = . (1.49)|v|+ cf,x
In cylindrical and spherical coordinate systems a constant opening angle in the φ-
and/or θ- directions means the innermost cells set the minimum timestep because
they are the smallest physically and these locations are also where the highest ve-
locities in an accretion disk are found. For the outer portions of the disk where the
evolution is slower, this spends unnecessary computational time integrating smaller
timesteps than are required which can greatly enhance the computational costs and
adds unnecessary numerical error. The most modern codes, although not the codes
used in this dissertation, work around this algorithmically by implementing adaptive
meshes to adjust the refinement in different regions of the grid mesh and by using
adaptive time stepping that will integrate different regions in comparable times, e.g.
evolving the inner region of an accretion disk simulation for several timesteps before
evolving the outer region.
A larger concern comes from simulating turbulence on a grid. When energy
is injected, it cascades to larger wavenumbers, and is ultimately deposited into
thermal energy on a dissipative scale. It is currently impossible to capture the full
separation of scales needed to properly model the inertial range of the cascade,
especially in global simulations like those presented in this dissertation. As such,
much effort has gone into understanding what resolutions are needed to produce
consistent macroscale behaviors. Local shearing box simulations, where a small
portion of a larger disk is modeled, have been invaluable in providing a more detailed
24
understanding of the small-scale turbulence that can’t otherwise be captured. A set
of convergence criteria have been established by many dedicated resolution studies
that serve as guidelines of “best practice” for resolvability in numerical models. The
resolution requirements needed to model MRI-driven turbulence and the robustness
of simulations that abide by these has nearly reached consensus, but it is an issue
that must constantly be revisited. Throughout the chapters studying MHD disks we
will discuss the previous resolution studies in more depth and rigorously demonstrate
that our models are well-resolved.
There are numerical worries in MHD simulations as well. The solenoidal con-
straint, ∇ ·B = 0, must be fulfilled at all times and grid-based codes do not do this
naturally. Several elegant solutions have been developed to rectify this shortcom-
ing and here we use the method of constrained transport (Evans & Hawley, 1988).
When considering a single cell in the grid mesh, there are several locations by which
we can define it, for instance the geometric center of the cell, the centers of the faces
that define it, or even the midpoints along the cell edges. Depending on the type of
variable (i.e. scalar or vector, density or magnetic field), certain positions are more
amenable. Constrained transport works by staggering the mesh and integrating the
magnetic fluxes from the induction equation directly using Stokes’ theorem,
∮
dbx
dA+ E · dl = 0. (1.50)
dt
For a cell face with an area dA and a direction normal to the surface x, the electro-
motive force E is calculated along the edges of the face and used to update the face
centered magnetic field, denoted as b rather than the cell-centered magnetic field
25
(B). Since each face acts as the interface between two cells in a certain direction,
there is a precise cancelation between the two cell’s outwardly directed field. This
cancelation ensures that any magnetic field initialized as divergence free will remain
divergence free to machine accuracy as it is evolved.
Turning to the actual calculation performed, the computational problem of
simulating an accretion disk can be boiled down to integrating a set of hyperbolic
partial differential equations of the form,
∂U
+∇ · f(U) = S(U), (1.51)
∂t
in the most abstract sense. This equation describes the changes in the conserved
quantities state vector, U, which describes the fluid’s mass, momentum, and energy.
The three terms denote the temporal change of U, the fluxes for each element in
U, f(U), and any source terms, S(U), which includes effects like artificial cooling.
To close the equations, an equation of state for the gas is included and in MHD the
induction equation is also needed.
The first type of algorithm used to solve this problem is the finite differencing
method used by the Zeus MHD code and used for the disk simulations presented
in Chapter 2. The second is the finite volume method used by the PLUTO code
and used in simulations presented in Chapters 3, 4, and 5. In discussing these
algorithms here, we mean to provide a brief overview of the important concepts and
general philosophies behind the simulation methods since specifics about each code
can be found in the supporting documentation, referred to in the relevant chapters.
Additionally, the details of each simulation presented in this dissertation (including
26
the equations evolved, grid configurations, initial conditions, quality checking, etc.)
is kept to the appropriate chapter.
The Zeus code is the older and simpler of the two. The basic approach taken
in a finite difference scheme is to evolve the values of U for a point in space and can
be illustrated by considering a one-dimensional cell, Ii spanning Ii = [xi−1/2, xi+1/2]
at a time tn with size ∆x = xi+1/2 − xi−1/2. In the simplest form the conservation
equations are discretized as,
d
U(x, t) = f(U(xi−1/2, t))− f(U(xi+1/2, t)). (1.52)
dt
Integrating in time by the timestep between tn and tn+1 according to the CFL
condition and rearranging, this equation becomes,
∫ tn+1
U(x, tn+1) = U(x, tn) +∫ f(U(xi−1/2, t))dt (1.53)tntn+1
− f(U(xi+1/2, t))dt.
tn
Defining ∫
1 xi+1/2
Uni = U(x, t
n)dt (1.54)
∆x xi−1/2
and ∫ tn+11
fi±1/2 = f(U(xi±1/2, t))dt (1.55)
∆t tn
the simple Eulerian form of the next timestep is given by
Un
∆t
1
i = U
n
i + (fi−1/2 − fi+1/2). (1.56)∆x
The actual algorithm used by Zeus uses more advanced and accurate than outlined
here, but we only mean to demonstrate the underlying principles of the method.
27
In practice, for each iteration/timestep in the integration Zeus solves the sys-
tem of equations in three general substeps. The first step is the “source” step which
takes the state U and solves the finite difference approximations to the momentum
and energy equations. The second step evolves the magnetic fields using the method
of characteristics within the framework of the method of constrained transport. The
method of characteristics accounts for the propagation of Alfvén waves by includ-
ing the magnetic tension. Not only is this an important physical component of the
MHD disk, but it improves stability by removing artificial large amplitude, high
frequency waves introduced by boundaries and discontinuities. The final step is the
“transport” step which handles the advection of the fluid through each zone. This
step solves the finite-difference approximations, but in integral form so that the flux
of material is evolved.
One of the benefits of Zeus is that its relative simplicity translates into algo-
rithmic speed. This performance means that computational resources are used very
efficiently so simulations can be run for long durations and at higher resolutions
than might be possible with other codes. For our application in studying the de-
velopment of propagating fluctuations in mass accretion rate, this was paramount
and made Zeus the ideal code for the investigation. However, this performance gain
comes at the cost of formal overall energy conservation. While mass is conserved
since step three evolves the integral form of the equations and there are no sources
terms for the density in step one, the total energy is not conserved because internal
energy is evolved which introduces grid errors from the source terms.
PLUTO is distinct from Zeus in many ways. Primarily, the conservation
28
Figure 1.2: Schematic of cell discretization in the Riemann problem with
first-order spatial reconstruction for a generic variable. The bar height
represents the cell’s value. The dashed line indicates the continuous
solution the cells are approximating. Overlaid in blue is a representation
of the propagation of the contact discontinuities, left and right going
shocks, and rarefactions induced by the sharp discontinuities. Here, the
vertical direction is meant to represent the forward propagation in time
between a time tn and tn+1.
29
Figure 1.3: Schematic of cell discretization in the Riemann problem with
second-order spatial reconstruction for a generic variable where height
represent its value. The dashed line indicates the continuous solution the
cells are approximating. Notice how the discontinuities are more mild
than the first-order accurate case and therefore better describe the true
solution.
30
equations are solved with a volume integrated approach, also known as a Godunov
method after Sergei Godunov who first developed this method in 1959, where the
focus is on the entire quantity within a cell, rather than an individual point. To
understand the finite volume method conceptually, we can again consider a one-
dimensional cell, Ii spanning Ii = [xi−1/2, xi+1/2] at a time t
n with size ∆x = xi+1/2−
xi−1/2. Now, we will consider the conservation equations in integral form,
∫
d i+1/2
U(x, t)dx = F(U(xi−1/2, t))− F(U(xi+1/2, t)), (1.57)
dt xi−1/2
with F(U(xi−1/2, t)) denoting the total flux through the cell surface on that side
summed over the area. Integrating in time and rearranging, this equation becomes,
∫ x ∫ ∫ n+1i+1/2 xi+1/2 t
U(x, tn+1)dx = U(x, tn)dx+∫ F(U(xi−1/2, t))dt (1.58)x − ni 1/2 xi−1/2 ttn+1
− F(U(xi+1/2, t))dt.
tn
Defining ∫
1 xi+1/2
Uni = U(x, t
n)dt (1.59)
∆x xi−1/2
and ∫ tn+11
Fi±1/2 = F(U(xi±1/2, t))dt (1.60)
∆t tn
the simple Eulerian form of the next timestep is given by
n ∆tU 1i = U
n
i + (Fi−1/2 − Fi+1/2). (1.61)∆x
The benefit of this volume integrated formulation is that it is conservative to machine
precision so that flow properties like the total energy are properly accounted for.
31
Like Zeus, the PLUTO code solves the conservation equations of HD or MHD
in three substeps during each iteration; however, the actual algorithmic methods
used by PLUTO are quite distinct and overall more rigorous. First, interpolation
routines reconstruct a piecewise polynomial approximation of the state from cell
averages. For our work we use a linear reconstruction which is second order accurate.
Next, the numerical fluxes are calculated. This is more challenging than in
the finite difference schemes because the discontinuity at each cell interface must be
accounted for. If the value at the interface is calculated by a simple average of the
neighboring cell values, for instance, unphysical oscillations appear in the solution
as the nonlinear development across the interfaces affects the linear approximation.
Instead, it must be treated as a Riemann problem and use approximate solvers to
capture effects from left and right moving shock waves, rarefaction fans, and contact
discontinuities that would develop. A schematic of these components are shown in
Figure 1.2 for a first order accurate spatial reconstruction to help visualize the role
of the different waves and the impact of implementing a linear reconstruction in
improving the accuracy of the solution. Figure 1.3 shows the second order accurate
spatial reconstruction. By evolving the change in the integrated amount of a given
quantity like energy and momentum contained within the cell volume based on the
total fluxes through the cell surfaces, the code is fully conservative.
In the final step, the state is marched forward in time. For this work we rely on
Runge-Kutta methods which improve accuracy by evaluating the timestep integral
32
in multiple stages, s, per step of size h. These methods can be generalized as,
∑s
yn+1 = yn + biki, (1.62)
i=1
where, ( ∑ )i−1
ki = hy
′ tm + cih, tm + aijkj (1.63)
j=1
where i goes from 1 to s and a, b, and c are coefficients.
Overall, the extra machinery of the finite volume method in PLUTO makes
it slower than the finite difference method used in Zeus. Still, it is fast enough to
enable the long-duration, high-resolution simulations that are performed in this dis-
sertation. With this slight performance sacrifice comes many crucial benefits. For
instance, we use PLUTO to model HD and MHD disks with a bistable cooling func-
tion to trigger thermal instability. It is essential to faithfully capture the thermally
driven dynamics in these simulations which mandates a fully conservative code is
used instead of a non-conservative finite difference type code.
1.5 The Dissertation in Context
Variability was one of the first distinguishing properties of quasars. Matthews
& Sandage (1963) found that 3C 48 showed 0.4 mag variation over 13 months. They
identified the emission as coming from an area of less than two-tenths of a parsec
based on the light travel time of the emission. At that time, it was not known that
this was a SMBH system so the large variability along with its unexplained Doppler
broadened Balmer lines made it an enigma. Other early studies of similarly odd
radio sources with large redshifts in their spectra also found more rapid variability
33
on timescales of days to weeks(e.g. Sandage, 1964; Sharov & Efremov, 1963; Smith
& Hoffleit, 1963). Modern X-ray studies of AGN find appreciable variability on
timescales of hours and even minutes. As an extreme example of variability, several
“changing-look” AGNs have been found where the accretion luminosity appears to
abruptly decrease.
Variability is also a strong characteristic of X-ray binaries. Due to their high
temperatures, they can only really be observed at high energies in the X-rays. As
such, observations must be done in space and it took slight longer to discover these
objects than it did for quasars. Once they were discovered, however, they provided
the first widely accepted proof for the existence of black holes with the discovery of
X-rays from Cygnus X-1 (Bolton, 1971, 1972) and have offered a wealth of knowledge
about accretion physics. The benefit of studying accretion in X-ray binaries is
that they grant access to variability on all important accretion timescales. With
satellites like the Rossi X-ray Timing Explorer (RXTE), accretion could be studied
from dynamical timescales, thanks to its millisecond timing resolution, up to feeding
events that caused it to go into outburst because of its 17-year long mission.
The minuscule apparent size of X-ray binaries and AGNs on the sky prevents
imaging these systems. In lieu of this direct method of studying the accretion physics
occurring within them, indirect methods must be used. The most powerful probe
we have of these perplexing objects is their variability in electromagnetic radiation
because signatures of the accretion physics are imprinted on the emission. In the
fifty years that variability has been studied, a rich phenomenology has developed,
yet a disconnect remains between the fundamental theoretical principles describing
34
accretion and the interpretation of astrophysical black hole observations.
This is not to say accretion theory is lacking - quite the contrary, in fact. The
analytic theory, which we have provided a brief introduction to here, is vast and well-
developed. The rise of robust simulations has been central to pushing forward on this
front and achieving many victories, including verifying the chief angular momentum
transport mechanism is MRI-driven turbulence. Local shearing-box simulations
have explored the development of this turbulence and global simulations investigated
the fundamental properties of both thick and thin accretion disk geometries. There
are now major pushes to develop spectral models from simulations, understand the
launching of jets and outflows from disks, properly include the effects from radiative
physics on the flow, and even model accretion disks around binary black holes.
We currently find ourselves in a unique time where time domain astronomy has
become a major focus and there several current and future telescopes like NICER,
TESS, and LSST whose primary goals are to survey the variable sky. Coupled with
the growing capability of simulations, there is a fortunate confluence of observational
and theoretical developments that will allow for an unprecedented advancement in
our understand of black hole accretion. To enable this, though, more reliable simu-
lations are needed that can be connected to empirical signatures. Many conclusions
drawn from spectral and timing studies are based on simple analytical arguments.
The primary goal of this dissertation is to help provide a theoretical grounding for
several common models of photometric variability observed from these systems. We
specifically target two accretion scenarios that are commonly invoked to explain
observations: propagating fluctuations in mass accretion Chapter 2 and truncated
35
accretion disks Chapters 3 and 4. Broadly speaking, we aim to establish the validity
of the scenario and study the variability and manifestation in MHD. Throughout
this work, we shed light on modifications that exist from standard α-disk models and
reveal the reoccurring role of the magnetic dynamo. The dynamo is so significant
in the disk evolution that we conduct a studied dedicated to studying its sensitivity
to disk height Chapter 5.
36
Chapter 2: Propagating Fluctuations In Mass Accretion Rate
2.1 Introduction
There is increasing evidence that the interplay between the local angular mo-
mentum transport from magnetohydrodynamic (MHD) turbulence and the more
global accretion flow shapes the behavior of the mass accretion (e.g. Sorathia et al.,
2010). The primary goal of this chapter is to connect MHD accretion disk theory
with one of the most widely discussed phenomenological models for global disk vari-
ability, the propagating fluctuations model. We explore how a stochastically varying
effective viscosity leads to the growth of so-called “propagating fluctuations” in our
long, global accretion disk simulation and relate the behavior of our simulation to
observable properties from accreting black holes.
The viscous behavior of the disk ultimately arises because of net internal stress
due to correlated fluctuations in the magnetic field and correlated fluctuations in
gas velocity (Balbus & Papaloizou, 1999). These correlated fluctuations lead to
the Maxwell stress (MRφ = −BRBφ/4π) and the Reynolds stress (RRφ = ρvRδvφ),
respectively. Following the α-prescription, the effective α from the net stress is thus
37
the ratio of the volume averaged stress to the volume averaged pressure,
〈MRφ +RRφ〉
α = . (2.1)
〈P 〉
This serves to parameterize the effectiveness of angular momentum loss by the
stresses acting on the gas in the disk. In this chapter we use angled brackets to
denote a volume average for a giv∫en quantity, U , where
U(t,∫r, θ, φ)r2sinθdrdθdφ〈U(t)〉 = . (2.2)
r2sinθdrdθdφ
Turbulent fluctuations in the disk will naturally lead to stochastic variability
in the effective α and provide an intuitive explanation for the existence of broadband
photometric variability observed in both galactic black hole binaries (GBHBs) and
active galactic nuclei (AGNs). The photometric variability is observed across several
decades in frequency, typically in the form of aperiodic “flicker” type noise. Em-
ploying a stochastic α, Lyubarskii (1997) developed a model for this flicker noise by
linearizing the standard geometrically thin, optically thick accretion disk evolution
equations. The model was limited to small perturbations in α, but it reproduced
the power spectrum of the fluctuations and laid the foundation for the subsequent
development of the propagating fluctuations model.
The propagating fluctuations model has since taken a more generalized form
and served as a phenomenological description for the structure and organization of
the broadband variability for GBHBs and AGNs. The evolution of a geometrically
thin, optically thick accretion disk (in units where G = M = c = 1) is governed by
a nonlinear diffusion equation (Pringle,[1981),∂Σ 3 ∂ ∂ ]1 1
= R 2 (νΣR 2 ) , (2.3)
∂t R ∂R ∂R
38
where R is the distance from the central engine, ν(Σ;R, t) is the local effective
viscosity in the disk, and Σ(R, t) is the local surface density. In this equation,
the diffusive redistribution of angular momentum depends on the product of the
effective viscosity and the local surface density. If the effective viscosity fluctuates,
the instantaneous mass accretion rate (Ṁ) at each point in the disk is thus the
product of the many prior stochastic events. A consequence of the multiplicative
combination is that fluctuations in Ṁ will tend to be preserved as they accrete as
higher density regions will typically have higher Ṁ and lower density regions will
typically have lower Ṁ . Additionally, since the viscous time increases with radii,
higher frequency fluctuations will be imposed on the accretion flow as material
moves inwards. This will lead to the development of a hierarchical structure in the
fluctuations with the smaller, high frequency fluctuations imposed upon the larger,
low frequency fluctuations.
The appeal of the propagating fluctuations model is that it explains three
properties of black hole variability that have been difficult to reproduce with other
variability models. First, GBHB and AGN light curves are log-normally distributed
(Gaskell, 2004; Uttley et al., 2005) which is accounted for through the multiplica-
tive combination of many independent fluctutations. Second, and related to the
log-normal distribution, the root-mean square (RMS) of the flux variability in the
light curves of both GBHBs and AGNs depends linearly on the flux level (Gaskell,
2004; Negoro & Mineshige, 2002), the so-called RMS-flux relationship. This in-
dicates the mass accretion rate variation has a constant fractional amplitude and
that the mechanism driving variability is, therefore, independent of the accretion
39
rate (Uttley & McHardy, 2001). Finally, coherent fluctuations in the emission at
different wavebands from GBHBs have frequency dependent time lags with variabil-
ity in higher energy bands lagging variability in lower energy bands (Nowak et al.,
1999a,b). This “hard lag” is believed to be a signature of fluctuations in the ac-
cretion flow moving to smaller radii in the disk where the emission temperature is
higher.
Despite the success the propagating fluctuations model has had in explaining
the phenomenology of GBHB and AGN variability, the model has yet to be exam-
ined in the context of modern MHD theory. To date, the model has been examined
in the context of geometrically-thin (vertically-integrated) α-disks in which some
stochasticity is imposed upon α. How well this viscous behavior translates into an
MHD disk is unknown. Particularly challenging is the timescales on which α fluctu-
ates and drives variability within the accretion flow. Lyubarskii (1997) prescribes α
fluctuations to be slow, of order the viscous time. However, if fluctuations in α are
solely due to turbulent fluctuations, we might expect them to be on the dynamical
timescale. Cowperthwaite & Reynolds (2014) show, using a semi-analytic model,
that propagating fluctuations only exist if fluctuations in α occur at sufficiently low
frequencies. If the frequency of α fluctuations is too high, they found that Ṁ fluctu-
ations are damped out and the nonlinear features of the variability do not develop.
Therefore, fluctuations in the effective-α on timescales longer than the dynamical
timescale are needed to reproduce the phenomenology seen in the variability.
In this chapter, we perform the first detailed analysis of propagating fluctua-
tions in the context of a long-duration, MHD accretion disk simulation. We show
40
that the well-known quasi-periodic dynamo induces slow fluctuations in the effec-
tive α that then drive propagating fluctuations. In Section 2.2 we introduce the
simulation and perform a convergence study. Section 2.3 presents a brief discussion
of the nature of the turbulence. In Section 2.4, we investigate the relationship of
the quasi-periodic disk dynamo and angular momentum transport, finding that it
imposes an intermediate-timescale modulation of the effective α. We then examine
the variability of both the mass accretion rate (Section 2.5) and a proxy for bolo-
metric radiative luminosity (Section 2.6) finding that both quantities display the
non-linear variability characteristics of real sources and that the coherence of these
quantities across the disk is exactly as expected in the propagating fluctuations pic-
ture. In Section 2.7, we place our results into a broader context, and then conclude
in Section 2.8.
2.2 Numerical Model and Convergence
In this section, we describe the construction and convergence properties of
our numerical model. In brief, we evolve a three-dimensional ideal MHD simulation
in order to study the local and global dynamics of a geometrically-thin (constant
opening angle h/r = 0.1) accretion disk. Given our focus on the time-variability of
the accretion flow, the crux of our problem is the evolution of the MHD turbulent
dynamics and how it influences the long-timescale behavior of the disk. Thus, we
concentrate our computational resources into the high-spatial resolution and the
long duration of the simulation. We demonstrate below that the numerical grid
41
is of high enough resolution that the macroscopic behavior of the disk turbulence
is converged and insensitive to increases in resolution. As a penalty, we simplify
the physics to the bare minimum. We employ non-relativistic ideal MHD, and
a pseudo-Newtonian gravitational potential to emulate the dynamical effects of a
general relativistic potential, including the presence of an inner-most circular orbit
(ISCO). We use a simple cooling function to keep the disk thin, but otherwise neglect
all radiation physics. Renderings of density and magnetic turbulent structure of our
fiducial disk simulation are shown in Figure 2.1.
2.2.1 Simulation Setup
Our simulation employs the finite-difference MHD code Zeus-MP v2 (Hayes
et al., 2006; Stone & Norman, 1992a,b). Zeus-MP solves the differential equations
of ideal compressible MHD,
Dρ
= −ρ∇ · v, (2.4)
Dt
Dv 1
ρ = −∇(P +) (∇×B)×B− ρ∇Φ, (2.5)Dt 4π
D e
ρ = −P∇ · v − Λ, (2.6)
Dt ρ
∂B
= ∇×(v ×B), (2.7)
∂t
where
D ≡ ∂ + v · ∇. (2.8)
Dt ∂t
to second order accuracy in space. The method of constrained transport is used
to maintain zero divergence in the magnetic field to machine precision. The ex-
42
plicit integration time step is set by the usual Courant conditions, and is first-order
accurate in time.
We begin by defining our fiducial simulation. The simulation domain is covered
by spherical coordinates (R, θ, φ) and spans R ∈ [4rg, 145rg], θ ∈ [π/2 − 0.5, π/2 +
0.5], φ ∈ [0, π/3), shown in Figure 2.1. The zone aspect ratio is approximately ∆R :
R∆θ : R∆φ = 2 : 1 : 2 and each scale height is resolved with 28 θ-zones. Outflowing
boundary conditions were used for the±R and±θ boundaries and periodic boundary
conditions were used in the φ direction. The most trivial implementation of the
outflowing boundary conditions in ZEUS-MP does not enforce the divergence free
condition required of the magnetic field at the boundary interface. Therefore, we
modified the standard ZEUS-MP boundaries to conserve ∇ · B = 0 across the R-
and θ-boundaries.
The simulation domain was broken into a well-resolved inner region (r < 45rg),
used for our analysis, and an outer region with poorer resolution to act as a gas
reservoir. The gas reservoir mitigates the effect of secular decay from the draining
of material from the disk over the course of the simulation. The radial spacing in
the inner region logarithmically increases in R from 4-45 rg. In all, the simulation
has NR × Nθ × Nφ = 512 × 288 × 128 = 1.89 × 107 zones in this region. The ∆R
spacing in the outer region logarithmically increases from 45− 145rg and a total of
NR ×Nθ ×Nφ = 128× 288× 128 = 4.72× 106 zones in this region.
We approximate the gravitational field around a nonrotating black hole with
43
(a) Gas Density
(b) |B2|
Figure 2.1: Snapshots of ρ (a) and |B2| (b) in the fiducial simulation
used in our analysis at t=0, the point we take to be the beginning of
science analysis after the initial transient behavior has died off and the
turbulence had saturated.
44
a pseudo-Newtonian gravitational potential (Paczyńsky & Wiita, 1980) of the form:
GM GM
Φ = − , rg ≡ . (2.9)
R− 2rg c2
With this potential, several important aspects of a general relativistic gravity field
are captured including an ISCO (at 6rg) and the qualitative change of radial shear
in the disk.
A γ = 5/3 adiabatic equation of state is used for the gas. An initially ax-
isymmetric thin disk was initialized with constant midplane density and radially
decreasing pressure corresponding to an isothermal vertical profile:
( )
cos2 θ
ρ(R, θ) = ρ0 exp − , (2.10)
2(h/r)2 sin2 θ
and
p(R, θ) = c2s ρ(R, θ) (2.11)
where,
GMR(h/R)2 sin2 θ
c2s = , (2.12)(R− 2r 2g)
locally.
The disk is initialized to a scale height of h/r = 0.1 and was maintained
by an ad hoc, optically thin cooling function similar to Noble et al. (2009). The
cooling function plays the role of the real physical processes and removes energy
at a rate equivalent to the thermal time of the disk (τcool = 10τorb ∼ τorb/α), as
was done in O’Neill et al. (2011). If the gas energy in a cell is above the target
energy, e 2targ ∝ ρvφ(h/r)2, it is cooled over the cooling time. The cooling function
used is Λ = f(e − etarg)/τcool, where f = 0.5[(e − etarg)/|e − etarg| + 1]. The switch
45
0.10
0.08
0.06
0.04
0.02
0.00 5 10 15 20 25 30 35 40 45
Radius rg
Figure 2.2: Radial profile of the time averaged disk scale height, h/r,
during the length of the simulation. The cooling function maintains an
aspect ratio of h/r ≈ 0.1 throughout the simulation. In the inner regions
(r ≤ 10rg) the ISCO is strongly felt by the gas and the disk is thinner,
the so-called “turbulent edge” (Krolik & Hawley, 2002). Throughout
this chapter we will calculate quantities within one scale height of the
disk. Since the cooling function maintains the aspect ratio fairly well,
these calculations are done within our target scale height of h/r ≈ 0.1.
function, f , acts as a threshold function and is zero when etarg > e. Figure 2.2 shows
the radial profile of the time averaged aspect ratio of the disk during the length of
the simulation and it can be seen that the cooling function effectively keeps the
disk near the target h/r value. Throughout this chapter we calculate quantities
within one disk scale height and, therefore, use our target value as h/r = 0.1 in
these calculations. A protection routine is implemented to ensure the density and
pressure values do not become artificially small and/or negative.
46
Scale Height (h/r)
The gas was initialized with a purely azimuthal velocity field and a weak
magnetic field. The local velocity is set so that the effective centripetal force balances
the gravitational force. A series of weak, large-scale poloidal magnetic field loops
are set for the initial magnetic field configuration (e.g. Reynolds & Miller 2009) to
seed the MRI. The loops were set from the vector potential A = (Ar, Aθ, Aφ) to
ensure the initial field was divergence free and had the form:
( )
1 2πr
Aφ = A0p 2f(r, θ) sin , Ar = Az = 0, (2.13)
5h
where A0 is a normalization constant, f(r, θ) is an envelope function that is 1 in the
body of the disk and smoothly goes to 0 at r = rISCO, rout and 3 scale heights above
the disk. The alternating field polarity is set by the final multiplicative term such
that loops have a radial wavelength of 5h. The A0 constant was set to initialize the
disk to an average ratio of gas- to magnetic-pressure of β = 500.
The following analysis treats the simulation after the MRI has saturated and
transient behavior from initialization has died away. B and e reach a quasi-steady
state in the well-resolved region of the disk and the MRI saturates, as measured by
the convergence metrics discussed in Section 2.2.2, after approximately 200 ISCO
orbits (12, 320 GM/c3). We take this point to be t = 0. The simulation was then
run for 1410 ISCO orbits (86, 856GM/c3). In the interest of data management, the
values of B, v, e, and ρ were output every two ISCO orbits (123.2GM/c3).
47
2.2.2 Convergence
The issue of convergence is forefront when considering the reliability of results
from any ab initio accretion disk simulation because the MHD turbulence that me-
diates angular momentum transport has structure on the smallest accessible scales.
With this type of simulation, the macroscopic behavior can have a resolution de-
pendence if the turbulent scales dominating the shear stresses are not adequately
resolved (Guan et al., 2009; Hawley et al., 2013; Sorathia et al., 2012). As a preface
to our study of variability, here we assess the degree to which we are adequately
resolving the turbulent dynamics using two measures. Firstly, we examine the re-
solvability of the fastest growing linear MRI mode through the commonly employed
quality factors introduced by Noble et al. (2010). Secondly, to characterize the
non-linear saturation of the turbulence, we use the magnetic tilt angle. Using these
diagnostics, we compare our fiducial (long) simulation with higher resolution com-
parison simulations.
The first set of diagnostics used to measure resolvability were the vertical and
azimuthal quality factors (Noble et al., 2010), Qθ = λMRI/R∆θ and Qφ = λC/R∆φ,
respectively. Qθ and Qφ measure resolvability through the number of grid cells per
fastest-growing MRI wavelength, λMRI , and critical toroidal field length, λC where
2π|B
√ θ
| 2π|Bφ|
λMRI = , λC = √ . (2.14)
ρ Ω(r) ρ Ω(r)
Using the quality factors, we characterized the global resolvability of the disk.
The local values of Qθ and Qφ were calculated in every voxel of the well-resolved
48
(a) Qθ
(b) Qφ
Figure 2.3: Plots of temporally and azimuthally averaged Qθ (a) and
temporally and vertically averaged Qφ (b). One scale height (1h) above
and below the disk is shown as dashed black lines in (a).
49
region (4− 45 rg) for every data dump after the transient behavior in the disk had
died away (t = 0 onwards). The volume-weighted average was calculated for each
quality factor by averaging over in the region within one scale height above and
below the disk mid-plane to get the instantaneous value and then over the entire
duration of the simulation to get a temporal average. The averaging was restricted
to the body of the disk because we want the most conservative estimate of the
resolvability of the simulation. In the highly magnetized, low-density coronal regions
the quality factors are very high and, if included, the quality factors would appear
artificially high, overestimating how well we resolve the MRI. We find 〈Qθ〉 = 7.0
and 〈Qφ〉 = 15.2, a bit below the nominal resolvability criteria of 〈Qθ〉 = 10 (Hawley
et al., 2011).
In addition to the global averages, we are interested in how the resolvability
depends on the position within the grid. Shown in Figure 2.3 is the time and
azimuthally averaged profile of Qθ, as well as the time and polar average profile of
Qφ. The average of Qθ was calculated as a function of r and θ by averaging over
entire φ domain. As we did in our global average, the vertical averaging over the
polar angle in our calculation of Qφ is only done in the region within one scale height
above and below the disk to exclude the coronal region. The spatial variation can
be seen in both quantities, but no apparent “dead zones” of gross under-resolution
are present.
These quality factors probe the ability of the simulation to resolve linear MRI
modes but do not address the structure of the (non-linear) saturated turbulence.
50
Figure 2.4: Evolution of magnetic tilt angle, ΘB, for long, fiducial disk
simulation (black line), high resolution wedge simulation (red line), and
high resolution 2π simulation (blue line). The t=0 time we set use in our
analysis is marked by the black dashed line. The evolution of the long
disk simulation used for the science run in our chapter has been truncated
to only show the initial saturation of the turbulence and a short time
period short after, but continues for 86, 856GM/c3. ΘB grows similarly
in all three simulations and saturates at the same value, ΘB ≈ 11− 12◦.
The black dashed line indicates t=0 used in our analysis.
Table 2.1: Simulation Parameters
Simulation ∆φ ISCO Orbits NR Nθ Nφ H/∆z
Fiducial π/3 1610 640 288 128 28.8
HiResWedge π/3 204 1280 576 256 57.6
HiRes2π 2π 110 1280 576 256 57.6
51
Table 2.2: Resolution Diagnostics
Simulation 〈Qz〉 〈Qθ〉 〈α〉 〈β〉 ΘB
Fiducial 8.1 20.5 0.12 23.4 10.5◦
HiResWedge 27.2 43.0 0.15 6.5 12.1◦
HiRes2π 18.4 11.8 0.09 7.9 11.6◦
Thus, we additionally measure the average in-plane magnetic tilt angle,
(〈 〉)
ΘB = −
Br
arctan , (2.15)
Bφ
the value of which is closely connected to the processes by which the MHD turbulence
saturates (Hawley et al., 2011, 2013; Pessah, 2010; Sorathia et al., 2012). We note
that, given the dominance of Bφ over the other field components, the tilt angle is
approximately given by
ΘB = arcsin(αMβ)/2, (2.16)
where αM is the Maxwell stress normalized by the magnetic pressure,
〈−2BRBφ〉
αM = , (2.17)〈P 〉
and β is the ratio of gas pressure to magnetic pressure,
P
β = . (2.18)
〈|B2|〉
The magnetic tilt angle is directly tied to the effectiveness of angular momentum
transport at a given field strength and can be used to probe whether a given simu-
lation has correctly captured the non-linear saturation of the MHD turbulence.
We calculate ΘB within the mid-plane regions of the disk, restricting to half
a scale height above and below the disk midplane in the well-resolved region. The
52
temporally averaged magnetic tilt (neglecting the first 200 ISCO orbits) is ΘB ≈
11.3◦ (Fig. 2.4), comparable to estimates from both analytic theory (Pessah, 2010)
and previous high-resolution local (Hawley et al., 2011) and global (Hawley et al.,
2013; Sorathia et al., 2012) simulations.
We further assess the fidelity of our fiducial simulation by performing two
additional high-resolution comparison runs. The first comparison simulation had
the exact same geometry as the fiducial simulation, but twice the resolution in each
of the three dimensions (hence voxels of 1/8th the volume). The second comparison
simulation also doubles the number of zones in each of the three dimensions, but
then extends the domain to include the full 2π in azimuth. Both of these comparison
simulations possess NR×Nθ×Nφ = 1280× 576× 256 = 1.89× 108 total zones with
a vertical resolution of 56 θ-zones per disk scale-height. Of course, the ∆R : R∆φ
ratio was larger in the 2π higher resolution simulation due to the extension of the
φ domain. The evolution of ΘB for the simulation used in our analysis and the two
test simulations is shown in Figure 2.4.
In the higher resolution simulations the evolution of ΘB is slightly faster than
our fiducial run, but settles at similar levels as the MRI-driven turbulence saturates.
One difference between the development of the turbulence in the comparison simu-
lations and our fiducial run is that our fiducial run has a transient spike in magnetic
tilt angle 5, 790GM/c3 (94 ISCO orbits) into the initialization. From visual inspec-
tion, we see the growth and break-up of a large channel flow in the disk body in the
fiducial run. This does not develop in the higher resolution comparison simulations
because the non-axisymmetric parasitic instabilities are more effective at breaking
53
up nascent channel flows. With this minor difference, the similarity in the evolution
and saturation of the turbulence between the three simulations indicates that our
fiducial run is adequately resolved.
In all, we can be confident in the convergence of our model. By standard
measures the simulation is adequately resolved and has reached a level of saturated
turbulence. The quality factors are slightly lower than what has been deemed well-
resolved by prior convergence studies (i.e. Hawley et al., 2011, 2013; Sorathia et al.,
2012), but the similarities between ΘB in our long science run and those of our
two shorter, high-resolution comparison simulations demonstrate the turbulence has
reached a saturated level.
2.3 Turbulence and the Effective α Description of the Model Disk
2.3.1 Fluctuating Effective α
We will now look at the behavior of the effective α in the disk. Of particular
interest is the variable behavior of the effective α as this is required to drive fluc-
tuations in Ṁ . Using Equation 2.1, we calculate the effective α by restricting the
domain over which we average to the disk midplane, θ = [π/2 ± h], where h is one
disk scale height. For each data output from our simulation (∆t = 123.2GM/c3) we
calculate the average stress and pressure within one scale height across the entire
azimuthal domain for each radial bin. The ratio of the stress to pressure was then
taken to provide the effective α as a function of radius and time.
Figure 2.5 shows the time variability of the effective α parameter in the disk
54
Figure 2.5: Time trace of the effective α parameter in the disk after
averaging over radius. The α parameter is highly variable and fluctuates
between α ≈ 0.04 and α ≈ 0.08 in a quasi-periodic manner.
after averaging over radius. Large variability occurs on an intermediate timescale,
longer than a dynamical time and shorter than a viscous time. The value of the
effective α in the disk typically fluctuates between α ≈ 0.03 and α ≈ 0.08 in a quasi-
periodic manner. On the lower end of the fluctuating range, α ≈ 0.03 seemingly acts
as a floor where the minima consistently returns. The upper range on the effective
α is less well-defined with maxima found at different levels as low as α = 0.07 and
as high as α = 0.13.
Figure 2.6 shows the histogram of effective α with normal (Gaussian) and log-
normal fits to the probability density function (PDF). Statistically, the PDF of the
55
Figure 2.6: Histogram of α fit with a normal distribution (red line) and
log-normal distribution (blue-line). The distribution is best fit by a log-
normal distribution.
56
effective α is better fit by a log-normal distribution. The best fit with a normal
distribution has χ2/D.o.F = 44.7/24 = 1.9 and the best fit with a log-normal
distribution has χ2/D.o.F = 29.6/24 = 1.2.
2.3.2 Characterizing Velocity and Length Scales of the Turbulence
Let us now characterize the velocity and length scales of the turbulence. Given
the range of time- and length-scales involved in a global disk simulation, we choose
to perform this exercise over a limited range of radii; we select the region R ∈
[15rg, 18rg], θ ∈ [π/2 − 0.35, π/2 + 0.35], φ ∈ [0, π/3). In physical coordinates, this
domain spans ∆R = 3rg, R∆θ = 10.5rg, R∆φ = 5πrg. The size of the subdomain
was chosen such that any localized turbulent structure would be well contained
within the box, but also so that any medium-scale structure would be captured by
our analysis if it is present.
2.3.2.1 Measuring Turbulent Velocity
Within this region, we first calculated the ratio of the turbulent speed to the
sound speed within the disk midplane as a function of time. The turbulent velocity
is defined as √
|v| = v2r + v2θ + (vφ − 〈v 〉)2φ (2.19)
and the sound is defined in its usual form√
γP
cs = . (2.20)
ρ
57
For each data dump, we calculated the volume weighted spatial average of |v|/cs to
serve as a diagnostic for the average velocity of the turbulence within the disk.
Figure 2.7 shows the time variability of 〈|v|/cs〉. This ratio is highly variable
and fluctuates within a range of |v|/cs ≈ 0.3 to |v|/cs ≈ 0.5. Comparing with the
time variability of α in Figure 2.5, the correlation between α and |v|/cs can be
readily observed. The average value is 〈|v|/cs〉 = 0.37.
2.3.2.2 Measuring Turbulent Scales
Measuring the spatial scales of the turbulence is significantly more involved
than measuring the turbulent velocity, but, nevertheless, provides a method for
characterizing the turbulence in simple terms. The spatial scales of the density,
magnetic field components and Maxwell stress of the turbulence were measured in
our r = 15 − 18rg subdomain using the autocorrelation the method of Guan et al.
(2009) & Beckwith et al. (2011) with some modifications. For each data dump after
the initialization of the simulation, the two-point autocorrelation of a given quantity
of interest, F (r, θ, φ, t), was calculated within this subdomain in the following way.
First, large-scale structure of F (r, θ, φ, t) was removed. For a given time step, t, the
average over ∆T = 1232 GM/c3 (11 data dumps) for each cell in the subdomain
was calculated. The time average was centered on the timestep of interest such that
the average for a single cell included its values 5 data dumps before and after. This
was then subtracted off of the present cell value:
∫
− 1
t+0.5∆T
Fsub(r, θ, φ, t) = F (r, θ, φ, t) Fnorm(r, θ, φ, t)dt (2.21)
∆T t−0.5∆T
58
(a) Gas Velocity
(b) Radial Correlation Length
Figure 2.7: Instantaneous values of vturb/cs. The fluctuations in vturb/cs
are correlated with the changes in α seen in Figure 2.5 indicating that
as the stress increases in the disk the gas is accelerated. Instantaneous
values of ∆x/h. ∆x/h fluctuates stochastically around l = 0.32 and is
not correlated with α. The turbulent spatial scales are independent of
the disk stress at all times.
59
leaving only the high-frequency perturbation. Since the average was determined for
an individual cell, this also removes vertical gradients.
Next, the physical coordinates were normalized by the local scale height to
express the turbulent scales in terms of h. This converts the wedge geometry into
a rectangular geometry and allows for easier comparison with prior work. In this
normalized geometry, r maps to x, φ maps to y, and θ maps to z.
After the normalization of the coordinates, the three-dimensional Fourier trans-
form was then taken of Fsub:
∫∫∫
F(k , k , k , t) = F (x, z, y)ei(kxx+kzz+kyy)x z y dxdzdy. (2.22)
From this, the autocorrelation, CF(∆x,∆z,∆y), was calculated according to:
∫∫∫
C (∆x,∆z,∆y) = |F(k , k , k , t)|2 × ei(kxx+kzz+kyy)F x z y dkxdkzdky. (2.23)
For the first part of our analysis into the spatial scales, we fit the time average
three-dimensional autocorrelations. Two-dimensional slices through the origin along
the x-y, x-z, and y-z planes are shown in Figure 2.8. To measure the shape of
the three-dimensional autocorrelations, we defined three surfaces corresponding to
where the value of the autocorrelation falls to CF(∆x,∆z,∆y) = 0.5, e
−1, and e−2.
The standard definition of the correlation length is the full-width, half-max of the
autocorrelation which is where the autocorrelation falls to 0.5. However, other works
use different definitions of the correlation length, so we present those values to aid
in comparing the turbulent scales between simulations.
A least-squares minimization was used to fit these surfaces with an ellipsoid
60
Table 2.3: Ellipsoid Fit Parameters
Variable Fit Parameters Fit Parameters Fit Parameters
θa
0.5 e−1 e−2
a - 0.32 a - 0.41 a - 0.66
ρ 16.7◦ b - 1.17 b - 1.67 b - 3.49
c - 0.24 c - 0.42 c - 1.08
a - 0.29 a - 0.38 a - 0.60
B ◦R 30.2 b - 0.75 b - 0.97 b - 1.79
c - 0.07 c - 0.10 c - 0.17
a - 0.20 a - 0.24 a - 0.36
Bθ 14.8
◦ b - 0.56 b - 0.74 b - 1.20
c - 0.14 c - 0.21 c - 0.38
a - 0.27 a - 0.35 a - 0.58
B ◦φ 22.2 b - 0.97 b -1.31 b - 2.52
c - 0.10 c - 0.14 c - 0.24
a - 0.20 a - 0.28 a - 0.47
B2 22.2◦ b - 0.78 b - 1.03 b - 2.03
c - 0.07 c - 0.10 c - 0.21
a - 0.23 a - 0.30 a - 0.53
BRBφ 26.4
◦ b - 0.70 b - 0.92 b - 1.64
c - 0.07 c - 0.10 c - 0.17
aInclination of autocorrelation with respect to the azimuthal direc-
tion.
Ellipsoid fit parameters to autocorrelations shown in Figure 2.8 of
turbulent quantities at CF = 0.5, e
−1, e−2. From the autocorrela-
tions, we find that the turbulent components of the magnetic field
are coherent on spatial scales smaller than the gas density (ρ).
61
of the form:
[x cos(θ)− y sin(θ]2 [y cos(θ)− x sin(θ)]2 z2
1 = + + . (2.24)
a2 b2 c2
where θ is the tilt angle introduced by the disk shear and a, b, and c are the axes
corresponding to the x, y, and z directions, respectively. The tilt angle and best fits
for the three different correlation lengths are given in Table 2.3.
The radial correlation length of the gas density, ρ, provides an estimate for
the characteristic length scale, l, of the turbulence. In the Shakura & Sunyaev α-
prescription the turbulent scales are less than the disk scale height. This requirement
does not strictly hold for the y- and z-directions, but it does hold for the x-direction,
which is the important direction for the angular momentum transport. Figure 2.7
shows the time variability of l calculated at individual data dumps. Unlike |v|/cs,
the variability of l is stochastic and lacks the structure seen in the time variability of
vturb/cs. Instead of varying in lock step with α, the time trace of l is flat and simply
fluctuates around l = 0.32. This means that increases in stress act to drive the
turbulence more quickly, rather than injecting energy at a larger scale and allowing
it to cascade down to smaller scales and that the turbulent eddy scales are preserved
regardless of the magnitude of the stress in the disk.
2.4 Disk Dynamo and Intermediate Timescale α Variability
We now turn to the disk dynamo and its influence on the variability of the
effective α in the simulation. The cyclical magnetic dynamo is a well-established
phenomenon in accretion disks and is seen in both local (Bodo et al., 2015; Branden-
62
(a) CF (∆x,∆z,∆y) in x-y plane
(b) CF (∆x,∆z,∆y) in x-z plane
(c) CF (∆x,∆z,∆y) in y-z plane
Figure 2.8: Slices of 3D autocorrelation function along x-y, x-z, and y-z
planes for ρ, B 2r, Bθ, Bφ, B , and BRBφ.
63
Figure 2.9: Azimuthally averaged Bφ at t = 3.7 × 104 GM/c3. Within
the disk midplane the fluctuations in Bφ are largely random. Large,
stratified regions of coherent magnetic field develop in the corona as
buoyancy lifts highly magnetized gas vertically and regions of similar
polarity grow through magnetic reconnection. The coronal regions are
symmetric across the midplane, but have opposite polarity.
64
burg et al., 1995; Davis et al., 2010; Hawley et al., 1996; Johansen et al., 2009; Stone
et al., 1996; Suzuki & Inutsuka, 2009; Turner, 2004) and global (Beckwith et al.,
2011; Flock et al., 2012; O’Neill et al., 2011; Parkin & Bicknell, 2013) MHD simu-
lations. It is characterized by rising bundles of toroidal magnetic field in which the
toroidal field flips polarity between two such events. The dynamo is quasi-periodic
in nature with ω ∼ 10−1 ωdyn, where ωdyn is the local dynamical frequency.
The influence the dynamo has on the disk is not fully understood. Since the
Maxwell stress and, thus, turbulent injection from the MRI is proportional to BRBφ,
we might imagine that the stress would be modulated by the large-scale toroidal
magnetic field generated by the dynamo. Indeed, a correlation between stress and
toroidal magnetic field has been found (Davis et al., 2010; Flock et al., 2012), how-
ever, the coupling of Maxwell stress to the low-frequency, periodic fluctuations of
the dynamo has not been well-investigated.
2.4.1 General Dynamo Behavior
We begin with an overview of the spatio-temporal properties of the magnetic
field in the disk. After the intialization of the simulation, a stratified magnetic
field geometry developed and was sustained. Figure 2.9 shows a snapshot of the
azimuthal average of Bφ at an arbitrary time (t = 3.7 × 104 GM/c3). Correlated
regions of field in the corona are seen spanning large portions of the disk. Like other
simulations, the large-scale magnetic field is generated in the disk and is then slowly
lifted by magnetic buoyancy (Davis et al., 2010; O’Neill et al., 2011).
65
(a) PSD of Bφ in midplane (b) PSD of Bφ at 2h
(c) PSD of Bφ at 3h (d) PSD of Bφ at 4h
Figure 2.10: PSDs of Bφ at different elevations above the disk with the
orbital frequency overlaid as the black, dashed line. Within the midplane
the standard αΩ dynamo behavior is seen. A band of power vertically
spanning ≈ 10 rg follows the shape of the orbital frequency at 10× lower
frequency. The radial correlation of Bφ increases with increasing height
above the disk due to the stratification that develops in the corona.
66
This growth can be seen in the PSDs of the azimuthally averaged Bφ. Shown
in Figure 2.10 are the PSDs of Bφ as a function of radius for the midplane and 2h, 3h,
and 4h above the disk midplane. The PSDs were calculated similarly to Reynolds &
Miller (2009) and are defined as P (ν) = |f̃(ν)|2 where f̃(ν) is the Fourier transform
of the time sequence of the variable of interest,
∫
f̃(ν) = f(t)e−2πiνtdt. (2.25)
In our case, the time sequence is the time variability of the azimuthal average of
Bφ in the disk mid-plane. The Fourier transform was calculated for each point on
the radial grid. The large gas reservoir supplies additional material to the disk
during the simulation which protected from secular changes due to the draining
from accretion. Therefore, there was no need “pre-whiten” before taking Fourier
transforms of the basic fluid variability, as Reynolds & Miller (2009) discuss the
need for in the case of significant secular evolution.
In the disk midplane the dynamo behaves similarly to the dynamo in Davis
et al. (2010). Rather than having a single characteristic frequency, a band of en-
hanced power parallels the orbital frequency at frequencies ≈ 0.1Ω. For a given
frequency, a band of power extends 10-15 rg, highlighting the radial coherence of
the dynamo action pointed out by O’Neill et al. (2011). For a given radius, the band
of power spans approximately a factor of three.
The PSDs of the azimuthal magnetic field above the disk show a larger radial
range of enhanced power. As the the height above the disk midplane increases,
the vertical bands of power at a given frequency stretches to smaller radii. At 4h,
67
power at the lowest frequencies is enhanced along the entire radial domain of the
simulation. All the while, the high frequency envelope seen in the PSD of the disk
midplane is preserved.
2.4.2 Modulation of Stress By Dynamo
Figure 2.11 shows the spatio-temporal evolution of the dynamo (probed by
the azimuthal average of Bφ) compared with the Maxwell stress for three distinct
radii (15 rg, 20 rg, & 25 rg). We will refer to these three radii in order to provide
a comparison of different properties related to the magnetic field throughout the
disk. As we saw in the previous section, the dynamo plays a central role in the
disk’s magnetic field evolution and, as we can see from the butterfly diagrams,
this translates into modulation of the Maxwell stress — when the magnetic field is
stronger at the maxima and minima of the dynamo cycle, the stress is largest. This
behavior is quite similar to that seen in the stratified shearing boxes of Davis et al.
(2010). Since the polarity of the global field flips during the dynamo cycle, the stress
is always positive. It is also worth noting that while the dynamo defines the trend
within a given cycle, the fluctuations in the stress are dominated by high-frequency
variability.
Shown in Figure 2.12 are the frequency-weighted PSDs of Bφ and BRBφ for 15
, 20, and 25 rg. The modulation of the Maxwell stress by the dynamo is more clearly
seen in the strong, multi-peaked bands of power in the PSDs of both Bφ and BRBφ
corresponding to oscillations of the low-frequency dynamo. However, the PSDs of
68
Figure 2.11: Spacetime diagrams of azimuthally averaged Bφ and BRBφ
for 15, 20, and 25 rg. The characteristic “butterfly” pattern is seen in the
spacetime diagrams of Bφ. When Bφ is stronger, BRBφ is stronger, al-
though the variability is mostly dominated by high-frequency variability.
69
Figure 2.12: Frequency-weighted PSDs (νP ) of Bφ (left panel) and BRBφ
(right panel) for 15 (top), 20 (middle), and 25 (bottom) rg. The strong,
multi-peaked bands of power correspond to the oscillations seen in Figure
2.11. A noticeable difference between the frequency-weighted PSDs of
Bφ and BRBφ is the strong power at high-frequencies in BRBφ.
70
BRBφ also have an additional high-frequency component to its variability from fluc-
tuations due to MRI-driven turbulence. This component is much more significant
than the high-frequency variability in Bφ. Interestingly, the bands of low-frequency
variability in the frequency-weighted PSDs of BRBφ are found around the local dy-
namo frequency, but the frequency of the variability is constant across the radii we
probe. The strong band of power in the PSDs of Bφ shows the expected shift to lower
frequency with increasing radii due to the decrease in orbital frequency. This dis-
crepancy seems to indicate that while the dynamo does introduce the low frequency
variability in α needed to drive the propagating fluctuations, the frequency at which
the effective α-parameter is modulated is subject to a more global interaction of the
magnetic field.
Fluctuations of the effective α-parameter on the viscous time were a priori
assumed by Lyubarskii (1997) and Cowperthwaite & Reynolds (2014). The fre-
quencies at which the Maxwell stress fluctuates in our simulation are higher than
that assumed in those previous models of propagating fluctuations in mass accretion
rate. Even though the frequency is a bit higher than previously assumed, the dy-
namo frequency is only a factor of a few longer than the inflow timescale, providing
the required “low”-frequency oscillations for this model.
71
Figure 2.13: Spacetime diagrams of Σ (top panel), Ṁ (middle panel),
and logarithm of the synthetic emission (bottom panel). Given the scale-
free nature of the simulation, units are arbitrary and dimensionless. The
preservation of fluctuations in the accretion flow that gives the appear-
ance of “propagating fluctuations” is readily observed in the spacetime
diagram of Σ and Ṁ . While a simple estimate of the radiation, our
emission proxy does track the other two quantities well, demonstrating
that the behavior of the accretion flow will be observable.
72
0.6
0.5
0.4
0.3
0.2
0.1
0.00 1 2 3 4 5 6 7 8 9
Time (GM/c3 ) 1e4
(a) Ṁ at ISCO
(b) PSD of Ṁ at ISCO
Figure 2.14: Ṁ at the ISCO (top panel) and its PSD (bottom panel).
The PSD is best fit by a single power law with a slope of Γ = −1, shown
by the blue line.
73
Ṁ  (M)
Figure 2.15: Histogram of Ṁ at the ISCO fit with a normal (red line) and
log-normal (blue line) distribution. The log-normal distribution provides
a superior fit.
74
2.5 Propagating Fluctuations in the Model Disk
2.5.1 Overview of Global Behavior
In studying the behavior of propagating fluctuations, we are interested in the
relationship between the instantaneous mass accretion rate,
∫
Ṁ(R) = ρvRRsin(θ)dφdθ, (2.26)
and other local disk variables, specifically the surface density and the effective α-
parameter. Additionally, we are interested in how evolution of the accretion disk
might appear if we could observe our simulated disk. If ZEUS was an energy con-
serving code, we could use the simulation’s cooling function as a measure of the
radiated energy. However, it is not, which means the turbulent losses are not fully
transferred into heat and captured by the cooling function. Therefore, we employ
a dissipation proxy to generate a synthetic, time-dependent measure of the disk lu-
minosity. In accretion disks, the outward transport of angular momentum liberates
gravitational potential energy from the disk gas. This is then converted to thermal
energy through the turbulent cascade of MRI driven turbulence and removed from
the system through radiative processes. Assuming the disk is radiatively-efficient,
the magnetic stress can be used as a tracer of energy dissipation since the energy
injected by the stress is quickly radiated away. Adopting Eqn 9 from Hubeny &
Hubeny (1998), the local flux at the photosphere of the disk (at one scaleheight)
75
due to dissipation is given by √ ( )∫
GM A h
F = B B dz (2.27)
r3
r φ
B 0
where A and B are relativistic correction factors given by
2GM
A = 1− (2.28)
rc2
and
− 3GMB = 1 . (2.29)
rc2
This approach has been used to produce synthetic light curves in other studies
of global accretion disks, e.g. Hawley & Krolik (2001) and Armitage & Reynolds
(2003).
Shown in Figure 2.13 are the spacetime diagrams of Σ, Ṁ and the emission
proxy. As can be seen in the figure, periods and regions of enhanced accretion (higher
Ṁ) are coincident with higher Σ. The correlation of Ṁ with surface density acts
to (partially) preserve the pattern of fluctuations in surface density as disk material
loses angular momentum and moves to smaller radii. Additionally, when Ṁ is
greater at a given radii, there is a corresponding increase in the emission proxy. It is
worth noting that the name “propagating fluctuations” is a slight misnomer as there
is no real propagation mechanism at play in the disk. Rather, the mass accretion
is a diffusive process that scales with Σ, providing the qualitative appearance of
“propagating” to smaller radii.
Figure 2.14 shows the time trace and PSD of Ṁ at the ISCO (6 rg). As
expected, there are stochastic accretion events when Ṁ dramatically increases (by
76
(a) Σ vs. Ṁ (b) Σ vs α
(c) α vs Ṁ
Figure 2.16: Correlations between Ṁ − Σ (top left panel), α − Σ (top
right panel), and Ṁ − α (bottom panel) at r = 15 rg in the simulation.
There is a clear positive, linear trend in the Ṁ dependence on both α
and Σ. Surprisingly, there is a negative trend in the Σ-α relationship
which we believe is a caused by magnetic buoyancy of more strongly
magnetized gas in the disk.
77
a factor of 3-5), and then returns to a stationary baseline value, a value of 0.1. The
PSD is featureless and is best fit with a powerlaw with index of Γ = −1.01± 0.07,
corresponding to flicker-type noise.
2.5.2 Log-normal Ṁ Distribution
The histogram of the instantaneous mass accretion rate at the ISCO in our
simulation is shown in Figure 2.15. The shape of the distribution resembles the
distribution of GBHB systems like Cygnus X-1 (Uttley et al., 2005) and is, similarly,
very well fit by a log-normal distribution o[f the form 2]
f(x;µ, σ) = P0exp −
(lnx− µ)
, (2.30)
2σ2
where P0 is the normalization, µ is the log of the mean, and σ is the standard
deviation. The best-fitting model parameters for the probability density function
(PDF) of the mass accretion rate at 6rg are µ = −1.84 and σ = −0.38 and gives a
χ2/D.O.F of 17.9/24. For comparison, the best fit with a normal distribution has
a χ2/D.O.F of 79.8/24. Statistically, the log-normal distribution is overwhelmingly
favored (∆χ2 = 62) and, visually, we can see the normal distribution fails to re-
produce the fast rise of the distribution at low Ṁ values and shallower tail of the
distribution at higher Ṁ values. The implication of this distribution is, hence, that
fluctuations in Ṁ combine multiplicatively, rather than additively in our MHD disk.
We can also quantify the asymmetry of the distribution by measuring skewness.
Skewness is the third momentum of t∑he distribution and is given by,N
(Xi − µ)3/N
γ1 =
i=1 (2.31)
σ3
78
where µ is the mean, σ is the standard deviation, and N is the number of data
points. The distribution of Ṁ at the ISCO has a value of γ1 = 1.08, indicating a
strongly positively skewed distribution.
2.5.3 Correlations Between α, Ṁ , & Σ
We will now look at the correlations in α, Ṁ , & Σ to verify that Ṁ does
indeed scale with both α and Σ. It is typically assumed α is independent of Σ thus,
according the canonical disk equation, Ṁ is expected to be higher when α and Σ are
larger. However, there is a degeneracy between these three parameters and we must
remove the stochasticity of the third in order to tease out relationships between
any two quantities. The easiest way to find the underlying trend is to simply bin
the data, which essentially averages out the stochasticity of the variable we are not
interested in. Since there are radial gradients in the disk, we focused on the behavior
at r = 15 rg. Ten bins were used with 70 data points per bin. The bars indicate the
standard deviation of the bin.
Figure 2.16 shows the Ṁ − Σ, α − Σ, and Ṁ − α correlations. We find that,
indeed, the mass accretion rate scales with both α and Σ, validating the a priori
assumptions that went into the phenomenological motivation of the propagating
fluctuations model. The Pearson correlation coefficient between Σ and Ṁ is 0.87 and
between α and Ṁ it is 0.70, indicating these are both statistically strong correlations.
Additionally, the correlation coefficient of α with Σ is r = −0.66, which is considered
a moderately strong anti-correlation. While the strong positive correlations of Ṁ
79
with α and Σ confirm the underlying assumptions of the propagating fluctuations
model, the negative α-Σ correlation is a slightly different than expected. We believe
the anti-correlation between α and Σ can be attributed to the magnetic buoyancy
in the disk. In the higher density regions, pressure balance in the disk displaces
magnetically dominated (lower β) gas upwards, decreasing the local field strength,
thereby decreasing α and causing the negative trend.
2.5.4 Radial Coherence
The strongest evidence for propagating fluctuations in our simulation comes
from radial coherence and frequency dependent phase shifts of the Ṁ variability. At
the heart of the propagating fluctuations model is the predication that modulations
in the accretion rate at larger radii will be seen at the inner radii with a time-lag
set by the viscous inflow time. The coherence function provides the cleanest way to
assess the causal connection in the disk.
Adopting the convention of Nowak et al. (1999a), we will consider Ṁ at two
radii, s1(t) and s2(t), with Fourier transforms S1(f) and S2(f), respectively. The
coherence of these two signals, given by
2 |〈S∗1(f)S2(f)〉|2γ = , (2.32)
〈|S (f)|2〉〈|S 21 2(f)| 〉
is a real-valued positive function that has a maximum of unity if and only if s2(t) is
related to s1(t) via simple a linear transfer function,
∫ ∞
s2(t) = Tr(t− τ)s1(τ)dτ (2.33)
−∞
80
(a) Coherence
(b) Phase
Figure 2.17: Coherence (top panel) and phase (bottom panel) maps. The
black dotted line indicates the local viscous frequency calculated from the
average radial velocity of the gas. The region of high coherence and net
phase shift is enveloped by the viscous frequency indicating fluctuations
above the viscous frequency at a given radii are quickly damped out
while fluctuations below the viscous frequency are preserved.
81
where Tr is some transfer function. In the other extreme, γ2 = 0 implies no linear
relationship between s1(t) and s2(t). In general, γ
2 can be considered a measure
of the fraction of variability in s2(t) that is coherently related to s1(t) and has
values between 0 (completely incoherent) and 1 (perfectly coherent). For non-zero
coherence, we can compute the cross-spectrum S∗1(f)S2(f). The complex phase Φ(f)
of the cross spectrum gives the frequency-dependent phase shift of the coherent parts
of s1(t) and s2(t).
The coherence function and phase shifts are shown in Figure 2.17. The coher-
ence function was calculated with the ISCO taken to be the inner, reference radii.
The Ṁ “signal” was broken into three segments for averaging and the coherence
function was calculated with each radial bin. Similar to Cowperthwaite & Reynolds
(2014), we find the coherent regions are found at frequencies below that of the local
viscous frequency,
2π(R− 6rg)
ωvisc(R) = . (2.34)〈vR(R)〉
The viscous time can hence be interpreted as a minimum timescale on which inward
communication of flow fluctuations can occur.
From the average phases, time-lags can be determined. The frequency-dependent
time lag at a given radii is τ(f) = Φ(f)/2πf . Figure 2.18 shows the coherence
function calculated between Ṁ at the ISCO and 10 rg. Given the length of our
simulation, only the inner regions have been run for a viscous time, so we only
show the structure of the phases and time lags calculated at these two radii. From
the two plots we can see fluctuations in Ṁ have phase shifts that increase with
82
(a) Phase Difference
(b) Time Lags
Figure 2.18: Frequency dependent phase differences and time lags be-
tween Ṁ at 6 rg and 10 rg. At frequencies greater than ω ≈ 3×10−3 the
coherence breaks down, as can be seen by the large scatter in the two
plots.
83
frequency, but a constant time lag up until the fluctuations become incoherent at
ω ≈ 3 × 10−3. The constant time lags of the fluctuation in Ṁ at different radii
was seen in the semi-analytic model of Cowperthwaite & Reynolds (2014). This
implies that the behavior of the propagating fluctuations in our MHD simulation is
consistent with those of the one-dimensional viscous disk.
2.5.5 RMS-Flux Relationship
We finish our analysis of Ṁ by looking at the RMS-flux relationship. While
the RMS-flux relationship is a staple of broadband variability (Heil et al., 2012), its
physical origins remain uncertain. This relationship is of great interest because it
appears to be an intrinsic property of the accretion process and, possibly, even more
fundamental than the PSD of the variability. Uttley & McHardy (2001) originally
hypothesized the RMS-flux relationship is evidence of Lyubarskii (1997) type prop-
agating fluctuations in Ṁ since the RMS-flux relationship is a ubiquitous property
from all black holes, regardless of mass.
If the RMS-flux relationship in the photometric variability is truly due to the
propagating fluctuations in the accretion disks of these sources, it should be seen in
the Ṁ of our simulation. To test this we calculated the mean,
1 ∑N
Ṁ = Ṁi, (2.35)
N
i=1
and RMS √√√√ ∑N1 ( )2σṀ = Ṁi − Ṁ , (2.36)N
i=1
of the mass accretion rate at the ISCO over a sliding window of 40 ISCO orbits
84
0.91e 1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.01.5 1.0 1.5 〈 〉 2.0 2.5 3.0
Ṁ 1e 1
Figure 2.19: Relationship between the RMS of Ṁ vs its average value
at the ISCO calculated over 40 ISCO orbit periods of the simulation. A
strong linear dependence is present in the variability, consistent with the
RMS-flux relationship observed from astrophysical black hole systems.
85
σṀ
(N=20 data dumps). Shown in Figure 2.19 is a plot of the raw 〈Ṁ〉 vs σṀ scat-
ter plot. The relationship was fit using a Bayesian, MCMC method with a linear
function of the form
σ̂ = k(〈Ṁ〉+ C) (2.37)
with best fitting parameters of k = 0.30± 0.03 and C = (−3± 3)× 10−3.
The linear correspondence of the RMS to the average mass accretion rate is
expected, as well as the intercept passing through the origin, i.e. when there is no
flux there should be no variability. However, the observed RMS-flux relationships
are typically offset and best described by a model with a constant baseline flux,
Fobs(t) = Fconst+Fvar(t) (Gleissner et al., 2004), that varies with black hole spectral
state. Our results suggest the variable component is related to the underlying MHD
turbulence of the disk and the flux offset can be attributed to disk processes not
captured by our simulation.
2.6 Proxy Light Curve Analysis
We now discuss the global behavior of our emission proxy and how the propa-
gating fluctuations in Ṁ might appear as photometric variability. The full details of
the radiation produced through the interaction of the accretion flow with the vari-
ous emission mechanisms can only be captured by including more realistic radiation
physics, which our simulation lacks. Nevertheless, we can use our emission proxy to
explore the emission variability in a broad sense. In particular, we want to deter-
mine if the structure of the Ṁ variability can be easily observed when the emission
86
(a) Synthetic Light Curve
(b) Average Disk Emission Profile
Figure 2.20: Synthetic light curve (top panel) and disk emission profile
(bottom panel, blue dots). The integrated synthetic emission from the
disk displays large, rapid fluctuations and behaves similarly to Ṁ across
the ISCO. Qualitatively, the variability is quite similar to the variability
of real astrophysical black hole systems. The rapid variability indicates
the inner regions of the disk have the largest contribution to the vari-
ability, which is confirmed by the radial profile of the synthetic emission
proxy.
87
is integrated over the entire disk. The flux we observe from GBHBs and AGNs is,
in effect, the integrated emission from a range of radii that evolve on timescales
proportional to their dynamical times. Consequently, there is the possibility some
of the variability could be smoothed out as lower-frequency variability from larger
radii can wash-out higher-frequency variability. Given how clear the nonlinear signal
is in astrophysical sources, we want to check it is reproduced by the variability of
the emission proxy from our simulation.
Figure 2.20 shows the synthetic light curve and radial profile of the average
emission for our disk. We created the light curve by integrating the dissipation from
the emission proxy over the well-resolved region of the simulation (4− 45 rg). Given
our radial range, the synthetic light curve is composed of signals from radii with
dynamical times that vary by a factor of 37.5, imitating observation with a broad-
band filter. Several aspects of the disk emission are apparent. First, the light curve
is similar to the time trace of Ṁ at the ISCO. The signal is aperiodic and flares
in the light curve correspond to large accretion events. Second, any flares in the
light curve quickly decay down to a well-defined, stationary average. The property
of mean-reversion is of statistical interest as a way to model its behavior. Finally,
and worth the most discussion, is the dominate role emission from the inner region
plays in driving variability. This is more clearly seen in the radial profile of the
emission per unit area of the disk, but can also be inferred by the high-frequency
flares. In Figure 2.20 the disk profile of Shakura & Sunyaev (1973) is shown for
comparison. The two profiles trace each other well at larger radii, but diverge in
the innermost regions. In the canonical α-disk the energy flux goes to zero at the
88
ISCO because it was believed at the time that no torques could be felt from the gas
in the inner plunging region. MHD simulations have shown that, in fact, torques
can be felt across the ISCO and radiation can originate from closer to the black
hole than the ISCO (Hawley & Krolik, 2001, 2002; Krolik & Hawley, 2002). In
addition to the physical effect of torques spanning the ISCO region, there may also
be a nonphysical contribution to our profile because we assume any energy from
stresses is immediately converted into radiated energy. In reality, there is a time
delay between the injection and the dissipation as the energy is transported through
the turbulent cascade. In the inner regions near the black hole the material could be
rapidly swept into the hole before it has had time to completely radiate the injected
energy. This would lead to an overestimation of the emission from the inner regions
by our emission proxy due to this advection of energy.
2.6.1 Signatures of Propagating Fluctuations
In our analysis of the mass accretion rate in the simulation we found the vari-
ability of Ṁ had near-identical structure to the universally observed flux variability
from black hole systems. Real disk emission is much more complicated than what
is captured by our emission proxy, however, the following analysis serves to provide
a proof of concept demonstration that the Ṁ fluctuations in our simulation will
translate into fluctuations in disk luminosity. While rudimentary, we will analyze
the synthetic emission using the diagnostics used in Section 2.5.
Figure 2.21 (upper and middle panels) shows the flux histogram and RMS-
89
(a) Histogram of Light Curve (b) Light Curve Coherence
(c) RMS-Flux Relationship
Figure 2.21: Histogram of light curve (top left), light curve coherence
(top right), and RMS-flux relationship (bottom). The structure of the
synthetic light curve variability is similar to that of Ṁ at the ISCO, as
is seen in Figures 2.15, 2.17, and 2.19.
90
flux relationship for the total (“broad-band”) proxy light curve shown in Fig. 2.20.
The preferred fit to the flux histogram is a log-normal distribution with parameters
µ = −5.7 and σ = −0.39. The χ2/D.O.F of this fit is 18.1/24 which is statistically
well-fit and much better than that of a normal distribution with χ2/D.O.F of this fit
is 98.9/24. The skewness of this distribution is γ1 = 1.28, indicating the histogram of
the light curve is more highly skewed than the histogram of the Ṁ and the log-normal
distribution is more pronounced. Naively, the increased skewness is surprising given
that the light curve is integrated over the disk and presumably additively combining
many random processes. According to the central limit theorem, this should drive
the distribution to be more Gaussian. However, since Ṁ is radially coherent, the
fluctuations of the emission proxy are not independent, as is assumed in a Gaussian
process.
The full synthetic light curve also has a linear relationship between the RMS
of the variability and the average flux value. As we did with Ṁ , the light curve was
broken into 20 sections and the standard deviation and mean values were calculated
for each. The distribution is well fit by a linear function with k = 0.34 ± 0.07 and
C = (−3± 3)× 10−4.
Examination of radial coherence in the emission proxy requires that we first
create two light curves to imitate observation in different energy bands (in a loose
sense this can be thought of as observing an accreting stellar mass black hole in
the hard and soft X-rays). The light curves we use were generated by integrating
the emission proxy in the regions spanning 4 − 10rg and 10 − 15rg. Then, as we
did with Ṁ , the coherence function was calculated by dividing the light curves
91
into three sections, resulting in the coherence function shown in Fig. 2.21 (bottom
panel). The coherence shows the same behavior as found for the mass accretion rate,
with highly-coherent signals at low-frequencies, and a falling coherence at higher-
frequencies. De-coherence is seen at ω ≈ 300 c3/GM which, while a relatively
high-frequency, agrees with the coherence and phase plots of Ṁ fluctuations with
the ISCO in Figure 2.17.
2.6.2 Timing Properties
Having established the signatures of propagating fluctuations in Ṁ are present
in the variability of our emission proxy, we will now look at the PSD and the
statistical behavior of the variability by fitting a simple Ornstein-Uhlenbeck (OU)
process.
Shown in Figure 2.22 is the PSD of the synthetic light curve fit with a broken
power law. The last 512 data points of our light curve were used to calculate the PSD
which ensures no phase dependencies were present. Following the method described
in Reynolds & Miller (2009), the PSD was fit with a power law plus white-noise
component and broken power law plus white-noise component of the form
Kω−Γ1 if ω < ωbrk
A(ω) = (2.38)
KωΓ2−Γ1 Γ2brk ω if ω > ωbrk
where K is a normalization constant, Γ1 is the low frequency slope, Γ2 is the high
frequency slope, and ωbrk is the break frequency, to determine which model provides
the best description of the PSD. The power density was assumed to have an ex-
ponential probability distribution with the probability of measuring a given power
92
between p and p+ dp expressed by,
1
P (p)dp = e−p/p0dp, (2.39)
p0
where p0 is the mean power. For an individual bin, the likelihood of a measuring
power pobs,i is
Li = (1/pmod,i)exp(pobs,i/pmod,i). (2.40)
For the entire PSD, the likelihood is the product of the individual likelihoods. Since
this can often be very large or very small, the standard statistic used is the log
likelihood, ∑
ln L = [−lnpmod,i − pobs,i/pmod,i]. (2.41)
i
A Markov Chain Monte Carlo (MCMC) algorithm was used to conduct the fitting
and find the parameters that maximize the log likelihood of each PSD model.
The PSD is well fit by both a broken power law with Γ1 = 0.5 ± 0.3, Γ2 =
1.8±0.2 , and ωbrk = (2.9±1.4)×10−4 and a single power law with Γ = 1.38±0.18.
However, the broken power law provides a much better statistical fit with ∆ln L =
5.08 at the expense of two degrees of freedom. This translates into a ∆χ2 = 10.16.
Hence, while the underlying PSD of Ṁ at the ISCO in our simulation has
Γ = 1, the integrated emission proxy has a steeper PSD. This is simply due to
the integrated contributions from across the accretion disk, with additional low
frequency power from large radii being added to the PSD. It is not immediately
clear what physics sets the break frequency, which corresponds to a timescale of
2000 − 6500 rg/c. It is faster than the nominal viscous time and approximately
matches the cooling/thermal timescale out to approximately r = 20 rg in the disk.
93
Figure 2.22: Synthetic light curve PSD with broken power law fit (blue line).
In order to ensure that the low-frequency break is not imposed by the truncation
of our emission proxy at r = 45rg, we generated light curves and the corresponding
PDSs truncating at r = 20, 30, 40rg. In all cases, the break persisted and the
frequency was unchanged.
In addition to the PSD, the statistical properties of the variability provide an
additional method of characterizing the synthetic light curve. Some recent work has
shown a damped random-walk, and more specifically an Ornstein-Uhlenbeck (OU)
process, provides a very good statistical model of GBHB and AGN variability (Kelly
et al., 2009, 2011; Koz lowski et al., 2010; MacLeod et al., 2010), although there are
others which suggest this may not be the case (Kasliwal et al., 2015; Kozl owski,
2017). The OU process is part of a class of statistical models called continuous
94
time, first order autoregressive, CAR(1), processes. CAR(1) processes can take
many forms and behave very different depending on their model parameters. The
OU process describes a noisy relaxation process where a system is stochastically per-
turbed from a stationary mean and decays back to the mean with a relaxation time
τ (Gillespie, 1996). The stochastic differential equation describing such a process is:
dX(t) = λ(µ−X(t))dt+ σdWt (2.42)
where λ = 1/τ , µ is the mean and σ is a measure of the average magnitude of
√
volatility per dt. This equation has the following√direct solution:
1− e−2λδ
X −λt −λδi+1 = Xie + µ(1− e ) + σ N0,1 (2.43)
2λ
where δ is the time step and N0,1 is a zero-centered, Gaussian distribution with a
variance of 1.
The even sampling of our light curve provides some simplicity in fitting the
OU process. Xi+1 is linearly dependent on Xi and takes the form:
Xi+1 = AXi + b+ , (2.44)
where  is a normal-random term that is independent and identically distributed.
From a simple linear fit, we can solve for λ, µ, and σ where
−ln a
λ = , (2.45)
δ
b
µ = √ ,& (2.46)1− a
−2 ln a
σ = sd() . (2.47)
δ(1− a2)
95
Since the OU process is inherently a Gaussian process we must therefore per-
form the fit to the synthetic light curve in log-space since our “flux” distribution
is log-normally distributed. Thinking in terms of real observations, this is equiva-
lent to fitting in magnitude space. For our light curve, the best fit parameters are
λ = (3.5 ± 1.2) × 10−3, µ = (3.3 ± 0.6) × 10−3 and σ = (6 ± 3) × 10−5. The λ
parameter corresponds to a decay time of t 3decay = 286GM/c .
2.7 Discussion
In our simulation we find that Lyubarskii (1997) type “propagating fluctu-
ations” quickly develop and are sustained throughout the entire simulation. The
multiplicative manner by which fluctuations in Ṁ combine and grow defines the be-
havior and subsequent evolution of the disk. The most significant consequence from
the multiplicative combination of Ṁ fluctuations is that the accretion history of a
given parcel of gas is retained, allowing for the apparent propagation of fluctuations
on a viscous time. The Ṁ variability at the ISCO in our simulation bears striking
resemblance to variability observed from astrophysical black holes and shares the
same phenomenological properties. In particular, it reproduces the log-normal flux
distributions, RMS-flux relationship, and radial coherence ubiquitously observed
from GBHBs and AGNs. These nonlinear features have typically been difficult for
variability models to account for, but are a consequence of a scenario where the
mass accretion rate is proportional to the multiplicative product of a stochastically
varying viscosity and the local surface density. In our simulated disk they develop
96
naturally from the MRI driven turbulence.
Additionally, we use an emission proxy to generate a disk-integrated synthetic
light curve to verify that the nonlinear characteristics of the Ṁ variability will
actually translate into observable features in the disk emission. While rudimentary,
our emission proxy follows the turbulent energy injected to the gas, which we assume
is quickly radiated away since we are considering a radiatively efficient disk. We find
that, indeed, the nonlinear features of Ṁ are present when the entire disk integrated
emission is considered and are not lost with the addition of low-frequency variability
originating from gas at large radii.
Our work was able to address several issues raised by Cowperthwaite & Reynolds
(2014) regarding the timescale on which the effective α parameter of the disk fluc-
tuates and how well an MHD disk can really be described by a strictly viscous
disk on long time scales. The previous efforts to model propagating fluctuations in
mass accretion rate of Lyubarskii (1997) using a linearized model and Cowperth-
waite & Reynolds (2014) using a one dimensional viscous disk model relied on the
a priori assumption that the low-frequency fluctuations in effective α occurred on a
viscous time. However, this is very long for the magnetic field to vary. When Cow-
perthwaite & Reynolds (2014) allowed α to vary on a dynamical time, the typically
assumed evolutionary timescale for the MRI, they found that the fluctuations were
too rapid and propagating fluctuations did not develop. We find the disk dynamo
can provide the necessary intermediate timescale modulation of the effective α to
drive propagating fluctuations in mass accretion rate. Dynamo action is universally
observed in modern accretion disk simulations and seems to be intimately tied to
97
the MHD physics of a differentially rotating plasma. Thus, the dynamo can provide
a natural mechanism to introduce variability in the effective α. While the ubiquity
of the dynamo has been established, its presence is typically neglected when con-
sidering accretion disk evolution. Our results highlight the significant impact the
dynamo can have on the disk and that it is a fundamental ingredient in the driving
of propagating fluctuations.
The properties of the Ṁ and light curve variability from our global, ab initio
MHD simulation are very similar to those from the simple viscous disk with stochas-
tic viscosity parameter of Cowperthwaite & Reynolds (2014). From this similarity,
we can conclude that on long timescales our MHD disk acts in a viscous manner.
This is not surprising, but there is not a trivial connection between the internal
MHD stresses that transport angular momentum and the expected viscous behavior
of the disk, even though it has largely been intuited from empirical results.
Our MHD simulation captures gross aspects of real disk variability, but it is
not without limitations. In particular, we use simple prescriptions to approximate
the affects of general relativity and radiative physics. As a first effort at modeling
propagating fluctuations in mass accretion rate in an MHD disk we are mainly
concerned with capturing the dynamics of the disk and the interplay of the accretion
flow with the stochastically fluctuating effective α. To do this, a long, high-resolution
simulation is required. Therefore, the priority of where computational resources are
used shifts away from more complex physics and into the duration and resolution
of the simulation.
While the focus of our analysis has been on understanding the signatures of
98
propagating fluctuations around black holes, we do not expect the effects of general
relativity and radiation physics to have a significant impact on the characteristics
of the disk evolution are primarily concerned with. In fact, the phenomenology we
seek to understand is also readily observed in cataclysmic variable (CV) systems
(Scaringi et al., 2012a,b, 2014) and young stellar objects (YSOs, Scaringi et al.,
2015). These systems are non-relativistic and not hot enough to be largely affected
by radiation pressure in the disk. The similarity of the broad-band variability be-
tween YSOs, CVs, GBHBs, and AGNs implies that propagating fluctuations are a
generic feature of highly ionized accretion disks. The fluctuations resulting from the
underlying accretion processes behave consistently across many orders of magnitude
of mass, temperature and evolutionary time implying that the growth of propagat-
ing fluctuations is not unique to a specific class of system, but rather intrinsically
tied to the angular momentum transport resulting from MHD turbulence. There-
fore, we believe that the results from this simulation are robust enough to provide a
qualitative description the accretion mechanism for highly ionized accretion disks.
2.8 Conclusions
In this chapter we have performed the first examination of the propagating
fluctuation picture using ab initio MHD turbulent models of disk accretion. The
propagating fluctuations model has served to explain the universally observed non-
linear features of photometric variability from accreting black holes of all masses.
In our simulation, propagating fluctuations in Ṁ form and behave in a manner con-
99
sistent with those of the viscous disk model of Cowperthwaite & Reynolds (2014).
The main results are summarized here:
1. The effective α-parameter in our simulated disk is highly variable. The vari-
ability has two components: a high-frequency component due to local fluctu-
ations from the MRI and a lower frequency component due to modulation of
the stress by the disk dynamo.
2. The lower frequency variability in the effective α-parameter from the disk dy-
namo drives fluctuations in Ṁ by modulating the internal disk stress on an
intermediate timescale. As was shown in Cowperthwaite & Reynolds (2014),
if fluctuations in α are too rapid, fluctuations will be damped out and propa-
gating fluctuations will not grow. The dynamo frequency is low enough that
propagating fluctuations can easily grow. Previous efforts directed at under-
standing the role of propagating fluctuations in Ṁ relied on ad hoc prescrip-
tions for the variability timescale. We show, for the first time, that the dynamo
fills this role and provides a natural driver for fluctuations in Ṁ .
3. The Ṁ in our simulation is positively correlated with Σ and the effective α.
In the analytic disk equations the mass accretion rate is proportional to the
product of the local surface density and the local viscosity. These correlations
are found in our ab initio simulation, suggesting that they naturally result
from the interaction of the MRI with accretion flow.
4. The Ṁ variability at the ISCO of our simulation resembles the photomet-
ric variability observed from GBHBs and AGNs. The Ṁ variability is log-
100
normally distributed, has a linear RMS-flux relationship, and displays radial
coherence. These features are empirically observed and have been attributed
to propagating fluctuations in Ṁ . We find that they are, in fact, properties of
accretion flow when propagating fluctuations in Ṁ are present.
5. Using an emission proxy to generate a disk integrated, synthetic light curve,
we find that the qualitative properties of the accretion flow are reflected in the
observable emission. While the nonlinear signatures are preserved in the light
curve, integrating over the disk adds additional low-frequency power from more
distant radii. However, since the inner region of the disk is brightest, the high
frequency variability is preferentially weighted and the nonlinear properties of
Ṁ are preserved.
101
Chapter 3: Truncated Accretion Disks In Hydrodynamics
3.1 Introduction
One of the most dramatic examples of variable behavior seen from black hole
binaries (BHBs) is the spectral state changes during outburst and the associated
hysteresis. BHBs are relatively rare, but the several dozen that have been identified
are found to have common evolutionary properties during their state transitions
(e.g. Cygnus X-1, 4U 1543-47, H1743-332, GRO J1655-40, GX 339-4).
Much uncertainty surrounds this behavior, but spectral timing studies find
that, in a broad sense, there are two primary BHB accretion states, low and high,
with distinct properties and additional transitionary intermediate states with com-
posite properties (Remillard & McClintock, 2006). Of the two main states, the
low state is less luminous and the BHB X-ray spectrum is dominated by a power
law with index ∼ 1.7. A thermal spectral component is present, but typically con-
tributes around 20% of the 2 − 20 keV flux. Radio emission originating from a
jet is usually observed and, in some objects with high luminosities, an outflowing
wind can be found (Homan et al., 2016), too. In the high state, a BHB is more
luminous. Thermal, blackbody emission dominates the X-ray spectrum and ac-
counts for > 75% of the unobscured 2 − 20 keV flux. The peak of this thermal
102
emission is around 1 keV, which is taken as evidence the geometrically thin disk
extends close to the inner-most stable circular orbit (ISCO). In this state, a jet is
absent, but signatures of outflowing winds are usually found in the X-ray spectrum.
Additionally, quasiperiodic oscillations (QPOs) are observed during accretion state
transitions, with high-frequency QPOs observed in certain intermediate states and
low-frequency QPOs found in some low-states.
While these common behaviors exist, every outbursting BHB is unique and has
its own peculiarities. Nevertheless, the gross differences between the low and high
state properties seem to naturally be explained with a model where the accretion
occurs through either a hot or cold mode. These different modes are associated with
different accretion flow structures and varying degrees of inner disk truncation (for
review see Done et al., 2007; McClintock & Remillard, 2006; Zdziarski & Gierliński,
2004). The preferred mode depends on the cooling efficiency of the inner region of
the accretion flow as the accretion rate evolves; specifically, the flow is in a radiatively
inefficient mode in the low state and a radiatively efficient mode in the high state.
In the low state, the inefficient cooling in the inner region effectively truncates the
cold, thin disk since the thermal energy of the gas cannot be radiated away. The
inner disk is replaced by a hot, diffuse flow, hence, the accretion flow should inflate
and be geometrically thick and optically thin. In the high state, radiation can
efficiently remove thermal energy, allowing the disk to take on an optically thick
and geometrically thin configuration like the α-disk of Shakura & Sunyaev (1973).
Since the disk can cool efficiently, there is no disk truncation and the thin disk
extends down to the ISCO.
103
The bifurcation of accretion states in active galactic nuclei (AGNs) is less clear,
but there is evidence that accretion in low-luminosity AGN (LLAGN), including Sgr
A* in our galactic center, is dominated by a hot mode (Narayan & Yi, 1995a; Rees
et al., 1982). In AGNs whose accretion rate is sufficiently low, less than 1% of their
Eddington limit, hot mode accretion (Ho, 2008; Yuan & Narayan, 2014) and inner
disk truncation (Nemmen et al., 2014; Trump et al., 2011; Yu et al., 2011) seems to be
common. Qualitatively, the properties of LLAGN resemble the BHB hard state. The
SED of a standard, unobscured, luminous Seyfert (1−30% Eddington), on the other
hand, typically has a large thermal component peaking in the far-UV, analogous to
the soft state of BHBs. The prevalence with which thick-to-thin disk transitions are
invoked to explain the phenomenology of both BHBs and AGNs warrants a rigorous
investigation of the behavior of the accretion flow at this interface to solidify the
theoretical underpinnings of the truncation model.
For gas to accrete onto the central black hole, it must lose angular momentum.
The seminal work of Shakura & Sunyaev (1973) parameterized the strength of the
internal stress relative to the gas pressure in the thin disk through the so-called
“α-parameter,” described in Chapter 1. The source of this anomalous viscosity
was assumed to be turbulence, but the driving mechanism was not elucidated until
the magnetorotational instability (MRI) was proposed as a viable instability (Bal-
bus & Hawley, 1991). The connection between MRI-driven turbulence and the
α-parameter has since been well-studied in global MHD accretion disk simulations
(e.g. Armitage, 1998; Beckwith et al., 2011; Flock et al., 2011; Hawley, 2000) and
thin disks have been shown to behave viscously.
104
The behavior, energetics, and dynamics of the hot phase accretion, on the
other hand, are not well constrained. Many types of hot flows have been suggested
to fill this role including advection dominated accretion flows (ADAFs; Narayan &
Yi, 1994, 1995a,b) and convection dominated accretion flows (CDAFs; Quataert &
Gruzinov, 2000). Simulations of non radiative, thick disks with varying levels of
sophistication have been used to shed light on the behavior of hot accretion flows.
Both HD (Igumenshchev & Abramowicz, 1999; Igumenshchev et al., 2000; Proga &
Begelman, 2003; Stone & Balbus, 1996; Yuan & Bu, 2010) and MHD (Hawley &
Balbus, 2002; Hawley et al., 2001; Igumenshchev et al., 2003; Stone & Pringle, 2001)
numerical models ubiquitously develop outflows. In the HD cases, and some early
MHD models, circulation often developed within the outflow and is interpreted as
convection. In more recent MHD models, though, the convection is found to be
stabilized (e.g. Narayan et al., 2012; Yuan et al., 2012, 2015).
Despite the thoroughness with which numerical models have investigated the
thin and thick disk configurations separately, the details of how the hot and cold
flow coexist are not understood. Several global accretion models have been proposed
and either explored analytically or with simple numerical models to determine how
a truncated disk might develop. For instance, a truncated disk could exist because
the inner disk evaporates due to the cool, Keplerian disk being heated by a hot
corona (Liu et al., 1999; Meyer & Meyer-Hofmeister, 1994; Narayan et al., 1996;
RóżaŃska & Czerny, 2000). Alternatively, the diffusion of energy from the interior
of the disk could also heat the gas beyond the viral temperature, thereby driving
the inner disk to a hot accretion solution and truncating the disk (Honma, 1996;
105
Manmoto & Kato, 2000; Manmoto et al., 2000). Local thermal instability can also
develop if the α-parameter is proportional to the magnetic Prandtl number (Potter
& Balbus, 2014, 2017). However, there is still much uncertainty underlying the
trigger mechanism.
Still, headway can be made in characterizing the flow dynamics of a truncated
disk. Of the outstanding issues, there is a particular need to understand the struc-
ture of the transition from the assumed thin disk to the thick disk and the influence
the truncation has on the disk evolution. For instance, how does the flow conserve
mass, angular momentum, and energy when it passes through an abrupt transition
zone? Can a steady-state configuration develop, or is time-variability a necessary
ingredient? On these fronts, numerical models provide a valuable tool and way to
glean insight into the properties of truncated disks.
Radiation HD (RHD; Das & Sharma, 2013; Wu et al., 2016a) simulations
have tried to directly address the viability of the state transitions of BHBs. They
find that when realistic radiation prescriptions which account for bremsstrahlung,
synchrotron, and synchrotron self-Comptionization are included in the models, disks
that are qualitatively similar to the BHB state transition model readily develop.
These models naturally form truncated, thin, cold disks embedded in a hot, diffuse
medium. Recently, Takahashi et al. (2016) used a simulation which accounted for
general relativity, radiation and MHD (GRRMHD) and found a truncated disk
developed where gas resided in a two-phase medium configuration. In the inner
region of their simulations, the gas is radiatively inefficient and the viscous time is
shorter than the cooling time. While it was not explicitly stated, this presumably
106
sets up an ADAF-like flow. Evidence for thermal, and possibly viscous instability,
has also been found in the GRRMHD simulations in Mishra et al. (2016).
The truncated disks found in the RHD and GRRMHD simulations suggest
these transitions can naturally happen, but relying on computationally intensive
algorithms comes at the sacrifice of resolution and simulation time. A pervasive
issue in these works is that the cold disk cools to the scale of the grid and the internal
dynamics are unresolved. This poses a problem because the defining evolution of
the accretion flow can occur on small scales and on long times, which makes a
proper simulation of a truncated accretion disk very difficult to do with current
computational capabilities. Thus, to conduct a detailed study of the accretion flow
itself and get a handle on how a truncated accretion disk will evolve, we must
make the problem tractable by forgoing some of the more computationally expensive
physics, like a full treatment of the radiation physics and the general relativistic
effects which may impact the evolution of the disk in the very inner region close to
the black hole. Here, we assume the truncation found in the simulations of Das &
Sharma (2013), Wu et al. (2016a), Takahashi et al. (2016), and Mishra et al. (2016)
can naturally occur and seek to fully characterize the accretion flow of a truncated
α-type viscous disk and understand the observational consequences.
In this chapter we discuss the behavior of the first sustained truncated accre-
tion disk that has been produced, i.e. a truncation that can be studied for thousands
of dynamical times. The simulation is well-resolved and allows for a detailed study
of the dynamics, energetics, and angular momentum transport in the disk. We do
not use a full prescription for the radiation, but rather use an ad hoc, bistable cool-
107
ing function to drive a thermal instability interior to some radius, thereby creating
a transition between a cold and hot disk that can be characterized. We empha-
size that the aim of this work is not to reproduce the state transitions observed in
BH systems, instead it is to study the putative truncated disk found in hard state
BHBs and LLAGNs. To isolate the behavior of the transition zone, we study the
most ideal scenario possible, an axisymmetric viscous HD accretion disk in a New-
tonian potential. The work presented here is a precursor to a study using a 3D
high-resolution MHD simulation with the same bi-stable cooling function that is
presented in Chapter 4.
The viscous formalism used here gives the freedom to set the viscosity and
balance the heating rate to the cooling rate according to an α-parameter. To better
match the accompanying MHD truncated disk simulation, we set α = 0.065. This
value was determined through a calibration MHD simulation, described in Chapter
4. In setting the HD α-parameter to that of an effective α-parameter from an MHD
model, we can provide a more direct association of the disk behaviors between the
HD and MHD regime.
By first establishing a benchmark with a simple α-disk, we hope to gain a
better understanding of the more complex disk model set in MHD, where heating
and angular momentum transport occur solely thorough MRI-driven turbulence,
and more honestly assess the viability of truncated disk models in the literature.
Much of the previous theoretical work is developed in HD, so the simplified model
considered here will help bridge the gap between expectations of models developed
in the literature and the more physical MHD model.
108
This chapter is organized as follows. In Section 3.2 we discuss our numerical
model, simulation setup, and bistable cooling function. In Section 3.3 we discuss our
analysis of dynamics, angular momentum transport, and energetics of the truncated
disk. In Section 3.4 we discuss the implications of our results and the caveats
associated with the simulation. In Section 3.5 we provide a summary and closing
remarks.
3.2 Numerical Model
The goal here is to emulate a truncated accretion disk and perform a detailed
study of its dynamics, angular momentum transport, and energetics. We therefore
idealize our test problem so that we can direct our computational resources to the
physics necessary to capture the hot-cold flow boundaries, run the model at high
resolution, and run for many evolutionary times. In this work, we consider a two-
dimensional, axisymmetric, viscous disk. The simulation is divided into two phases:
an initialization period before the switch to the bistable cooling function and the
latter portion used in our analysis where the bistable cooling function drives thermal
instability and creates a sustained truncation. For numerical reasons that become
especially important in the MHD case, we initially evolve the simulation as a thin
disk with constant aspect ratio h/r = 0.1 where the disk thickness is maintained
by a cooling function with a single target disk thickness. Once a quasi-steady state
develops, the cooling is switched to a bistable cooling function where the gas can
cool efficiently with a target scale height h/r = 0.1 or inefficiently with a target scale
109
height of h/r = 0.4, depending on its temperature. If the gas temperature is above
the transition temperature, TT , it cools according to the inefficient prescription; if
it is below it cools according the the efficient prescription. The analysis focuses on
the truncated disk.
The additional computational expense of accounting for full GR in this hy-
drodynamic simulation is manageable; however, it would be very restrictive for our
follow-up simulation in MHD and hinder our ability to achieve the goal of performing
a detailed study of the flow behavior. Therefore, we opt to consider the simplified
Newtonian case here to aid in comparing between the HD and MHD scenarios. Ad-
ditionally, these transitions are expected to occur at 30 − 100 rg (Nemmen et al.,
2014), which is far enough from the black hole that relativistic effects are minimal,
so there is observational motivation for neglecting GR as well.
3.2.1 Simulation Code
For this study we use the second-order accurate PLUTO v4.2 code (Mignone
et al., 2007) to model the gas dynamics. The equations of hydrodynamics are
evolved,
∂ρ
+∇ · (ρv) = 0, (3.1)
∂t
∂
(ρv) +∇ · (ρvv + PI) = −ρ∇Φ +∇ · σ, (3.2)
∂t
∂
(E + ρΦ) +∇ · [(E + P + ρΦ)v] = ∇ · (σ : v)− Λ, (3.3)
∂t
110
where ρ is the gas density, v is the fluid velocity, P is the gas pressure, I is the unit
rank-two tensor, σ is the viscous stress tensor, E is the total energy density of the
fluid,
1
E = u+ ρ|v|2, (3.4)
2
and Λ accounts for radiative losses through cooling. PLUTO was used in the
dimensionally-unsplit mode with the hllc Riemann solver (a three-wave, four state
solver) with piecewise linear spatial interpolation and second-order Runge-Kutta
time integration. Since the Godunov scheme conserves the total energy of simu-
lation to machine accuracy, the only energy losses come from outflow across the
domain boundary and the cooling function of the simulation.
3.2.2 Simulation Setup
In our simulation we adopt an axisymmetric, two-dimensional, spherical co-
ordinate system (r, θ, φ) for our computational grid. The domain spans r ∈
[10rg, 1000rg] and θ ∈ [π/2− 1.0, π/2 + 1.0] with NR×N 5θ = 512× 512 = 2.62× 10
zones. The radial spacing of the grid increases logarithmically so that ∆r/r is
constant. In the θ-direction, the simulation uses 320 uniformly grid zones in the
midplane region from θ = π/2 ± 0.5 and 96 geometrically stretched grid zones in
each of the coronal upper regions (π/2 + 0.5 ≤ |θ| ≤ π/2 + 1.0).
Outflowing boundary conditions are used for the inner and outer r-boundaries
and the ±θ boundaries, but were modified to prevent the inflow of material. Peri-
odic boundary conditions are used for the axisymmetric φ-direction. A density and
111
pressure floor is implemented to protect against values becoming artificially small
or negative. All fluid variables (e.g. ρ, p, T ) are reported in normalized units from
the simulation.
We consider an ideal gas with a γ = 5/3 adiabatic equation of state in a
Newtonian gravitational potential where,
−GMΦ = . (3.5)
r
Initially, the simulated disk is thin (h/r = 0.1) and in vertical hydrostatic equilib-
rium. The steady state α-disk solution(is used for)in[itiali(zatio)n:1]
2
z2 R∗
ρ(R, θ) = ρ R−3/20 exp − 1− (3.6)
2c2sR
3 R
where ρ0 is a normalization constant to set the maximum disk density to unity, R is
the cylindrical radius (R√= r cos θ), z is the vertical disk height (z = r sin θ), and cs
is the sound speed (cs = P/ρ = (h/r)vK). R∗ is taken to be just inside the inner
simulation domain (R∗ = 11GM/c
2) to account for the torque-free inner boundary
condition.
In astrophysical accretion disks, the effective viscosity originates from MRI
driven turbulence. In our hydrodynamic simulation, however, we approximate this
by treating the angular momentum transport with an α-type viscosity according to
Shakura & Sunyaev (1973). We set the shear viscosity term to be,
2 2
ν = αPΩ−1K = αρTΩ
−1
3 3 K
, (3.7)
where ΩK is the Keplerian angular velocity. A dimensionless α-parameter is set to
α = 0.065 to match the test MHD run discussed in Chapter 4, as we described in
112
Section 3.1.
3.2.3 Cooling Function
Our goal is to construct a model accretion disk that undergoes a radial tran-
sition from a geometrically-thin radiatively-efficient state to a geometrically-thick
inefficient state as the matter flows inwards. Once such a state has been constructed,
we can then characterize the dynamics of this transition region. Here, we describe
the toy optically-thin cooling function that we use to achieve this goal.
The volumetric heating due to the r − φ shear stresses is
H 3= αRρTΩK , (3.8)
2
where ΩK is the Keplerian angular velocity, and R is the gas constant. In a steady-
state, radiatively efficient disk we will have H = Λ. We seek to construct a cooling
function Λ(ρ, T, r) that (i) is positive-definite everywhere (in contrast to many of the
“target thickness” cooling functions employed in previous work; Noble et al. 2009),
(ii) yields a disk model that is viscously stable, (iii) produces a hot, geometrically-
thick solution at small radii, (iv) produces a cool, geometrically-thin solution at
large radii. We adopt a cooling function of the form,
Λ = Λ0(T )ρT
2f(r), (3.9)
where f(r) shall be specified later to maximize convenience. The linear dependence
on density ensures that the thermal balance is independent of density, removing the
possibility of viscous instability. Noting that c2s = RT = (h/r)2Ω2K , we see that the
113
condition H = Λ gives √
h 3αR2
= . (3.10)
r 2Λ0(T )r2ΩKf(r)
We choose f(r) = R2r−2Ω−1 ∝ r−1/2K so√that
h 3α
= , (3.11)
r 2Λ0(T )
and, by construction, the disk has a constant aspect ratio (i.e., fixed h/r) if Λ0 is a
constant.
We induce thermal instability and hence a radial transition into the flow by
introducing an abrupt decrease inΛ0(T ) at a prescribed transition temperature TT ,Λc (T < TT )Λ0(T ) =  (3.12)Λh (T ≥ TT )
where Λc and Λh are constants, and we will assume here that Λc > Λh. With Λ0 so
defined, the thermal balance condition H = Λ delineates three radial regions within
the disk. Inside of a radius rtr,i = 3αGM/2Λ R2c TT (the inner transition radius),
local thermal balan(ce r)equires√that the flow is hot with geometric thickness
h 3α 3αGM
= for r < rtr,i := (3.13)
r hot 2Λ
2
h ΛcR TT
Outside of a radius r 2tr,o = 3αGM/2ΛhR TT (the outer transition radius), thermal
balance requires th(at t)he flow√is cold with geometric thickness
h 3α 3αGM
= for r > rtr,o := (3.14)
r cold 2Λc Λ
2
hR TT
Between these two transition radii, the flow is bistable and thermal balance can be
achieved either in th(e co)ld or the h(ot)branch with geometric thickness
h h
or for rtr,i < r < rtr,o. (3.15)
r cold r hot
114
Note that, due to the form of our cooling law, the ratio of the cold/hot geometric
thickness is related to the ratio of the inner/outer transition radii,
[ ]2
rtr,o (h/r)hot
= (3.16)
rtr,i (h/r)cold
In this work, we set
( )
h
( ) cold = 0.1 (3.17)r
h
hot = 0.4 (3.18)r
rtr,i = 100rg. (3.19)
which implies rtr,o = 1600rg. As we shall see, this toy cooling function results in a
successful transition from a cold/thin to a hot/thicker disk close to the inner tran-
sition radius. Interestingly, the flow in the bistable region remains predominantly
in the cold/thin state.
3.3 Results
We will now discuss our analysis of the hydrodynamic disk simulation. We
show that a truncated disk does indeed develop and investigate the dynamics, ener-
getics, and angular momentum transport associated with the accretion flow in the
truncated state. Interestingly, we find that a sustained stream of sub-Keplarian gas
flows inwards from the thick-to-thin disk transition and undergoes delayed heat-
ing. As the gas heats to TT and beyond, pockets of hot gas form in the dense disk
which buoyantly rise and cool, inducing a circular flow in the coronal regions of the
disk and CDAF-like behavior. As the disk refills at the transition radius, the cycle
115
Figure 3.1: Snapshots of the gas density during transient heating event at
the end of the disk initialization. The top left panel shows the simulation
when the cooling is switched to the bistable cooling function at t =
1.99 × 106 GM/c3 (10, 000th inner orbit). The following three panels
show the simulation evolution at a 3.0 × 104 GM/c3 (150 inner orbit
times) interval.
repeats quasi-periodically.
3.3.1 The Truncated Disk
We begin by discussing the initial development of the truncated disk in the
final phase of the initialization and the general properties of the truncated disk. As
discussed in Section 3.2.3, the simulation is first evolved with constant h/r = 0.1,
from t = 0 GM/c3 to t = 1.98 × 106 GM/c3 (the first 10, 000 inner orbits) before
the cooling is switched to include both a hot and a cold branch. Immediately after
the switch, there is a large, transient heating event within r ≈ 200 rg that lasts
roughly ∆t = 1.4× 104 GM/c3 and then cools over ∆t = 8.6× 104 GM/c3. At 100
rg, tdyn ∼ 6.3 × 103GM/c3 so the heating event lasts approximately 2.5 dynamical
116
times and the cooling event occurrs on roughly the thermal time of the inner disk.
Figure 3.1 shows four snapshots of ρ during the heating event.
The gas in the inner region of the disk responds to the change in cooling by
inflating since it is beyond TT and has an excess of thermal energy which it cannot
rapidly dissipate on the inefficient cooling branch. Additionally, viscous stresses
continue to do work on the gas, further increasing the temperature. A thin, dense
stream of material plunges into the inner region between t = 2.01× 106 GM/c3 and
t = 2.04× 106 GM/c3. Since this gas is near TT , it ablates away, ultimately leaving
a truncated disk. The final remnants of the transient can be seen in a very thin
stream of dense gas in the inner region of the disk at t = 2.01 × 106 GM/c3 and
large, buoyant bubbles rising away from the transition radius.
The abrupt change in the cooling law is, obviously, unphysical and the heating
event is the reaction of the gas to the sudden switch. Following this, however, the
disk quickly settles into a new, sustained configuration with a “transition zone”
from r = 80 − 100 rg. In this transition zone, the region is cyclically filled with
material through accretion, but then evacuated as the gas heats to TT . Unless
explicitly stated, the analysis we detail is conducted following the implementation of
the bistable cooling function.
Figure 3.2 shows two representative snapshots of the gas density and scale
height temperature during initialization and during the analysis. Before the change
in cooling function, the disk has a uniform scale height ratio and very little hot gas
is present in the simulation. In the density and temperature maps of the truncated
disk, wispy regions of cold, dense gas are seen surrounding diffuse, hot bubbles. The
117
Figure 3.2: Density and temperature structure of the disk. In the top
two rows, the logarithm of the instantaneous density (left column) and
temperature scale height (right column) are shown of the inner simula-
tion region during the initialization at the switch to the bistable cooling
function t = 1.99×106GM/c3 and at an arbitrary time when the disk is
truncated (t = 5.96×105GM/c3). In the bottom row, the time-averaged
density and temperature maps are shown of the portion of the simula-
tion over which we the used in our analysis (t > 1.99 × 106 GM/c3).
Overlaid is the average geometric scale height (hg/r, black line) and the
temperature scale height (hT/r, white dashed line).
118
cold features are very dynamic and short lived, as they are close to the transition
temperature and can spontaneously transition to the hot phase.
Figure 3.2 also shows the time averaged density of the disk with two measures
of the characteristic disk scale height overlaid. The first measure is that of the
geometric scale height ratio, 〈√∫ 〉
hg(r) (θ(r)∫− θ̄(r))2ρdΩ= , (3.20)
r ρdΩ
where dΩ = sin θdθdφ is the solid angle e∫lement in spherical coordinates and ,
θ∫(r)ρdΩθ̄(r) = (3.21)
ρdΩ
is the average polar angle of the gas. The second measure is the density weighted,
temperature scale height ratio, 〈∫ √ 〉
hT (r) ∫ TrρdΩ= , (3.22)
r ρdΩ
where T is the gas temperature, T = P/ρ. The values of hg/r and hT/r were
calculated for every data dump and time averaged, denoted by the angled brackets.
Throughout this chapter we take gas in the “hot phase” to be that with T > TT and
gas in the “cold phase” to have T < TT .
Spatially, the time-averaged gas density shows a distinct truncation at r = 100
rg, which also corresponds to location where the disk becomes thicker, as shown by
hg/r and hT/r. In the inner region of the disk ρ is lower than the disk body by
approximately an order of magnitude. The averaged scale height ratio of the inner
disk is only twice as thick as the rest of the disk and does not reach the “thick”
scale height target of hh/r = 0.4 due to a significant amount of advective cooling.
119
Figure 3.3: Top panel- Radial profiles of the time-averaged fc (blue
line) and fh (red line). Bottom panel- Radial profiles of time-averaged
Σc (blue line) and Σh (red line). The surface density at the end of the
initialization prior to the switch to the bistable cooling function is shown
as the black dashed line.
120
The signature of the thick-to-thin transition is also present in the thermal
profile of the disk. Figure 3.3 shows the time averaged radial profile of the fraction
of gas in the hot and cold phases. The fractions do vary in time, but, nevertheless,
from this profile, we can see hot phase gas typically resides in the inner region of the
disk and cold phase gas resides beyond the density transition. It is also worth noting
that the transition from a predominately hot to cold phase, i.e. where the fraction
gas in the hot phase equals the fraction of gas in the cold phase, does not occur at
r = 100 rg as we would expect from the cooling function, but rather just inwards of
this radius at r = 80 rg. In brief, this is due to delayed heating as the gas is close to,
but not yet at, TT as it crosses r = 100 rg. At the hot/cold flow interface it moves
inwards and then makes the transition within the expected transition radius.
The truncation is also present in the time averaged surface density,
∫
1
Σ = ρr2dΩ. (3.23)
2πr
The hot and cold gas phases as a function of radius are also shown in Figure 3.3.
Compared to the surface density at the end of the initialization, the inner disk
surface density is lower by a factor of five. Further out, beyond the transition radius
where the gas is predominantly in the cold phase, the overall surface density is very
near the intialization level. Thus, through the launching of the convective cells by
the transitioning gas, there is essentially an evacuation of the inner disk relative to
the untruncated disk.
121
(a) 〈vφ〉/vK
(b) 〈vr〉/vK
Figure 3.4: Maps of 〈vφ〉/vk and 〈vr〉/vk. For reference, hg/r is overlaid.
The 〈vφ〉/vk map shows the gas in the coronal regions of the disk and
within the transition radius is sub-Keplerian. The stratified radial veloc-
ity structure of the gas is seen in the 〈vr〉/vk map. Within the cold disk
body, the gas has very little radial motion. The inflowing gas is seen as
a thin boundary layer at the geometric scale height. Large convective
cells are found the coronal regions of the disk, which are seen here as
alternating positive velocities at lower altitudes and negative velocities
at higher altitudes.
122
3.3.2 Disk Dynamics
The disk dynamics in a truncated disk are of interest since hot and cold ac-
cretion flows should behave quite differently, but the truncated disk model used to
explain astrophysical sources expects the two flows to exist in close proximity. In
the canonical thin accretion disk, the disk is rotationally supported and thermal
motions are a small component of the overall velocities, i.e. h/r = cs/vK where vK
is the Keplarian velocity (Shakura & Sunyaev, 1973).
In Figure 3.4 the time averaged ratios of 〈vφ〉/vK and 〈vr〉/vK are shown as a
function of position in the simulation domain. In the normalized simulation units
(i.e. G = M = c = 1), the Keplerian velocity is:
√1vK = . (3.24)
r
The disk establishes four dynamically distinct regions. First is the outer disk
body where the gas follows Keplerian orbits with very small radial velocity, as in a
standard thin accretion disk. Second, are the coronal regions, where the gas is sub-
Keplerian with an average azimuthal velocity of ≈ 15% below vK . For comparison,
previous hydrodynamic studies of hydrodynamic CDAF-like flows (e.g. Stone et al.,
1999), typically find that the azimuthal velocities are nearly-Keplerian and greater
than we find here. Additionally, as can be seen in the plot of 〈vr〉/vK , there is a
substantial outflow with vr ≈ 10% of the local vK immediately above and below the
disk, and a comparable fallback velocity at higher altitude. This is due to a single
convective cell that occupies each of the coronal regions. Third is the inner disk
123
Figure 3.5: Launching of outflow as pockets of hot gas spontaneously
transition to the hot phase. This sequence of images shows the gas
density in the inner region at t = 2.48×106GM/c3 (12,500 inner orbits),
2.50× 106 GM/c3 (12,600 inner orbits), 2.52× 106 GM/c3 (12,700 inner
orbits), and 2.54×106GM/c3 (12,800 inner orbits) and is representative
of the general behavior observed in the simulation.
124
body found within r = 100 rg and one scale height above and below the disk. In
Figure 3.2 we saw that this region was less dense, but contained a fair amount of
cold gas. In Figure 3.4 we can see that this cooler region is modestly sub-Keplerian
and inflowing at 1% of vK . Finally, there is a thin layer of gas enveloping the disk
at one scale height where the 〈vr〉/vK map reveals that there is a prominent inflow
directly on the disk body. It is in this region that the bulk of the mass accretion
from the outer disk occurs.
The launching mechanism is interesting, especially considering this is a sim-
ple hydrodynamic simulation so a magnetocentrifugally driven wind (Blandford &
Payne, 1982) or a radiation driven wind (Proga & Kallman, 2004; Proga et al., 2000)
cannot form. Here, we find the outflow in the simulation is launched by pockets of
hot gas that spontaneously undergo a transition from the cold phase to the hot phase
in the cool stream. As the gas expands, it becomes less dense and, thus, buoyantly
unstable according to the the Solberg-Høiland criterion describing the stability of
a rotating fluid. When the gas is launched, there is already an angular momentum
deficit, so it has little rotational support at a large radius. Unlike a pure CDAF, the
gas accretes when it falls back because, while the cooling of the hot gas is inefficient,
it is not zero and Bernoulli parameter is not conserved.
The launching processes is shown in a sequence of images in Figure 3.5. The
cold, sub-Keplerian stream can be seen in the dense gas. As the gas spontaneously
transitions to the inefficient cooling branch, it heats and expands. In doing so,
these pockets of newly heated gas are surrounded by denser, cooler material so they
buoyantly rise which disrupts the cold stream and disturbs the flow. Once the gas
125
Figure 3.6: Gas fractions (fc and fh) at r = 100 rg following the switch
to the bistable cooling function. The initial heating event is seen as a
spike in the hot gas fraction and subsequent decay at early times. The
gas fraction oscillations are found after this transient behavior dies away.
reaches a low density region, at approximately one density scale height, it moves
radially against the temperature-gravity gradient.
Figure 3.6 shows a time trace of the fraction of the gas in the hot phase,
fh, and the fraction of gas in the cold phase, fc, at r = 100 rg. The transient
heating event is seen at early times in time traces of fh and fc. Once the disk
relaxes into its truncated configuration, fh and fc oscillate as heating events quasi-
periodically occur in the cold streaming material and hot gas bubbles rise. These
fluctuations are relatively small, but rapid with the fluctuations on average taking
t = 2.65×104GM/c3, or four thermal times. At r = 100rg the average gas fractions
in each phase are fc = 0.84 and fh = 0.15. The oscillations in the gas fraction are
asymmetric and have a standard deviation of 0.03. As the oscillations occur, fh rises
rapidly and then experiences a slower decay. Consequently, fc decreases sharply and
gradually increases before the cycle repeats.
126
Figure 3.7: Ṁh (top panel) and Ṁc (bottom panel).
3.3.3 Mass Accretion and Angular Momentum Transport
Previous numerical studies of convective-like, hydrodynamic accretion flows
have found that very little gas actually accretes onto the black hole (Igumenshchev
& Abramowicz, 1999, 2000; Narayan et al., 2000; Stone et al., 1999). Here, we report
conflicting results, as we find that the hot gas does indeed accrete in our simulation.
One possibility for this divergence from previous studies of hot flows is that the
convective cells in our simulation are continually fed at the truncation of the thin
disk so the gas originates from deeper in the the gravitational potential, rather than
being a hot flow throughout the entire domain and overall being less gravitationally
bound.
In our 2D, axisymmetric disk, the instantaneous mass accretion rate in the
127
simulation is given by, ∫
Ṁ(R) = −ρv r2r dΩ. (3.25)
Using our definition of the gas phases from Section 3.3.1, we show the time trace of
the accretion rate for the hot (ṀH) and cold phase (ṀC) gas at the inner boundary
of our simulation in Figure 3.7. The pre-analysis disk mass accretion rate of cold
gas, 〈Ṁ 〉 = 6.65× 10−3pre , is also shown for reference.
We find, somewhat surprisingly, the mass flux across the inner boundary is
slightly higher in the truncated disk than in the initialization phase. Additionally,
there is a striking disparity between the Ṁh and Ṁc. Overwhelmingly, by approxi-
mately three orders of magnitude, hot gas is accreted rather than cold gas. Initially,
the overall mass accretion rate is higher during the transient phase of the heating
event, but it settles into a sustained accretion rate of 〈Ṁ〉 = 1.0 × 10−2 for the
remaining length of the simulation.
Since an appreciable amount of gas is accreted, there must be efficient angular
momentum transport in the simulation. In our disk, the angular momentum can be
transported three ways: through advection, stresses arising from the radial shear of
the disk, and stresses from shear in the atmosphere of the disk as it rotates within the
sub-Keplerian convective outflow. The conservation of angular momentum equation
takes the following form,
( ) ( ( ) ( ) )
∂ ∂ vφ ∂ vφ
ρrvφ = ∇ · ηr2 êr + η sin θ êθ − ρrvφv ,
∂t ∂r r sin θ ∂θ sin θ
where the three components appearing in the divergence term are, respectively, the
advection term, the r − φ viscous stress term, and the θ − φ stress term. The
128
Figure 3.8: Maps of r−φ viscous stress (top panel), θ−φ viscous stress
(middle panel), and advection of angular momentum (bottom panel).
Overlaid is the average geometric scale height (black solid line).
129
time-averaged maps of these terms shown in Figure 3.8.
The importance of the transport terms at different locations in the disk is
quite clear. The r − φ viscous stress is negative everywhere and largest within the
disk body, closer in towards the inner domain. This structure is expected since
we consider an α-type viscosity where the viscous stress is proportional to the gas
pressure and φ-shear and the pressure is largest in the inner regions of the disk.
Above and below the disk, at approximately one scale height, there are localized
layers of θ − φ stress where the Keplerian disk interacts with the sub-Keplerian
outflow. Also, within the very inner regions, the sign of the stress in each respective
hemisphere reverses where the convective cell closes, reflecting the opposite signs
of the vertical vφ/ sin(θ) gradient. The largest θ − φ stresses occur on the upper
and lower edges of the truncated disk where the cold, dense disk is buffeted by the
outflowing, hot, buoyant gas. The stress gradually decreases at larger radii as the
strength of the interaction between the disk and outflow decreases.
The time averaged map of the angular momentum advection terms is also
stratified, reflecting the structure seen in Figure 3.4 of the radial velocities. It is
worth noting that within the disk body meridional circulation establishes a very
small, but positive, net radial velocity at the disk midplane. This behavior was
originally predicted by Kippenhahn & Thomas (1982) and Urpin (1984) and has
since been reproduced numerically for viscous accretion disks (Kley & Lin, 1992;
Rozyczka et al., 1994). Moving away from the disk midplane to higher latitude, the
circulation reverses the flow and gas moves inwards.
The radial profiles of the time averaged angular momentum fluxes resulting
130
Figure 3.9: Top panel- Radial profiles of angular momentum advection
(dashed line), r − φ viscous stress term (solid line), and θ − φ term
(dotted line) integrated through a shell at a given radius. Blue indicates
a negative quantity and red indicates a positive quantity. Bottom Panel-
Radial profile of time averaged specific angular momentum.
131
Figure 3.10: Time variability of Σ (top panel), T (second panel), r − φ
stress (third panel), and advection (bottom panel) at r = 80 rg.
from the terms in Equation 3.26 are shown in Figure 3.9. On average, the viscous
r−φ stress is balanced by advection, at least within the region that has established a
quasi-steady state and viscous equilibrium (out to ≈ 500 rg). In a purely convective
flow, angular momentum is advected inwards, rather than transported outward (Ryu
& Goodman, 1992; Stone & Balbus, 1996). However, in this model, the viscous
stresses continually act on the gas, allowing for the loss of angular momentum.
Consistent with the removal of angular momentum by viscous torques, the specific
angular momentum profile, also shown in Figure 3.9, is Keplerian, l ∝ r1/2, when
averaged over the duration of the simulation.
The angular momentum transport at the hot-cold phase interface is complex
and depends on the time variability of Σ, T , the r − φ stress component, and the
132
advective stress component. Figure 3.10 shows these quantities at r = 80rg, in
the heart of the transition zone, over several of the bubble launching cycles. In
Figure 3.10, Σ, T , the r − φ stress, and advective stress have been averaged over
one thin-disk scale height (h = π/2± 0.1).
At the truncation, a cyclical building and expulsion of material occurs with a
period of t ≈ 2.65 × 104 GM/c3. In this cycle, peaks in T occur when the density
is lowest and vice versa. The angular momentum terms vary with Σ and T , but
there is a slight delay, ∆t ≈ 2.05× 103 GM/c3, in the maxima and minima. For the
majority of the cycle, approximately 80%, the surface density increases. Eventually,
a critical threshold is reached when the heating rate exceeds the cooling rate, causing
the temperature to increase and the cooling to become inefficient as pockets of gas
reach Tc. This causes a runaway heating event, seen in the rapid increase of T , and
the outflow to be launched which can be seen as turn over in the advection term to
positive values.
A key feature throughout this cycle is that the gas pressure remains relatively
constant as the gas expands to maintain a pressure equilibrium, a facet which is
discussed to greater depth in Section 3.3.4. This, in turn, means that the specific
angular momentum of the lower density, transitioning gas at the truncation is more
rapidly lost to the neighboring dense gas of the disk body through the r−φ viscous
stress since the amplitude of the viscosity is proportional to the pressure. Addition-
ally, as seen in Figure 3.5, the hot transitioning gas is in contact with the dense gas
of the cooler stream which cannot lose angular momentum as well. As the specific
angular momenta of these two regions changes due to the differences in the viscous
133
Figure 3.11: Radial profiles of terms in energy equation. Blue indicates
negative values, red indicates positive values. The combined viscous
dissipation (W , θ − φ and r − φ terms) are shown by the dashed line,
the advection of kinetic energy (AK) is shown by the dotted line, and
the advection of thermal energy (AT ) is shown by the dot-dashed line.
Since the cooling term (C) is positive everywhere in our notation and
the change in gravitational potential energy (G) is negative everywhere,
both are shown as solid lines.
134
Figure 3.12: Time variability of EK (top panel), EI (middle panel), and
EG (bottom panel) at r = 80 rg.
efficiency, a vertical shear is introduced. This shear allows the θ−φ viscous stress to
transfer angular momentum from the dense gas in the stream to the diffuse, outflow-
ing bubble where it is carried away. This changes the shear rate of the dense gas,
resulting in a spike in the r − φ stress component. This rapid exchange of angular
momentum drives the angular momentum transport in the stream which then flows
to smaller radii, as evidenced by the sharp negative dip in the advective angular
momentum transport term corresponding to the peak in the r − φ stress term.
After this involved transitory event, the disk refills and the cycle repeats.
135
3.3.4 Disk Energetics
3.3.4.1 Energy Terms
In our hydrodynamic model, the conservation of energy equation is,
∂E
+ C +AK +AT + G −W = 0, (3.26)
∂t
where E is the total disk energy, C is the “radiative” cooling done by the cooling
function, AK is the advection of kinetic energy, AT is the advection of thermal
energy, G is the advection of gravitational potential energy, andW is the work done
by viscous torques. We will consider these terms in a spherical shell integrated form,
such that they are [ ∫ ( )]
∂E ∂ 1 P
= dΩ ρv2 + +∫ρΦ , (3.27)∂t ∂t 2 γ − 1
[ ∫ C(= )dΩΛ], (3.28)
A 1 ∂ 1K = [ r2∫ dΩ( ρv2)vr], (3.29)r2 ∂r 2
A 1 ∂ PT = r2[ d∫Ω ( )vr], (3.30)r2 ∂r γ − 1
G 1 ∂={ ∫ r2 [dΩ ρ(Φ vr ), (3.31)r2 ∂r
W 1 ∂ 2 ∂ v( )φ= r dΩηvφ r (3.32)r2 ∂r ∂r sin]θ}
∂ vφ
+ sin θ .
∂θ sin θ
The time averaged, radial profiles of these terms are shown in Figure 3.11. The
presence of a truncated disk influences every energy term, with the exception of G.
Given that there is sustained accretion and the average specific angular momentum
profile is Keplerian, this is not necessarily surprising in the time average profile. Es-
136
sentially, the smooth, time averaged G indicates that, globally, material is accreting
and there is no mass build-up anywhere in the disk. Even though material may
temporarily be caught in convective cells, it is eventually accreted so gravitational
potential energy is liberated.
In the outer disk the AT term, which is positive, is suppressed and accounts
for very little of the overall energy transport of the gas. This changes within r < RT ,
however, as AT and AK become major contributors in the truncated disk region.
Advection plays a dominate role in the energy transport within the transition radius,
overtaking the cooling. The dynamical influence of the convective cells acts to advect
energy across the inner boundary as material gets swept inwards.
There is a slight deficit in the AK term and a significant deficit in theW term
inside of r < Rt because of the disk truncation. The impact on the AK term is
twofold. In the inner disk the density is lower and the flow is sub-Keplerian, thus
overall advection of kinetic energy in this region is slightly less what it might oth-
erwise be. Regardless, the sign does not change as the bulk of the kinetic energy is
swept inwards in the dense gas that streams inwards through the viscous interaction
of the hot outflow. In a general, time averaged sense, the energy dissipation of the
W term is suppressed because the density is lower and the fluid motions are more
chaotic in this region. However, as seen in the variability of the viscous stress com-
ponents, there are brief periods of enhanced viscous stress and, therefore, heating.
Nevertheless, the disruption of the flow by the hot gas bubbles makes it more diffi-
cult to establish the sustained radial and vertical gradients in vφ that generate the
viscous stresses and heating in the outer disk, thereby diminishing the contribution
137
of the W term in the inner region of the disk.
The transition has the strongest affect on the AT term, as its sign actually flips
at the disk transition. Within the inner disk, material is generally swept inwards,
either in the streaming material or through the return of the convective cells. The
sign of AT flips at the transition, however, because the majority of the thermal
energy resides in the buoyant, outflowing gas. The apparent suppression of this
term is somewhat artificial because it accounts for two competing components; the
net inflow of thermal energy through the accretion in the dense, thin disk and the
outflow of the diffuse, hot gas.
Further insight into the periodic launching of the buoyant bubbles can be
gained from examining the individual components of E , i.e. the kinetic energy
(EK = 1/2ρv), the internal energy (EI = P/(γ− 1)), and the gravitational potential
energy (EG = ρΦ) of the gas. The variability of these energy components at r = 80rg
is shown in Figure 3.12. Similar to Figure 3.10, these terms have been averaged over
h = π/2± 0.1.
The cyclical behavior observed in Σ, T , the r − φ stress, and the angular
momentum advection is also seen in the gas energy. Unlike the involved angular
momentum transport interaction observed in this region, the temporal variability
of the gas energy terms is simply due to the removal of gas in the buoyant bubbles,
which can be seen in the corresponding dips in Σ in Figure 3.10 and in the variability
of EG since Φ is a constant at r = 80rg. The removal of the gas is tied to an increase
in the gas energy as it heats (i.e. increased EI). Throughout the launching of the
hot bubbles, the internal energy and gas pressure have very little variability (≈ 2%).
138
When gas transitions in the flow, the bubble expands adiabatically as it buoyantly
rises vertically along the density gradient.
3.3.4.2 Bound Outflow
Throughout the simulation, the Bernoulli parameter,
1
Be = v2 + Φ +W, (3.33)
2
where W is the specific enthalpy
γc2
W = s (3.34)
γ − 1
of the gas, is negative. When Be is negative, the kinetic and thermal energy of the
gas is less than the gravitational potential energy, so the gas is bound and cannot
escape.
Shown in Figure 3.13 is the distribution of outflowing mass fraction (Ṁout), i.e.
gas with vr > 0, at r = 100 rg as a function of Be. On the surface, it looks like Ṁout
is nearly independent of Be. When we dig deeper into the distribution, we find that,
in fact, there are two components to the outflow: one component that is diffuse gas
with high velocities and other that is dense gas with low radial velocity. Also shown
in Figure 3.13 are the average density of the outflow and velocity as a function of
Be. When a bubble of hot gas is launched from the inner region of the disk, the gas
has a large, positive radial velocity, but low density. As it rises, higher density gas
gets entrained in the wake of the bubble and pulled along to larger radius, although
at a lower velocity.
139
(a) Histogram of Be of Outflowing Gas at 100 rg
(b) Be vs. ρ of Outflowing Gas at 100 rg
(c) Be vs. vr of Outflowing Gas at 100 rg
Figure 3.13: Histogram of the outflowing gas’s Be at r = 100 rg during
the analysis portion of the simulation. This is decomposed into the av-
erage density of the outflow as a function of Be and the average velocity
as a function of Be.
140
3.3.4.3 Synthetic Light Curves
To finish our analysis, we use the cooling in our simulation to generate a
synthetic light curve. Since PLUTO conserves total energy in the simulation, the
“radiative losses” our cooling function imitates can be used as a proxy for the
broadband emission. There are obvious caveats to this exercise, namely that we
use an ad hoc cooling function to approximate the real, detailed radiation physics of
the disk. Regardless, it will provide a sense of what type of photometric variability
could originate from this disk.
Observationally, in astrophysical black hole systems, the higher energy emis-
sion is attributed to the hot, radiatively inefficient gas. The lower energy, blackbody
emission is attributed to the cooler, radiatively efficient, dense gas. Following this
paradigm, we use our previously defined temperature criteria to generate “hard,”
Lh, and “soft,” Ls, band light curves, shown with the volume integrated cooling as
a function of time of the hot and cold gas, respectively, in Figure 3.14.
During the initialization, the large heating event causes an immediate spike in
Lh that decays down to a level of Lh = 3.0 × 10−3 which it maintains for the rest
of the simulation. This is higher than the total pre-analysis emission level, Lp, of
2.8× 10−3. Ls is approximately a quarter less than the Lh, due in part to the fact
that the hot gas fills a large portion of the simulation volume. Interestingly, there
is very little sign of the oscillatory behavior seen Figure 3.3 of the gas fraction at
rtrans. Instead, the light curves of each gas phase qualitatively resemble the the time
trace of the respective mass accretion rates, shown in Figure 3.7.
141
3.4 Discussion
By introducing a bistable cooling law to a simple hydrodynamic viscous disk,
we find that gas transitioning from the hot phase to the cold phase develops signifi-
cant time variability which affects the evolution of the disk. The transition between
efficient and inefficient gas cooling leads to disk truncation and causes large, convec-
tive cells to develop in the inner region of the disk. The presence of sub-Keplerian,
outflowing gas in close proximity to the accretion disk generates a large θ − φ vis-
cous stress which has previously been neglected, but is influential here. The inner,
truncated region of the disk is occupied by hot gas that cools inefficiently. Beyond
the transition, the bulk of the gas resides in the cool phase in a configuration sim-
ilar to a standard Shakura & Sunyaev thin disk. In general, the truncated disk in
our simulation is qualitatively similar to the phenomenological model invoked to
explain the spectral properties of BHBs in the low/hard and intermediate states
like Cygnus X-1 (Done & Zycki, 1999), XTE J1650-500 (Done & Gierliński, 2006),
H1743-322, (Sriram et al., 2009), MAXI J1543-564 (Rapisarda et al., 2014), and GX
339-5 (Plant et al., 2015), which was expected by construction. There are several
notable differences between the simulation and the standard truncated disk model,
which could manifest as observable signatures in BHB and LLAGN systems.
In the canonical HD truncated disk model, the inner accretion flow is assumed
to be an ADAF and many of the observational results are interpreted with this
assumption (e.g. Gammie et al., 1999; Ho et al., 2000; McClintock et al., 2003;
Shields et al., 2000; Ulvestad & Ho, 2001; Yuan et al., 2005). Hence, it is interesting
142
that a flow which closely resembles a CDAF naturally forms. Even though there are
qualitative similarities, the convective flow is ultimately distinct from the standard
CDAF. The primary way in which it differs is that the gas does not stay in the
convective flow indefinitely as it is constantly cooling. Once it is launched, the gas
rises to larger radii, but then stalls and falls back. The accretion rate is higher
following the transition because of the interaction of the outflowing gas and the thin
disk body. It is, in fact, primarily the gas in the hot phase that accretes with very
little cold gas making it to the inner boundary of the simulation.
One might hope the oscillatory gas fraction could be connected to observable
variability, like a QPO, but that is not currently supported by our simulation. QPOs
are present in the intermediate states between the low-hard state and the high-soft
states when the disk is assumed to be truncated, similar to what we modeled.
However, neither the mass accretion rate nor our simple cooling prescription show
appreciable variability. If the buoyant bubbles we capture in our simulation can be
physically realized and related to QPO behavior, their signature must ultimately
arise from more complex radiative processes than we have included.
Despite the obvious issues with connecting this HD accretion disk model to
real black hole disks, it still plays an important role in providing a needed bench-
mark against which future MHD models of truncated disks can be compared, as we
originally motivated. In particular, the surprising complexity in the disk structure,
angular momentum transport, and energetics demonstrate how poorly the behavior
of disks in this state is understood, even in what should be a simple case. The
results from this simulation help to provide a bridge between the simple disk model
143
Figure 3.14: Disk integrated synthetic light curve from the hot gas cool-
ing (Lh, top panel) and cool gas cooling (Ls, blue line). The cooling level
during initialization of Lp = 2.8×10−3 is shown as the black dashed line
for reference.
144
currently used to understand state transitions in BHBs and LLAGNs and the more
complex MHD physics that actually govern their evolution.
Of the many caveats and pitfalls associated with using an HD model to describe
a real accretion disk, we will briefly discuss two. The first concern is that the
behavior is sensitive to the strength of the anomalous viscosity. As the viscosity
of the disk changes, the ability of the fluid to develop small scale structure and
buoyantly rise also changes, which has been shown to affect the size and behavior
of hydrodynamic, convective accretion flows (Igumenshchev & Abramowicz, 1999).
To test the sensitivity of this model we reran the full simulation with α = 0.04.
The development of this low α-parameter was similar. Granted, the amplitudes of
the stress, Ṁ , etc. changed because of the lower viscosity, but the distinguishing
flow properties like the rapid inflow of gas along the skin of the disk, a single large
convective eddy in the atmosphere, and the quasi-periodic development of buoyant
bubbles at the truncation were all reproduced. In all, we can be confident that the
behavior we present in this chapter is robust for reasonable α-parameter values.
The biggest concern is the role MHD will play in altering the behavior of the
disk. One of the most conspicuous features of our simulation is the presence of a
“cold” stream of gas extending from the transition radius down to near the inner
boundary. Observationally, there is evidence that cool gas may be present near the
black hole, even when the disk is thought to be truncated (Miller et al., 2002a,b).
While it is tempting to make a connection between what develops in our simulation
and what is empirically observed, this must be done cautiously. In an astrophysical
accretion disk the gas motions are dominated by vigorous, MHD turbulence with
145
correlation lengths of order the disk scale height (Beckwith et al., 2011; Guan et al.,
2009; Hogg & Reynolds, 2016). The stream of material in this simulation is smaller
than that nominal scale, hence, it is reasonable to expect that even if this stream
could form, it would quickly be disrupted by the turbulence and unable to exist in
the way it does in our simulation.
3.5 Conclusions
Using a viscous, hydrodynamic simulation of an accretion disk with a bistable
cooling function, we conduct a detailed analysis of the dynamics, angular momentum
transport and energetics. We find that a truncated configuration develops and that
convective cells are launched from the evacuated region as cold gas flows inwards
and undergoes delayed heating. Pockets of hot gas form within the cold gas stream
and buoyantly rise along the surface of the cooler disk which introduces a θ − φ
viscous stress component through a vertical shear between the coronal atmosphere
and the disk itself. This viscous stress component has been neglected in previous
hydrodynamic treatments of accretion disks, but it plays in important role in the
evolution of the disk by transferring additional angular momentum from the dense
disk to the outflow and enhancing accretion in a thin layer long the surface of the
disk. In this truncated disk, the bulk of the accreted material is in the hot phase,
rather than the cold phase, and the accretion rate is slightly elevated compared to
that of the thin disk during the initialization. The spontaneous transition of gas
to the hot phase in the cold stream of material causes a quasi-periodic oscillation
146
in the hot/cold gas fraction at the transition radius, but this does not translate in
any appreciable fluctuations in the mass accretion rate, nor the volume integrated
cooling ratio. With this baseline hydrodynamic model of a truncated disk, we can
begin to understand more complex future MHD models.
147
Chapter 4: Truncated Accretion Disks In Magnetohydrodynamics
4.1 Introduction
Accreting stellar mass black hole systems in black hole binaries (BHBs) display
a characteristic hysteresis during outburst as they undergo state changes (Done
et al., 2007; Remillard & McClintock, 2006; Zdziarski & Gierliński, 2004). In the
low/hard state the system is less luminous and the spectrum is harder, while in
the high/soft state the luminosity is dramatically increased and the spectrum is
dominated by soft emission from thermal, black body emission with a peak around
≈ 1 keV. Accreting supermassive black holes in active galactic nuclei (AGNs) seem
to similarly fall into spectral classes like the BHB states, although interpreting them
as true analogs is treacherous because the long evolutionary time scales prevents the
direct observation of any possible hysteresis cycle. Low-luminosity AGNs (LLAGNs)
are observed to have low accretion rates, low luminosities, and hard spectra which is
interpreted as evidence of accretion through a hot, diffuse flow (Ho, 2008; Narayan
& Yi, 1995a; Rees et al., 1982; Yuan & Narayan, 2014). The standard Seyfert or
quasar, on the other hand, has a higher accretion rate, higher luminosity, and a
softer spectra dominated by a thermal continuum which peaks in the ultraviolet.
The canonical model behind the two accretion state paradigm in BHBs and
148
AGNs attributes the distinct properties to varying degrees of disk truncation (Esin
et al., 1997; Nemmen et al., 2014; Trump et al., 2011; Yu et al., 2011). The putative
truncation is believed to arise as the gas transitions from a radiatively-efficient,
geometrically-thin flow in the outer regions of the disk to a radiately-inefficient,
geometrically-thick flow in the inner region. When the disk is truncated and the
inner region of the disk is occupied by gas in the hot phase, the system should be
less luminous and have a harder spectrum as free-free emission and Comptonization
dominates the radiation. When the optically-thick, geometrically-thin accretion disk
extends down to the innermost stable circular orbit (ISCO), the system should be
more luminous and the spectrum should be dominated by quasi-blackbody thermal
radiation.
Additional evidence for disk truncation in BHBs in the high/soft state and
LLAGNs comes from X-ray monitoring of the 6.4 keV Fe Kα line. In BHB systems
like GX 339-4 (Tomsick et al., 2009) and H1743-322 (Shidatsu et al., 2014), the line
profile is seen to evolve as the outburst evolves. In the low/hard state, the line is
narrower and is best fit if the line originates from large radii, while in the high/soft
state, the line is broader and best fit if the bulk of the emission originates close to
the central black hole. Narrow Fe Kα lines are also observed from LLAGNs and
associated with a truncated accretion disk in systems like M81 (Dewangan et al.,
2004), NGC 4579 (Dewangan et al., 2004), NGC 4593 (Brenneman et al., 2007;
Markowitz & Reeves, 2009; Reynolds et al., 2004), and NGC 4258 (Reynolds et al.,
2009). This seemingly direct evidence of truncation lends credibility to the model,
but it is not quite so clear. In fact, broadened Fe Kα lines from relativistic blurring
149
have been reported in a number of sources when they are in the (luminous) hard
state, including GRS 1739-278 (Miller et al., 2015), Cygnus X-1 (Parker et al., 2015)
and even GX 339-4 (Miller et al., 2006), suggesting the optically-thick disk extends
to the ISCO even in the hard state.
A transition in the radiative efficiency makes it particularly challenging to
understand the flow dynamics. Separately, much effort has gone into studying
cooler accretion flows in radiatively-efficient, geometrically-thin disks and hot flows
in non-radiative, geometrically-thick disks. However, merging the two models is not
straightforward and has a strong dependence on whether the disk is considered in a
hydrodynamic (HD) or magnetohydrodynamic (MHD) regime.
Computational and algorithmic advances have recently enabled the numer-
ical exploration of BHB state transitions. HD models (Das & Sharma, 2013; Wu
et al., 2016b) which self-consistently account for radiation from bremsstrahlung, syn-
chrotron and synchrotron self-Comptonization demonstrate one of the basic tenets
of the current state transition model: that a cold, dense disk forms out of the hot,
diffuse medium once the accretion rate exceeds a critical threshold. As the accre-
tion rate and density increase, cooling becomes more efficient in the inner region of
the disk, thereby decreasing the radius of the truncation. Using a three-dimensional
MHD simulation which accounts for general relativity and radiation, Takahashi et al.
(2016) corroborate this finding.
Despite the success simulations have had in demonstrating the viability of
the truncated disk model to explain the phenomenology observed from BHB and
AGN systems, they have yet to yield a detailed understanding of the gas dynamics
150
and disk evolution in a truncated disk system. Properly modeling a truncated disk
requires a simulation grid capable of resolving the internal dynamics of the thin-disk,
a large computational volume, and a long enough integration that global processes
can develop over several viscous timescales. Fundamentally, these requirements are
at odds with each other, requiring sacrifices in various aspects of the simulation.
This chapter is a continuation of the work in Chapter 3 where we implemented
an ad hoc bistable cooling prescription to force a truncation in a two-dimensional,
axisymmetric, HD, viscous accretion disk. Here, we extended the model to a three-
dimensional, MHD accretion disk, which better represents a physical system since
MRI driven turbulence provides an effective viscosity and heats the gas.
In both of these simulations, we prioritize resolution and evolutionary time over
less influential physics. For instance, we simulate both disks with non-relativistic
physics in a Newtonian potential since empirical measurements place the trunca-
tion at large radii where GR effects should be negligible. Additionally, the cooling
function used in these simulations is computationally simple, but still captures the
essence of the true radiative physics, albeit in an approximate manner.
The HD case provides a benchmark against which we can compare the behavior
of the MHD accretion disk. In Chapter 3, several HD features significantly affect
the evolution. The launching of buoyant bubbles within the flow as gas transitioned
from the efficient cooling branch to the inefficient cooling branch was crucial in
shaping the global disk dynamics. This cyclical process involved the interplay of
the gas density and temperature, and introduced time variability to the angular
momentum transport efficiency and energetics. Buoyant bubbles drove an outflow,
151
leading to a circulation pattern in the disk atmosphere. Gas was expelled from the
inner disk, but fell back and was eventually accreted. Additionally, a thin stream of
cooler gas reached from the disk body into the truncated region, almost to the inner
boundary. Since we considered a full viscous prescription with an isotropic viscous
stress tensor, the θ−φ stress component was influential where vertical velocity shears
were present, for instance between the Keplerian disk body and the sub-Keplerian
outflow in the disk atmosphere.
Considering an MHD accretion disk introduces additional complexity, most no-
tably the role of turbulence and the large-scale magnetic dynamo. The chaotic disk
turbulence, by its nature, acts to disrupt coherent structure on small scales. Much
of the behavior in the HD accretion disk that developed on large scales originated
on much smaller scales. The large-scale magnetic dynamo induces low-frequency
variability in the accretion disk. Dynamos, in some form, seem to be a natural
occurrence in rotating plasmas. While magnetic activity cycles with long periods
have been known to exist in stars for many decades (Wilson, 1978), we have only
just begun to uncover the role dynamos play in moderating the evolution and time
variability of black hole accretion disks. Numerically, dynamos are ubiquitously
observed in shearing box and global simulations of MRI driven turbulence. The
large-scale dynamo is characterized by oscillation of the global magnetic field orien-
tation and amplitude. Except for extreme circumstances, like a strong magnetic field
(Salvesen et al., 2016), the dynamo frequency is regularly found to be Ωd ≈ 5−20ΩK ,
where ΩK is the orbital frequency. This becomes important because the dynamo
modulates the local Maxwell stress (Davis et al., 2010; Flock et al., 2012; Hogg &
152
Reynolds, 2016), which consequently influences variability in the angular momentum
transport and, presumably, gas heating.
In this chapter we begin with a summary of the numerical model in Section
5.2 and present an overview of the initial setup, bistable cooling function, a test
simulation and a resolution assessment. In Section 5.3 we present our analysis of the
truncated disk’s dynamics, angular momentum transport, energetics, and magnetic
field properties. We summarize our analysis and discuss the implications of our
results in Section 5.5 and provide closing remarks in Section 5.6.
4.2 Numerical Model
In this chapter we consider a three-dimensional, semi-global accretion disk
with a bistable cooling curve. The disk was constructed to match the HD model,
but extended to MHD and three dimensions. An overview of the bistable cooling
function we used is given here, as a more rigorous discussion and motivation is
presented in Chapter 3. The cooling prescription is calibrated through a pre-chosen
effective α-parameter, which we determined with a calibration simulation. The
fiducial model is evolved in two stages. The first stage is an initialization where the
model is evolved with a constant disk aspect ratio (h/r = 0.1) for 1.99× 105GM/c3
to allow the MRI driven turbulence to fully develop in the simulation volume. At
the inner-edge of the simulated disk (r = 10rg), this is 1000 orbital timescales where
t = 199 GM/c3orb,10 . Then the cooling function is switched to the bistable cooling
function described below and the model evolves for an additional 1.99× 105 GM/c3
153
to allow the truncated disk to fully develop and the unphysical transient behavior
to die away before we begin our analysis. The second stage is the portion of the
simulation used in our analysis and lasts t = 8.14× 105 GM/c3 (4096 torb,10).
4.2.1 Simulation Code
We use the second-order accurate PLUTO v4.2 code (Mignone et al., 2007)
to solve the equations of ideal MHD,
∂ρ
+∇ · (ρv) = 0, (4.1)
∂t
∂
(ρv) +∇ · (ρvv −BB + PI) = −ρ∇Φ, (4.2)
∂t
∂
(E + ρΦ) +∇ · [(E + P + ρΦ)v −B(v ·B)] = −Λ, (4.3)
∂t
∂B
= ∇×(v ×B),
∂t
where ρ is the gas density, v is the fluid velocity, P is the gas pressure, B is the
magnetic field, I is the unit rank-two tensor, E is the total energy density of the
fluid,
1 B2
E = u+ ρ|v|2 + , (4.4)
2 2
and Λ accounts for radiative losses through cooling. The equations were solved
using the hlld Riemann solver with the code in the dimensionally unspilt mode.
A piecewise parabolic reconstruction is used in space and time integration is done
with a second-order Runge Kutta algorithm. PLUTO is a Godunov code, so the
total energy in the simulation is conserved except for losses across the boundary and
energy removed through our cooling function. The method of constrained transport
is used to enforce the ∇ ·B = 0 condition.
154
4.2.2 Simulation Setup
The simulation grid is the same as in the HD model, but expanded in the
φ-direction, rather than being treated axisymmetrically. Spherical coordinates are
used in a computational domain that spans r ∈ [10rg, 1000rg], θ ∈ [π/2− 1.0, π/2 +
1.0], and φ ∈ [0.0, π/2] with NR ×Nθ ×Nφ = 512 × 512 × 128 = 3.35 × 107 zones.
Logarithmic spacing is used in the radial coordinate so that ∆r/r is constant. In
the θ-direction, the simulation uses 320 uniformly grid zones in the midplane region
from θ = π/2± 0.5 and 96 geometrically stretched grid zones in each of the coronal
regions (i.e. π/2 − 1.0 ≤ θ ≤ π/2 − 0.5 and π/2 + 0.5 ≤ θ ≤ π/2 + 1.0). Outflow
is allowed through the r and θ boundaries while the the φ boundaries are periodic.
To prevent artificially low or negative values, floors are imposed on density and
pressure.
Since the truncation is expected to occur at 30−100rg (Nemmen et al., 2014),
we can consider a scale free Newtonian potential,
GM
Φ = − . (4.5)
r
Similarly, all fluid variables (e.g. ρ, p, T ) are considered and reported in a normal-
ized, scale free form. The gas is considered ideal with a γ = 5/3 adiabatic equation
of state.
To initialize the simulation, we us(e the stead)y[state(α-di)sk ]solution:1
2 2z R∗
ρ(R, θ) = ρ0R
−3/2 exp − 1− (4.6)
2c2 3sR R
where ρ0 is a normalization constant to set the maximum disk density to unity, R
155
is the cylindrical radius (R = r cos θ), z√is the vertical disk height (z = R sin θ), and
cs is the isothermal sound speed (cs = P/ρ = hvK). R∗ accounts for the torque-
free inner boundary condition and is set just inside the inner simulation domain
(R∗ = 11GM/c
2).
The gas starts with a purely azimuthal velocity that is set such that the effec-
tive centripetal force balances the gravitational force. The initial magnetic field is
weak (〈8πpgas/B2〉 = 200) and set from a vector potential,
Ar = 0, (4.7)
Aθ = 0, ( ) (4.8)
1 4
A = A p e−(z/h) R sin π log(R/2)φ 0 2 , (4.9)h
where A0 is a normalization constant. Beyond |z| = 2.5h and within r = 15 rg, Aφ
is set to zero. Because of these field loops, the gas is unstable to the MRI.
To help manage the data volume, simulation data was dumped every 2 torb,10
(∆t = 397.4GM/c3).
4.2.3 Test Simulation
The bistable cooling function used in the simulation assumes a balance between
the global heating and cooling and requires an effective α-parameter be known a
priori. To determine this value, we ran a test simulation using the same grid and
initial conditions that are used in our fiducial MHD model. Instead of the bistable
cooling function, a simple Noble et al. (2009) style cooling function enforces a target
scale height ratio, h/r = 0.1 in this case, such that Ptarg = ρv
2
φ(h/r)
2/γ. In terms of
156
pressure the cooling function is, Λ = f(P −Ptarg)/τcool, where f is a switch function
(f = 0.5[(P − Ptarg)/|P − Ptarg|+ 1]) and τcool is the cooling time. We take τcool to
be the orbital time to strictly enforce the scale height ratio.
The simulation was run for t = 1.01×105GM/c3 (512 torb,10) to allow the MRI
driven turbulence to fully develop and saturate in the inner third of the simulation
domain. Averaged over the final 50 orbits of the simulation, we find 〈α〉 = 0.065
which we use to set the cooling rates of the bistable HD and MHD simulations.
4.2.4 Cooling Function
Our overarching goal is to characterize the dynamics of a truncated accretion
disk. To this end, a thermodynamic transition between an efficient cooling state
and an inefficient cooling state must arise at some radius to produce the trunca-
tion. We accomplish this by treating the gas cooling with an ad hoc optically-thin
cooling function with two branches reflecting the efficient and inefficient cooling, a
similar technique to that employed by Fragile et al. (2012) to study how accretion
disk geometry influences jet power. Starting from the simple, viscous formalism of
Shakura & Sunyaev (1973), the volumetric heating rate is
H 3= αRρTΩK , (4.10)
2
where R is the gas constant. In a steady-state disk, H = Λ. Taking as an ansatz
that the cooling function has the form,
Λ = Λ0(T )ρT
2f(r), (4.11)
157
we can construct a cooling law that is positive-definite everywhere, viscously stable,
and produces a geometrically-thick solution at small radii and a geometrically-thin
solution at large radii.
We seek to cool to a local equilibrium temperature. This is related to a target
scale height ratio by c2s = RT = (h√/r)2Ω2K . The H = Λ condition gives
h 3αR2
= . (4.12)
r 2Λ0(T )r2ΩKf(r)
For convenience, we take f(r) = R2r−2 −1√Ω ∝ r−1/2K so that
h 3α
= . (4.13)
r 2Λ0(T )
The disk will have a constant aspect ratio if Λ0 is constant. If there is an abrupt
decrease in Λ0(T ) at a prescribedtemperature TT ,
Λ0(T ) = Λc (T < TT ) (4.14)Λh (T ≥ TT )
where Λc, Λh are constants and Λc > Λh, we can induce a thermal instability and
consequently a radial transition into the flow.
In this formulation, the H = Λ condition dictates three regions exist in our
disk. An inner truncation radius occurs at r = 3αGM/2Λ R2tr,i c TT inside of which
thermal balance req(uir)es the fl√ow to be hot with geometric thickness
h 3α 3αGM
= for r < rtr,i := (4.15)
r hot 2Λh ΛcR2TT
Outside of r 2tr,o = 3(αG)M/2ΛhR√TT the flow will be cold with geometric thickness
h 3α 3αGM
= for r > rtr,o := (4.16)
r cold 2Λc Λ
2
hR TT
158
The flow is bistable in between these two radii and thermal balance is satisfied on
either the hot or cold branch with geometric scale height
( ) ( )
h h
or for rtr,i < r < rtr,o. (4.17)
r cold r hot
The ratio of the cold/hot geometric thickness is related to the ratio of the
inner/outer transition radii,
[ ]2
rtr,o (h/r)hot
= (4.18)
rtr,i (h/r)cold
In this work, we set
( )
h
( ) cold = 0.1 (4.19)r
h
hot = 0.4 (4.20)r
rtr,i = 100rg. (4.21)
which implies rtr,o = 1600rg. As we shall see, this toy cooling function results in a
successful transition from a cold/thin to a hot/thicker disk close to the inner tran-
sition radius. Interestingly, the flow in the bistable region remains predominantly
in the cold/thin state.
4.2.5 Resolution Diagnostics
When interpreting results from numerical MHD accretion disk models, the
issues of convergence and resolvability are paramount. As a result, much effort has
gone into developing criteria and diagnostics to judge the fidelity of these models.
Here, we apply a standard suite of tools to demonstrate that this model is adequately
159
resolved. All measurements are averaged over time and volume. Time averages are
taken during the first initialization stage before the change in the cooling law from
t = 1.79 − 1.99 × 105 GM/c3 (900-1000 torb,10). Volume averages are taken within
the main disk body (h/r < 0.1) and within the region where the MRI turbulence
has saturated (r < 500 rg) during this time. These can be taken as lower bounds
on the resolution because the disk inflates in the analysis stage of the simulation to
h/r > 0.1 and the effective disk resolution increases.
At the minimum target scale height ratio of h/r = 0.1, the simulation grid
covers one disk scale height with 32 zones. This value has been shown to be a
crude threshold where the macroscopic disk behavior and simulation results converge
(Hawley et al., 2013; Sorathia et al., 2012). Quality factors provide a way to assess
resolvability of the simulation by comparing the characteristic MRI wavelength in
each direction, λMRI = 2πvA/Ω, where vA is the Alfvén speed, to the grid zone size,
i.e. the θ component in the vertical direction and φ component in the azimuthal
direction (Hawley et al., 2011). Strictly speaking, they only measure the linear MRI
amplitude, but they serve as a probe of the fully nonlinear turbulence. The quality
factors are,
λMRI,θ
Qθ = (4.22)
R∆θ
and
λMRI,φ
Qφ = . (4.23)
R∆φ
We find that 〈Qθ〉 = 16.5 and 〈Qφ〉 = 19.5. This is considered“well-resolved” as
Qθ > 6− 8 has been shown to properly capture the growth rate of the MRI (Flock
160
0.26
0.24
0.22
0.20
0.18
0.16
0.14
0.12
0.10
0 200 400 600 800 1000
Radius (rg)
Figure 4.1: Average hg(r)/r (solid line) and hT (r)/r (dashed line). Av-
eraging has been done over time as described in the text.
et al., 2010; Hawley & Stone, 1995; Sano et al., 2004). The resolvability criteria for
capturing the nonlinear behavior of the MRI driven turbulence is somewhat stricter,
Qθ > 10 and Qφ > 20 (Hawley et al., 2011, 2013; Sorathia et al., 2012), although
they are not independent of each other. Nevertheless, we either meet or exceed these
guidelines.
The saturation level of the anisotropic turbulence can be measured through
the characteristic orientation of the magnetic field, given by the average in-plane
magnetic tilt angle, (〈 〉)
Br
ΘB = − arctan . (4.24)
Bφ
Theoretical estimates predict ΘB ≈ 15◦ (Guan et al., 2009; Pessah, 2010) and it is
161
Scaleheight (h/r)
empirically found to be between ΘB ≈ 11 − 13◦ in high resolution simulations of
local (Hawley et al., 2011) and global (Hawley et al., 2013; Hogg & Reynolds, 2016;
Sorathia et al., 2012) MRI driven turbulence. We measure 〈ΘB〉 = 12.3◦, which is
within the bounds of previously measured saturated turbulence.
4.3 Results
In our analysis we are primarily focused on the dynamics of the truncated
disk and any sources of variability that could potentially be observable. We strive
to understand the drivers behind the disk behaviors and aim to dissect the details
of the angular momentum transport and disk energetics. To this end, we take a
broad approach and consider all aspects of the accretion flow by investigating how
properties like the gas temperature, motions, magnetic field strength, and density
vary in both space and time and their interdependencies. The analysis is specifically
interested in understanding how the presence of the truncation alters the components
of the accretion flow interior to, in, and beyond the truncation. In many cases the
analysis considers both the average and instantaneous properties to develop the
most complete understand of the truncated disk behavior.
4.3.1 The Truncated Disk
We begin our analysis with an overview of the truncated disk in the simulation.
The truncated disk we study has hot, diffuse gas in the inner region and a cooler,
Keplerian main disk body. The inner region thickens since it is hot and the gas has
162
(a) ρ at t = 1.99× 105 GM/c3 (b) hT /r at t = 1.99× 105 GM/c3
(c) ρ at t = 3.98× 105 GM/c3 (d) h /r at t = 3.98× 105T GM/c3
(e) 〈ρ〉 (f) 〈hT /r〉
Figure 4.2: Top row- 3D renderings of density (left) and hT/r (right) at
t = 1.99 × 105 GM/c3 (1000 torb,10). The upper surface of the density
rendering is set by the ρ = 0.001 contour. Middle row- 3D renderings of
density (left) and hT/r (right) at t = 3.98×105GM/c3 (2000 torb,10). The
upper surface of the density rendering is set by the ρ = 0.001 contour.
Bottom row- Density (right) and hT/r (left) averaged over azimuth and
time. Both 〈hg/r〉 (white dashed line) and 〈hT/r〉 (black solid line )is
overlaid in for reference.
163
additional thermal support. Figure 4.1 shows two diagnostics of the time averaged
scale height ratio. The first meas〈ur√e is the geometric sca〉le height,∫
hg(r) (θ(r)∫− θ̄(r))2ρdΩ= , (4.25)
r ρdΩ
where dΩ = sin θdθdφ is the solid angle element in spherical coordinates and,
∫
θ∫(r)ρdΩθ̄(r) = (4.26)
ρdΩ
is the average polar angle of the gas. Second, is the density weighted, temperature
scale height ratio, 〈∫ √ 〉
hT (r) ∫ TrρdΩ= (4.27)
r ρdΩ
(where we have taken R = GM = 1). For each data dump, hg/r and hT/r were
calculated and the time average was taken, denoted by the angled brackets. At
small radii, r < 50 rg, hg/r and hT/r are ≈ 0.25. This is below the target scale
height of the upper branch of the cooling law (h/r = 0.4) and similar to the peak of
the average disk scale height in the HD model. Like the HD model, advection acts
as a non-radiative cooling term which helps to keep the disk thinner than the hot
phase h/r target. At larger radii, the gas resides on the efficient cooling branch and
maintains a thin geometry.
Figure 4.2 shows 3D volume renderings of ρ and the local scale height temper-
ature at t = 1.98× 105 GM/c3 and t = 8.94× 105 GM/c3, to highlight the changes
the bistable cooling function introduces from the untruncated disk in the initializa-
tion. Of the general differences, the decreased density and increased temperature
of the gas in the inner disk are most evident. While the inner region is evacuated,
164
the density in the coronal regions increases. Additionally, the gas in the atmosphere
is significantly hotter and entirely in the hot phase, with temperature scale heights
at or above hT/r = 0.4. Throughout the simulation chaotic turbulence dominates
the disk and at no point do coherent, small scale structures form. Qualitatively, the
truncation resembles the phenomenological model invoked to explain the spectral
features from putative truncated accretion disks in BHB and AGN systems, e.g. the
schematics in Zdziarski & Gierliński (2004) and Done et al. (2007), albeit with a
hotter atmosphere.
Time and azimuthal averages of ρ and the local hT/r are also shown in Figure
4.2. The gas density peaks in the disk midplane between r = 150 − 200 rg. In the
truncated region the gas density is appreciably lower, but the disk is inflated so the
density is enhanced vertically for a given radius. Buoyancy introduces a stratified
temperature structure in the outer disk. The thickness of the cold, intermediate,
and hot layers changes based on the radial heating profile. The cold gas layer thins
with decreasing radius as the gas interior to the truncation zone has hT/r ≥ 0.2.
Figure 4.3 shows profiles of the time averaged fraction of gas in each phase
as a function of radius. The hot gas is taken to have T > TT . In the inner disk
(r < 100 rg), the gas is entirely above TT . In the outer disk (r > 300 rg), ≈ 10%
of the gas resides in the hot phase. The gas fractions gradually change in between
these two regions as fluctuations allow varying amounts of the disk to reach TT .
Quantitatively we take the “truncation” to be at r = 170 rg where the two gas
fractions are equal, but the region between 130− 200 rg should really be considered
a truncation zone between the outer, cold thin disk and the inner, hot thick disk.
165
Figure 4.3: Top panel- Radial profiles of the time-averaged fc (blue line)
and fh (red line). Bottom panel- Radial profiles of time-averaged Σc
(blue line) and Σh (red line). The surface density during the initialization
immediately prior to the switch to the bistable cooling function is shown
for reference as the black dotted line. In the inner regions of the disk
Σc ≈ 1× 10−7 and is off the plot axes.
166
Figure 4.4: Time averaged mass flux (ṁ) profiles along the polar angle
(θ) at r = 100 rg (left), r = 200 rg (middle), and r = 300 rg (right). The
blue dashed lines mark one disk scaleheight above and below the disk.
We adopt the convention that inflow is positive and outflow is negative.
This region corresponds to the tapering of the cold gas seen in the averaged ρ and
hT/r in Figure 4.2. The surface density,
∫
1
Σ(r) = ρr2dΩ, (4.28)
2πr
of each respective gas phase is also shown in Figure 4.3. The integration has been
done over the entire θ and φ domain at each radius. In the inner disk Σ is reduced
by 70− 80% compared to the thin disk in the initialization stage.
4.3.2 Disk Dynamics and Variability
Three dynamically distinct regions develop in the simulation’s global flow pat-
tern. The first is the thin, rotationally supported main disk body. Radial motions
are dominated by the chaotic flow, but are slightly negative on average as material
slowly sinks deeper into the gravitational potential and accretes. Inside of the tran-
sition zone, the flow behavior is slightly modified and vφ is modestly sub-Keplerian
167
(by 10%) because thermal support becomes more influential to the dynamics. The
radial velocity is greater than the slow diffusive drift found in the main disk body,
but vr still remains less than a few fractions of a percent of the local Keplerian veloc-
ity. In the hot gas filled disk atmosphere, however, the dynamics are quite different
from the other two regions. There, thermal support plays its biggest role and the
gas is 10− 15% below the local Keplerian velocity. The real distinguishing feature
of this region, and most dynamically interesting, is a thermally driven outflow that
underlies the turbulent fluctuations.
The vertical structure of the bulk gas dynamics is revealed through the mass
flux as a function of polar angle
∫
ṁ(θ)r = − ρvrr sin θdφ, (4.29)
where r is the radius of interest where the mass flux is calculated. Figure 4.4
shows the averaged mass flux at r = 100, 200, & 400 rg. At r = 100 rg, there is
a time-averaged inflow as material and thermal energy is advected into the inner
boundary, similar to an ADAF. The profiles at r = 200 & 400 rg show that this
changes beyond the truncation. Accretion still occurs in the main disk body where
the mass is concentrated, but directly above and below the cold disk, starting at one
disk scaleheight on either side, ṁ is negative, indicating a net outflow of material.
This outflow is not a persistent, laminar wind. Rather, it is a bulk motion in the
highly turbulent gas from the thermal instability (compared to the strong wind
in weak turbulence seen in Yuan et al. (2015), for instance). Figure 4.5 shows
the spacetime diagram of the azimuthally averaged radial velocity at r = 300 rg.
168
Figure 4.5: Spacetime diagram of vr at r = 300 rg. Red denotes positive
radial velocities and blue denotes negative radial velocities. The disk
midplane is found at θ = π/2.
Consistent with the mass flux profiles, the outflow appears as the typically positive
vr values directly above the main disk body, which is seen as the smaller amplitude
fluctuations within θ = π/2 ± 0.1. Moving away from the disk to higher altitudes,
the fluctuations become more stochastic as the outflow strength diminishes.
The radial structure of the inflow in the inner disk and the outflow beyond
the truncation zone is shown in the time averaged profiles of T , vr, and ρ in Figure
4.6. We have taken a slice along the heart of the outflow from θ = π/2 − 0.3 to
θ = π/2 − 0.325. The temperature profile reveals a near perfect powerlaw scaling
with index Γ = −0.992 ± 0.003, consistent with the T ∝ r−1 solution proposed
169
by Narayan & Yi (1995a). The profiles of vr and ρ display different behaviors on
either side of transition zone. In the inner disk the gas velocities are negative and
accelerating inwards as material advects, and the density profile from r = 20−100rg
is described by a Γ = 0.69±0.01 powerlaw. In the transition zone the average radial
velocity is vr = 0 and the density profile rolls over. Beyond the transition, the radial
velocity of the outflow is positive and the gas feels a positive acceleration away from
the black hole. The acceleration of the outflow is due to buoyant forces and not the
magnetic forces, which are quite small (a 10% contribution). The density profile
steepens and falls off with a powerlaw index of Γ = −2.13± 0.01.
The turbulent dynamics of the disk also display differences in the inner disk,
transition zone, and outer disk. A simple diagnostic of the turbulent density fluc-
tuations is the turbulent intensity. This is the time averaged, fractional standard
deviation of the density as a function of radius, δ(r), shown in Figure 4.7. For each
data dump, we calculate,
√
〈|ρmid(r)− 〈ρmid(r)〉|2〉
δ(r) = , (4.30)
〈ρmid〉
with 〈·〉 representing azimuthal averages and subscripts denoting that we only con-
sider the disk midplane. Considering the turbulent intensity offers several advan-
tages for comparing the global behavior of the turbulence in the disk. First, this
essentially normalizes away any long-term variability in the density. Second, turbu-
lence is inherently a multiplicative process (Kowal et al., 2007; Passot & Vázquez-
Semadeni, 1998) and, under constant conditions, the amplitude of fluctuations scales
with the density so we need to remove effects from radial density gradients in the
170
(a) 〈T (r)〉 of Outflow
(b) 〈vr(r)〉 of Outflow
(c) 〈ρ(r)〉 of Outflow
Figure 4.6: Radial profiles of the average gas temperature (a), radial
velocity (b), and density (c) along the outflow (θ = π/2 − 0.3 to θ =
π/2−0.325). The profiles have been averaged over azimuth, height, and
the duration of the simulation. The blue-dashed line marks v = 0, for
reference.
171
Figure 4.7: Radial profile of the average turbulent intensity, δ(r), mea-
sured in the density fluctuations in the φ-direction at the disk midplane.
disk.
Throughout most of the disk, δ = 0.13, meaning that density fluctuations
are on average 13% of the mean. As gas reaches the temperature transition in
the truncation zone, δ dramatically increases from 0.14 at r = 200 rg to 0.22 at
r = 130 rg. There is then a precipitous drop in the inner disk and δ falls to δ ≈ 0.06
as the turbulence is suppressed. Close to the inner boundary (r < 30 rg), the
turbulence increases again and approaches the intensity of the main disk body.
Searching for time variability is a goal of our analysis because it could offer
a way to observationally probe the accretion flow and test the model predictions.
One key source of variability in a truncated accretion disk on timescales longer than
the turbulent fluctuations is from appreciable changes in the gas fraction. If this
172
Figure 4.8: Gas fractions fc (blue line) and fh (red line) at r = 200 rg,
just beyond the truncation.
results from large changes in the thermal instability, it should appear on the local
thermal time and potentially be observable. Figure 4.8 shows the time evolution of
the gas fraction in the hot phase, fh, and the cold phase, fc, at r = 200 rg. Both
fh and fc are featureless and meander around their average values of 0.29 and 0.71,
respectively. There is no obvious indication of an instability occurring on a thermal
time in this simulation, where t 3 3th ≈ 2×10 GM/c at r = 200rg. The only prominent
behavior is a gradual heating event in the latter half of the simulation. At the peak
of the heating event fh and fc are equivalent, but then the disk begins to cool and fh
decreases. Small amplitude, high frequency variability from turbulent fluctuations
are superimposed on top of the long term trend.
4.3.3 Mass Accretion and Angular Momentum Transport
The mass accretion rate,
∫
Ṁ(r) = ρv r2r dΩ, (4.31)
173
Figure 4.9: Ṁh (top panel) and Ṁc (bottom panel) at the inner boundary
over the course of the simulation. The Ṁini at the inner boundary is
shown as the black dashed line in the top panel for reference.
is highly variable on short timescales. The variability in the simulation is from
rapid fluctuations due to the disk turbulence, and there is a marked absence of
any long-term, coherent variability that might be tied other accretion processes.
Figure 4.9 shows Ṁ at the inner boundary separated into the hot phase gas and
cold phase components. The hot gas is accreted at a rate of 〈Ṁh〉 = 2.5× 10−3 with
a characteristic variability of σṀ = 7.0× 10−4, or 28%. The cold gas accretion rate
is negligible at 〈Ṁ〉 = 3.4 × 10−9. There are brief excursions from this very low
value, but it rarely exceeds more than a 1× 10−7. For reference, the mass accretion
rate of the thin disk during initialization was 〈Ṁinit〉 = 2.8 × 10−3 is also shown.
The slight suppression in the truncated disk mass accretion is from the removal of a
small amount of material by the weak outflow, which prevents it from reaching the
inner boundary.
174
Figure 4.10: Radial profiles of the shell integrated angular momentum
advection (blue dotted line), Maxwell stress (solid red line), and the
Reynolds stress (dashed black line). The profiles are shown as the log-
arithm of the absolute value with the original sign of the stress, i.e.
sign(x)log(|x|). Additionally, they have been scaled by r2.
The transport of angular momentum enabling the mass accretion has three
components: the Maxwell stress, the Reynolds stress, and advection. Conservation
of angular momentum is given by,
∂
(ρrvφ) +∇ · (ρrvφv − rBφB) = 0. (4.32)
∂t
The dominant torque, rBrBφ, is exerted by the Maxwell stress. The ρrvφv term
can be further separated into an average velocity term representing the advection
of angular momentum, ρrvφ〈vr〉, and its fluctuating component, ρrvφδvr, which
accounts for torques from the Reynolds stress. The radial profiles of the three terms
are shown in Figure 4.10 after summing over shells and taking the time average.
Throughout the disk the Maxwell and Reynolds stress transport angular mo-
175
mentum outwards to larger radii. The Maxwell stress is ≈ 5× larger than the
Reynolds stress and provides the majority of the angular momentum transport.
This is consistent with the 4-5:1 ratio broadly seen in other numerical models of
MHD accretion disks (Blackman et al., 2008; Hawley et al., 1995; Pessah et al.,
2006). Interior to the transition zone all of the angular momentum transport terms
are enhanced by factors of 2 over what they are in the outer disk.
Spatially, there is a partitioning of the Reynolds and Maxwell stress enhance-
ments and the increased stresses occur in discrete zones. Figure 4.11 shows the two-
dimensional structure of the average Reynolds stress, Maxwell stress, and advection.
The Reynolds stress is more intense within the disk body where transitioning gas
causes the disk to taper. This region coincides with the larger turbulent intensity,
δ. Together, this indicates that the turbulent fluctuations not only have a larger
amplitude, but they are also correlated.
The Maxwell stress enhancement forms an envelope around the truncation
zone. Given that it occurs where the outflow originates and flows along the disk,
it seems the outflow amplifies the field. The likely mechanism behind this amplifi-
cation is the conversion of kinetic energy into magnetic energy by nonlocal energy
transfers through shells in Fourier space. For a wavenumber of the velocity field kv,
and wavenumber in the magnetic field, km in a turbulent flow, the magnetic field
will receive energy from the velocity field for all shells with kv < km (Mininni, 2011;
Moffatt, 1978). Comparing the small turbulent scales in this region to the outflow-
ing gas scale, the flow meets this criterion. As the gentle outflow moves through
the turbulent accretion flow, field lines are stretched and the field is amplified. This
176
Figure 4.11: Maps of ρrvφδvr (top panel), −rBrBφ (middle panel), and
ρrvφ〈vr〉 (bottom panel). Overlaid is the average geometric scale height
(black solid line).
177
phenomenon has been invoked in a number of contexts, including geophysical dy-
namo action of the terrestrial magnetic field (Peña et al., 2016), but is less well
studied in accretion flows. Lesur & Longaretti (2011) showed that these non-local
transfers in spectral space operate in MRI-driven turbulence, but there has yet be
an investigation into the global energy transfer properties of an accretion disk.
Several distinct regions form in the advective term. The greatest inwards
advection of angular momentum occurs on the surface of the disk, coincident with
the largest Maxwell stresses. Disk material typically accretes along the skin of the
disk and then in a channel like region in the disk midplane stemming from the
truncation. It is important to bear in mind that the appearance of this channel-like
region is simply the average advection and that entire region is turbulent, despite
it being suppressed relative to the rest of the disk. The inwards drift of material is
still very slow and the gas dynamics are dominated by the turbulent velocity. Above
this inflowing region, the outflowing gas is launched in the transition zone as hotter
gas rises to larger radii. Interior to this region (r < 100 rg), all of the gas is in the
hot phase and the Maxwell and Reynolds stresses are large which drives material
inwards to the inner boundary.
4.3.4 Disk Energetics
In this MHD accretion flow, the conservation of energy equation is,
∂E
+ C +AK +AT +AM + G −W = 0, (4.33)
∂t
178
where E is the total disk energy, C is the “radiative” cooling done by the cooling
function, AK is the advection of kinetic energy, AT is the advection of thermal
energy, AM is the advection of magnetic energy, G is the advection of gravitational
potential energy, and W is the work done by magnetic torques. Here, we will
consider a spherical shell[in∫tegra(ted form of the terms, such)t]hat they are
∂E ∂ 1 2 P B2= dΩ ρv + + ρΦ +∫ , (4.34)∂t ∂t 2 γ − 1 2
[ ∫ C(= )dΩΛ], (4.35)
A 1 ∂= [ r2
1
K ∫ dΩ( ρv2)vr], (4.36)r2 ∂r 2
A 1 ∂ 2 PT = r[ ∫dΩr2 ∂r γ(− 1)vr], (4.37)
1 ∂ 2A = [r2
B
M ∫ dΩ( )vr], (4.38)r2 ∂r 2
G 1 ∂= [r2 ∫ dΩ ρΦ vr ], (4.39)r2 ∂r
W 1 ∂= r2 dΩvφBφBr . (4.40)
r2 ∂r
The time averaged radial profiles of C, AK, AT , AM, G, andW are shown in Figure
4.12.
In the inner disk the kinetic, thermal, and magnetic energy fluxes are nega-
tive, indicating there is an inwards advection with the accretion flow. Most of the
advection is in AK, followed by thermal energy (AT ≈ 0.5AK) and finally the mag-
netic energy (AM ≈ 0.03AK). Beyond the truncation in the thin disk, AK remains
constant, but AT actually switches sign as the hot gas in the outflow dominates
this energy term. AT is slightly suppressed since there is competition between the
inwards flow of cooler gas in the disk body and the hot gas with positive radial
velocity, but ultimately the hot gas is main contributor. Additionally, the AM term
179
10 2
10 3
10 4
10 5
-10 5
-10 4
-10 3
-10 2
101 102 103
Radius (rg)
Figure 4.12: Radial profiles of terms in the energy equation. The profiles
are shown as the logarithm of the absolute value with the original sign
of the energy, i.e. sign(x)log(|x|). Additionally, they have been scaled
by r2. The work done by magnetic torques (W) is shown by the black
dashed line, the advection of kinetic energy (AK) is shown by the blue
dotted line, the advection of thermal energy (AT ) is shown by the black
solid line, the advection of magnetic energy AM is shown by the red
dotted line, the cooling term (C) is shown as the red solid line, and the
change in gravitational potential energy (G) is shown by the blue solid
line.
180
sign(Value) * r2 * Value r2 * Value
Figure 4.13: Disk integrated synthetic light curve from the hot gas cool-
ing (top panel) and cool gas cooling (bottom panel).
switches sign because buoyancy lifts organized patches of overmagnetized gas as the
disk dynamo cyclically builds magnetic field in the outer disk. The role of magnetic
buoyancy is minimal in the inner disk because the dynamo is suppressed, an aspect
of the simulation discussed on Section 4.3.5. The contribution of C to the energy
budget increases further out in the disk, reflecting the change in gas fraction and
cooling efficiency. In the inner disk, C ≈ 0.3 G. In the outer region, though, the
disk approaches the standard thin disk ratio of C ≈ 1.5 G.
Using the ad hoc cooling as a proxy for emitted radiation, we create synthetic
light curves, shown in Figure 4.13. The cooling is separated into that from the hot
and cold gas phases, which can loosely be considered “hard”-band and “soft”-band
emission, following observational convention. The cool gas light curve has been
renormalized to correct for the draining of cool gas from the simulation. Over the
course of the simulation, the cool gas depletes by ≈ 20%. We scale the density by
181
the ratio of cool gas mass at the beginning of our analysis to the current cool gas
mass at each point. This scaling was not applied to the hot gas because we have
reached an approximate inflow equilibrium so the rate at which it is accreted is the
same rate at which cold gas transitions. This keeps the total hot gas mass roughly
constant throughout the analysis portion of the simulation.
The variability in these light curves predominately occurs on long timescales,
i.e. longer than a dynamical or thermal time. The main feature is a brightening
event that peaks in both light curves at t = 7×105GM/c3. This event increases the
cool gas light curve by only 10%, but the hot gas light curve by 50%. The variability
of the gas fraction near the transition zone seen in Figure 4.8 does not translate to
the light curve. The amplitude of stochastic fluctuations is much greater in the hot
gas light curve. Several flaring events occur when the hot gas light curve rapidly
increases by 30% and then decays to the lower trend level. These events occur
randomly and do not have any distinguishing periodicity or even a characteristic
shape.
4.3.5 Disk Dynamo
The importance of the magnetic dynamo action on the evolution of a black
hole accretion disk cannot be overstated. On small scales an effective MHD dynamo
is essential for sustaining the MRI driven turbulence because it provides a continual
source of the magnetic field regeneration against dissipation and reconnection (first
shown by Hawley et al., 1995). On larger scales, spacetime diagrams of Bφ in
182
(a) 〈Bφ〉 at r = 100 rg
(b) 〈Bφ〉 at r = 200 rg
Figure 4.14: Spacetime diagrams of the azimuthally averaged Bφ at r =
100rg (top panel) and r = 200rg (bottom panel). The time axis in the top
panel has been rescaled to show r1.5 1.5100/r200 of the time to better match the
differences in the evolutionary timescales of the dynamo process and aid
in the direct comparison of Bφ at each radius. These two disk locations
were chosen to represent the more general behavior on either side of the
truncation zone.
numerical simulations reveal a quasiperiodic reversal of the magnetic field orientation
and lifting of the field which forms the so-called “butterfly” pattern. Global PSDs
show that, in fact, the organization of the field is a large-scale phenomenon and
bands of power in the oscillation span several radii (O’Neill et al., 2011). Nominally,
the frequency of the oscillation, Ωd, is taken to be one tenth of ΩK .
Figure 4.14 shows the spacetime diagram of the azimuthally average magnetic
field on either side of the truncation zone at r = 100rg and r = 200rg. The time axis
of the r = 100 rg butterfly diagram has been rescaled to show a similar evolutionary
183
time (i.e. only a portion corresponding to r1.5100/r
1.5
200 of the analysis time). Beyond
the transition zone, the Bφ pattern is organized with a coherent, quasi periodic
oscillation at a frequency of f = 3.7 × 10−6 c3/GM (15× the orbital frequency).
At the midplane the field strengthens, diminishes, alternates sign, strengthens and
the cycle repeats. The region is dominated by field of similar polarity spanning the
midplane. A cycle is present inside the truncation, but the pattern is disturbed and
stochastic fluctuations account for much of the variability. Additionally, the mag-
netic field is less ordered and the field oscillations are intermittent, thus preventing
the development of strong field throughout the entire midplane. Rather than sus-
tained values that are typically near the maximum amplitude, weak Bφ is common.
Nevertheless, this does not lower the average Maxwell stress which is actually higher
within the truncation zone, as we saw in Figure 4.10. At higher altitudes above and
below the disk the cycle recovers and is more clearly defined.
A consequence of disrupting the dynamo is that the net field strength is lower
than it would be otherwise. Coupled with the increased gas pressure from the
transitioning gas, the ratio of the gas pressure to the magnetic pressure,
Pg
β = , (4.41)
Pm
is high in the inner disk and the transition zone. Figure 4.15 shows β averaged over
time and azimuth. The increase in β is most prominent in the inner region of the
disk body where thermal pressure inflates the disk. Extending away from the inner
regions are wings of high-β gas that trace the outflow.
The increased gas pressure also acts to decrease the effective α-parameter in
184
the inner disk. Figure 4.15 shows the time averaged radial profile of the effective α.
For each data dump we calculated the volume averaged stress and volume averaged
pressure within the geometric scale height (hg/r) and took the ratio, according to
Eqn 2.1. Beyond the transition in the region that has reached a quasi-steady state,
the effective α parameter is α = 0.055. This is 15% below the average effective α
in our calibration due to the additional gas pressure support from small patches
of transitioning gas triggered by turbulent fluctuations. Moving inwards from the
truncation zone, the effective α-parameter decreases from 200 rg to 110 rg which
corresponds spatially with the tapering of the cold disk body seen in Figure 4.2.
In the very inner regions where the gas is entirely in the hot phase, the effective
α-parameter settles to a value of α ≈ 0.02. There is a rapid recovery in the effective
α at the inner edge of the simulation which could be due to boundary effects and
thus artificial.
The Fourier transform of the Maxwell stress,
∫
M̃rφ(f) = M (t)e
−2πift
rφ dt. (4.42)
reveals periodicity in its variability. Shown in Figure 4.16 is the power spectrum
of the integrated Maxwell stress in the truncation zone between r = 130 rg and
r = 200 rg. The total stress has been summed over θ =
π ± 0.2. Overall, the power
2
spectrum scales as f−1 and fM˜rφ(f) is flat. On top of the powerlaw spectrum,
though, there is a band of power where the variability is strongly periodic. This
feature is centered at f = 4.0 × 10−5 c3/GM , coherently spans five spectral bins,
and is ≈ 4× greater than the noise spectrum. Given its strength and relatively low
185
frequency, this develops from an interaction with the disk dynamo. Interestingly, it
is five times faster than the local dynamo frequency at f = 7.2 × 10−6 c3/GM for
r = 170 rg.
4.4 Discussion
The MHD model presented here and the viscous HD model in are targeted to
study the dynamics of truncated accretion disks. Compared to the rich literature
on hot, geometrically thick accretion flows and cool, geometrically thin accretion
flows, the truncated disk model is poorly explored, despite the frequency with which
this flow configuration is invoked to explain the spectral properties of hard state
BHBs and LLAGNs. This work has characterized the general flow properties, but
further work is required to fully capture the subtleties of flow at the transition
zone. Broadly, the HD and MHD models were similar even though they were set
in very different fluid regimes. Both models preferred a cool solution until near
the imposed rT , launched some sort of thermally driven outflow, and formed a thin
layer of enhanced stress as the hot gas moved along the surface of the main disk
body. In the viscous HD model the enhanced stress was due to the θ − φ viscosity
component and in the MHD model this arose through the stretching of field lines
which intensified the Maxwell stress. These are two fundamentally distinct origins,
but they ultimately have the same effect. Furthermore, the inner disk regions in
both models are evacuated, which is seen as deficits in the surface density profiles
interior to rT compared to what it would be as a standard thin, cold disk. This is
186
(a) r − θ map of 〈β〉
(b) Radial 〈α〉 profile
Figure 4.15: r − θ map of 〈β〉 (a) and the radial 〈α〉 profile (b). The
r − θ map of 〈β〉 has been averaged over azimuth and time. The radial
〈α〉 profile is the time averaged effective α value calculated within the
local geometric scale height (hg).
187
Figure 4.16: Power spectrum of the Maxwell stress, fM̃rφ, in the trun-
cation zone. The local dynamical frequency at r = 170 rg is shown as
the dashed blue line.
generally consistent with observational results where the inner regions of hard state
BHBs and LLAGNs are measured to be less dense.
In the HD and MHD models the flow takes on a configuration where the cooler
thin disk is embedded within the hot gas. Typically, phenomenological models of
truncated disks envision the hot gas occupying the inner region of the disk in a spher-
ical bulge whose extent is roughly the truncation radius (e.g. Shapiro et al., 1976).
Our results, along with other numerical models of truncated disks (e.g. Mishra
et al., 2016; Takahashi et al., 2016), suggest that the hot gas may fill the entire
region surrounding the black hole and take on a structure akin to that presented in
Cui et al. (1997); Esin et al. (1997); Liang & Nolan (1984). Additional scattering
resulting from “sandwiching” a disk between between the two hot coronal regions
can produce telling observational signatures in the X-ray spectrum (Takahashi et al.,
2016) and in polarization (Schnittman & Krolik, 2010).
While the presence of an outflow is shared between the HD and MHD truncated
disk models, the manifestations are distinct. In the HD model convection appears in
188
the flow atmosphere, reminiscent of a CDAF-type flow, except that the gas in these
convective eddies eventually fell back and was accreted. In the MHD model convec-
tion is stabilized and, even though the instantaneous gas motions are dominated by
MHD turbulence, there is a thermally driven bulk outflow. Scaling the system to
physical values, there appears to be only limited promise in observing the outflow.
Taking a characteristic inner disk temperature to be virial at T = 1012 K (Shapiro
et al., 1976), the outflow would have temperatures between T ≈ 5 × 109 − 1012 K.
Scaling the dimensionless velocity by the speed of light, c, we find that the velocity
range spans vr ≈ 0−300 km/s. The velocity is in the range of empirically measured
outflows, but its observational prospects may be hindered by the high temperature
and the large turbulent broadening since the turbulent velocity is three times greater
than the bulk flow (∆v ≈ 1, 000 km/s).
Considering the truncated disk in MHD has elucidated several unique disk
features that have not previously been considered. First is the interplay between
stochastic temperature fluctuations and the thermal instability. MHD turbulence
driven by the MRI has been understood as a key way to transport angular mo-
mentum and heat the disk through turbulent dissipation, and we find it is also
important in triggering the thermal instability. At a given radius, the gas has a
range of temperatures with some fraction of the distribution lying beyond the tran-
sition condition. This can impact the disk geometry because local pockets of gas can
transition at radii which would otherwise be too cool to satisfy the thermodynamic
criterion. As gas sinks to smaller radii, the fraction of gas on the inefficient cooling
branch will increase. The natural result of this radial variation is a disk tapering
189
in a transition zone like that produced in our simulation, rather than a transition
edge. Measurements of truncation radii from spectral fits may benefit from fitting
a smoother transition rather than a sharp inner disk cutoff, as is often used.
The details of the transition rests on the precise transition mechanism. Here,
we treated the cooling with a simple threshold, but a more physical prescription is
needed before a complete understanding of the dynamics in a truncated disk can
be achieved. This demands that the theory underlying the thermal instability be
developed. So far, several global accretion models have been proposed as candidates
to fill this role. Inner disk evaporation of the cool, Keplerian disk by heating from
a hot corona (Liu et al., 1999; Meyer & Meyer-Hofmeister, 1994; Narayan et al.,
1996; RóżaŃska & Czerny, 2000) could lead to disk truncation. The interior of the
disk could also heat the gas beyond the virial temperature because of the diffusion
of energy from through the disk (Honma, 1996; Manmoto & Kato, 2000; Manmoto
et al., 2000). Alternatively, local thermal instability can develop if the α-parameter
is proportional to the magnetic Prandtl number (Potter & Balbus, 2014, 2017).
Work has gone into understanding these mechanisms analytically and with simple
numerical models, but their viability in a global MHD accretion disk is untested.
Numerical models must rigorously test these mechanisms to determine which are
the most feasible and to tie them to observational signatures that could originate
at the transition. The latter goal could provide a fruitful avenue for informing the
large, time-domain surveys scheduled for the near future.
The second unique insight that comes from studying the MHD truncated disk
is the interaction with the magnetic dynamo; specifically, that the dynamo was
190
disrupted in the truncated disk region and that the outflow strengthened the field
surrounding the cooler disk body. Some form of low-frequency magnetic dynamo is
universally observed in three-dimensional MHD accretion disk models. Usually, the
dynamo appears as quasi-periodic oscillations of the large-scale magnetic field in a
standard thin accretion disk. However, there is a developing narrative that certain
conditions can hinder the organization of the large-scale field by the dynamo and
impede the regular strengthening of the dynamo cycle. A few instances from nu-
merical models include quenching in a magnetically dominated disk (Bai & Stone,
2013; Salvesen et al., 2016) and the mixing of field from hydrodynamic convection
(Coleman et al., 2017). Global simulations of super-Eddington flows that include
radiative effects find that the dynamo period remains regular, but is slower when
than what it should normal be (Jiang et al., 2014). Nonphysical causes like resolu-
tion effects from sensitivities to the simulation domain aspect ratio have also been
shown to quench the dynamo (Walker & Boldyrev, 2017). The sporadic dynamo
oscillations in the inner regions of the truncated disk provide another example of
how the dynamo behavior can be modified, which we expand on in Chapter 5.
Interior to the transition the magnetic field might have been expected to scale
with the increase in gas pressure. Instead, we find that β increases and there is a
steep drop in the effective α-parameter to α = 0.02, or ≈ 35% of its pretransition
value. The inability of the magnetic field to adjust and increase to maintain the
higher α-parameter is most likely due to the outflow launched from the gas transi-
tioning between the two cooling branches in the transition zone. This modifies the
flow dynamics and disrupts the organization of the large-scale field.
191
Oddly enough, the same outflow that suppresses the dynamo in the inner
disk produces a thin layer of enhanced magnetic field that envelopes the transition
zone. This occurs as the outflowing gas stretches the magnetic field when it flows
over the tapered disk body, which converts kinetic energy into magnetic energy. In
addition to demonstrating a secondary pathway that can amplify field, this feature
is interesting because of its observational prospects. In particular, it could appear
as a quasiperiodic oscillation (QPO). QPO phenomenology is rich and complex,
but generally low-frequency QPOs appear in the PSDs of hard-state BHB light
curves as strong and narrow peaks whose central frequency can evolve with the
black hole outburst. The magnetic dynamo has been suggested as a driver of this
peculiar behavior (Begelman et al., 2015; O’Neill et al., 2011) and our results offer an
extension to this model that may bolster its theoretical underpinnings. Specifically,
the additional field strengthening occurs at the innermost part of the truncated disk,
which is already the most luminous region of the disk. Given that this emission is
quite localized, it will preferentially weight a small range of frequencies and could
explain the high quality factors that are typically observed. Frequency evolution in
the signal is naturally accounted for in this model because the QPO frequency is
tied to the orbital and dynamo frequencies, and will therefore change as the secular
changes in the mass accretion rate change the truncation radius of the disk.
Headway can be made in more accurately modeling the dynamics of truncated
accretion disks by including a more realistic radiation prescription, which we have
purposefully treated with a simple optically thin approximation. The accretion flow
and transition zone behavior could be altered from the model analyzed here because
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radiation pressure could modify the flow dynamics and the thermal transition could
evolve differently. Our aim was to conduct a generalized investigation that is appli-
cable to black holes across the mass scale. Uniquely targeting truncated accretion
disks in either outbursting stellar mass black holes in the hard-state or LLAGNs
requires accounting for the detailed thermodynamics, which are quite different be-
tween the two regimes. The ability of the radiation field to impart momentum to
the gas and help launch an outflow (Proga, 2000) is likely to be a significant con-
tributor to the thermal acceleration we find here. In AGNs, line driving could also
lead to truncation (Laor & Davis, 2014) and iron opacity may be important to the
disk structure (Jiang et al., 2016).
4.5 Conclusions
We investigate the dynamics of a truncated accretion disk around a black hole
using a well-resolved, semi-global MHD model. An ad hoc bistable cooling function
is implemented to mimic the radiative transitions thought to be responsible for the
distinct spectral states in stellar mass and supermassive black hole systems. This
cooling function produces a truncated accretion disk where the outer disk cools
efficiently and resembles the standard Shakura & Sunyaev thin disk. Within the
truncation the gas is hot and diffuse, and the disk thickens. Between these two
regions lies a “transition zone” from r = 130 − 200 rg where the cool, Keplerian
gas disk tapers into the hot flow. A thermally driven outflow is launched from this
region, slightly depleting the accretion flow. As the outflow buoyantly rises along
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the disk body, we find it amplifies the magnetic field on the edge of the main disk
body. The turbulent intensity (δ) is boosted in the transition zone as a result of
the transitioning gas and the increase in the Maxwell stress from the outflow; in
stark contrast to the inner disk where δ is at a minimum. In addition to the gas
temperature and density, we find that the inner accretion flow of the truncated disk
is differentiated from the outer cool disk in other ways as well. In particular, the
magnetic dynamo is suppressed and the effective α-parameter is greatly diminished.
Despite these changes between the inner and outer disk, hot gas is able to efficiently
accrete throughout the duration of the simulation. We do not find that the accretion
is inhibited by the presence of the truncation, with the exception of a slight siphoning
of gas by the outflow. The accretion rate through the inner simulation domain is
essentially set by the feeding from the outer disk with the truncated region filling
the role of an intermediary between the outer Shakura & Sunyaev-like thin disk and
the black hole.
Two aspects of the simulation are worth highlighting because they are perti-
nent to the interpretation of spectra from systems with a truncated accretion disk:
the effectiveness of the hot phase gas to fill the volume of the coronal regions above
the cold, thin disk and the influence of turbulent fluctuations in introducing a radial
taper. In this simulation, the upper atmosphere of the disk is full of hot, radiatively
inefficient gas and the density is greater because of the mass loaded outflow. A
large effort has gone into constraining the coronal geometry empirically and models
typically assume the hot gas lies within the truncation. Our results, in conjunction
with prior studies, suggest it may not be the case that hot gas is only confined to
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a spherical region with a size set by the truncation radius. A configuration where
the hot gas freely fills the volume around the black hole above and below the cold,
thin disk may be more realistic. Additionally, consensus has been reached that MRI
driven turbulence is the chief mediator of the angular momentum transport and disk
heating. Incorporating the modification of the truncation from a sharp interface to
a more gradual radial dependence with a vertical temperature stratification because
of the turbulent fluctuations may lead to more accurate measurements of truncation
radii in these systems. Since the truncation zone accounts for an appreciable portion
of the disk, it is unclear where iron reflection measurements would place the trunca-
tion edge in such a system. Adding sophistication to a well-resolved, long-duration,
global MHD truncated accretion disk model by including more accurate radiative
physics will help alleviate outstanding uncertainty and advance the understanding
of the dynamics of truncated accretion disks.
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Chapter 5: Towards A More Complete Understanding Of The Large-
scale Magnetic Dynamo
5.1 Introduction
The vigorous MRI-driven turbulence and shear expected in an accretion disk
provide conditions that make it prime territory for the development of a magnetic dy-
namo (Tout & Pringle, 1992). Indeed, the self-organization of the magnetic field on
large-scales has been universally observed in simulations of moderately mangetized,
thin accretion disks. Despite the ubiquity with which dynamo behavior develops,
the phenomenon remains an enigma and much of the fundamental theory behind its
global growth remains undeveloped.
The large-scale dynamo often presents itself in simulations as a quasiperiodic
reversal of the azimuthal field polarity. Bundles of field rise as the overmagnetized
regions feel a buoyant force, and spacetime diagrams show a characteristic “butterfly
pattern,” akin to that observed in the migration of sunspots on the Sun. This
behavior is often interpreted as the evolution of the mean field within the framework
of the αΩ dynamo model, which has two ingredients. First is the α effect which
is sourced in the induction of azimuthal field from the movement of radial and the
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vertical magnetic field in a helical fluid flow (Krause & Raedler, 1980). This acts to
generate large-scale field from small-scale turbulent motions. Second is the Ω effect
which arises through differential rotation and grows azimuthal magnetic field back
from radial and vertical fields. This seeds the MRI from the large-scale field and
further sustains turbulence allowing the cycle to continue. For the sake of clarity,
we denote the α effect parameter as αd henceforth to prevent confusion with the
effective Shakura & Sunyaev α-parameter.
Shearing box simulations have been instrumental in allowing for the detailed
exploration of the relation between the accretion flow and large-scale magnetic field,
thus enabling the assembly of many of the pieces in the dynamo puzzle. Early
simulations demonstrated the sensitivity of the magnetic stresses to the net field
spanning the domain (Hawley et al., 1995; Pessah et al., 2007; Sano et al., 2004) and
that large-scale magnetic cycles with periods of roughly ten times the orbital period
readily develop (Brandenburg et al., 1995; Davis et al., 2010; Guan & Gammie,
2011; Lesur & Ogilvie, 2008; Oishi & Mac Low, 2011). Considering only a local
patch of an accretion disk allows for the rigorous investigation of the structure of
the turbulence, including its spectral properties (Gogichaishvili et al., 2017; Murphy
& Pessah, 2015) and saturation (Bodo et al., 2008; Latter et al., 2009; Obergaulinger
et al., 2009; Pessah, 2010; Pessah & Goodman, 2009), which is important because
the large-scale field grows from turbulent fluctuations.
Ultimately, though, the disk dynamo is a global phenomenon which requires
models with large domains that properly account for the vertical and radial gradients
in the accretion flow, as well as the coupling of radii with different evolutionary
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times. Like their local counterparts, global models find dynamo periods of 10−20×
the local orbital period (e.g. Beckwith et al., 2011; Flock et al., 2012; Hawley et al.,
2013; Hogg & Reynolds, 2016; O’Neill et al., 2011; Parkin & Bicknell, 2013). These
simulations are thin disks with typical thermal scaleheight ratios of h/r = 0.07−0.1.
They find that the dynamo coherently spans roughly ∆r = 10rg in radius, modulates
the accretion disk stresses, and collects into sheets of azimuthal magnetic field of
the same orientation in the coronal region.
In this chapter, we aim to understand how the timing properties of the dynamo
and large-scale magnetic field evolution depend on accretion disk geometry, i.e. the
disk scaleheight ratio, and if there is a threshold of either large or small thickness
beyond which the large-scale dynamo cannot be excited. Exploring this is crucial
because dynamos have been directly invoked to explain time variability signatures
from accreting black holes (King et al., 2004; Mayer & Pringle, 2006), as well as
in several peripheral contexts. Examples include dynamo cycles as a source of the
low-frequency quasi-periodic oscillation (QPO O’Neill et al., 2011), the driver of
“propagating fluctuations” in mass accretion rate (Chapter 2), and as the trigger
behind the secular evolution in the spectral state transitions (Begelman et al., 2015).
Furthermore, while numerical simulations have established the robustness of
the dynamo in a standard thin accretion disk, several cases have been found where
the well-ordered oscillations are altered or vanish in atypical disk conditions. For
instance, using global simulations of a super-Eddington accretion flow, Jiang et al.
(2014) found that the dynamo period is regular, but slower. The dynamo can be
quenched in a magnetically dominated disk (Bai & Stone, 2013; Salvesen et al.,
198
2016) or if hydrodynamic convection acts to mix the field (Coleman et al., 2017).
Additionally, the transition in the flow geometry of a truncated disk and its associ-
ated flow dynamics has been shown to impede the dynamo and lead to a sporadic,
intermittent oscillation of the large-scale magnetic field in the inner hot disk (Chap-
ter 2). Of course, a lingering concern is always that simulations are affected by
nonphysical sensitivities like resolution and domain aspect ratios, which have also
been shown to halt the dynamo (Walker & Boldyrev, 2017).
To delve into the scaleheight dependence of the magnetic dynamo, we con-
structed a suite of global, MHD disk models with varying scaleheight ratios. In
Section 5.2 we describe the numerical simulation of these models including the code
details, initial conditions, and resolution properties. In Section 5.3 we present an
analysis of the large-scale dynamo properties, measure αd values for each simulation,
briefly look at the evolution of the large-scale helicity in one of our simulations, and
discuss potential observational characteristics of each simulation. We discuss our
results and their broader implications in Section 5.5 and provide closing remarks in
Section 5.6.
5.2 Numerical Model
In this chapter we consider four well-resolved MHD simulations of model ac-
cretion disks in a pseudo-Newtonian potential, Equation 5.5, with scaleheight ratios:
h/r = {0.05, 0.1, 0.2, 0.4}. The goal of this chapter is to use these simulations to
understand how the accretion disk geometry affects the properties of the large scale
199
magnetic field. Hence, we strive for consistency in our models and to evolve the
models for long enough to fully sample the global dynamo for several cycles. To
this end, these simulations were initialized with the same initial conditions and each
were allowed for evolve for t ≈ 4× 104 GM/c3, or roughly 650 ISCO orbits.
5.2.1 Simulation Code
This work uses the second-order accurate PLUTO v4.2 code (Mignone et al.,
2007) to solve the equations of ideal MHD,
∂ρ
+∇ · (ρv) = 0, (5.1)
∂t
∂
(ρv) +∇ · (ρvv −BB + PI) = −ρ∇Φ, (5.2)
∂t
∂
(E + ρΦ) +∇ · [(E + P + ρΦ)v −B(v ·B)] = −Λ, (5.3)
∂t
∂B
= ∇×(v ×B),
∂t
where ρ is the gas density, v is the fluid velocity, P is the gas pressure, B is the
magnetic field, I is the unit rank-two tensor, E is the total energy density of the
fluid,
1 | |2 B
2
E = u+ ρ v + , (5.4)
2 2
and Λ accounts for radiative losses through cooling. All fluid variables (e.g. ρ, P , T )
are evolved and reported in a scale-free, normalized form. The hlld Riemann solver
was used to solve the MHD equations in the dimensionally unsplit mode. Linear
reconstruction is used in space and the second-order Runge Kutta algorithm is used
to integrate forward in time. To enforce the ∇ · B = 0 condition, the method of
constrained transport is used.
200
5.2.2 Simulation Setups
To remain as consistent as possible between models, the simulation grids are
designed under the following strategy. Each model is set in spherical coordinates
with θ ∈ [π/2 − 5h/r, π/2 + 5h/r]. As we vary the scaleheight ratio, we keep the
number of grid cells in this direction (Nθ = 248) the same. Uniform grid spacing
is used within ±3 h/r around the disk midplane with 28 zones per scaleheight. A
stretched grid is used beyond 3h/r with 40 zones in each of the coronal regions where
there is less small scale structure. The only exception being the thickest h/r = 0.4
disk, which we only model with θ ∈ [π/2− 2.5h/r, π/2 + 2.5h/r] and halve the grid
cells, accordingly. The radial range stays the same between models (r ∈ [5rg, 145rg])
and is divided into an inner well resolved region used for the analysis (r ∈ [5rg, 45rg])
and an outer less resolved region (r ∈ (45 rg, 145 rg]) that acts as a gas reservoir
to mitigate artificial effects from draining of the disk material that could introduce
secular changes in the accretion flows. Logarithmic spacing is used to keep ∆r/r
constant along the grid. The azimuthal (φ ∈ [0, π/3]) domain is also fixed between
models and has uniform spacing in φ. When varying the scaleheight ratio by factors
of 2 between models, we change the number of radial and azimuthal grid zones to
preserve the cell aspect ratio (∆r : ∆r sin θ∆θ : r∆φ ≈ 1 : 1 : 2). This helps
remove any resolution dependencies that could influence the results and adjusts
the integration timestep, which is set by the Courant condition. The validity of
restricting the φ-domain to φ ∈ [0, π/3] between our models rather than adjusting
it in a similar manner to that used in the θ-domain is addressed in Section 5.4. The
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Table 5.1: Simulation Parameters
Simulation Total Duration ISCO Orbits Nr Nθ Nφ H/∆θ
hr 005 3.58× 104 GM/c3 581.5 1232 248 256 28
hr 01 3.88× 104 GM/c3 630.5 616 248 128 28
hr 02 4.19× 104 GM/c3 680 308 248 64 28
hr 04 4.15× 104 GM/c3 675.5 154 248 32 28
full details of the simulation grids are given in Table 5.1. Outflow is allowed through
the r and θ boundaries while the φ boundaries are periodic. Density and pressure
floors are imposed to prevent artificially low or negative values.
A psuedo-Newtonian potential of the form,
− GM ≡ GMΦ = , rg (5.5)
R− 2r c2g
is used to approximate the dynamics of a general relativistic flow around a non-
rotating black hole. This captures features like the shear profile and the presence
of an innermost stable circular orbit (ISCO) at r = 6 rg without the computational
expense of fully including general relativity. A γ = 5/3 adiabatic equation of state
is used for the gas; the internal energy density of the gas is hence given by u =
P/(γ − 1).
As the turbulence decays, it deposits energy in the gas in the form of heat
so the disk would tend to become thicker. To enforce the target disk aspect ratio
for each of the simulations, a Noble et al. (2009) style cooling function is used to
emulate radiative losses. The local target gas pressure is set to,
ρv2K(h/r)
2
Ptarg = , (5.6)
γ
202
0.5
0.4
0.3
0.2
0.1
0.0
0 1 2 3
Time (GM/c3) ×104
Figure 5.1: Time variability of disk scaleheight ratios for hr 005 (dotted
line), hr 01 (dot-dash line), hr 02 (dashed line), and hr 04 (solid line).
where vK is the local Keplerian velocity given by
√
GM r
vK = . (5.7)
r − 2rg
The excess heat is cooled according to,
f(P − Ptarg)
Λ = , (5.8)
τcool
where f is a switch function,f = 0.5[(P − Ptarg)/|P − Ptarg| + 1], and τcool is the
cooling time which we set to the local orbital period.
The maintenance and stability of the target disk scaleheights is of paramount
importance in our simulations so that we can clearly isolate any disk height depen-
dences. Figure 5.1 shows the time variability in of average scaleheight ratio for the
simulations. For each radial element in each data dump we calculate the geometric
203
h/r
scaleheight ratio 〈√∫ 〉
h(r) (θ(r)∫− θ̄(r))2ρdΩ= , (5.9)
r ρdΩ
where dΩ = sin θdθdφ is the solid angle e∫lement in spherical coordinates and,
θ∫(r)ρdΩθ̄(r) = (5.10)
ρdΩ
is the average polar angle of the gas. The total disk-averaged scaleheight is calcu-
lated for each instant by weighting every radial bin by its width to account for the
nonuniform spacing of the grid in the r-coordinate. This shows the cooling function
keeps our target scaleheight throughout the duration of each simulation. Turbulence
introduces small fluctuations in the disk scaleheight ratio, but no worrisome trends
are present in the hr 01, hr 02, and hr 04 simulations. In the time trace of the
hr 005 simulation there is a residual transient from initialization artificially inflates
the disk, but it decays quickly. It has no significant influence on our study, but
does appear in synthetic light curves presented in Section 5.3.5 where we discuss it
further.
5.2.3 Initial Conditions
The simulations are initialized to a stead(y-state α-d)isk like solution:
2
ρ(R, θ) = ρ R−3/2 − z0 exp (5.11)
2c2 3sR
where ρ0 is a normalization, R = r sin θ is the cylindrical radius, and z = r cos θ
is the vertical disk height. The gas pressure is set from the density, P = c2sρ. The
velocity field is set such that the azimuthal velocity has the local Keplerian value
(vK) and vr = vθ = 0.
204
We set a weak magnetic field (〈β〉 = 〈Pgas/Pmag〉 = 200) from a vector poten-
tial to guarantee the divergence free condition is satisfied to machine level precision.
The initial field configuration is set to be logarithmically spaced loops with a vertical
taper function in the upper atmosphere with the form,
Ar = 0, (5.12)
Aθ = 0, ( ) (5.13)
1 4
Aφ = A p e
−(z/h)
0 2 R sin
π log(R/2) . (5.14)
h
In the early evolution of each model, these field loops feel the radial shear of the disk
and the gas is destabilized by the MRI, which goes nonlinear and drives turbulence.
5.2.4 Convergence
To assess the overall convergence and consistency between the models, we ap-
ply several resolvability and convergence metrics. The results of these diagnostics
are presented in Table 5.2. All averages are taken over the portion of the simulations
used the analysis and taken within the volume of the simulation domain contained
within two scaleheights above and below the disk where the dynamo behavior orig-
inates.
The resolution of the grid in simulation zones per disk scaleheight provides
the simplest measure of the resolvability. Several studies (e.g. Hawley et al., 2013;
Sorathia et al., 2012) have shown around 30 vertical zones per disk scaleheight offers
a crude threshold near where the turbulence is resolved well enough to capture the
small-scale evolution, and hence motivated our grid design. We use the correspond-
205
ing “quality factors” to probe how well the simulation grid samples characteristic
MRI wavelength, λMRI = 2πvA/Ω, where vA is the Alfvén speed. For each simula-
tion, we calculate the average quality factor in the θ and φ directions,
λMRI,θ
Qθ = (5.15)
R∆θ
and
λMRI,φ
Qφ = . (5.16)
R∆φ
Values of Qθ > 6− 8 have been shown to properly capture the linear growth of the
MRI (Flock et al., 2010; Hawley & Stone, 1995; Sano et al., 2004) and a stricter
threshold of Qθ > 10 and Qφ > 20 (Hawley et al., 2011, 2013; Sorathia et al., 2012)
has been established as adequate to capture the nonlinear growth of the instability.
Finally, we measure the saturation of the anisotropic MRI driven turbulence through
the average in-plane magnetic tilt angle,
(〈 〉)
− BrΘB = arctan . (5.17)
Bφ
This quantifies the characteristic orientation of the magnetic field, a value that
theoretical estimates predict to be near ΘB ≈ 15◦ (Guan et al., 2009; Pessah, 2010).
Measurements in prior accretion disk simulations (e.g. Hawley et al., 2011, 2013;
Hogg & Reynolds, 2016; Sorathia et al., 2012) find the turbulence saturates at a
somewhat lower tilt of ΘB ≈ 11− 13◦.
By all measures, our models are well-resolved and have similar properties,
with the exception of the hr 04 model. Given the remarkable consistency of the
other models, it is difficult to attribute the sudden decrease in the quality factors,
206
Table 5.2: Resolution Diagnostics
Simulation 〈Qθ〉 〈Qφ〉 〈ΘB〉 〈αSS〉 〈β〉
hr 005 15.9 37.7 12.3 0.065 9.1
hr 01 11.8 30.1 12.4 0.057 10.1
hr 02 11.9 28.7 12.4 0.052 10.4
hr 04 5.7 16.6 11.8 0.026 11.8
effective α-parameter, and increase in plasma β to the changes in the grid, although
we are simulating half of the physical domain (i.e. only ±2.5 h/r). One clue into
the discrepancy is that the inconsistent diagnostics all depend on the strength of
the magnetic field, which appears to be lower by roughly half compared to the
thermal energy. The ΘB value, on the other had, is similar to the other three
models, suggesting the saturation of the turbulence is the same, but the field doesn’t
naturally grow to the same relative level. A key result of this chapter is that the
organization of the dynamo in the thicker disks is impeded and less efficient, so
the resolution metrics could be biased by this phenomenon. Nevertheless, when
interpreting the following results, it is important to remain aware of what could be
a decrease in the effective resolution of the hr 04 model.
5.3 Results
The analysis we conduct is restricted to the final ∆t = 3.15× 104 GM/c3 (512
ISCO orbits) of each simulation. By allowing the simulations to evolve for at least
100 ISCO orbits, we avoid transient nonphysical behaviors that only exist as the
simulated disks relax into a quasisteady state. The simulation data is written out
every ∆t = 30.7812GM/c3, or every half of an orbital period at the ISCO, providing
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Figure 5.2: Spacetime diagrams of the azimuthally averaged Bφ at
r = 15 rg for hr 005(top), hr 01 (second from top), hr 02 (second from
bottom), and hr 04 (bottom). Positive values (red) indicate orientation
of the field in the positive φ-direction while negative (blue) indicates
an opposite orientation. Color intensity corresponds to the averaged
magnitude.
1024 snapshots for the analysis.
5.3.1 The Global Dynamo
We begin by first presenting spacetime diagrams of the toroidal magnetic field
in Figure 5.2. The spacetime diagrams were calculated by taking azimuthal averages
of Bφ at each time step at r = 15 rg in each of our four simulations. The large-scale
dynamo organizes itself into global, vertically stratified sheets of field of similar
polarity. These diagrams effectively trace the evolution of the cross section of this
208
pattern at a chosen radii, which typically reveals a vertically propagating pattern
that has a characteristic acceleration, evidenced by the increasing slope. Snapshots
of the normal pattern’s structure are shwon in Figure 2.9.
The different behaviors between the models are immediately apparent with
the most striking difference being the breakdown of the regular, periodic oscillation
pattern with increasing disk thickness. The thinnest disk in the hr 005 model shows
a cyclical building of the field and decay with a reversal of the orientation. Near
the midplane it is fairly stochastic, but in the atmosphere the organization develops
as the overmagnetized regions buoyantly rise. In the hr 01 model, a pattern is still
present, but it is less well organized compared to the hr 005 simulation. In hr 02
and hr 04 there are hints the organized oscillations attempt to develop, but patchy,
chaotic fluctuations dominate the behavior. However, there is still field amplification
and regeneration of the field throughout the simulation, despite the lack of a regular,
ordered pattern.
Further inspection of the hr 005 and hr 01 models reveal other interesting fea-
tures of the butterfly pattern. There are periods of the simulation where the dynamo
seemingly fails to reverse, for instance t = 1.5×104GM/c3 and t = 1.7×104GM/c3
in hr 005. Additionally, the large-scale field generated by the dynamo can evolve
independently in the upper and lower coronal regions of each disk. At some points
in the simulations the magnetic fields are aligned in the upper atmospheres, like at
t = 6.0× 103GM/c3 in the hr 005 simulation, but then at a later time the fields are
antialigned, like t = 2.1 × 104GM/c3. This is the same change in parity observed
in Flock et al. (2012). Several factors contribute to this, including irregularities in
209
Figure 5.3: PSDs of Bφ at 1.5h above the disk midplane for hr 005 (top
left), hr 01 (top right), hr 02 (bottom left), and hr 04 (bottom right).
Darker colors (black) represents greater power in that frequency bin for
a given radius. The orbital frequency is shown with the red line and one
tenth the orbital frequency is shown with the blue line.
210
the local evolution of the dynamo cycle and a global influence from the coupling of
field since the field buoyantly rises as coronal sheets of field with similar polarity
(Beckwith et al., 2011; Hogg & Reynolds, 2016). These butterfly diagrams can be
compared to those observed from local shearing box simulations, e.g. Davis et al.
(2010) and Simon et al. (2012), which tend to have more more regular periods.
Turning to the thicker disks in the hr 02 and hr 04 models, we see the am-
plification of the field is typically localized. Enhanced regions of strong field form,
but they are disrupted before they can collect in the midplane. In both of these
simulations we see that even though the pockets of strengthened field do not trace
out a butterfly pattern per se, they still typically originate near the midplane and
are expelled into the disk atmosphere, presumably due to their magnetic buoyancy
like before. In the hr 02 model, there are periods when the butterfly pattern almost
takes hold, but it often only in one hemisphere and then traces out an inconsistent
rise.
The power spectral density (PSD) at one scaleheight above the disk midpane,
shown in Figure 5.3, more clearly show the presence or absence of periodicity in the
dynamo. The PSDs were calculated as P (ν) = ν|f̃(ν)|2 where f̃(ν) is the Fourier
transform of the time sequence of the variable of interest,
∫
f̃(ν) = f(t)e−2πiνtdt. (5.18)
Here, we consider the power spectra of the azimuthally averaged Bφ at each radial
grid point. We also average Bφ over the θ direction from π/2 − 1.25 h/r to π/2 −
0.75 h/r to get a better sense of the dominating mean field.
211
The hr 005 and hr 01 simulations show a distinct band of power at one tenth
of the orbital frequency. This band of enhanced power spans roughly a factor of
three in frequency space and extends radially approximately 10 rg, consistent with
the PSDs seen in thin accretion simulations, like those presented in O’Neill et al.
(2011) and Figure 2.10. At lower frequencies there is very little power and at higher
frequencies there is residual power up to a dissipative scale.
The power spectra of the hr 02 and hr 04 simulations tell a different story,
though. In these simulations there are no discernible bands of power indicating
they lack any unique timescales where the field has significant oscillations. There
is power at all frequencies below the orbital frequency, indicating the large-scale
dynamo is operating; however, it is neither organized nor confined to a specific
timescale. This confirms the seeming randomness and disorder seen in the respective
butterfly diagrams, suggesting there is no single characteristic scale on which energy
is injected, rather the disorder allows the flow to inject energy over a range of scales,
which then decay.
5.3.2 Measuring αd
Next, we seek to probe the heart of the dynamo mechanism by measuring the
parameterization of the “α-effect.” The large-scale dynamo is typically interpreted
through a mean field theory. To produce the large scale toroidal field, there must
be a net electromagnetic force (EMF) to induce a magnetic field in the azimuthal
212
direction, which is predominately governed by:
〈E ′φ〉 = αd,φφ〈Bφ〉. (5.19)
Like other works (e.g. the local shearing box simulations of Brandenburg et al.
(1995), Brandenburg & Donner (1997), Ziegler & Rüdiger (2001), and Davis et al.
(2010) and global disk model of Flock et al. (2012)), we neglect contributions to the
α effect from the less influential αd,rr component and also higher order derivatives
of the magnetic field from the diffusivity tensor (usually denoted as η̃), thereby
allowing us to approximate αφφ through a simple correlation between the turbulent
EMF,
E ′ ′ ′ ′ ′φ = vrBθ − vθBr, (5.20)
where X ′ of a flow variable X indicates its fluctuating component taken by subtract-
ing off its azimuthal average (X ′ = X − 〈X〉), and the average toroidal magnetic
field.
We calculate the volume averaged E ′φ and Bφ in the upper and lower hemi-
spheres of each simulation for each data dump. The regions in which we take
these averages spans the entire azimuthal domain, radially from r = 15 rg to
r = 15 + 15(h/r), and poloidally from h to 2h in the upper hemisphere and −h
to −2h in the lower hemisphere. The correlation of 〈E ′φ〉 and 〈Bφ〉 is then measured
by fitting a simple line of the form,
〈E ′φ〉 = αd〈Bφ〉+ C, (5.21)
where C is an allowed vertical offset. The fitting was done using a Bayesian MCMC
method. Figure 5.4 shows the data, best fit, and parameter contour plots of the fit
213
(a) Upper Hemisphere hr 005
(b) Upper Hemisphere hr 01
(c) Upper Hemisphere hr 02
(d) Upper Hemisphere hr 04
Figure 5.4: Scatter plots of instantaneous values of 〈Bφ〉 vs 〈E ′φ〉 in the
upper coronal regions of the disks (black dots) with the best fitted lines
(left column) and the best parameter fits from our MCMC modeling with
1σ and 2σ contours (right column). The color coding in the lefthand
panels shows the density distribution of the points, estimated from a
Gaussian kernel.
214
(a) Lower Hemisphere hr 005
(b) Lower Hemisphere hr 01
(c) Lower Hemisphere hr 02
(d) Lower Hemisphere hr 04
Figure 5.5: Scatter plots of instantaneous values of 〈Bφ〉 vs 〈E ′φ〉 in the
lower coronal regions of the disks (black dots) with the best fitted lines
(left column) and the best parameter fits from our MCMC modeling with
1σ and 2σ contours (right column). The color coding in the lefthand
panels shows the density distribution of the points, estimated from a
Gaussian kernel.
215
Table 5.3: Upper Hemisphere Dynamo Coefficient Fits
Simulation αd,uh Offsetuh
hr 005 (−9.5± 0.1)× 10−5 (−0.4± 1.2)× 10−8
hr 01 (−2.1± 0.2)× 10−4 (−1± 7)× 10−8
hr 02 (−2.2± 0.4)× 10−4 (−4± 2)× 10−7
hr 04 (−2.3± 0.5)× 10−4 (−2.1± 1.0)× 10−7
parameters for the upper hemisphere of the different models, and Figure 5.5 shows
the corresponding results for the lower hemispheres. Tables 5.3 and 5.4 summarize
the best fits.
We find that αd consistently has an amplitude of |αd| ≈ 1 − 2 × 10−4 in all
of these disk simulations with the upper hemisphere having a negative sign and the
lower hemisphere having a positive sign, similar to Brandenburg et al. (1995) and
Davis et al. (2010), but different than other works. Furthermore, they are a slightly
weaker than other values reported in the literature. The increase in the scatter of
the 〈E ′φ〉 and 〈Bφ〉 correlations with increased disk thickness indicates the diffusive
term becomes a larger contributor.
5.3.3 Magnetic Helicity
As an exploratory exercise to bore into the organization, or lack thereof, in the
dynamo pattern, we sought to measure the large- and small-scale magnetic helicities,
216
Table 5.4: Lower Hemisphere Dynamo Coefficient Fits
Simulation αd,lh Offsetlh
hr 005 (1.0± 0.1)× 10−4 (0.4± 1.5)× 10−8
hr 01 (1.7± 0.2)× 10−4 (−1± 7)× 10−8
hr 02 (2.2± 0.4)× 10−4 (−1.9± 1.8)× 10−7
hr 04 (1.7± 0.4)× 10−4 (−1.5± 1.2)× 10−7
current helicities, and kinetic helicities in the four simulations. Teasing out the time
variability of these quantities from the chaotic turbulence ultimately proved not to
be feasible with these models, and we were not able to overcome the inherent noise
from differencing the stochastically fluctuating fluid variables. However, we were
able to find a correlation between the large-scale magnetic helicity
Hm = 〈Ā · B̄〉 (5.22)
with the average 〈Bφ〉 in the midplane of the hr 005 simulation, shown in Figure
5.6. Since the large-scale helicity is a volume integrated quantity, it was calculated
in a subdomain of the global simulation. We chose a reference radius, rref = 15 rg,
and then defined the subdomain to span radially from rref to rref (1+h/r), vertically
from the midplane (θ = π/2) to one disk scaleheight (θ = π/2−h/r), and in azimuth
over the entire φ-domain. The subdomain was designed this way for several reasons.
The radial location was selected to be far enough from the ISCO that turbulent
217
×10 3 ×10 5
2 1
0 0
2 1
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Time (GM/c3) ×104
Figure 5.6: Average Bφ (dark black line) and Hm (thin blue line) for
hr 005 at r = 15 rg. The two time traces have a Pearson-r statistic of
0.4.
edge effects (Krolik & Hawley, 2002) are miniscule. Moving away from the inner
boundary also has the additional benefit that the dynamo period is longer, so our
effective time resolution is increased by approximately a factor of 4, but it is still
short enough we can still study the behavior for many evolutionary times. We
are primarily interested in where the dynamo pattern originates and first organizes
itself, which drives us to the midplane of the disk. However, since the helicity
production is roughly asymmetric about the midplane (Blackman, 2012), we must
only sample one hemisphere so that there is not cancelation, and we choose the
upper hemisphere by default. We tested several vertical extents and determined
that the general trends held no matter where we placed the upper vertical boundary
of the subdomain within one disk scaleheight above the midplane, but the noise was
minimized at the upper limit.
Typically, Hm evolves with 〈Bφ〉. When 〈Bφ〉 oscillates and changes sign, Hm
seems to vary with it. There are periods when the cycle seems to stall, like from
218
B
Hm
t = 1.0 × 104 GM/c3 to t = 1.3 × 104 GM/c3, where Hm similarly shows no real
evolution. This is either because the helicity is not produced or because there is
no net production due global effects like cancelation with neighboring radii. Noise
dominates much of the signal, but the linear correlation ofHm and 〈Bφ〉, as measured
with the Pearson-r statistic, is r = 0.38 which indicated a moderate correlation.
Over different periods of the simulation it changes, though. For instance, if the
correlation coefficient is only calculated over the last half of the simulation, it is
higher at r = 0.58.
This hint that the large-scale magnetic helicity is tied to the organization of the
dynamo should help motivate further study of the role that the flow helicity plays in
regulating the global dynamo in future investigations. There is a wealth of literature
showing a connection between the dynamo and its quenching to the flow helicity from
analytic studies (Blackman & Brandenburg, 2002; Brandenburg & Subramanian,
2005; Vishniac & Cho, 2001). Explaining the large-scale accretion disk dynamo
through this lens offers great prospect and could yield a greater understanding of the
global disk evolution. Of particular interest is how the helicity produced at different
radii, and consequently on different timescales, interacts. Unlike the magnetic field
which decays through turbulence to larger wave numbers, the magnetic helicity
undergoes an inverse cascade and relaxes to smaller wave numbers, which could
effectively couple radii as the helicities add or cancel, depending on the interference
of the production patterns. Unraveling these connections may explain some of the
radial coherence, as well as the intermittencies and irregularities observed in global
dynamos.
219
5.3.4 Mass Accretion Rates
Having seen the role of organization on the magnetic field, we can begin to
ask how it effects the disk evolution. One of the clearest ways to detect the dynamo
influence is in the mass accretion r∫ate
Ṁ = ρvRRsin(θ)dφdθ, (5.23)
where we take R to be the inner simulation boundary, calculated from the instan-
taneous values of density and velocity for each data dump. In Chapter 2 we showed
that the large-scale dynamo drove propagating fluctuations which appear as a telling
log-normal distribution in the large-scale accretion rate due to the modulation of
the effective disk viscosity. Figure 5.7 shows the histograms of Ṁ for each of the
models. Note, that in the hr 005 model, there is a residual transient in Ṁ from
the initialization so the first 100 orbits of the analysis which has been removed to
prevent contamination of the histogram. In all four of our models the histograms
have a characteristic skewed distribution. We fit both normal
−(x−µ)2/2σ2P (x)N = P0e (5.24)
and log-normal,
P0
P (x) = e−(lnx−µ)
2/2σ2
LN (5.25)
x
functions to the Ṁ distributions, where x is the data count rate in each bin, σ is
the distribution width, and µ is the peak. In every case they are better fit by the
fast rise and slow decay in the high valued tail of the log-normal distribution. The
normal distributions fail to match the shapes of the distributions at high and low
220
Ṁ and consistently are shifted to the right of the peak in the distribution. As a
reminder, the simulations are evolved in a scale free form and the amplitude of the
mass accretion rate is largely set by the density in the model. We are primarily
interested in the consistent qualitative behavior here, since all of the models are
initialized with the same density normalization. This means the integrated “mass”
of the disk is larger with increasing scaleheight, which, when coupled with the shorter
evolutionary timescales and shorter accretion timescales, should give higher Ṁ with
progressively larger scaleheights.
5.3.5 Observational Signatures
On their own, the distinct dynamo behaviors we find are interesting, but
greater physical meaning can be found by connecting the unique manifestations
to observables. The rich photometric variability seen from accreting black holes
encodes information about the accretion process, with the imprint of the dynamo
being a likely component of this signal. Since we neglect detailed radiative physics
in order to save computational resources, we explore this with an emission proxy.
We use a scheme employed by other global accretion disk simulations (e.g. Hawley
& Krolik (2001), Armitage & Reynolds (2003) & Hogg & Reynolds (2016)) based
on the internal disk stress. Adopting Eqn 9 from Hubeny & Hubeny (1998), the
local flux at the photosphere of√the disk(to)dissipation is given by∫
3 GM A h
F = BrBφdz, (5.26)
2 r3 B 0
221
Normal Fit Normal Fit
100 Log-normal Fit 100 Log-normal Fit
80 80
60 60
40 40
20 20
0 0
0.000 0.002 0.004 0.006 0.008 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
M M
(a) hr 005 Ṁ Distribution (b) hr 01 Ṁ Distribution
Normal Fit Normal Fit
Log-normal Fit Log-normal Fit
80 80
60 60
40 40
20 20
0 0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.15 0.20
M M
(c) hr 02 Ṁ Distribution (d) hr 04 Ṁ Distribution
Figure 5.7: Mass accretion rate distributions at the inner boundary of
hr 005 (a), hr 01 (b), hr 02 (c), and hr 04 (d). The distributions have
been fit by fit by a Gaussian function (red) and log-normal function
(blue lines), and they are all better fit by a log-normal function. As
noted in the text, we are primarily concerned with the qualitative shape
rather than the quantitative amplitude since all simulations were ini-
tialed with the same density normalizations and the total disk “mass”
scales increases with increasing thickness.
222
Occurance Occurance
Occurance Occurance
where A and B are relativistic correction factors given by
− 2GMA = 1 (5.27)
rc2
and
3GM
B = 1− . (5.28)
rc2
For each data dump we integrate F within r = 25 rg to calculate a bolometric “lu-
minosity,” L. The region beyond r = 25 rg is excluded because of residual transient
behavior from the initialization of the disk which affects the very early part of our
analysis phase. This also contaminates the very early part of the hr 005 lightcurve,
which we ignore in our calculation of its standard deviation. We normalize by the
mean luminosity to express the variability in a fractional form and because the mod-
els are evolved dimensionless, scale free form. The lightcurves are shown in Figure
5.8.
The most obvious difference in the synthetic light curves is the level of orga-
nization provided by the relative coherence of the dynamo. Prior numerical studies
have shown the stress in the disk is modulated by the dynamo (Davis et al., 2010;
Flock et al., 2012; Hogg & Reynolds, 2016), which provides a link to the disk heat-
ing as the energy injected by the dynamo is ultimately deposited as heat. In our
simulations, we see that the thinner disks with a more organized dynamo display
slower undulations and a smaller fractional amplitude. As measured by the standard
deviation, √√√√ 1 ∑N ( )2σ = Li − L , (5.29)
N
i=1
223
2 h/r = 0.05
1
2 h/r = 0.1
1
2 h/r = 0.2
1
2 h/r = 0.4
1
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Time (GM/c3) ×104
Figure 5.8: Synthetic light curves calculated from the disk cooling for
hr 005 (top), hr 01 (second to top), hr 02 (second to bottom), and
hr 04 (bottom). The cooling has been normalized to the mean value of
the lightcurve to express the amplitude in terms of a fractional ampli-
tude.
224
L L L L
the typical fractional amplitude of the hr 005 model is σ = 0.14 and for the hr 01
model it is σ = 0.20.
The thicker disks display rapid, incoherent variability with a much larger frac-
tional amplitude. In the hr 02 model, σ = 0.24 and in the h 04 model, σ = 0.35.
In these models, the fluctuations are much more “flarey” and the light curves show
rapid brightening episodes and subsequent troughs. For instance, in the hr 02 model
from t = 1.1× 104 GM/c3 to t = 1.3× 104 GM/c3 when the disk luminosity would
appear to diminish by 36%, only to quickly rebrighten by a factor of three. Later,
from t = 1.7× 104GM/c3 to t = 1.9× 104 GM/c3, the disk dips to half its baseline
value, only to return quickly. The hr 04 model shows less structure than the hr 02
model, with fast, stochastic fluctuations of roughly factors of 2.5, i.e. fluctuations
between 0.6 and 1.5 on several occasions. At two points there are large excursions
from the mean with flares that peak at over twice the mean value.
5.4 Extended Domain Tests
To confirm the results from our analysis, we ran two additional simulations of
the thickest accretion disks, hr 02 pi and hr 04 pi. These simulations were con-
structed identically to their ∆φ = π/3 counterparts, but with a φ-domain extended
to ∆φ = π. The resolution element is preserved by tripling the number of zones in
the azimuthal direction so that hr 02 pi has NR×Nθ×Nφ = 308×248×192 zones
and hr 04 pi has NR ×Nθ ×Nφ = 154× 248× 96 zones. The hr 02 pi model was
integrated for 680 ISCO orbits, or t = 4.19× 104 GM/c3. The hr 02 pi model was
225
Figure 5.9: Spacetime diagrams of the azimuthally averaged Bφ at r =
15 rg for the extended π-domain hr 02 (top) and hr 04 (bottom) tests.
Positive values (red) indicate orientation of the field in the positive φ-
direction while negative (blue) indicates an opposite orientation. Color
intensity corresponds to the averaged magnitude.
also integrated for 680 ISCO orbits, or t = 4.19 × 104 GM/c3. As before, only the
final ∆t = 3.15× 104 GM/c3 (512 ISCO orbits) is used in the analysis.
Extending the φ-domain places these models within a similar regime to hr 005
and hr 01 in terms of the number of disk scale heights per azimuthal expanse.
Since the large-scale dynamo depends on the self-organization of the MRI-driven
turbulence, a smaller domain might not fit the global flow structure that leads to
this behavior. However, these models would allow similar butterfly patterns to
develop in these thicker disk simulations if our previous results were solely due to a
failure to capture the global modes.
Figure 5.9 shows spacetime diagrams of the azimuthally averagedBφ of hr 02 pi
and hr 04 pi at r = 15 rg. As seen in hr 02, the dynamo attempts to establish a
butterfly pattern, but it is much more irregular and disorganized than that seen in
the hr 005 and hr 01 models. Sometimes an organized field dominates for a long
226
Figure 5.10: PSDs ofBφ at 1.5h above the disk midplane for the extended
π-domain hr 02 (left), and hr 04 (right) tests. Darker colors (black)
represents greater power in that frequency bin for a given radius. The
orbital frequency is shown with the red line and ten times the orbital
frequency is shown with the blue line.
portion of the simulation, i.e. t = 1.3 − 1.5 × 104 GM/c3 in the upper hemisphere
of the simulation, while at others it is only momentary present. The hr 04 pi
simulations shows no organization, like hr 04. Fluctuations in the field are rapid,
disorganized, and chaotic which resembles nothing more than turbulent fluctuations.
Figure 5.10 shows the PSDs of hr 02 pi and hr 04 pi. Like Figure 5.3, the thick
disks show broad bands of power with no evidence of the typical band of power
found at one tenth the orbital frequency. The similarity between the PSDs offers
secondary evidence the behavior of the magnetic field evolution does not depend on
the domain size.
These ∆φ = π simulations offer compelling evidence that the breakdown in the
dynamo pattern with increasing disk scaleheight is a real effect and not a nuance
that arises from the grid scheme. In Section 5.5 we offer several refinements for
future studies that will help alleviate any remaining concern of numerical artifacts.
227
Incorporating these improvements with additional features, like a test-field method
to probe the field evolution, will provide valuable insights into the dynamo process.
5.5 Discussion
The results from the suite of accretion disk simulations we present here high-
light how poorly the dynamo mechanism in accretion disks is understood and the
additional work that is needed to fully leverage this phenomenon as an observational
probe. The crux of this work is that the large-scale dynamo can fail to organize into
the low-frequency, quasiperiodic butterfly pattern typically seen in time traces of
the vertical, azimuthally averaged toroidal magnetic field (〈Bφ〉). Instead, in the
thicker accretion disks the field is amplified in a stochastic way and on small-scales
with no obvious structure or order, but still at low frequencies. This challenges the
ubiquity of a self-organized large-scale dynamo appearing as a feature of black hole
disks. Consequently, this could translate into observable signatures as the synthetic
lightcurves we generate reflect the degree of order in the magnetic field.
Unfortunately, we cannot pin down a detailed reason for why the dynamo fails
to organize when the disk thickness is increased. However, these results suggest a
separation between turbulent fluctuating scales and the dominant scale of the flow
is needed for the dynamo organization. Considering the MHD turbulence spectrally
has led to “shell” models of the interactions in Fourier space where it has been shown
that the magnetic and velocity fluctuations both decay locally to larger wavenum-
ber. However, the interactions between the two occur non-locally (Mininni, 2011;
228
Moffatt, 1978) as long wavelength velocity fluctuations are needed to stretch the
magnetic field lines and long wavelength magnetic field fluctuations are needed to
apply a net Lorentz force. Understanding the details of how the spectral behaviors
and interactions of the turbulence depend on disk thickness may help to tie the
development of the large-scale dynamo into this framework.
Related to this, the different turbulent scales may change the effective mag-
netic Prandtl number, Prm. This could be a viable explanation since the effective
diffusivity changes because of the larger turbulent eddies, which we find evidence
for here in the increased scatter of the 〈E ′φ〉 and 〈Bφ〉 correlations. Guan & Gammie
(2009) explored the role of the turbulent Prm in MHD accretion disks, and found
it increased with increasing scaleheight. The excitation of the MRI has been shown
to depend on Prm (Fromang & Papaloizou, 2007; Fromang et al., 2007), and simu-
lations of forced turbulence have shown that the large-scale dynamo excitation also
has a dependence on Prm, depending on how the turbulence is injected (Branden-
burg, 2014). Since the effective viscosity, as measured by αss, is roughly constant,
this could show the role of the larger eddies increasing the turbulent magnetic diffu-
sivity and connect to disappearance of they dynamo cycle at low Prm (Riols et al.,
2015).
Investigating the role of helicity in the disk may help in deciphering the dif-
ferences between the thick and thin disks. The evolution and influence of the flow
helicities is expected to be intimately linked to global field behavior, and could be
important in the excitation and organization of a large scale dynamo (e.g. Blackman,
2012, 2015; Frick et al., 2006; Käpylä & Korpi, 2011). However, it is unclear the role
229
that it plays in the global behavior of an accretion disk. The clear case of a relation
between the large-scale magnetic helicity with the azimuthally averaged toroidal
field in the hr 005 simulation offers a glimpse into the the behavior. Pursuing this
connection more may help to explain the curious changes in the parity between the
two disk hemispheres, as well as abnormal features like the intermittency and “failed
reversals” in the field oscillations because they may be related to the ability of the
disk to assemble the large-scale field into an ordered pattern and its ability to shed
the helicity generated by the turbulence.
Fundamentally, all of these scenarios are related and could all be contributing
to the disorganization of the dynamo on some level. Moving forward, there are sev-
eral routes that could clarify why the large scale dynamo failed to organize. Using
test field methods, first applied to the geodynamo (Schrinner et al., 2005, 2007), to
measure the mean-field dynamo coefficients in these global runs is possibility the
most important next step as it has been effectively used to characterize the dynamo
in local shearing box simulations (Gressel, 2010; Gressel & Pessah, 2015). Addition-
ally, expanding the simulations to encompass the full 2π azimuthal domain could be
significant for fully capturing of the large scale structure. To run the simulations for
our planned duration, we were restricted to a truncated azimuthal domain out of
necessity. Our simulations were already very computational expensive, with hr 01
requiring 2 million CPU hours and hr 005 requiring over 7 million CPU hours, so
extending the domain with proper resolution like we had in these models was im-
practical. A number of dedicated resolution studies have shown that the properties
of the MRI turbulence can depend on the domain and it is possible we are simply
230
seeing an analogous sensitivity in the dynamo. In Section 5.4 we present two test
simulations of hr 02 and hr 04 that have φ-domains extended to ∆φ = π. These
simulations show identical behavior to their compliments used in this analysis, which
offers encouraging evidence the grid design is not the root cause of the unstructured
pattern. This bolsters these results, but subtle numerical effects remain a linger-
ing concern. Guided by this first attempt to explore the scaleheight dependence
of the dynamo, we can tailor our simulations to target the two interesting regimes.
Furthermore, we were intentionally very conservative with the length of the initial-
ization phase in these simulations to prevent transients from distorting our results,
and it could likely be shorted to redirect computational resources in the future.
As a note, in Chapter 4 we studied an MHD model of a “truncated” accretion
disk where there is believed to be a transition region between a hot, radiatively inner
accretion flow and a cooler, radiatively efficient disk. In the outer thin disk (h/r =
0.1), we find the large scale dynamo readily develops and is sustained. However,
in inner region where the disk thickens (h/r > 0.2), we find it fails to develop,
similar to this study. In the truncated disk the flow had the added component of
an outflow originating from the truncation zone, so it is not an isolated system like
those presented here, but the failed organization of the dynamo in the thicker disk is
reminiscent of that detailed in this chapter. The truncated disk scenario essentially
provides a composite accretion flow and demonstrates that it truly is a scale height
dependence since the two distinct flow behaviors develop in close proximity and can
even interact.
The observational consequences of changing disk height are of great interest.
231
The lightcurves we present in Figure 5.8 show that the temporal behavior of the
photometric variability from emission should be distinct, which meshes with empir-
ical results. Accreting stellar mass black holes in black hole binaries (BHBs) and
supermassive black holes in active galactic nuclei (AGNs) display a bifurcation in
spectral states that is typically attributed to the radiative efficiency of the system.
In the low-hard state of BHBs and in low-luminosity AGNs (LLAGNs) the accretion
flow is presumed to be radiatively inefficient and hot. The accretion flow should,
therefore, take on a thicker disk geometry since it cannot radiate away its thermal
energy. High-soft state BHBs and the typical Seyfert-like AGNs and quasars, on
the other hand, should be able to radiate efficiently, so the accretion is expected to
occur through a thin disk.
Making a direct association between the lightcurves from our simulations to
astrophysical black hole systems is not straightforward since the disk emission is
produced by different processes and in different wavebands for the thin and thick
disk cases, but it nevertheless seems to hold. Indeed, this scheme matches the trends
from BHBs as they change states during outburst (Remillard & McClintock, 2006).
Variability in AGNs is poorly understood, but the different timing properties of
LLAGNs compared to Seyfert-like AGNs may be a clue about their ability to host a
large-scale dynamo in the disk. LLAGNs typically have rapid variability e.g. NGC
4258 (Markowitz & Uttley, 2005) and NGC 3226 (Binder et al., 2009), but they
are less well studied than their higher Eddington ratio counterparts. Extensive
monitoring campaigns have been completed across the electromagnetic spectrum
for Seyfert-like AGNs which show variability on a thermal time, ttherm ≈ 1/αΩ,
232
(Kasliwal et al., 2017; Kelly et al., 2009; MacLeod et al., 2010), roughly the same
timescale that the dynamo oscillations might present themselves.
The results we present here may have additional observational impacts as a
number of other variable processes could stem from the presence of a well-ordered
large scale dynamo. Features like dynamo driven low-frequency QPOs could be
impacted by the departure from the assumed cyclical behavior, so it is prudent to
revisit their utility as probes of the central black hole mass and spin (e.g. Gierliński
et al., 2008; Pan et al., 2016; Pasham et al., 2014; Zhou et al., 2010) to verify the
assumptions that serve as the foundation of the mass scaling remain valid since they
will not appear if they dynamo is unordered.
In Chapter 2 we presented a detailed analysis of a simulated thin disk where
propagating fluctuations in mass accretion rate naturally developed from the tur-
bulence. At its core, the telling nonlinear signatures that are commonly observed,
i.e. log-normal flux distribution, linear relations between the RMS and flux level of
the variabiltiy, and interband coherence where the harder emission lags the softer
emission, arise from the multiplicative combination of stochastic fluctuations in the
mass accretion rate. As we emphasized the name “propagating fluctuations” is a
misnomer since the phenomenon is fundamentally just the preservation of the fluc-
tuation pattern as angular momentum is diffusively redistributed according to the
canonical disk equation (Pringle, 1981),
∂Σ 3 ∂ [ ]1 ∂ 1
= R 2 (νΣR 2 ) . (5.30)
∂t R ∂R ∂R
A key component in the growth of propagating fluctuations in mass accretion rate
233
is that the effective viscosity must be modulated at low enough frequencies that
turbulent fluctuations do not wipe out the growth of the structure in the accre-
tion flow (Cowperthwaite & Reynolds, 2014). The results presented here show that
this condition is met, even when the dynamo is not ordered into its standard cycle
because low-frequency power is still seen in the 2D PSDs. The log-normal Ṁ distri-
butions we find in these simulations further confirms that the growth of propagating
fluctuations is hardy enough to occur in an accretion disk regardless of the dynamo
organization, the dynamo just simply needs operate.
There has been a recent push to understand the dynamo beyond the standard
thin accretion disk, and from these efforts a narrative is developing that it is possible
to impede or alter the growth of the large-scale dynamo. Here, we present one more
example of modification of the large-scale dynamo. Further investigation into the
peculiarities of the dynamo are needed and warranted to get a better handle on when
it is a reliable source of variability and when it cannot contribute to the observational
signatures. This discussion is meant to highlight several ways in which commonly
observed features could be impacted by the breakdown of the dynamo organization,
but it is by no means complete. There are additional ramifications beyond these
that could be significant for interpreting the accretion flow dynamics around a black
hole.
234
5.6 Conclusion
With this study, we sought to clarify how the dynamo fits into the larger puzzle
of black hole accretion. Using a suite of four global, MHD accretion disk simula-
tions with scale height ratios h/r = {0.05, 0.1, 0.2, 0.4}, we expose a scaleheight
dependence in the ability of an accretion disk to organize the large-scale dynamo.
In summary, our top-line results from these simulations are:
1. Low-frequency, ordered oscillations in the azimuthal magnetic field from the
large-scale dynamo are present in the hr 005 (h/r = 0.05) and hr 01 (h/r =
0.1) models, but are increasingly absent in the hr 02 (h/r = 0.2) and hr 04
(h/r = 0.4) models.
2. When the organized large-scale dynamo is present in the thinner disks, there
is a coherent band of power in the PSD of azimuthally average toroidal mag-
netic field, 〈Bφ〉, at approximately 10× the local orbital period. In the thicker
accretion disks where the large-scale dynamo is unorganized, the PSD is fea-
tureless with power on all timescales, down the the lowest frequencies we can
probe with the duration of our simulations.
3. Calculation of αd through correlations between 〈Bφ〉 and the turbulent elec-
tromotive force, 〈E ′φ〉, in the coronal regions of the simulations yield similar
values across our models. In the upper hemisphere of the simulations αd is
negative, while in the lower hemisphere it is positive.
4. In synthetic light curves produced through a proxy, the presence of a large-scale
235
dynamo is related to the level of order and amplitude of the fluctuations. The
light curves of the thicker disks display large amplitude, stochastic fluctuations,
which reflects the lack of organization in the accretion flow. The thinner disk
simulations, in comparison, show a dearth of large amplitude variation and
smaller-scale, slower undulations instead.
As we continue to assemble the accretion puzzle and interpret variability from
accreting black holes, exploring the details of the underlying physics is an important
pursuit. The spectrotemporal behavior provides a crucial window into these systems,
but a first principles understanding of their origin remains elusive. This hinders the
ability to leverage the observational signatures to their full potential and offers an
opportunity for future advancement. The odd dynamo behavior we detail here has
the immediate implication that the large-scale dynamo shows markedly different
appearance in thicker accretion disks compared to its well characterized behavior
in thin disks, which could have broader importance in the global accretion disk
evolution.
236
Chapter 6: Conclusions
With this work, we sought to confront phenomenological models of black hole
accretion with MHD theory. We investigated the growth of “propagating fluctua-
tions” in mass accretion rate and how they lead to nonlinear variability, the dynamics
of truncated accretion disks, and finally the development of dynamo action in thick
and thin accretion disks. With minimal input, the simulations we conducted re-
semble the scenarios called upon in interpreting observations. This is encouraging
and suggests the models commonly used in the literature are likely to be realized in
nature. However, at each point in this dissertation research, we discovered impor-
tant revisions to these models that better reflect the physical reality of black hole
systems. To summarize each of the main areas of focus:
1. Propagating Fluctuations In Mass Accretion Rate - The main driver
of angular momentum transport in a thin accretion disk around a black hole
is MRI-driven turbulence. This turbulence causes an internal stress in the
disk and also has the interesting property that fluctuations multiplicatively
combine. If there is sufficiently slow modulation of the disk stress, fluctua-
tions will grow hierarchically and be preserved as they accrete. This imparts a
distinguishing nonlinear structure on the mass accretion rate fluctuations, and
237
subsequent variability of the emitted radiation, in the form of log-normal dis-
tributions, a linear relation between the RMS of fluctuations and the average
level, and radial coherence. Analytic and semi-analytic studies demonstrated
the feasibility of this variability model, but we show it naturally develops
in MHD turbulence. Most significantly, we identify the large-scale magnetic
dynamo as the source of the low-frequency stress modulation for the first
time. Prior studies provisionally assumed the modulation occurred on a vis-
cous timescale, but we were able to uncover the true physical driver.
2. Truncated Accretion Disks - Spectral evidence of a population of lower
luminosity X-ray binaries and AGNs implies that the disks in these systems
are truncated and the inner regions are evacuated. The current paradigm
dictates that hot gas is confined to an inner region and the transition is sharp
and well-defined. Here, we attempted to bridge the gap between this simple
picture and a full MHD theory by initially running a hydrodynamic model and
then following it up with a matched MHD model. These studies taught us that
rather than the currently envisioned truncated disk model where gas gets to
a certain radius and changes phase, it really should be considered more like a
boiling. Because of turbulent fluctuations, there is a range of gas temperatures
and a certain fraction of the distribution will be above the instability criterion
at a given radius. As gas moves inwards, the temperature increases so the
fraction above the instability threshold increases until eventually all of the gas
occupies a hot phase. This leads to a “tapering” in the disk in a truncation
238
zone, rather than an edge. Like boiling a pot of water, once gas is in the hot
phase, an outflow is launched as it buoyantly rises against gravity and fills the
volume around it, rather than being confined to a small region. Schematics
of the usually accepted disk configuration and our revised model are shown
in Figure 6.1. Interestingly, this outflow amplifies the magnetic field and
enhances the stress in a thin sheath enveloping the cool disk body. This
interacts with the dynamo to introduce a periodicity that could conceivably
appear as a strong quasiperiodic oscillation.
3. Large-scale Magnetic Dynamo Action - Dynamo activity is critical for
maintaining magnetic field against dissipation in an accretion disk and also
seems to be the puppet master behind many different forms of variability. To
understand the dynamo activity better, we studied its dependence on the disk
scaleheight. Despite the simplicity of our experiment, the dynamo showed
puzzling behavior. At the outset, we naively hypothesized that changing the
disk scaleheight might change the dynamo period. Instead, we found that in
thin disks the period is the same and that in thick disks it fails to organize
into its ordered pattern. Regardless of the pattern organization, the dynamo
strength is consistent across disk thickness. In the thinnest disk simulation,
we find tantalizing, yet inconclusive, evidence that the dynamo cycle may be
tied to the flow helicity, but further investigation must be done before any
firm claims can be made.
Looking forward, we find ourselves at an exciting time. Immense efforts are
239
(a) Standard Truncated Disk Model
(b) Revised Truncated Disk Model
Figure 6.1: Illustrations of the current truncated disk configuration (a)
and our revised model (b). In both drawings cooler gas is shown by redder
colors an hotter gas is shown by bluer colors. Motivated by the results
from our simulations, we advocate for a truncation zone rather than a
sharp transition. Turbulent fluctuations allow for pockets of gas to enter
the hot phase within the disk body, represented by the blue bubbles, which
then buoyantly rise outflow, shown with the darker blue regions hugging
the disk body. Our results suggest that hot gas more easily fills the volume
surrounding the black hole instead of just being confined to radii within
the truncation and is fed by the wind.
240
underway to understand black hole variability with the next-generation of ground-
and space-based telescopes, and through massive simulation efforts. Since beginning
this dissertation, there have been rapid advancements to bring theoretical predic-
tions and observations together, which we have contributed to.
There are several technical ways to expand the work we have presented. Firstly,
including radiative physics may be important, especially when it comes to driving
the outflows from truncated disks. Including radiative physics is computationally
expensive and should not dramatically change the disk dynamics that we have stud-
ied, but its inclusion could aid in connecting to observables. These methods have
reached a point that they can be considered mature and robust enough to gain
meaningful insight, so this is an important next step in adding realism to these
simulations. Secondly, we could potentially rerun this work with recently released
algorithms that have performance enhancements to run for longer duration. We
were careful in identifying numerical transients and understanding how they could
affect our results, but running simulations long enough that they are no longer issues
may ensure they did not affect our work in subtle ways.
More fundamentally, pinning down the truncation mechanism is a pressing
matter. The bistable cooling function used in studying truncated accretion disks was
simply that, a bistable cooling law triggered by the gas temperature. Relating this
to more physical properties of the flow could provide deeper insight and including a
proper triggering mechanism in simulations will more faithfully reproduce the flow
dynamics.
We are well-poised to unravel the curious dynamo behavior. Adding test-field
241
machinery to our simulation codes and rerunning models like those presented here
will provide a deeper look into the dynamo mechanics. In Chapter 5 we speculate
on what may be causing the dynamo pattern to not organize, but we do not have a
satisfactory explanation. Being able to directly measure properties like the flow he-
licity in a global simulation, rather than trying to calculate them in post-processing,
may be the key to make headway on this problem.
The future of accretion theory is bigger than just these topics, though. As
mentioned in Chapter 1, the stabilization of an accretion disk against thermal and
viscous instability in radiation dominated accretion disks like those expected to re-
side in the interiors of AGNs has not been resolved. Much work is needed to under-
stand how they might be stabilized, or if they can be. If they cannot, it may explain
the large variability this is observed and therefore be a useful accretion diagnos-
tic if we can characterize it. Furthermore, understanding the peculiarities of black
hole variability may come from including kinetic effects in simulations. The MRI
has been demonstrated to work in a kinetic shearing box (Kunz et al., 2016), but
much uncertainty surrounds the putative “corona” that dominates X-ray variability
in AGNs. Allowing for particle-wave interactions opens numerous instabilities that
can add additional heating and particle acceleration to drive variability. Also, as
we are now in the era of gravitational wave astronomy and on the search for elu-
sive SMBH binaries with optical surveys, modeling efforts are underway to simulate
these systems. Taking variability as a marker for these systems is a tricky business
(Vaughan et al., 2016), so using it as reliable probe requires the accretion physics
and their signatures are reasonably well understood.
242
There is also ample room to learn about accretion by better leveraging obser-
vational data. The prevalence of advanced classification and regression algorithms
in modern life has changed the way we shop, diagnose illness, and conduct business.
Machine learning models have become so advanced they can even identify cancer
cells far more accurately than expert epidemiologists and drive cars. This technol-
ogy has begun to creep into the astrophysical community, but there is much more
that could be done. As we are tasked with interpreting seemingly random events
like stochastic variability and changing look AGNs where the emission from the disk
“turns off,” we have to take a more comprehensive look at the data and use all the
tools at our disposal. Conceivably, there must be some sort of distinguishing signal
from the accretion physics that projects its future actions. As large surveys like the
Large Synoptic Sky Telescope come online, we will be inundated with information.
If we can understand the complex relationships between the many parameters that
impact the accretion behavior, we can get a handle on this difficult physical problem
and perhaps even predict interesting phenomena rather than simply react.
243
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