ABSTRACT Title of dissertation: MAGNETIC ISLANDS PRODUCED BY RECONNECTION IN LARGE CURRENT LAYERS: A STATISTICAL APPROACH TO MODELING AT GLOBAL SCALES Raymond Luis Lachica Fermo, Doctor of Philosophy, 2011 Dissertation directed by: Professor James F. Drake Department of Physics Magnetic reconnection is a process responsible for the conversion of magnetic energy into plasma ows in laboratory, space, and astrophysical plasmas. A prod- uct of reconnection, magnetic islands have been observed in long current layers for various space plasmas, including the magnetopause, the magnetotail, and the solar corona. In this thesis, a statistical model is developed for the dynamics of magnetic islands in very large current layers, for which conventional plasma simulations prove inadequate. An island distribution function f characterizes islands by the ux they contain and the area they enclose A. An integro-di erential evolution equation for f describes their creation at small scales, growth due to quasi-steady reconnection, convection along the current sheet, and their coalescence with one another. The steady-state solution of the evolution equation predicts a distribution of islands in which the signature of island merging is an asymmetry in r phase space. A Hall MHD (magnetohydrodynamic) simulation of a very long current sheet with large numbers of magnetic islands is used to explore their dynamics, speci cally their growth via two distinct mechanisms: quasi-steady reconnection and merging. The results of the simulation enable validation of the statistical model and benchmarking of its parameters. A PIC (particle-in-cell) simulation investigates how secondary is- lands form in guide eld reconnection, revealing that they are born at electron skin depth scales not as islands from the tearing instability but as vortices from a ow instability. A database of 1,098 ux transfer events (FTEs) observed by Cluster between 2001 and 2003 compares favorably with the model?s predictions, and also suggests island merging plays a signi cant role in the magnetopause. Consequently, the magnetopause is likely populated by many FTEs too small to be recognized by spacecraft instrumentation. The results of this research suggest that a com- plete theory of reconnection in large current sheets should account for the disparate separation of scales { from the kinetic scales at which islands are produced to the macroscale objects observed in the systems in question. MAGNETIC ISLANDS PRODUCED BY RECONNECTION IN LARGE CURRENT LAYERS: A STATISTICAL APPROACH TO MODELING AT GLOBAL SCALES by Raymond Luis Lachica Fermo Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial ful llment of the requirements for the degree of Doctor of Philosophy 2011 Advisory Committee: Professor James F. Drake, Chair/Advisor Dr. Michael M. Swisdak Professor Thomas M. Antonsen Professor Daniel P. Lathrop Professor Eve C. Ostriker ? Copyright by Raymond Luis Fermo 2011 A limerick (and a plea) to Jim, Marc, Tom, Dan, and Eve: A scientist lacking direction Studied magnetic reconnection But to get his degree He?d never be free From fear of his advisor?s rejection RLF Composed on August 24, 2009 in response to a friend?s challenge to write a poem about my research. May the work I?ve done here set me free. Dedication This thesis is dedicated to Telesforo Nacionales Lachica, or as I have always called him, Lolo. My grandfather was born April 15, 1912 in rural northern Philip- pines. He never surpassed the sixth grade because his mother { who encouraged him to study faithfully in school { passed away, leaving him to a father who preferred an extra body for the farm. I doubt anyone would?ve envisioned at the time the highest levels of education in the world just two generations after he left the sixth grade. But the educational values instilled by his mother remained, and were passed on to his children, who all attended college: Magdalena, Orlando, Potenciana, and Alejandro. Potenciana Perez Lachica received her degree in accounting from the University of the East, found a job in New York City, married Reymundo Almalvez Fermo, and then bore me. But it was Lolo who walked me to school daily, stayed with me while both parents worked, and encouraged me throughout my studies. I keep just one framed picture in my bedroom. In it, I?m standing in a grad- uation gown just outside of the Stephen C. O?Connell Center at the University of Florida. I appear distracted, looking around, perhaps at another camera or maybe for my friends. But the center of the photo features a man standing next to me, attentively xed upon the camera: si Lolo. He framed this photo for me along with a handwritten prayer he composed and a certi cate for the President?s Honor Roll that I had received at UF. He died on February 7, 2006, shortly after I started graduate school at the University of Maryland. Nevertheless, this photo of a man proudly standing with me is a reminder, not of a long-ago past, but of a present still happening. Talagang salamat po. Mahal kita. iii iv Acknowledgments First, I thank my advisor, Prof. James F. Drake. Even with his many ac- complishments, Jim has been a sel ess advisor, helping me to grow as a scientist, showing me the ropes, and always putting my career needs rst. I also thank Dr. Marc Swisdak, who has acted as a co-advisor in many respects, patiently helping me through my (many) daily problems. I am grateful to Prof. Tom Antonsen, Prof. Dan Lathrop, and Prof. Eve Ostriker for sitting on my thesis committee. To all of the members of our group, past and present, Prof. Paul Cassak (and Julie), Dr. Hai- hong Che, Dr. Yi-Hsin Liu (and Shania), Kevin Schoe er, Tak-Chu Li, Joel Dahlin, Kalman Knizhnik, and Lora McMurtrie, I?ve truly enjoyed our shared learning expe- riences. I appreciate everyone at NASA Goddard who contributed the observational data for this research, welcomed me to their meetings, and gave helpful advice, es- pecially Dr. David Sibeck, Dr. Yongli Wang, and Dr. Kyoung-Joo Hwang. I thank all the professors, scientists, and fellow students with whom I?ve interacted at GEM, SHINE, AGU, DPP, etc. I especially thank Prof. Merav Opher for the upcoming opportunity to work with her and her group at Boston University as a postdoc. There are many faculty and sta at the Physics Department and IREAP wor- thy of my gratitude, but I name in particular Prof. Adil Hassam, Ed Condon, Mohini Kaul, Linda O?Hara, Jane Hessing, and Margaret Hess. I am also thankful for my fellow students in the Ph.D. program who helped me through it all: David Norris, Kevin and Ellie Twedt, Aron Wall, Brandon Anderson, Jupiter Bagaipo, Nick and Shaela Mecholsky (as well as Abram and my godson Roman). I?d be remiss not v to acknowledge those at the University of Florida who inspired me into a career in physics, especially Prof. Steven Detweiler for teaching my rst university Physics course PHY 2048 (I subsequently changed my major to Physics) and Prof. Bernard Whiting for my undergraduate research experience with LIGO. The many friends and family who have loved and supported me throughout are indispensable to me. My gratitude for the support of my mother and father knows no bounds. Graduate school was by no means a certainty in early 2005, but knowing where my heart lay, they encouraged me to apply with their blessing. My sister Christine has also been a huge part of my life, even if I don?t tell her often enough. I?m proud to be her Kuya. I?ve received so much support from my family all across the globe, from Florida to New York to California to Germany and of course the Philippines. I truly miss the pinsans-night-outs with the cousins. Much love to all the Lumucsos, Medinas, Aquinos, Albays, and Cardinezes. Thanks also to Ate Clarisse \Bing" Andong, Ate Aileen Fermo Fernando (and family { I long to hear Isabella say Ninong), Kuya Rob Rebodos, Ate Grace Villanueva (and Aileen), and all my cousins everywhere. I have the best family ever. My achievements are not just my own but a credit to Fermos and Lachicas the world over. I am truly fortunate to have been given wonderful friends; I can?t imagine any- thing I?ve done to merit such a gift as these. I cannot possibly list by name everyone here, across the nation, and around the world, who has impacted my life, but I can start to try. First, a shout out to my fellow Gators, namely Palani Rathinasamy, Ko- mal Kirtikar, Ryan Isaac, Atchar Sudhyadhom, Dana and Alison Ward, bonds which have lasted even as we?ve spread from UF throughout the country. I?m proud to vi have known the families of Kevin and Jessica Hake, Joe and Daina Currano, Justin and Bernadette McClain, and their beautiful children Mercedes, Orlando, Marija, and John-Paul. I cannot forget my compatriot at the Little Sisters of the Poor, Bienvenida San Juan, who has been both a friend and my Filipina bonus-auntie-in- DC. I am profoundly grateful for Fr. George Wilkinson?s spiritual guidance, and for having befriended Rod Cruz at Holy Redeemer and Ashwini Sebastian at Memorial Chapel. I can only begin to think of everyone in Communion and Liberation who has helped me to live with a new awareness and openness to life, but I?ll just start with the CLU at UMD and CUA: Ellie Twedt, Caterina Poggi, Kevin Schoe er, Abby Holtz, Chiara Tanzi, Meghan Cassidy, Ann Marie and Matt McPherson, Tyra Villadiego, Diana Liddi, Giacamo and Franny Fornasini, Emily Yang, John and Lauren Martino, Ellen and Marie Claire Roderick, Annie Devlin, Amy Sapeno , Katrina Ten Eyck, Ryan Norris, Fr. Antonio Lopez, Fr. Pietro Rossotti, Fr. Paolo Prosperi, and in a particular way Dr. Liliana Maggioni, leader of my rst SoC. I cherish what I?ve met through Msgr. Luigi Giussani and the faces of these friends. In thanking all these family and friends \outside" of Physics, I emphasize that these relationships are not extraneous to my career. The acknowledgments here are not just academic. I am a better scientist because these people help me to seek, recognize, and love the truth in my life. Accompanied as I am, I can embrace my studying, my research, even my thesis-writing with the certainty that life is a gift, and neither my family nor my friends nor my work are extraneous to that life. Lastly, I thank the God who has given me that life, and who, in Physics as in everything else, reveals to me every day a glimpse of the Mysteries of the universe. vii Table of Contents List of Figures x List of Abbreviations xvi 1 A history of magnetic reconnection research 1 1.1 The pursuit of a theory for solar ares . . . . . . . . . . . . . . . . . 1 1.2 The advent of magnetic reconnection theory . . . . . . . . . . . . . . 6 1.3 The time-scale problem . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Reconnection in the magnetosphere . . . . . . . . . . . . . . . . . . . 19 1.5 Reconnection in laboratory, space, and astrophysical plasmas . . . . . 22 2 Magnetic islands in theory, simulations, and observations 27 2.1 The tearing instability . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Magnetic islands in nature . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2.1 Flux transfer events on the magnetopause . . . . . . . . . . . 34 2.2.2 Plasmoids in the magnetotail . . . . . . . . . . . . . . . . . . 37 2.2.3 Supra-arcade down ows in the solar corona . . . . . . . . . . . 39 2.3 Recent theories and simulations of islands . . . . . . . . . . . . . . . 43 2.3.1 Islands and fast reconnection in MHD . . . . . . . . . . . . . 43 2.3.2 Secondary islands in collisionless guide eld reconnection . . . 46 2.3.3 Particle energization by the Fermi mechanism . . . . . . . . . 50 2.4 The dynamics of many magnetic islands . . . . . . . . . . . . . . . . 52 3 Formulation of a statistical model 56 3.1 De ning the statistical distribution function f( ;A) . . . . . . . . . . 56 3.2 Rules governing island behavior . . . . . . . . . . . . . . . . . . . . . 58 3.2.1 Creation, growth, and convection of islands . . . . . . . . . . . 59 3.2.2 Properties of island coalescence . . . . . . . . . . . . . . . . . 60 3.3 An equation for f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3.1 Mathematical formalism for the merging rules . . . . . . . . . 63 3.3.2 The evolution equation . . . . . . . . . . . . . . . . . . . . . . 65 3.3.3 A non-dimensional form . . . . . . . . . . . . . . . . . . . . . 65 4 Steady state solutions 67 4.1 The no-merging solution . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Numerical solution with merging and the r asymmetry . . . . . . 70 4.3 Moments of the evolution equation . . . . . . . . . . . . . . . . . . . 73 4.3.1 Total island number N . . . . . . . . . . . . . . . . . . . . . . 73 4.3.2 Total island area AT . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 Solutions at large A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 viii 5 A Hall MHD simulation of many islands 82 5.1 F3D computational details . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2 Simulation results and analysis . . . . . . . . . . . . . . . . . . . . . 86 5.2.1 Characterizing the ux and size of islands . . . . . . . . . . . 87 5.2.2 Tracking islands in r phase space . . . . . . . . . . . . . . 90 5.3 Matching to the numerical solution . . . . . . . . . . . . . . . . . . . 92 6 Island formation in a PIC simulation of guide eld reconnection 95 6.1 P3D computational details . . . . . . . . . . . . . . . . . . . . . . . . 96 6.2 A non-tearing mechanism for island generation . . . . . . . . . . . . . 98 6.2.1 Vortical ow in a secondary island . . . . . . . . . . . . . . . . 99 6.2.2 Maturation of a de-scale vortex into a magnetic island . . . . . 101 6.3 Implications for future work . . . . . . . . . . . . . . . . . . . . . . . 105 7 Cluster observations of ux transfer events 107 7.1 Dissecting the bipolar BN signature . . . . . . . . . . . . . . . . . . . 107 7.1.1 Extracting r and from the bipolar BN . . . . . . . . . . . . 108 7.1.2 Magnetic eld normalization . . . . . . . . . . . . . . . . . . . 109 7.1.3 Dependence on spacecraft trajectory . . . . . . . . . . . . . . 110 7.2 A study of 1,098 FTEs . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.2.1 An exponential tail in r . . . . . . . . . . . . . . . . . . . . . 113 7.2.2 Asymmetry in r phase space . . . . . . . . . . . . . . . . 114 7.3 Interpretation of the results . . . . . . . . . . . . . . . . . . . . . . . 116 8 Conclusions 118 8.1 Summary of the model and relevant simulations . . . . . . . . . . . . 118 8.2 Comparing statistical studies of FTEs and SADs . . . . . . . . . . . . 120 8.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A Solar are energy calculations 127 B Area conservation in merging terms 128 Bibliography 130 ix List of Figures 1.1 (a) A solar prominence eruption photographed on October 20, 1891 at 2:30 P.M. by George Hale in the hydrogen (H) and potassium (K) lines. Reprinted with permission from Hale (1892) [71]. ?1892 by the Insitute of Physics. (b) A solar prominence eruption imaged on March 30, 2010 by the Solar Dynamics Observatory?s AIA (Atmospheric Imaging Assembly) shortly after its launch, showing a wavelength band centered around 304 A, an extreme ultraviolet (EUV) emission line from He II at a temperature of approximately 50,000 K. Courtesy of NASA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The magnetic elds within a plane connecting two sunspots, with a neutral point near the center, as proposed by Ronald Giovanelli. Reprinted with permission from Giovanelli (1947) [67]. ?1947 by the John Wiley & Sons, Inc. . . . . . . . . . . . . . . . . . . . . . . 7 1.3 (a) The initial setup of oppositely directed magnetic elds separated by a sheet of current into the plane. (b) As the current sheet thins, upstream plasma is pulled into the x-line at the center, where the magnetic eld lines cross-connect and accelerate away from at the center at the Alfv en speed (dragging plasma along with it) cA. . . . 8 1.4 (a) Two bipolar magnetic elds reconnecting at the point N , accord- ing to the model proposed by Peter Sweet. (b) The reconnection con guration with a long, thin neutral line N . (c) An idealized hy- drodynamic model, with external forcing F on two plates forcing the uid in between to ow out the sides at high speeds. Reprinted with permission from Sweet (1958) [174]. ?1958 by the International As- tronomical Union. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 The dissipation region of a reconnection site (in orange), de ned by its dimensions L . The reconnecting magnetic eld (in blue) is characterized by the upstream horizontal magnetic eld Bx. The plasma ow (in green) enters the dissipation region with speed vin and ows out with speed vout. . . . . . . . . . . . . . . . . . . . . . . 12 1.6 (a) The magnetic eld con guration for Sweet-Parker reconnection, with a long dissipation region of length 2L. (b) The magnetic eld con guration for Petschek reconnection, with a much smaller dissipa- tion region of length 2y 2L. The bold lines that de ne the bound- ary layer are slow shocks. Reprinted with permission from Petschek (1964) [127]. ?1964 by NASA Science and Technical Information Division. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.7 The Hall magnetic eld signature predicted by Sonnerup [168]. Reprinted with permission from Cassak (2006) [20]. ?2006 by Paul Cassak. . . 17 x 1.8 The reconnected magnetic ux as a function of time t for four di erent simulation methods: resistive MHD, Hall MHD, hybrid, and PIC. Reprinted with permission from Birn et al. (2001) [11]. ?2001 by the American Geophysical Union. . . . . . . . . . . . . . . . . . . . 18 1.9 The Earth?s magnetosphere, and interactions with the interplanetary magnetic eld BIMF . Lines 1-9 outline the convection of magnetic ux known as the Dungey cycle. Reconnection occurs at the mag- netopause (lines 1 and 1?) and in the magnetotail (lines 6 and 6?). The inset shows the footpoints of the respective eld lines on Earth. Reprinted with permission from Hughes (1995) [84]. ?1995 by NASA Science and Technical Information Division. . . . . . . . . . . . . . . 21 1.10 The heliospheric current sheet undergoing the Parker spiral. Reprinted with permission from Jokipii and Thomas (1981) [88]. ?1981 by the Institute of Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1 (a) A 2-D magnetic island enclosed by the separatrices of two x-lines. (b) A ux tube, the 3-D analogue of a magnetic island. . . . . . . . 28 2.2 The Harris sheet Bx0 (solid line) and Jz0 (dashed line) described by Eqs. (2.1)-(2.2), for w = 1. . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 A magnetic island generated by the tearing mode, withBx = B0 tanh(y=w) and By = Bye t cos(kx) with B0 = 1, w = 1, By = 0:1, t = 0, and k = 0:5. The thick lines are the separatrices that de ne the borders of the magnetic island. . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 (a) A magnetopause crossing by ISEE-1 and ISEE-2 featuring the bipolar signatures in BN now identi ed as ux transfer events. (b) A qualitative sketch by Russell and Elphic of how they interpreted this event as a ux tube on the surface of the magnetopause. Reprinted with permission from Russell and Elphic (1979) [142]. ?1979 by the American Geophysical Union. . . . . . . . . . . . . . . . . . . . . . . 34 2.5 Three di erent models for FTEs. (a)-(b) The Russell and Elphic el- bow reconnection model [141, 142]. (c)-(d) The Lee and Fu patchy reconnection model [99]. (e)-(f) The Scholer [151] and Southwood [169] bursty reconnection model. The top row (a), (c), and (e) shows views along the magnetopause normal; the bottom row (b), (d), and (f) shows views tangential to the magnetopause. Black curves repre- sent closed magnetosphere eld lines, red curves the magnetosheath eld lines, blue curves reconnected eld lines, and green curves the borders of the FTE. Reprinted with permission from Fear et al. (2008) [61]. ?2008 by the European Geosciences Union. . . . . . . . . . . . 36 2.6 A plasmoid structure in the magnetotail owing tailward. Based on the ow speed of 500 1000 km/s and the 30 min delay from observed auroral activity, it was estimated that the plasmoid was formed at XGSM & 70RE before owing past ISEE-3 at XGSM 220RE. Reprinted with permission from Hones et al. (1984) [80]. ?1984 by the American Geophysical Union. . . . . . . . . . . . . . 38 xi 2.7 The canonical single x-line model for solar ares. Reprinted with permission from Shibata et al. (1995) [158]. ?1995 by the Institute of Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.8 A 195 A ltergram of an April 21, 2002 are on the west limb, as recorded by TRACE. The vertical dimension of each panel is ap- proximately 117 Mm. The post are loops visible in the lower left are emitting in Fe XII 195 A. The di use plasma cloud in the center (cor- responding to the Fe XXIV 192 Aline) is penetrated by down owing \tadpole"-shaped voids, highlighted by the arrow in successive im- ages. The fourth panel is a di erence image, indicating the change from 0147:37 UT to 0148:37 UT. Reprinted with permission from Sheeley et al. (2004) [156]. ?1995 by the Institute of Physics. . . . 42 2.9 Magnetic island formation in (a) a resistive MHD simulation with S = 3 106 [82], and (b) a full PIC simulation with Fokker-Planck collisions and S = 5000 11700 [36]. In both simulations, the color signi es out-of-plane current density Jz and contour lines correspond to magnetic eld lines. Reprinted with permission from Huang and Bhattacharjee (2010) [82] and Daughton et al. (2009) [36], respec- tively. (a) ?2009 by the American Institute of Physics. (b) ?2010 by the American Physical Society. . . . . . . . . . . . . . . . . . . . 44 2.10 Schematic of quadrupolar density perturbation in reconnection with a guide eld showing the density asymmetry across the dissipation region. Reprinted with permission from Drake and Shay (2007) [42]. ?2007 by Cambridge University Press. . . . . . . . . . . . . . . . . 47 2.11 Contours of (a) the out-of-plane current density Jy (Jz in the co- ordinates used here) and (b) the ion density ni for a simulation of component reconnection. Reprinted with permission from Pritchett and Coronoiti (2004) [133]. ?2004 by the American Geophysical Union. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.12 Fermi acceleration of a test particle within a PIC simulation of a contracting magnetic island. (a) The test particle trajectory; the electromagnetic elds are frozen, but are still present, so the test par- ticle still feels the island \contracting" with each re ection. (b) The particle?s kinetic energy E as it bounces in x. (c) The particle?s par- allel and perpendicular velocities vk and v? in time. (d) A sketch of reconnection by many volume- lling islands. Reprinted with permis- sion from Drake et al. (2006) [45]. ?2006 by Macmillan Publishers Limited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.13 The out-of-plane electron current density Jez in a PIC simulation of reconnection with a guide eld Bg = Bx0, shown at times (a) t = 11 1ci , (b) t = 14 1 ci , (c) t = 20 1 ci , and (d) t = 24 1 ci . Reprinted with permission from Drake et al. (2006) [46]. ?2006 by the American Geophysical Union. . . . . . . . . . . . . . . . . . . . 54 xii 3.1 The results of a PIC simulation of island merger, showing the mag- netic eld lines at (a) t = 0 1ci , (b) t = 0:3 1 ci , (c) t = 4:0 1 ci , and (d) t = 8:0 1ci , where ci is the ion cyclotron frequency. The smaller island initially has 75% of the ux and 25% of the area of the larger island. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 The maximum ux m and combined area Am of the coalescing is- lands in time t from the start of the simulation. The variations in and A have respective standard deviations of 2% and 7% from the mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.1 A November 18, 2003 CME current sheet and helmet streamer imaged by the white light coronagraph LASCO C3 on the SOHO spacecraft. Times shown are in UT. Reprinted with permission from Lin et al. (2005) [100]. ?1981 by the Institute of Physics. . . . . . . . . . . . 69 4.2 The time evolution of F ( ; r) with S N = 4000 at (a) t = 1000 1 ci , (b) t = 2000 1ci , (c) t = 7000 1 ci , and (d) t = 17000 1 ci . The contours are smoothed and on a logarithmic, such that F decreases by a factor of 4 for each contour going outwards. . . . . . . . . . . . . . . . . . 71 4.3 The steady state distribution function F1( ; r) for (a) S N = 40, (b) S N = 400, (c) S N = 4000, and (d) S N = 40000. The contours are logarithmic as in Fig. 4.2. . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4 The number of islands N vs. t in the numerical solution (solid line; S N = 4000, as in Fig. 4.2) and predicted by Eq. (4.10) (dashed line; ts 89 1 ci , Nf = 88). . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.5 (a) The steady state distribution function in r, given by F1(r) as de ned in Eq. (4.28) for various SN . (b) The average magnetic eld strength B as a function of island radius r for various SN . . . . . . . 80 5.1 Snapshots of Jz in the Hall MHD simulation showing island forma- tion, growth, and merging at (a) t = 420 1ci , (b) t = 520 1 ci , (c) t = 620 1ci , and (d) t = 686 1 ci . Values of Jz shown are capped at Jz = 0:6n0ecA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2 Location of all o-lines in time. Horizontal dashed lines correspond to snapshots of Jz in Fig. 5.1. . . . . . . . . . . . . . . . . . . . . . . . 87 5.3 (a) A simple cartoon example of how magnetic islands can be de- ned by an x-line, o-line, and separatrix, all of the same color for a particular island. The corresponding cut(x) is shown below its cur- rent sheet, x-lines aligned with local minima and o-lines with local maxima. (b) A similar cartoon for two islands with a reconnecting x-line in between them pushing them apart. (c) Another example of two islands, now with a merging x-line in between. The lesser island shrinks as it is consumed by the dominant island. . . . . . . . . . . . 89 xiii 5.4 The distribution of islands in r phase space in the Hall MHD simulation overlaid upon the the integrated numerical solution shown in red. Each island is marked by a green x { overlaid on top of a numerical solution to Eq. (3.11) at times (a) t = 420 1ci , (b) t = 520 1ci , (c) t = 620 1 ci , and (d) t = 686 1 ci . The numerical solution in red shows the evolution of Eq. (3.11) on a logarithmic scale with the parameters L = 1638:4di, " = 0:055, SN = 1:5 ci, and Kmrg = 6. This gure is also available as a movie in the online auxiliary material in Fermo et al. (2011) [62]. . . . . . . . . . . . . . . . . . . . . . . . 91 6.1 The out-of-plane current density Jz at t = 87 1 ci depicting the tilted electron current sheet. Compare with the simulation by Pritchett and Coroniti (2004) [133] in Fig. 2.11(a). . . . . . . . . . . . . . . . . . . 98 6.2 (a) The out-of-plane current density Jz and contours of the magnetic eld near the x-line at t = 98 1ci . (b) A zoom-in of the region within the green box of (a), with arrows showing the electron ow ve. . . . 100 6.3 (a) The out-of-plane current density Jz and contours of the magnetic eld near the x-line at t = 90 1ci . (b) A zoom-in of the region within the green box of (a), with arrows showing the electron ow in the frame of the out ow, ve + 2cA. (c) The reconnection electron out ow vex. (d) Vertical cuts of vex (solid curve) and Jz (dashed curve) through the center of the green box in (c), with dashed vertical lines denoting the top and bottom boundaries of the green box. . . . 102 6.4 (a) The out-of-plane current density Jz and contours of the magnetic eld near the x-line at t = 91 1ci . (b) A zoom-in of the region within the green box of (a), with arrows showing the electron ow in the frame of the out ow ve + 2cA. . . . . . . . . . . . . . . . . . . . . . . 103 6.5 (a) The out-of-plane current density Jz and contours of the magnetic eld near the x-line at t = 92 1ci . (b) A zoom-in of the region within the green box of (a), with arrows showing the electron ow in the frame of the out ow ve + 2cA. . . . . . . . . . . . . . . . . . . . . . . 104 7.1 A sample ux transfer event observed by Cluster 4 at 0420 UT on 19 February 2001. The event occured at (5:8; 4:7; 9:0) RE in GSM coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.2 (a) Jz for a sample magnetic island from the Hall MHD simulation. (See Fig. 5.1.) (b) The bipolar BN signature a spacecraft would measure from passing through this magnetic island traveling along the solid white horizontal line in (a), a distance 10di from the center of the main current sheet. To the right are (c) the scale size rmeas and (d) the magnetic ux meas that a spacecraft would measure by horizontally passing through the island a distance y from the current sheet. The dashed lines denote the actual scale size rtrue = (Atrue= )1=2 and magnetic ux true. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 xiv 7.3 A distribution of the scale sizes of 1,098 ux transfer events detected by Cluster between 2001 and 2003. The dashed curve ts an expo- nential tail with r^ = 5277 km to the tail of the distribution (beyond its peak). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.4 (a) The distribution of ux transfer events in r phase space, over- laid on a numerical steady state solution to Eq. (3.11) for appropriate parameters. Flux transfer events above the = B0r diagonal are in green, whereas those in the prohibited region below the diagonal are marked in red. (b) The (normalized) average in-plane magnetic eld within the FTE versus the absolute (non-normalized) Bt;surr. The horizontal dashed line corresponds to the diagonal dashed line of (a). Most of the outlier events have small Bt;surr. . . . . . . . . . . . . . 115 8.1 Histograms for the (a) areas and (b) uxes of SADs and SADLs ob- served by SXT (S), XRT (X), and TRACE (T), accompanied on the left by quartile plots for each instrument?s measurements of areas for SADs (left) and SADLs (right). Reprinted with permission from Sav- age and McKenzie (2011) [148]. ?2011 by the Institute of Physics. . 122 8.2 (a) Histograms for the areas and uxes of SADs observed by SXT, shown with a tted log-normal distribution (dash-dotted) and for the latter an exponential distribution (dotted). (b) Cumulative distribu- tion functions of SAD areas and uxes with corresponding theoretical CDFs. Reprinted with permission from McKenzie and Savage (2011) [113]. ?2011 by the Institute of Physics. . . . . . . . . . . . . . . . 124 xv List of Abbreviations AAS American Astronomical Society ACE Advanced Composition Explorer ACR anomalous cosmic ray AGU American Geophysical Union AIA Atmospheric Imaging Assembly CDF cumulative distribution function CME coronal mass ejection DPP (American Physical Society) Division of Plasma Physics ESA European Space Agency EUV extreme ultraviolet FTE ux transfer event GEM Geospace Environment Modeling GSM Geocentric Solar Magnetospheric coordinate system HXR hard x-ray ICME interplanetary coronal mass ejection IMF interplanetary magnetic eld ISEE International Sun-Earth Explorer ISM interstellar medium IREAP Institute for Research in Electronics and Applied Physics LASCO Large Angle and Spectrometric Coronagraph LIGO Laser Interferometer Gravitational-Wave Observatory MESSENGER Mercury Surface, Space Environment, Geochemistry and Ranging MHD magnetohydrodynamics MMS Magnetospheric Multiscale Mission MST Madison Symmetric Torus MRX Magnetic Reconnection Experiment NASA National Aeronautics and Space Administration PIC particle-in-cell (simulation) RHESSI Reuven Ramaty High Energy Solar Spectroscopic Imager SAD supra-arcade down ow SADL supra-arcade down owing loop SDO Solar Dynamics Observatory SGR soft -ray repeater SHINE Solar, Heliospheric, and Interplanetary Environment SOHO Solar and Heliospheric Observatory SSX Swarthmore Spheromak Experiment SUMER Solar Ultraviolet Measurement of Emitted Radiation SXR soft x-ray SXT Soft X-ray Telescope THEMIS Time History of Events and Macroscale Interactions during Substorms TRACE Transition Region and Coronal Explorer UT Universal Time VTF Versatile Toroidal Facility XRT X-Ray Telescope xvi Chapter 1 A history of magnetic reconnection research 1.1 The pursuit of a theory for solar ares The original impetus for the development of magnetic reconnection theory was a desire to understand the nature of solar ares. Among the rst recorded obser- vations of a solar are1 was the famed Carrington Event of 1859, independently observed by English amateur astronomers Richard Carrington and Richard Hodg- son. In describing this experience, the former wrote, \While engaged in the forenoon of Thursday, Sept. 1, in taking my customary observation of the forms and positions of the solar spots, an appearance was witnessed which I believe to be exceedingly rare.... My rst impression was that by some chance a ray of light had penetrated a hole in the screen attached to the object-glass, by which the general image is thrown into shade, for the brilliancy was fully equal to that of direct sun-light; but ... I saw I was an unprepared witness of a very di erent a air. I thereupon noted down the time by the chronometer, and seeing the outburst to be very rapidly on the increase, and being somewhat urried by the surprise, I hastily ran to call some one to wit- ness the exhibition with me, and on returning within 60 seconds, was morti ed to 1A Chinese record dated December 9, 1638 possibly suggests a much earlier observation of a white light are. The observer writes, \Within the Sun there was a black spot, and black and blue white vapors." [7] 1 nd that it was already much changed and enfeebled." [19] In an article titled \On a curious appearance seen in the sun," Hodgson expressed a similar astonishment in front of what had just happened: \While observing a group of solar spots on the 1st September, I was suddenly surprised at the appearance of a very brilliant star of light, much brighter than the sun?s surface, most dazzling to the unprotected eye, illuminating the upper edges of the adjacent spots and streaks, not unlike in e ect the edging of the clouds at sunset; the rays extended in all directions; and the centre might be compared to the dazzling brilliancy of the star Lyr when seen in a large telescope with low power. It lasted for some ve minutes, and disappeared instantaneously about 11.25 A.M." [78]. To this date, that very are, the rst one to be observed, was also probably the most powerful [32]; the subsequent magnetic storms generated auroras as far south as Cuba. Fig. 1.1(a) shows a photograph similar to what Carrington and Hodgson might have seen. This image from 1891 was one of the rst photographic images of a solar prominence by American solar astronomer George Hale [71]. Of course, the technology that enabled the scientists of that era to marvel at this newly discovered phenomenon (for Hodgson, a nineteenth century telescope with a 6-inch aperture) pales in comparison to the instruments used today to observe these events. So one could only imagine what awe might have been inspired in Carrington and Hodgson had they seen a solar prominence through the eyes of the recently launched Solar Dynamics Observatory (SDO) as in Fig. 1.1. Even following this rst observation, however, an explanation for solar ares eluded scientists for decades. Hale in 1908 showed that sunspots exhibit ow vor- 2 Figure 1.1: (a) A solar prominence eruption photographed on October 20, 1891 at 2:30 P.M. by George Hale in the hydrogen (H) and potassium (K) lines. Reprinted with permission from Hale (1892) [71]. ?1892 by the Insitute of Physics. (b) A solar prominence eruption imaged on March 30, 2010 by the Solar Dynamics Ob- servatory?s AIA (Atmospheric Imaging Assembly) shortly after its launch, showing a wavelength band centered around 304 A, an extreme ultraviolet (EUV) emission line from He II at a temperature of approximately 50,000 K. Courtesy of NASA. 3 tices [73] and are accompanied by a strong magnetic eld [72]. Nonetheless, because the coupling between these processes was poorly understood, models for the solar atmosphere relied solely upon hydrodynamics. In fact, writing about a \great mag- netic storm" that immediately followed the Carrington Event of 1859, Carrington himself suggested that \while the contemporary occurrence may deserve noting, he [Carrington] would not have it supposed that he even leans towards hastily con- necting them. ?One swallow does not make a summer.?" [19] Without a proper understanding of how magnetic elds interact with a plasma, and how these e ects can propagate across the interplanetary medium, it was understandably di cult to connect two events separated by 17 hours when light takes only eight minutes to reach the earth from the sun. In an address to the Royal Society, Lord Kelvin proclaimed, \Guided by Maxwell?s electromagnetic theory of light ... it seems as if we may also be forced to conclude that the supposed connexion between magnetic storms and sun-spots is unreal, and that the seeming agreement between the periods has been a mere coincidence." [92] The causal connection between solar phenomena and terrestrial magnetic storms was nally established in 1905 by British astronomer Edward Maunder, on the basis of a 27-day recurrence of magnetic storms correlated with the sun?s 27-day rotation period [109]. Even still, an ultraviolet light the- ory of magnetic storms proposed in 1929 asserted that solar ares were no more than a burst of ultraviolet radiation that excited and ionized atoms in the upper atmosphere [108]. It would eventually be shown that the exorbitant amount of energy released in a are (3 1032 erg) can only come from the magnetic eld; neither thermal 4 nor gravitational energy would be adequate [131]. (The relevant calculations can be seen in Appendix A.) Therefore, the key needed to unlock this mystery was the connection between electromagnetics and hydrodynamics. The rst step was de n- ing the notion of a plasma. In 1918, the British mathematician Sydney Chapman proposed a theory that geomagnetic storms could be the result of a solar beam of either protons or electrons reaching the Earth?s atmosphere [23]. When Frederick Lindemann rightly questioned whether repulsive e ects would prevent such a beam from reaching Earth [103], Chapman revised his theory to a neutral beam consist- ing of both protons and electrons [24]. American chemist and Nobel laureate Irving Langmuir dubbed this new phase of matter a plasma, simply because it reminded him of the blood plasma [96]. The second and decisive step came in 1942 with the emergence of Hannes Alfv en?s theory of magnetohydrodynamics (MHD) [3]. For a quasineutral plasma (number density n ni ne for ions and electrons) in a homogeneous magnetic eld B, the Swedish physicist showed that a perturbation to the density generates a perturbation in the magnetic eld, and the eld line tension acts as the restoring force. The resulting wave propagates along the magnetic eld line with speed cA = B p 4 nmi (1.1) where mi is the ion mass. This rst magnetohydrodynamic wave (since dubbed the Alfv en wave) inspired the ideal magnetohydrodynamics equations, which (true to the name given to them) combine Maxwell?s equations from electromagnetism and the 5 inviscid Navier-Stokes equations (a.k.a. the Euler equations) from hydrodynamics: @n @t +r (nv) = 0 (1.2) nmi @v @t + v rv = 1 c J B rp (1.3) @ @t 1 2 nmiv 2 + B2 8 + p 1 +r 1 2 nmiv 2 + p 1 v + c 4 E B = 0 (1.4) r B = 0 (1.5) @B @t = 1 c r E (1.6) J = c 4 r B (1.7) E + v B c = 0: (1.8) These equations are, respectively: the continuity equation, the momentum equation, the energy equation2, Gauss? law for magnetism (absence of magnetic monopoles)3, Faraday?s law, Amp ere?s law4, and Ohm?s law. In these equations, t is time, n the number density, v the bulk ow velocity, mi the ion mass, J the current density, B the magnetic eld, p = nT the pressure, T the temperature, the ratio of speci c heats, c the speed of light, and E the electric eld. As essentially the founding father of plasma physics, Hannes Alfv en was awarded the 1970 Nobel Prize in Physics. 1.2 The advent of magnetic reconnection theory Armed with this new tool, physicists could now couple the plasma ejected by the prominences in Fig. 1.1, and the latent magnetic eld observed in solar 2The heat ux Q is ignored to complete closure of the moment equations. 3Gauss?s law for electricity r E = 4 is not used. The electric eld E is given by Ohm?s law. 4The displacement current term (1=4 )@E=@t is ignored. 6 Figure 1.2: The magnetic elds within a plane connecting two sunspots, with a neutral point near the center, as proposed by Ronald Giovanelli. Reprinted with permission from Giovanelli (1947) [67]. ?1947 by the John Wiley & Sons, Inc. active regions (regions containing sunspots). With this in mind, Australian physicist Ronald Giovanelli proposed in 1947 that ares could be the result of a current sheet containing a magnetic neutral point, as seen in Fig. 1.2 [67]. The magnetic eld vanishes at this neutral point, implying that it must reverse direction on either side of the current sheet. Famed British astronomer Fred Hoyle discussed this problem with Giovanelli [81], and subsequently suggested it as a project for his graduate student, James Dungey. In the coming years, Dungey would study oppositely directed magnetic eld lines separated by a thin current sheet, as in Fig. 1.3(a). The picture in Fig. 1.3(b), however, would be impossible in ideal MHD, i.e., when the plasma has a high col- lision frequency , low resistivity , and relevant length scales L much larger than the ion skin depth di = c=!pi (where c is the speed of light, !pi is the ion plasma 7 Figure 1.3: (a) The initial setup of oppositely directed magnetic elds separated by a sheet of current into the plane. (b) As the current sheet thins, upstream plasma is pulled into the x-line at the center, where the magnetic eld lines cross-connect and accelerate away from at the center at the Alfv en speed (dragging plasma along with it) cA. frequency !pi = p 4 nie2=mi and e is the electron charge). This is because for ideal MHD, substituting the ideal Ohm?s law Eq. (1.8) into Faraday?s law Eq. (1.6) gives the magnetic induction equation @B @t = r (v B) (1.9) where v is the bulk plasma ow. Helmholtz?s second theorem states that when a eld obeys this induction equation, the plasma remains \frozen" to that eld. In other words, the bulk plasma ow carries not only the particles but also the magnetic eld B along with it, so that two particles connected by a single magnetic eld line at one time t would remain connected by the same eld line at some later time t+ t. This \frozen-in" condition therefore prohibits any changes in magnetic eld line topology, as exhibited in Fig. 1.3(b). On the other hand, Dungey proposed that by introducing a nite resistivity 8 into Eq. (1.8), E = v B c + J (1.10) where J is the current density, the topology of magnetic eld lines would be able to change near a magnetic neutral point. This would enable magnetic eld lines to cross-connect at the x-line in the center of Fig. 1.3(b). The tension in the newly formed eld lines would propel those eld lines away from the x-line. Once outside of the dissipation region near the x-line, the frozen-in condition holds once again, and so the magnetic eld drags plasma along with it downstream. Dungey coined the term magnetic reconnection to describe this process [48, 49]. Soon thereafter, at the 1956 International Astronomical Union Symposium on Electromagnetic Phenomena in Cosmical Physics in Stockholm, Sweden, the En- glish physicist Peter Sweet presented his theory, which modeled solar ares by the reconnection of two bipolar magnetic elds as in Fig. 1.4(a) [174]. Sweet proposed that, with a nite resistivity and su ciently low pressure, a hydrostatic equilibrium such as that in Fig. 1.4(b) is unstable to reconnection as the magnetic eld attens along the neutral line N (where the magnetic pressure is weakest). He draws an analogy with \the attening of a motor tyre when loaded" in which, as long as the internal pressure P is not too strong, an external driving force F can force the gas to ow out at high speeds. Fig. 1.4(c) illustrates the same idea using an idealized hydrodynamic analogy, in which an external force pushes together two plates, forc- ing a high-speed out ow of the uid between them. (It should be noted that the external driving force is not actually necessary for reconnection. Once reconnec- 9 Figure 1.4: (a) Two bipolar magnetic elds reconnecting at the pointN , according to the model proposed by Peter Sweet. (b) The reconnection con guration with a long, thin neutral line N . (c) An idealized hydrodynamic model, with external forcing F on two plates forcing the uid in between to ow out the sides at high speeds. Reprinted with permission from Sweet (1958) [174]. ?1958 by the International Astronomical Union. 10 tion begins and plasma is expelled from the dissipation region, the density void in the dissipation region draws in new plasma upstream. Therefore, reconnection is a self-driven process.) At this same conference, American astrophysicist Eugene Parker witnessed Sweet?s talk and, on his way home, performed the scaling analysis of Sweet?s the- ory using resistive MHD. Substituting Eq. (1.10) and Amp ere?s law Eq. (1.7) into Faraday?s law Eq. (1.6) gives the resistive induction equation @B @t = r (v B) + c2 4 r2B: (1.11) In a steady state, the magnetic di usion term must balance the dynamical term r (v B). If we consider this for the dissipation region diagrammed in Fig. 1.5, with width corresponding to the characteristic length of the gradient in Bx, then balancing the right hand side of Eq. (1.11) gives us a relation for the in ow velocity vin c2 4 : (1.12) We consider the continuity equation @n=@t +r (nv) = 0 for a steady state and an incompressible plasma (r v = 0) in order to ensure mass conservation. If the dissipation region has length L as in Fig. 1.5, we can relate the in ow and out ow speeds to the aspect ratio: vin = vout L : (1.13) Likewise, if the energy required to push that out ow vout comes from the upstream magnetic eld Bx, then we can calculate the out ow speed explicitly from energy conservation: 1 2 nmiv 2 out = B2x 8 =) vout = Bxp 4 nmi = cA: (1.14) 11 Figure 1.5: The dissipation region of a reconnection site (in orange), de ned by its dimensions L . The reconnecting magnetic eld (in blue) is characterized by the upstream horizontal magnetic eld Bx. The plasma ow (in green) enters the dissipation region with speed vin and ows out with speed vout. Thus, the reconnection out ow speed is equal to the upstream Alfv en speed de ned in Eq. (1.1). Substituting for vout and from Eq. (1.13) into Eq. (1.12), we get vin = r c2cA 4 L (1.15) E 0 L = vin vout = s c2 4 cAL = S 1=2 (1.16) where E 0 is a dimensionless reconnection rate and S = 4 cAL= c2 is the Lundquist number (also known as the magnetic Reynolds number). The reconnection rate is denoted as E 0 because it is proportional to the out-of-plane electric eld Ez, which by the integral form of Faraday?s law can be related to the rate at which magnetic ux enters the x z plane. 12 1.3 The time-scale problem However, Sweet and Parker?s model for reconnection had a major aw: it was still too slow to explain observations. For the systems in question, L can be very large, the resistivity quite low, and subsequently the Lundquist number S is exceptionally large. For example, in the corona, L 109 cm, = 10 16 s, S 1014, and so E 0 10 7. Although the time-scales for Sweet-Parker reconnection ( SP L=vin 1:4 107 s 4 months) are much faster than di usive time-scales ( R 4 L2= c2 1014 s 4 million years), this model is still far too slow to explain the solar are that lasted but a disappointing 5 minutes for Carrington [19, 78]. In light of this limitation, Sweet, Parker, and others continued to seek a mech- anism to enable faster reconnection { Joule dissipation, ambipolar di usion, various forms of Sweet?s mechanism { but without immediate success [124]. Finally, in 1963, Harry Petschek, an American physicist5, presented a theory that employed standing slow shocks outside of the di usion region to convert the magnetic energy into plasma ow [127]. In Sweet-Parker reconnection as in Fig. 1.6(a), nothing will prevent the dissipation region from expanding into the system size 2L. In contrast, Petschek reconnection as in Fig. 1.6(b) can have plasma accelerated by the slow shocks (thick black lines) without having to go through the dissipation region, now 5\American physicist" is a rather lacking description. Harry Petschek was born in Prague, Czechoslovakia, to Jewish parents who ed the Nazi invasion. Besides his noteworthy contributions to physics, he invented an automated device for extracting DNA samples and collaborated on the invention of the intra-aortic balloon pump, a device for treating heart failure. He was also a civil rights activist, responsible for staging 1960s rallies in Lexington, Kentucky. 13 Figure 1.6: (a) The magnetic eld con guration for Sweet-Parker reconnection, with a long dissipation region of length 2L. (b) The magnetic eld con guration for Petschek reconnection, with a much smaller dissipation region of length 2y 2L. The bold lines that de ne the boundary layer are slow shocks. Reprinted with permission from Petschek (1964) [127]. ?1964 by NASA Science and Technical Information Division. much smaller (2y 2L). As a result, Petschek calculates a reconnection rate E 0 that only weakly (logarithmically) depends on S, and coronal time-scales P 100 s, fast enough to explain solar ares. When Petschek presented this theory at the AAS- NASA symposium, Peter Sweet enthusiastically proclaimed to Petscheck, \I am in favor of your theory, which I thoroughly approve. Dr. Parker and I have been living with this problem for several years and have got the feel of it. Your solution struck me at once as the solution for which we have been seeking." [127] It would not prove to be so simple. Petschek reconnection was widely accepted as having solved the problem, but ultimately would fail to be validated by numerical simulations except with a locally enhanced resistivity [145]. Uniform resistivity failed to produce the slow-mode shocks predicted by Petschek [13]. Turbulence and 14 certain collisionless instabilities [83, 47] have been proposed as mechanisms for such an anomalous resistivity, but a more general theoretical explanation is still lacking. An alternative to anomalous resistivity came about with the consideration of collisionless e ects. In particular, it was discovered that the Hall e ect, by which ions are decoupled from electrons, can trigger su ciently fast reconnection rates to explain ares. The Hall e ect is encapsulated by the J B term in the generalized Ohm?s law [139] E = v B c + J + J B nec r Pe ne + me e2 d dt J n : (1.17) The generalized Ohm?s law adds to the resistive Ohm?s law, Eq. (1.10), three addi- tional terms on the right hand, which describe the Hall e ect, the electron pressure, and electron inertia, respectively. The Hall term introduces the rst intrinsic length scale into the MHD equations: the ion skin depth di, or the ion Larmor radius i = cs= ci = p di (where cs = p T=mi is the sound speed, ci = eB=mic the ion cyclotron frequency, and = 8 nT=B2 the plasma beta parameter, the ratio between thermal and magnetic energy). Ideal MHD breaks down near the x-line precisely for this reason. As the ions near the magnetic eld reversal, their gyro- orbits reverse direction mid-orbit, and the frozen-in condition is no longer valid. Meanwhile, the electrons remain frozen-in because they have a smaller Larmor ra- dius on account of their much smaller mass me mi. They eventually decouple as well within their own electron dissipation region, but at even smaller scales, within the electron skin depth de = c=!pe = p mec2=4 ne2 di [42]. It is worth noting that the Hall e ect does not in itself enable reconnection. 15 In fact, at the smallest scales near the x-line within the electron dissipation region, the Hall term vanishes because the electrons are decoupled from the magnetic eld as well as the ions [42]. Some other dissipation mechanism is required to break eld lines. Rather, the role of the Hall e ect is to enhance the reconnection rate by opening the out ow region into a Petschek-like con guration, similar to Fig. 1.6(b) but without the requirement of standing slow shocks. In place of slow shocks, the Hall e ect employs whistler waves to open the out ow nozzle (or a kinetic Alfv en wave if there is a guide eld, a component of the magnetic eld Bz perpendicular to the reconnecting plane) [43]. Suppose for instance that we treat the reconnected magnetic eld (say, the leftmost and rightmost blue curves in Fig. 1.5) as a sine- wave perturbation to By with wave number ky. Whistlers are dispersive waves (dispersion relation ! = k2ycAydi), meaning that the out ow speed !=ky increases proportionally as the scales get smaller (ky gets larger). This is in contrast with non-dispersive waves (e.g., the Alfv en wave with dispersion relation ! = kycAy), for which the out ow speed is a constant cAy, forcing a bottleneck at small scales. Because whistler waves are circularly polarized, the Hall e ect also tends to pull the magnetic eld up and out of the plane in z. The result is a quadrupolar magnetic eld as seen in Fig. 1.7, the primary signature for Hall reconnection [168]. This quadrupole structure has since been observed in the magnetopause [114, 152], magnetosheath [129], magnetotail [119, 118, 140], and laboratory plasmas [33, 136]. A major test for the Hall e ect was the GEM Reconnection Challenge, an attempt to simulate reconnection using di erent types of simulation methods { re- sistive MHD, Hall MHD, hybrid, and full particle-in-cell { on the same initial con- 16 Figure 1.7: The Hall magnetic eld signature predicted by Sonnerup [168]. Reprinted with permission from Cassak (2006) [20]. ?2006 by Paul Cassak. ditions [11]. The resistive MHD simulation simply solves Eqs. (1.2)-(1.7) along with Eq. (1.10) numerically. Hall MHD adds the Hall term J B=nec to Eq. (1.10). The hybrid simulation treats the electrons as a uid but simulates the ions as particles. The particle-in-cell (PIC) simulation treats both ions and electrons as particles. Particles in hybrid and PIC simulations are stepped forward in time using dx dt = v (1.18) and the relativistic Lorentz force law d( mv) dt = q E + v B c (1.19) where here, = 1= p 1 v2=c2 is the Lorentz factor (as opposed to the ratio of speci c heats in Eq. (1.4)). The latter three types of simulations all include the Hall term in one form or another. Fig. 1.8 summarizes the conclusion of this study, that while resistive MHD only produced slow Sweet-Parker reconnection, the other three simulations that include the Hall e ect gave similarly fast reconnection rates. One participant in the GEM reconnection challenge, Michael Shay, has gone 17 Figure 1.8: The reconnected magnetic ux as a function of time t for four di erent simulation methods: resistive MHD, Hall MHD, hybrid, and PIC. Reprinted with permission from Birn et al. (2001) [11]. ?2001 by the American Geophysical Union. so far as to suggest that the reconnection rate is a universal constant of order E 0 0:1 independent of system parameters such as the mass ratio mi=me, the resistivity , or the system size L [153]. In the corona, this would give an in ow speed vin E 0cA 700 km/s and a time-scale for reconnection of 14 s, certainly fast enough to explain are energy release times, and orders of magnitude faster than Sweet-Parker reconnection ESP 10 7, SP 4 months. Nonetheless, a theory for why the reconnection rate should take on this partic- ular value of 0:1 has remained elusive and controversial. Alternative theories have sought fast reconnection through an anomalous resistivity, magnetic islands (to be discussed in Sec. 2.3), or various other terms in the generalized Ohm?s law, Eq. (1.17). 18 1.4 Reconnection in the magnetosphere The conversion of magnetic energy into plasma ows is not just con ned to solar ares. One important application is in the Earth?s magnetosphere, the region of space dominated by the Earth?s magnetic eld, which de ects the solar wind. The concept of a magnetosphere originated with Sydney Chapman and Vincenzo Ferraro in 1930 trying to develop a theory for the magnetic storms that Maunder had correlated with solar phenomena [109]. Chapman and Ferraro suggested that a \cloud" of plasma from the sun occasionally envelops the Earth, and the Earth?s magnetic eld would carve out a \cavity" in that cloud [25, 26]. In truth, the streams of particles from the sun are not from isolated clouds, but rather from a steady solar wind (as suggested by the German astronomer Ludwig Biermann in 1951 [9] and by Eugene Parker in 1958 [123], then veri ed observationally by the Soviet satellite Luna 1 in 1959 on its way to the moon), and the resulting cavity is not a temporary response to transient events but the magnetosphere. The intermittency of magnetic storms is due to coronal mass ejections (CMEs), a burst of plasma and magnetic eld originating from the corona, often associated with solar ares. Even as Dungey was rst considering the applicability of his ideas to ares, he was already considering reconnection also in the context of the Earth?s magne- tosphere. In particular, Dungey speci ed two candidate locations in the magneto- sphere where reconnection would take place: the magnetopause and the magnetotail [50]. The magnetopause is the border between the magnetosphere and the interplan- etary magnetic eld (IMF) facing the sun; the magnetotail is the extended region 19 of the magnetosphere opposite the sun. The Earth?s dipole eld points northward, so when the IMF points southward, the oppositely directed eld lines reconnect at the magnetopause. (See eld lines 1 and 1? in Fig. 1.9.) The magnetic eld lines are suddenly opened (having only one footpoint in the Earth) and ung anti-sunward by the solar wind (lines 2-5 and 2?-5?). Eventually, those eld lines pinch together in the magnetotail (lines 6 and 6?) and reconnect a second time (lines 7 and 7?). The newly reconnected elds (lines 7 and 7?) retreat Earthward back into a dipole con guration (line 8), then convect around the Earth (through the dusk-side) back to the dayside magnetosphere (line 9). The transfer of magnetic ux described here is called the Dungey cycle. As in the sun, magnetic reconnection in the magnetosphere can also lead to particle energization. In the tail, Syun-Ichi Akasofu and Sydney Chapman suggested that magnetic disturbances (dubbed substorms by Chapman) generate auroras [2, 1], the result of accelerated plasma colliding with the atmosphere, and whose spectac- ular light shows in the skies have inspired great awe and wonder throughout human history. The tie-in with reconnection and the Dungey cycle was rmly established in 1966 by Donald Fair eld, when he showed that substorms are closely associated with the IMF Bz (the north-south component). Southward IMF triggers subsolar recon- nection and generates frequent substorms, whereas northward IMF was associated with quiet intervals [59, 58]. De nitive evidence for magnetospheric reconnection came in in the 1970s, starting in 1971 with a Bengt Sonnerup paper on a magnetopause crossing by Explorer-12 [167], and later in 1978 from the ISEE spacecraft, from which Christo- 20 Figure 1.9: The Earth?s magnetosphere, and interactions with the interplanetary magnetic eld BIMF . Lines 1-9 outline the convection of magnetic ux known as the Dungey cycle. Reconnection occurs at the magnetopause (lines 1 and 1?) and in the magnetotail (lines 6 and 6?). The inset shows the footpoints of the respective eld lines on Earth. Reprinted with permission from Hughes (1995) [84]. ?1995 by NASA Science and Technical Information Division. 21 pher Russell and Rick Elphic identi ed ux tubes generated by reconnection in the magnetopause [141, 142]. (The nature of these ux tubes shall be discussed in further detail in ?2.2.1.) Similar structures were observed soon thereafter in the magnetotail [80, 56]. The relationship between reconnection in the magnetotail, current disruptions nearer to the Earth, and auroral events remains a signi cant area of current research and the primary mission of the THEMIS spacecraft [107, 4]. 1.5 Reconnection in laboratory, space, and astrophysical plasmas Although solar and magnetospheric plasmas served as the primary motivations for research in reconnection, the conversion of magnetic energy into plasma ows is in fact a ubiquitous process in many varied laboratory, space, and astrophysical plasmas. Following the success of Dungey?s reconnection model in space plasmas, Harold Furth, John Killeen, and Marshall Rosenbluth in 1963 wanted to study this new con- cept for applications in laboratory plasmas. In so doing, they determined conditions under which a collisional plasma with a magnetic eld reversal would be linearly unstable to the formation of x-lines and subsequent magnetic islands [64]. They named this mode the tearing instability (to be discussed in greater detail in ?2.1). This would prove to be important for the fusion community. The high tempera- tures required for fusion necessitated some means of con ning this very hot plasma. Tokamaks are one such attempt to accomplish this con nement using a helical mag- netic eld wrapped around a torus. However, early tokamaks exhibited instabilities 22 that would disrupt con nement, such as the sawtooth crash (so named because the temperature would steadily increase, then suddenly crash to a much lower temper- ature before rising again, forming the shape of a sawtooth when plotted in time) [182]. In 1975, Soviet scientist Boris Kadomtsev identi ed the sawtooth crash as a reconnection event [89]. A number of experiments have delved into the reconnection problem for ap- plications to both fusion and space plasmas. Some of these include the Madison Symmetric Torus (MST) at the University of Wisconsin [130], the Swarthmore Spheromak Experiment (SSX) at Swarthmore College [16], the Magnetic Recon- nection Experiment (MRX) at the Princeton Plasma Physics Laboratory [186], and the Versatile Toroidal Facility (VTF) at the Massachusetts Institute of Technology [54]. Reconnection may also play a role in the coronal heating problem. The problem was rst brought to light by the solar eclipse of August 7, 1869, during which Harkness and Young observed an inexplicable green emission line spectrum at 530:3 nm in the coronal spectrum [105]. Initially, this wavelength did not correspond to any known elements, so a new element called coronium was proposed [31]. Fi- nally, in the 1930s and 1940s, German astronomer Walter Grotrian [70] and Swedish physicist Bengt Edl en [52, 53] identi ed this wavelength with multiply-ionized iron, Fe13+. The discovery originally was not widely accepted because ionization to this level would require temperatures in the millions of kelvins, while the photosphere, much closer to the fusion energy source at the center of the sun, was observed at just 5000 K. One theory proposed by the French astrophysicist Evry Schatzman 23 suggested that Alfv en waves and magneto-acoustic waves launched by turbulence could carry energy into the corona, and shock waves dissipate that energy as heat into the corona [150]. A competing theory put forth by Eugene Parker in 1987 asserted that random footpoint motion could lead to numerous small reconnection events called nano ares [125, 126]. The magnetic energy is converted into plasma ows, but at such small scales, turbulence and viscosity quickly transforms it into heat. The power law behavior observed in statistical distributions of observed are sizes is consistent with coronal heating by this mechanism [27]. Nonetheless, this remains a wide open question. The planned spacecraft Solar Probe Plus, a NASA mission schedule for a 2015 launch, will approach the sun to within 8:5R (where the solar radius R = 6:955 105 km) from the surface of the sun and hopefully answer many of these lingering questions [65]. Although most in situ observational evidence for reconnection exists in the magnetosphere, where spacecraft can be directed towards the regions where recon- nection is predicted to occur, observations of reconnection exhausts in the solar wind recently have become prevalent. In a 2006 paper by Phan et al., three sepa- rate spacecraft { ACE, Cluster-3, and Wind { measured a magnetic eld reversal and reconnection exhaust in the solar wind [128]. The reconnection signatures in the three spacecraft were nearly identical despite spanning an astounding 390RE (where the Earth radius RE = 6378:1 km) in separation in the z direction (us- ing the coordinate system of Fig. 1.5). The observations suggest that a very large and nearly two-dimensional reconnection event can operate at a quasi-steady state. Other observations in the solar wind have suggested that reconnection can occur in 24 conjunction with interplanetary coronal mass ejections (ICMEs, or CMEs that have escaped the corona into the interplanetary medium) [69, 68]. Reconnection is expected to be important for other planetary magnetospheres just as on Earth. MESSENGER ybys of Mercury have shown that the Hermean magnetosphere is perhaps even more sensitive to reconnection, on extreme occasions possibly even having all of its dayside ux depleted by reconnection [162, 163]. In Jupiter, both the Galileo and Ulysses spacecraft found reconnection signatures in the Jovian magnetotail and magnetopause [143, 117]. Likewise, the Cassini mission has shown that reconnection occurs in Saturn?s magnetosphere [87, 110]. Further away from home, reconnection may also be taking place in the outer heliosphere. As the sun rotates, its dipole magnetic eld twists into the famed Parker spiral [123], causing the heliospheric current sheet to ap like a ballerina skirt as in Fig. 1.10. One recent theory proposes that out past the termination shock, where the solar wind transitions from supersonic to subsonic, these aps pile up and compress as they approach the heliopause, the border between the sun?s magnetic eld and that of the interstellar medium (ISM). The result is a sectored magnetic eld and thin current sheets that can undergo reconnection. This has been proposed as a possible source of anomalous cosmic rays (ACRs) [98, 41, 122]. The dissipation of magnetic energy can easily be generalized to various astro- physical plasmas as well. Stellar ares have been observed on a number of sun-like stars such as Kappa Ceti [138], binary star systems such as II Pegasi [40], and even on magnetars such as SGR 1806-20 [85]. Reconnection is also a possible mechanism for supernova shocks to release their magnetic energy to generate X-ray gas in the 25 Figure 1.10: The heliospheric current sheet undergoing the Parker spiral. Reprinted with permission from Jokipii and Thomas (1981) [88]. ?1981 by the Institute of Physics. galaxy [177]. Black hole accretion disks can also develop coronae with strong mag- netic elds, which naturally would also be prone to aring [14, 66]. Lastly, MHD models of the jets associated with rapidly spinning black holes suggest that the e ect of reconnection on the global topology of magnetic elds can hinder jet formation [6]. 26 Chapter 2 Magnetic islands in theory, simulations, and observations An important byproduct of reconnection is the topological structure called a magnetic island (or plasmoid in some literature), the result of the magnetic eld between two x-lines reconnecting into a closed structure. Islands can be a result of patchy reconnection, meaning the current sheet breaks up into multiple x-lines [99]. Fig. 2.1(a) exempli es a simple 2-D magnetic island. In our 3-D reality, magnetic islands take the form of ux tubes (or ux ropes), as in Fig. 2.1(b). Most reconnection simulations historically have been performed in 2-D, and observational evidence of extended x-lines in the solar wind [128] and of azimuthally long ux tubes in the magnetopause [61] suggests that, at least in some cases, the 2-D picture is a fair representation. The 3-D dynamics of ux tubes could be an important factor in other systems (e.g., the corona) but the 3-D nature of reconnection and of ux tubes is not yet well understood. 2.1 The tearing instability As mentioned in ?1.5, the theory of the tearing instability was rst developed by Furth, Killeen, and Rosenbluth [64]. In this derivation of the tearing mode, we start with a magnetic eld con guration called the Harris sheet [74], consisting of a 27 Figure 2.1: (a) A 2-D magnetic island enclosed by the separatrices of two x-lines. (b) A ux tube, the 3-D analogue of a magnetic island. current sheet separating a magnetic eld reversal: B0(y) = B0 tanh y w x^ (2.1) J0(y) = c 4 r B0 = cB0 4 w sech2 y w z^ (2.2) where B0 is the asymptotic magnetic eld and w the width of the current sheet. Both Bx and Jz0 are shown in Fig. 2.2. An important feature of this con guration is that it is a Vlasov equilibrium, i.e., a Maxwellian distribution of particles is a steady state solution to the Vlasov equation @f @t + v rf + q m E + v B c @f @v = 0: (2.3) Other similar con gurations (e.g., the current slab Bx0 = B0y=w or the sine sheet Bx0 = B0 sin( y=2w) for jyj < w, Bx0 = B0jyj=y otherwise) will give similar results. We shall also assume that a pressure gradient or guide eld (an out-of-plane compo- nent Bz0) exists to maintain pressure balance (i.e., n0T0 + B20=8 = constant) and ensure that the initial plasma is in equilibrium. 28 Figure 2.2: The Harris sheet Bx0 (solid line) and Jz0 (dashed line) described by Eqs. (2.1)-(2.2), for w = 1. Because the Harris sheet is a one-dimensional equilibrium (@=@x = @=@z = @=@t = 0), a quantity perturbed from its equilibrium value 0 can be written as (x; y; t) = 0 + ~ (y)e ikx i!t (2.4) for kz = 0 modes as we shall assume here. If we apply such a perturbation to the Harris sheet con guration in resistive MHD and linearize the induction equation, Eq. (1.11), we get @ ~B @t = r (~v B0) + c2 4 r2 ~B: (2.5) In the resistive layer, the narrow region of y in which ~By varies sharply, we write r2 @2=@y2, in which case the y-component of Eq. (2.5) becomes i! ~By = ikBx0~vy + c2 4 @2 ~By @y2 : (2.6) 29 This equation shows why, even when is very small for many of the plasmas we consider, the resistivity is essential for the tearing instability to occur. Without the resistivity term, in the reversal plane y = 0 where Bx0 = 0, Eq. (2.5) can be satis ed only if ~By = 0, and the ideal MHD equations will produce an Alfv en wave but no instability. The resistivity allows a non-zero ~By to develop across the current sheet and gives the plasma more freedom to nd ways to dissipate the magnetic energy in Bx0 through a ~By. We now derive the instability criterion for the linear tearing mode from the 2-D reduced MHD equations. We de ne the ux function such that B = z^ r +Bzz^ (2.7) and the stream function such that v = z^ r : (2.8) The former equation is valid since r B = 0; the latter likewise assumes that r v = 0. This assumption of incompressibility is broadly valid as long as the reconnection time scale is short compared with the magnetosonic propagation time across the system. Under these conditions, the magnetosonic wave smooths out the total pressure as reconnection proceeds so that local compressions do not take place and r v = 0. This assumption is most accurate during reconnection with a guide eld where compressions of guide eld rapidly propagate away from reconnection sites. With these de nitions, Amp ere?s law Eq. (1.7) becomes Jz = (c=4 )r2 , Faraday?s law Eq. (1.6) becomes cEz = @ =@t, and substituting into the resistive 30 Ohm?s law Eq. (1.10), we get for the z-component @ @t + v r = c2 4 r2 (2.9) The other reduced MHD equation comes from taking the z-component of the curl of the momentum equation, Eq. (1.3): 4 nmi @ @t r2 + v rr2 = B rr2 : (2.10) Linearizing Eqs. (2.9)-(2.10) with respect to perturbations ~ and ~ gives @ ~ @t Bx0 @ ~ @x = c2 4 r2 ~ (2.11) 4 n0mi @ @t r2 ~ = Bx0 @ @x r2 ~ d2Bx0 dy2 @ ~ @x : (2.12) We then multiply Eq. (2.12) by ~ , then integrate over all space: Z d3x4 n0mi ~ @ @t r2 ~ = Z d3xBx0 ~ @ @x r2 ~ Z d3x d2Bx0 dy2 ~ @ ~ @x : (2.13) Integrating by parts for each term and assuming that ~ and ~ vanish at 1 gives us Z d3x4 n0mir~ @ @t r~ = Z d3xBx0 @ ~ @x r2 ~ + Z d3x d2Bx0 dy2 @ ~ @x ~ : (2.14) From Eq. (2.11) we solve for @ ~ =@x: @ ~ @x = 1 Bx0 @ ~ @t c2 4 r2 ~ ! (2.15) and substitute into the latter two terms of Eq. (2.14) to get Z d3xn0mir~ @ @t r~ = 1 4 Z d3x @ ~ @t c2 4 r2 ~ ! r2 ~ 1 4 Z d3x 1 Bx0 d2Bx0 dy2 @ ~ @t c2 4 r2 ~ ! ~ : (2.16) 31 The rst term on the right hand side can be integrated by parts as before, with ~ ! 0 at 1. Then collecting the @=@t terms (using, for example, ~ @ ~ =@t = 12@j ~ j2=@t) gives an equation for the energy integral: @ @t Z d3x 1 2 n0mi r~ 2 + 1 8 r ~ 2 + 1 8 1 Bx0 d2Bx0 dy2 ~ 2 = Z d3x c2 16 2 r2 ~ 2 1 Bx0 d2Bx0 dy2 ~ r2 ~ : (2.17) The rst term on the left hand side is the kinetic energy of the perturbation. The latter two terms are the potential energy due to the magnetic eld perturbation WB = 1 8 Z d3x r ~ 2 + 1 Bx0 d2Bx0 dy2 ~ 2 : (2.18) The tearing mode is unstable when the perturbation releases magnetic energy, i.e., when WB < 0. The rst term in WB is positive-de nite and therefore stabilizes the tearing mode; the second term is negative, since for the Harris sheet described by Eq. (2.1), 1 Bx0 d2Bx0 dy2 = 2 w2 sech2 y w : (2.19) Therefore, the instability is driven by a strong current gradient. A simple scaling analysis on Eq. (2.18) readily produces the tearing instability criterion. The rst term scales as r2 k2, whereas (for our Harris current sheet con guration) the latter scales as B 1x0 d 2Bx0=dy2 1=w2, so the tearing mode is unstable when kw . 1: (2.20) In other words, the Harris sheet is more unstable to long-wavelength (small k) per- turbations. The physical reason for this criterion is that it requires energy to bend 32 Figure 2.3: A magnetic island generated by the tearing mode, with Bx = B0 tanh(y=w) and By = Bye t cos(kx) with B0 = 1, w = 1, By = 0:1, t = 0, and k = 0:5. The thick lines are the separatrices that de ne the borders of the magnetic island. eld lines, and short-wavelength perturbations are highly bent. On the other hand, for su ciently long wavelengths, the energy released by magnetic eld annihila- tion more than compensates for the energy required to bend the eld lines to the perturbation. When the tearing mode goes unstable according to Eq. (2.20), the perturbation will grow exponentially as long as it remains in the linear phase. If we write the real part of ~By at some particular time as ~By = Bye t cos (kx) (where By is approximately independent of y near the resistive layer y = 0), we can plot magnetic eld lines for this perturbation added to the Harris equilibirium Eq. (2.1). Fig. 2.3 shows how an elongated (kw = 0:5) magnetic island can grow from the tearing instability. 33 Figure 2.4: (a) A magnetopause crossing by ISEE-1 and ISEE-2 featuring the bipolar signatures in BN now identi ed as ux transfer events. (b) A qualitative sketch by Russell and Elphic of how they interpreted this event as a ux tube on the surface of the magnetopause. Reprinted with permission from Russell and Elphic (1979) [142]. ?1979 by the American Geophysical Union. 2.2 Magnetic islands in nature 2.2.1 Flux transfer events on the magnetopause One of the rst in situ observations to evidence magnetic reconnection was published by Russell and Elphic [141, 142], who inferred reconnection by the presence of magnetic ux tubes. The ISEE-1 and ISEE-2 spacecraft crossed the magnetopause on November 8, 1977, and they found bipolar behavior in the component of the magnetic eld perpendicular to the surface of the magnetopause, BN , highlighted in Fig. 2.4(a). Russell and Elphic called these structures ux transfer events (FTEs) and interpreted them as ux tubes on the surface of the magnetopause, as sketched in Fig. 2.4(b). 34 In the original Russell and Elphic FTE picture, the ux tube on each side of the magnetopause is aligned with the local magnetosphere or magnetosheath mag- netic eld, respectively, and the reconnected ux tube kinks at the reconnection site, forming an elbow. A few years later, Lee and Fu expanded on this picture, arguing that FTEs are the result of multiple x-line (patchy) reconnection at the magne- topause, possibly due to the tearing instability [99]. Another idea independently developed in 1988 by Scholer [151] and Southwood [169] proposed that a single x- line with a bursty (time-dependent) reconnection rate can also generate ux tubes. These three descriptions are visually summarized in Fig. 2.5 [61]. A recent study of ux transfer events observed by ESA?s Cluster spacecraft suggests that these ux tubes have a very long azimuthal extent, substantiating a 2-D picture of magnetic islands in opposition to the elbow reconnection model [61]. THEMIS observations of a 2007 ux transfer event found evidence for x-lines on both sides of the FTE [75]. The magnetic islands we shall consider in this thesis therefore shall be of the form proposed by Lee and Fu [99]. The interpretation of the bipolar signature of FTEs as a ux tube (or a mag- netic island) is consistent with a spacecraft trajectory in Fig. 2.3 along x, where the magnetopause surface is basically the y = 0 plane. Even though the spacecraft crosses the magnetopause, the FTE is in fact traveling along the current sheet much faster than the spacecraft. E ectively, ISEE is standing still while the FTE ows through it. We imagine the spacecraft at the magnetopause traveling towards posi- tive x and measuring By, corresponding to BN in Fig. 2.4(a). Initially, at the rst x-line near x = , By = 0. Then, as it passes through the island, the spacecraft 35 (a) (b) (c) (d) (e) (f ) Figure 2.5: Three di erent models for FTEs. (a)-(b) The Russell and Elphic elbow reconnection model [141, 142]. (c)-(d) The Lee and Fu patchy reconnection model [99]. (e)-(f) The Scholer [151] and Southwood [169] bursty reconnection model. The top row (a), (c), and (e) shows views along the magnetopause normal; the bottom row (b), (d), and (f) shows views tangential to the magnetopause. Black curves represent closed magnetosphere eld lines, red curves the magnetosheath eld lines, blue curves reconnected eld lines, and green curves the borders of the FTE. Reprinted with permission from Fear et al. (2008) [61]. ?2008 by the European Geosciences Union. 36 encounters a positive By. (Note that the magnetic eld is clockwise within the is- land to match Bx > 0 for y > 0 as required by Eq. (2.1).) Then, By goes to 0 at the o-line in the center of the island and proceeds to a negative By at the other end of the island. Finally, By returns to 0 at the right-hand x-line. The result is a bipolar By similar to those highlighted in Fig. 2.4(a). This will be discussed in greater detail in ?7.1. Flux transfer events remain an area of emphasis in space research because of their role at the beginning of the Dungey cycle. As spacecraft capabilities improve, we are starting to see FTEs as small as 200 km, generated within the Hall elec- tromagnetic eld region [178]. The upcoming Magnetospheric Multiscale Mission (MMS), with capabilities of resolving these inertial scales, will shine more light on the microphysics of reconnection and FTE formation. Meanwhile, simulations and observations continue to explore their creation [135, 38, 178], their internal structure [60, 188, 161], their evolution [121, 189], and the conditions under which they form [184, 185]. 2.2.2 Plasmoids in the magnetotail In 1979, Edward W. Hones, Jr. predicted similar structures in the magnetotail generated by a magnetic x-line 10 25RE from the Earth [79]. This prediction was borne out a few years later in 1984, when ISEE-3 detected these plasmoid structures owing tailward (anti-sunward) at roughly XGSM 220RE [80]. (In GSM coordinates, the origin is the Earth, XGSM points towards the sun, YGSM 37 Figure 2.6: A plasmoid structure in the magnetotail owing tailward. Based on the ow speed of 500 1000 km/s and the 30 min delay from observed auroral activity, it was estimated that the plasmoid was formed at XGSM & 70RE before owing past ISEE-3 at XGSM 220RE. Reprinted with permission from Hones et al. (1984) [80]. ?1984 by the American Geophysical Union. roughly duskward, and ZGSM roughly northward, so that the X Z plane contains the dipole axis.) This is illustrated in Fig. 2.6 [80]. As with the FTEs discussed in ?2.2.1, plasmoids in the magnetotail can be identi ed by a bipolar signature in BZ [116, 29]. These tailward owing plasmoids are likely generated by reconnection in the tail at 20 30RE [115]. Of course, earthward owing plasmoids have also been seen [190], even as a result from reconnection in the near tail XGSM 10RE [164]. More recently, spacecraft have been able to explore the small-scale x-line structure of the magnetotail plasma sheet, in particular the Hall electromagnetic eld and the 38 formation of secondary islands in the extended electron current sheet [51, 28]. An important result related to these islands in the magnetotail was the discov- ery that energetic electrons peaked in ux within them [29]. The observed energies of these highly energetic electrons (as high as 100 keV) cannot be explained by simply invoking the traditional reconnection out ow cA, even at its largest possible values corresponding to only about 2 keV. The data suggests that some other mecha- nism, likely involving magnetic islands, is responsible for electron acceleration. One such theory will be discussed further at the end of ?2.3. 2.2.3 Supra-arcade down ows in the solar corona For quite a while, the canonical picture for solar ares has appeared something like Fig. 2.7 [158, 180]. In the soft x-ray (SXR) loop, the observed radiation mostly comes from the spectral lines of thermal plasma heated to millions of kelvins. In the hard x-ray (HXR) region in Fig. 2.7, the radiation is dominated by brehmsstrahlung from nonthermal (accelerated) electrons decelerating as they come into contact with thermal ions. According to this model, the accelerated plasma gains its energy at a reconnection x-line above the loop, as observed by Lin et al. (2005) [100]. The downward reconnection out ow generates the hard x-rays, while upward reconnec- tion out ow can escape the corona as a coronal mass ejection. How is the plasma accelerated to such high energies? Observed emissions al- lude to the fact that the bulk of the magnetic energy goes into energetic (10 100 keV) electrons [101, 102, 57, 95]. RHESSI has observed electron energies > 39 Figure 2.7: The canonical single x-line model for solar ares. Reprinted with per- mission from Shibata et al. (1995) [158]. ?1995 by the Institute of Physics. 40 300 keV (and up to 10 MeV) and ions > 30 MeV (up to 1 GeV), meaning that an adequate theory must include a common acceleration mechanism for both species [160]. Yet as with the energetic electrons in the magnetotail [29] discussed in ?2.2.2, the energies observed here cannot be attained from a simple Alfv enic re- connection out ow jet. Neither is the parallel electric eld a suitable mechanism for particle acceleration because 3-D PIC simulations of guide eld reconnection show that it remains localized within density cavities along the separatrices [133]. Mount- ing evidence suggests that the single x-line model does not explain observations of energetic particles. An alternative to the single x-line model of Fig. 2.7 is one where bursty or patchy reconnection generates many magnetic islands. Observational evidence for this kind of picture is presented in Fig. 2.8, which shows a 2002 TRACE image of \tadpole"-like structures owing down into the post are loop [156]. Comparing these results with MHD simulations of localized reconnection on a current sheet, Linton and Longcope interpreted these \tadpoles" as reconnected ux tubes de- scending from the reconnection site higher in the corona [104]. McKenzie had seen similar structures using lower resolution Yohkoh data and came to a similar inter- pretation of these dark voids [111], later using an automated algorithm to perform a wide-ranging statistical study of supra-arcade down ows (SADs) seen by TRACE, SOHO/LASCO, SOHO/SUMER, and Hinode/XRT [112]. Other structures seen in post-CME arcades, seen as down owing shrinking loops called supra-arcade down- owing loops (SADLs), have been identi ed as instances of SADs from a perpendic- ular viewing angle [149]. 41 Figure 2.8: A 195 A ltergram of an April 21, 2002 are on the west limb, as recorded by TRACE. The vertical dimension of each panel is approximately 117 Mm. The post are loops visible in the lower left are emitting in Fe XII 195 A. The di use plasma cloud in the center (corresponding to the Fe XXIV 192 Aline) is penetrated by down owing \tadpole"-shaped voids, highlighted by the arrow in successive images. The fourth panel is a di erence image, indicating the change from 0147:37 UT to 0148:37 UT. Reprinted with permission from Sheeley et al. (2004) [156]. ?1995 by the Institute of Physics. 42 SOHO/LASCO images also have revealed very long current sheets left behind in the wake of an ejected CME [94, 173]. Large, bright, blob-like structures seen on these current sheets could also be interpreted as magnetic islands generated by bursty reconnection [100]. These current sheets are observed to be very thick, much thicker than the predictions of reconnection models employing classical or anomalous resistivity [30]. A possible explanation for the observed thickness is the presence of many magnetic islands along the course of the current sheet. 2.3 Recent theories and simulations of islands 2.3.1 Islands and fast reconnection in MHD Tearing mode notwithstanding, the theory of island formation remains incom- plete. The tearing mode is only a linear theory; it allows a small perturbation to grow, but the theory breaks down once the system escapes the linear phase of the instability. MHD simulations have shown that islands form for su ciently high Lundquist number S = LcA= > Scrit, empirically estimated at Scrit 104 [13, 97, 144, 22, 82]. Fig. 2.9(a) shows islands generated in a resistive MHD simulation with Lundquist number S = 3 106 [82]. A more recent theory by Loureiro et al. (2007) based on the reduced MHD equations predicts a plasmoid instability with growth rate S1=4cA=L (where L is the length of the current sheet) [106]. The theory is supported by MHD simulations [144], but again, only through the linear growth stage, before other dynamics such as island convection and coalescence become 43 Figure 2.9: Magnetic island formation in (a) a resistive MHD simulation with S = 3 106 [82], and (b) a full PIC simulation with Fokker-Planck collisions and S = 5000 11700 [36]. In both simulations, the color signi es out-of-plane current density Jz and contour lines correspond to magnetic eld lines. Reprinted with per- mission from Huang and Bhattacharjee (2010) [82] and Daughton et al. (2009) [36], respectively. (a) ?2009 by the American Institute of Physics. (b) ?2010 by the American Physical Society. 44 important. In 2009, Cassak et al. considered how a rescaling of Sweet-Parker reconnection that accounts for the presence of magnetic islands could enhance the reconnection rate by p N , where N is the number of islands [22]. This is because the di usion region (of length LSP ) is divided into N regions of length L0 LSP=N . If each segment can be described by the Sweet-Parker theory of ?1.2, then from Eqs. (1.13), (1.14), and (1.15), the thickness of each segment scales as = s c2L 4 cA (2.21) meaning that SP= p N where SP is the classical Sweet-Parker thickness [36]. Consequently, the global reconnection rate scales as E =L ESP p N . Fluid sim- ulations with high Lundquist number S supported this scaling [22]. It was thought that this could give fast reconnection if large current sheets produced many islands. The Loureiro et al. (2007) theory predicted that the number of islands would scale as N S3=8 [106], and S is generally very large for the systems in question, but his theory again applies only to the linear regime. A subsequent theory suggested that in the nonlinear regime, plasmoids dominate until the e ective Lundquist number for the shortened current sheet between plasmoids becomes small enough to sup- press the plasmoid instability Slocal . Scrit 104 [82]. According to this theory, the number of plasmoids would scale as N S=Scrit, predicting a reconnection rate weakly dependent on the Lundquist number: E S 1=2crit 0:01 [82, 181]. Despite the recent studies in how islands could boost the e ective reconnec- tion rate, Eq. (2.21) presents a signi cant drawback to these theories. All theories 45 of fast reconnection by islands requires the inter-plasmoid current sheets to thin out according to Eq. (2.21). However, Daughton et al. in 2009 showed using PIC simulations with a Fokker-Planck collision operator that as the inter-plasmoid cur- rent sheet thickness approaches the ion inertial length di, the transition to kinetic scales triggers Hall physics [36]. The reconnection electric eld escalates beyond the runaway limit, forming an electron current layer unstable to the continual forma- tion of new plasmoids. Substituting coronal parameters to Eq. (2.21), even using a global length scale L 109 cm, the Sweet-Parker current sheet thickness comes to 1 m, already smaller than di 4 m. Therefore, the inter-plasmoid cur- rent sheets should transition immediately into Hall physics before they even have a chance to thin out according to Eq. (2.21). This also suggests that secondary island formation probably involves kinetic e ects, and that a complete theory of island formation should include kinetic scale physics. 2.3.2 Secondary islands in collisionless guide eld reconnection Accordingly, island formation has also been seen in full particle-in-cell sim- ulations. For example, Drake et al. (2006) showed that component reconnection (including a guide eld Bz) becomes bursty [46]. In a collisionless plasma, the nature of component reconnection is quite di erent from the anti-parallel Hall re- connection discussed in ?1.3. The key di erence is that the reconnection electric eld Ez now has a component parallel to the magnetic eld [42]. The resulting par- allel acceleration of electrons produces a strong Jez, but the in-plane components of 46 Figure 2.10: Schematic of quadrupolar density perturbation in reconnection with a guide eld showing the density asymmetry across the dissipation region. Reprinted with permission from Drake and Shay (2007) [42]. ?2007 by Cambridge University Press. the Ek acceleration results in electron streaming along the newly reconnected eld lines. The electron ow across the current layer results in a density depletion on one side of the current sheet and a density enhancement on the other, as shown in Fig. 2.10. An alternative way to describe Fig. 2.10 is to consider that the guide eld Bz is now complemented by the quadrupolar Hall magnetic eld. If even a small Hall magnetic eld ~Bz perturbs the guide eld, the change in magnetic pressure pm = (Bz + ~Bz)2 8 B2z 8 Bz ~Bz 4 (2.22) 47 must be counteracted by a corresponding change in the plasma density ~n given by ~nT = Bz ~Bz 4 (2.23) in order to maintain pressure balance. (Note that in the absence of a guide eld, the change in magnetic pressure is second order in ~Bz, and so this density asym- metry does not occur.) Thus, the regions where the guide eld is enhanced by the quadrupolar Hall magnetic eld are accompanied by a density depletion, whereas the regions where the guide eld and quadrupolar Hall magnetic eld are of opposite sign show a density enhancement [20]. The con guration in Fig. 2.10 is characteristic of the kinetic Alfv en wave, which replaces the whistler wave described in ?1.3 as the mediator of fast reconnection. A consequence of this is that the separatrix with the enhanced density (the upper-right and lower-left of Fig. 2.10) carries most of the current, and so the electron current sheet with Jez = neevez becomes tilted along the density enhancement. This twist in the current sheet has been observed in simulations of component reconnection [176, 132, 133, 44], such as in Fig. 2.11. Furthermore, simulations have shown that the electron current layer becomes very narrow (width less than de) [77] and long in the vicinity of the x-line, ripe for the formation of secondary islands seen in Drake et al. (2006) [46]. Other kinetic simulations have seen secondary island formation without a guide eld as well [35, 63, 90, 93]. Fig. 2.9(b), for example, shows islands generated in a full PIC simulation with Fokker-Planck collisions with Lundquist number ranging from S0 = 5000 (based on the initial resistivity) to Smax = 11700 (due to electron 48 Figure 2.11: Contours of (a) the out-of-plane current density Jy (Jz in the coordi- nates used here) and (b) the ion density ni for a simulation of component reconnec- tion. Reprinted with permission from Pritchett and Coronoiti (2004) [133]. ?2004 by the American Geophysical Union. 49 heating in the current layer) [36]. Recently, 3-D PIC simulations exploring the electron physics of turbulent reconnection show that 3-D ux ropes develop from the tearing mode [37]. 2.3.3 Particle energization by the Fermi mechanism An important application of magnetic islands is the role they could play in par- ticle acceleration. In 2006, Drake et al. suggested that electrons could gain energy by a classic Fermi mechanism within contracting magnetic islands [46]. The process is analogous to a ball bouncing between two converging walls, gaining energy with each re ection. An elongated magnetic island will contract until its magnetic tension is released, with the ends closing in typically at the Alfv en speed cA. Consequently, super-Alfv enic particles within the island would potentially have the chance to tra- verse between both ends of the island multiple times. This process is illustrated for a test particle in a PIC simulation in Fig. 2.12(a). Fig. 2.12(b) shows how the particle gains parallel energy Ek = mv2k=2 with each bounce. When the island nally crosses the separatrix, the parallel energy is scattered into perpendicular energy E? as in Fig. 2.12(c). If the system consists of many volume- lling islands as sketched in Fig. 2.12(d), scattered particles can repeat this process inde nitely and attain very high energies.1 The rate of energy gain dEk dt = 2Ek vxB2x xB2 (2.24) 1Such a picture is possible in 3-D, for which resonance surfaces of the tearing instability (those where k B = 0) are not constrained to the y = 0 symmetry line. 50 Figure 2.12: Fermi acceleration of a test particle within a PIC simulation of a con- tracting magnetic island. (a) The test particle trajectory; the electromagnetic elds are frozen, but are still present, so the test particle still feels the island \contracting" with each re ection. (b) The particle?s kinetic energy E as it bounces in x. (c) The particle?s parallel and perpendicular velocities vk and v? in time. (d) A sketch of reconnection by many volume- lling islands. Reprinted with permission from Drake et al. (2006) [45]. ?2006 by Macmillan Publishers Limited. 51 is independent of mass, so this could be an e cient heating mechanism for both ions and electrons. If volume- lling islands dominate coronal current sheets as suggested in ?2.2.3, the Fermi mechanism could be a factor in electron acceleration. In the outer he- liosphere, where ACRs were unexpectedly observed well beyond the termination shock by Voyager 1 and Voyager 2 [170, 171], Opher et al. have suggested that sec- tored magnetic elds could undergo reconnection, generate islands, and accelerate particles into ACRs [122, 41]. Other mechanisms for particle energization within magnetic islands have been proposed as well. One theory asserts that electrons can be accelerated at a merging x-line, the site where two magnetic islands coalesce with one another [120]. The energetic electrons associated with plasmoids in the magne- totail discussed in ?2.2.2 also support particle energization in islands [29]. Power law spectra of energetic particles in the magnetotail observed by Wind are consistent with Fermi acceleration [118], although this remains an open question [55]. 2.4 The dynamics of many magnetic islands It is now apparent that a complete understanding of current sheets in the magnetosphere, solar corona, and other relevant space and astrophysical plasmas requires a proper understanding of magnetic islands. As computational capabili- ties increase, we have begun to see that secondary islands form at kinetic scales [46, 35, 63, 90, 93, 36]. Observations of islands within an extended electron current sheet in the Hall region also suggest that kinetic physics are key to their formation 52 [51, 28, 178]. Yet in the magnetopause, the magnetotail, and in the solar corona, what we most frequently observe are macroscale objects. Magnetic islands on the magnetopause can be as large as several RE [137, 146, 61] or even larger in the mag- netotail [56, 166]. Likewise, the down owing coronal voids in Fig. 2.8 are measured in megameters (di in the corona is on the order of meters) [156]. The large blob-like structures seen on post-CME current sheets are visible on SOHO/LASCO C3, a white light coronograph that images from 3:7 32R . An important question then is: How do magnetic islands grow from small kinetic scales into macroscale objects? To answer this question, we need to understand the dynamics of magnetic is- lands. To this end, we observe the secondary islands generated by PIC simulations of guide eld reconnection performed by Drake et al. [46]. The simulation is initialized with a perturbation that allows for just a single x-line on each current sheet. In Fig. 2.13(a), we see an elongated electron current layer form. The current layer becomes unstable to the formation of secondary islands in Fig. 2.13(b). As reconnection proceeds, the islands gain ux and grow larger in Fig. 2.13(c). By the end of the simulation in Fig. 2.13(d), the islands have convected away from the original x-line and have started to merge with the primary island. We also see a new secondary island starting to form in the lower current sheet. The dynamics of the magnetic islands observed in this simulation include their formation, growth, convection, and coalescence. Although these dynamics of magnetic islands have been investigated in preexisting literature (for example, formation [64, 46, 35, 106], growth [76, 5, 86], convection [34, 12, 159], and coalescence [134, 39]), in this thesis we begin to study these processes together in a single model. 53 Figure 2.13: The out-of-plane electron current density Jez in a PIC simulation of reconnection with a guide eld Bg = Bx0, shown at times (a) t = 11 1 ci , (b) t = 14 1ci , (c) t = 20 1 ci , and (d) t = 24 1 ci . Reprinted with permission from Drake et al. (2006) [46]. ?2006 by the American Geophysical Union. 54 However, the dynamics of islands in very large current layers are not yet well understood. On the one hand, even the largest particle-in-cell (PIC) simulations come nowhere near the system sizes of these current layers. In the magnetopause L 4000di, L 6000di for the magnetotail plasma sheet, and L 104 km 107di in the corona. On the other hand, L = 64di in Fig. 2.13 [46], and even the largest, most recent 2-D PIC simulations, only reach L . 1000di [36]. Although global uid simulations can encompass these large scales[135, 38], they fail to capture the small scale dynamics of reconnection within the dissipation region. This presents a challenge because it is extraordinarily di cult to perform su ciently large scale simulations of many magnetic islands while still capturing the separation of scales and kinetic e ects. In light of these limitations, this thesis will consider a novel statistical method for treating magnetic islands in large current layers. We shall de ne a distribution function that describes islands in the whole current layer and develop an evolution equation for that distribution based on the dynamics observed in Fig. 2.13. We then present steady state solutions, which show how the merger of smaller islands drives the growth of large islands. To validate this model, we shall perform large Hall MHD simulations of many islands and compare the resulting distributions of islands to the predictions of our model. We shall also explore the formation of secondary islands in PIC simulations similar to those shown in Fig. 2.13 and investigate how they are formed at small scales. Lastly, we continue the validation of our model with a large database of FTEs observed by the Cluster spacecraft. 55 Chapter 3 Formulation of a statistical model In order to be able to describe systems as large as the magnetopause, the magnetotail, or especially coronal current sheets, it is bene cial to try to reduce the number of degrees of freedom. To make an analogy with thermodynamics, it can be quite cumbersome to try to follow the state of every individual particle; it is much easier (and more illuminating) to deal with distribution functions of particles or even macro-state parameters such as temperature and density. In this case, it is very di cult to achieve through simulations the separation of scales necessary to understand island dynamics. Consequently, we might consider a distribution of islands, characterized by a few key but simple parameters, that describes islands over the whole current sheet of length L. 3.1 De ning the statistical distribution function f( ;A) Our choice of parameters derives from the relatively simple picture of a mag- netic island in Fig. 2.3. We choose to describe this island by the in-plane magnetic ux it contains and the area it encloses A. To be more precise, both of these quantities can be de ned in terms of the ux function such that B = z^ r .1 1In Ch. 2, the ux function was denoted by . Henceforth, we shall use for the ux function in order to distinguish it from the island parameter . 56 The in-plane magnetic ux of the island is just the di erence in from the x-line to the o-line: = Z o-line x-line dxBy = j X Oj: (3.1) The area A is simply the area enclosed by the separatrix (the emboldened eld line in Fig. 2.3). In reality, even the 2-D picture is never quite as simple as Fig. 2.3, because the separatrices for the two x-lines in general might not coincide. A more precise de nition in this more general case is discussed for our Hall MHD simulations in ?5.2, but the more relevant point is that regardless of the particular de nition, both and A are easily quanti able in simulations of magnetic islands such as those in Fig. 2.9. How do these parameters translate to the case of 3-D ux tubes? The area A is of course simply the cross-sectional area of a 2-D cut through the ux tube. With regards to the ux , it is worth di erentiating between the magnetic ux through such a cross-section R dA B (or in Fig. 2.3, R dABz) and the in-plane magnetic ux we use here. The former de nition of ux is used in some contexts, but for our purposes, the in-plane ux is the more fundamental parameter. Whereas the magnetic ux through a cross-section of the ux tube is related to the guide eld, the in-plane magnetic ux is a better indicator of the amount of reconnected ux. To a large extent, and A are su cient to characterize the state of an island because many of the important characteristics of the island can be described in terms of these two quantities. A simple case is the characteristic scale-size for the 57 island, which can be de ned as r = r A : (3.2) (We will eventually use r interchangeably with A as our dependent variable. The island need not be circular, however; r merely acts as a proxy for the more physical quantity A.) We assume that the plasma is basically incompressible, so that the total mass of plasma in the island is basically nA. A typical in-plane magnetic eld strength within the island is given by B = =r = p =A: (3.3) Lastly, we can calculate the magnetic energy within the island as the magnetic energy density multiplied by the island size W = (B2=8 )A 2=8: (3.4) Thus, we con dently model the islands using a distribution function f in terms of the island?s ux and area A. De ned in this manner, dN = f( ;A)d dA signi es the number of magnetic islands in the entire system that have ux in the range [ ; + d ] and area in [A;A+ dA]. The total number of islands is then N = Z 1 0 d Z 1 0 dAf( ;A): (3.5) 3.2 Rules governing island behavior We can expand the distribution function to include a time-dependence. Now, f( ;A; t) evolves in time in the phase space of ( ;A). We now establish rules for the behavior of magnetic islands, based on what is observed in simulations such as 58 in Fig. 2.13, and formalize mathematically how these rules a ect the distribution. Ultimately, our goal is to write down an equation for how f( ;A; t) evolves in time. In ?2.4, we described four dynamical process that we would like to describe in our model: island creation, growth, convection, and coalescence. We now consider those four processes here. 3.2.1 Creation, growth, and convection of islands Islands rst form at kinetic scales between the electron de and ion di skin depths in the current layers near x-lines [46, 35, 93, 36, 51, 28, 178]. The evolution equation therefore should include a source term S( ;A) at this scale. The most important feature of the source term is its magnitude SN = Z 1 0 d Z 1 0 dAS( ;A); (3.6) which describes the total number of islands generated at de to di scales per unit time. If there is a su cient separation of scales between kinetic and global scales (i.e., L di), SN is in fact the only feature of the source term of global importance. Although our choice of S( ;A) treats islands down to the de scale, the model remains valid even if the classical tearing mode without kinetic e ects acts as the source term. Furthermore, although we will use a log-normal distribution in and A, the particular shape of S( ;A) gets washed out by the time the distribution reaches global scales. The model must also account for the growth of magnetic islands due to recon- nection. Magnetotail observations reveal that plasmoids grow as they ow tailward 59 [76, 5, 86]. PIC simulations have shown that once reconnection reaches a nonlinear stage, the normalized reconnection rate plateaus at roughly " 0:1 [154]. Given this quasi-steady reconnection rate and an asymptotic reconnection magnetic eld of B0, the magnetic ux of an island increases at the rate _ = vinBx = "cAB0. Likewise, if the plasma is largely incompressible, the island?s characteristic radius r increases at a constant rate _r = "cA, and so the area increases at the rate _A = 2 rvin = 2"cA p A. We wish to model the convection of islands along the current sheet at the Alfv en speed [159]. However, f( ;A) describes the distribution islands over the whole current sheet of length L without regard for their positions in the current sheet. Ultimately though, those islands convect outwards and leave the system. For example, FTEs generated at the subsolar magnetopause convect poleward, eventu- ally colliding with the cusps, where they disintegrate due to secondary reconnection [121]. In the magnetotail, islands meet their end at the Earth or in the distant tail. The down owing coronal voids seen by Sheeley et al. are probably ux tubes that have convected down from a reconnection site higher up [156]. Therefore, as islands convect outwards at the Alfv en speed, a sink term models the convection of islands out of the system at the rate cA=L. 3.2.2 Properties of island coalescence We now describe the rules for island coalescence: the merger of two islands yields an island with an area A equal to the sum of the two initial areas and a ux equal to the higher of the two initial uxes. The area adds assuming incompress- 60 Figure 3.1: The results of a PIC simulation of island merger, showing the magnetic eld lines at (a) t = 0 1ci , (b) t = 0:3 1 ci , (c) t = 4:0 1 ci , and (d) t = 8:0 1 ci , where ci is the ion cyclotron frequency. The smaller island initially has 75% of the ux and 25% of the area of the larger island. ibility. The ux does not add because magnetic reconnection does not increase or decrease magnetic ux but simply changes its connectivity. Thus, eld lines from the island with more ux reconnect with those from the island with less ux until all of the latter?s ux is depleted. The simplicity of these rules re ect our choice of and A as the variables de ning our phase space. Fig. 3.1 shows results from a PIC simulation that demonstrates this in the simple case of two isolated ux bundles. The simulation was performed using the code P3D [187].2 Magnetic eld contours are shown at various times during the merging process, with the outermost eld line 2The computational methodology for P3D will be discussed in greater detail in ?6.1. 61 Figure 3.2: The maximum ux m and combined area Am of the coalescing islands in time t from the start of the simulation. The variations in and A have respective standard deviations of 2% and 7% from the mean. representing the boundary of the island for the purposes of computing area. Initially, while the two islands are separate, we de ne the maximum ux m = maxf 1; 2g and the combined area Am = A1 + A2. At some point, the merging x-line and the o-line of the smaller island meet, and the merging process is essentially complete, after which we simply de ne m and Am as the ux and area of the resultant merged island. Fig. 3.2 shows that both m and Am remained nearly constant throughout the simulation. These merging rules reveal why the merging process is energetically favorable: the dissipation of magnetic energy in the reconnection process. We can write the magnetic energy in an island as according to Eq. (3.4). Before the merger the energy is given by Wi / 21 + 2 2, and after the merger the energy (supposing 1 2 62 without loss of generality) is Wf / 22 < Wi. 3.3 An equation for f We now derive an equation for the island distribution f . The equation will parallel the collisional Boltzmann equation [15] @f @t + p m rf + F @f @p = @f @t coll (3.7) with island growth by quasi-steady reconnection playing the roles of p and F and island coalescence acting like a collision operator. As such, the growth terms will have the form _ @f=@ and _A@f=@A. 3.3.1 Mathematical formalism for the merging rules Modeling the merging terms will prove to be more complicated. As islands convect through the current sheet, we hypothesize that the probability of two islands merging with one another over some time interval t is roughly v t=L, where v is some merging velocity which is characteristically the Alfv en speed in the MHD limit. More generally, for islands larger than di, we want v to resemble the pre- dicted out ow speed for asymmetric reconnection, that is, the hybrid Alfv en veloc- ity v2 B1B2=4 nmi where constant density n is assumed [21, 175]. For smaller islands, down to the electron skin depth de, v2 1 2d2i =4 nmir 2 1r 2 2 k 2d2i c 2 A in agreement with the dispersion relation !2 = k4d4e 2 e = k 4d2ec 2 Ae = k 4d2i c 2 A of the whistler dynamics that dominate this regime [43]. Lastly, v ! 0 as r ! 0 would be desirable. We therefore write an ad hoc merging velocity of two islands with uxes 63 1; 2 and areas A1; A2 as v2( 1; A1; 2; A2) = 1 2r1r2(r21 + d 2 i ) 1=2(r22 + d 2 i ) 1=2 4 nmi(r21 + d2e)3=2(r 2 2 + d2e)3=2 (3.8) where the dependence on A1 and A2 is implicit in r1 and r2 according to Eq. (3.2). This expression is consistent with the prescribed limits for the cases r di, de r di, and r de, respectively. Now we consider the number of islands N jmrg+( ;A) formed by merging. A merged island with ux arises from an island of ux that has merged with another island with ux 0 . Likewise, an island with area A must come from an island with A0 < A that has merged with an island of area A A0. The probability of those two islands merging within a time t is given by v( ;A0; 0; A A0) t=L, so N jmrg+ = Z A 0 dA0f( ;A0) Z 0 d 0 v t L f( 0; A A0) A: (3.9) A similar analysis yields the number of islands lost through merging N jmrg ( ;A). Such an island is lost if it merges with any island of nite ux 0 and area A0. As before, the probability of merger depends on v( ;A; 0; A0) t=L, and so N jmrg = Z 1 0 dA0 Z 1 0 d 0 v t L f( 0; A0)f( ;A) A: (3.10) The proposed rate of merging is only an approximation, however, so we will intro- duce a dimensionless coe cient to the merging terms Kmrg. A value of Kmrg = 1 corresponds most similarly to a current sheet in which the islands are like point- particles and the current sheet is a string-of-beads. Of course in reality, islands occupy a nite volume, so Kmrg > 1 is likely. In Ch. 5, we will t the parameter Kmrg to simulation results. 64 3.3.2 The evolution equation The change in f due to merging is simply N jmrg= A t, with N jmrg given by adding Eqs. (3.9)-(3.10) multiplied by Kmrg. Combining this expression in di erential form with the change in f due to quasi-steady reconnection, the source term S( ;A), and the convective loss term proportional to cA=L, we get our evolution equation: @f @t + @ @ _ f + @ @A _Af = S( ;A) cA L f + Kmrg L Z A 0 dA0f( ;A0) Z 0 d 0v( ;A0; 0; A A0)f( 0; A A0) Kmrg L f( ;A) Z 1 0 dA0 Z 1 0 d 0v( ;A; 0; A0)f( 0; A0): (3.11) On the left-hand side, the time-derivative of f results from the growth in ux _ and area _A due to reconnection. On the right-hand side, we have the source term S( ;A) responsible for island creation, a sink term proportional to cA=L representing island convection out of the system, and the merging terms. Consistent with the merging rules, the merging terms in Eq. (3.11) preserve total area. (This is demonstrated in Appendix B.) 3.3.3 A non-dimensional form Although Eq. (3.11) appears to contain several parameters (the current sheet length L, the reconnection rate ", the merging coe cient Kmrg, and the source term strength SN), in reality it only has one free parameter. By normalizing time to t=t = L=cA, island ux to = = "B0L, island area to A=A = ("L)2, the merging 65 velocity to v=v = cA, and the distribution function to f=f = 1=Kmrg("L)3B0, then Eq. (3.11) reduces to a non-dimensional evolution equation @f @t + @ @ _ f + @ @A _A f = KmrgL cA SNS ( ; A ) f + Z A 0 dA 0f ( ; A 0) Z 0 d 0v ( ; A 0; 0; A A 0)f ( 0; A A 0) f ( ; A ) Z 1 0 dA 0 Z 1 0 d 0v ( ; A ; 0; A 0)f ( 0; A 0) (3.12) where S is a non-dimensional source term of unit magnitude: R1 0 d R1 0 dA S = 1. As suggested in ?3.1, so long as there is a su cient separation of scales { math- ematically, if S is non-zero only in a localized region of 1 and A 1 { the particular form of S (e.g., delta function, Gaussian, log-normal etc.) does not a ect the global solution f . It is now apparent that the sole free parameter in Eq. (3.12) is the coe cient of S , which we de ne as the normalized source amplitude S N = KmrgL cA SN : (3.13) The signi cance of this statement is that we can e ectively model any system solely by varying SN . For example, we might set L = 4000di in order to model the magnetopause, but if we want to model a coronal current sheet of length L = 106di with otherwise the same parameters (i.e., the same reconnection rate, etc.), the solution obtained by increasing SN by three orders of magnitude will su ce. 66 Chapter 4 Steady state solutions We seek steady state solutions to Eq. (3.11). We expect such a steady state solution to occur when the island source S( ;A) balances the sink cA=Lf . Such a solution could then be used to predict the distribution of islands in long current sheets such as the magnetopause, the magnetotail, coronal current sheets, and vari- ous other space and astrophysical systems. This is not to say that the current sheet is itself at a steady state. Rather, we suppose that the system after some time has reached a point where the distribution of islands is representative of the distribution at any other time, a sort of statistical steady state. The distribution of islands at the steady state shall be denoted f1( ;A). 4.1 The no-merging solution The complexity of the merging terms make Eq. (3.11) impossible to solve ana- lytically. We therefore rst consider the case where the merging terms are negligible. This is valid if there are few islands in the layer, e.g., if S( ;A) is small. For sim- plicity, we now change variables from the area A to the characteristic radius r, with A = r2, and consider the statistical distribution function F ( ; r). Island number must be conserved under this transformation, so dN = f( ;A)d dA = F ( ; r)d dr, 67 and F ( ; r) = f( ;A) dA dr = 2 rf( ; r2): (4.1) Substituting f = F=2 r, _A = 2 r _r, and @=@A = (2 r) 1@=@r into Eq. (3.11), the evolution equation without merging becomes @F @t + @ @ _ F + @ @r ( _rF ) = S( ; r) cA L F (4.2) where _r = "cA. The form of _r reveals why we have transformed from A to r. Whereas _A / p A, for our assumption of quasi-steady reconnection both _ and _r are constants. With a delta function source S( ; r) = SN ( 0) (r r0) (4.3) the steady state solution FG is given by the Green?s function FG( ; r; 0; r0) = SN "cA exp r r0 "L H(r r0) (( 0) B0(r r 0)) (4.4) where H(r) is the Heaviside function, which simply ensures that islands are no smaller than those generated by the delta function source. (No mechanism exists for making islands smaller.) The delta function in (( 0) B0(r r0)) ensures that all islands in this solution have a characteristic in-plane magnetic eld strength B0, equal to the ambient magnetic eld (assuming 0 = B0r0). The solution in r phase space is a decaying exponential along the line = B0r, starting at = 0 and r = r0. A key feature of Eq. (4.4) is the characteristic island size r^ = "L in the exponential. This length scale arises from balancing the rate of island growth _r = 68 Figure 4.1: A November 18, 2003 CME current sheet and helmet streamer imaged by the white light coronagraph LASCO C3 on the SOHO spacecraft. Times shown are in UT. Reprinted with permission from Lin et al. (2005) [100]. ?1981 by the Institute of Physics. "cA with a system transit time of L=cA. For example, with a reconnection rate of " 0:1 on the magnetopause where L 30RE, this simple model predicts islands of size 3RE. A survey of ux transfer events along the magnetopause [137, 146] determined that typical scale sizes are 0:5RE 2RE, for which r 1RE. Such island sizes are also consistent with those seen in direct observations of current sheets formed by CMEs. Fig. 4.1 shows what is believed to be a post-CME current sheet, featuring macroscale \blobs," which could be interpreted as magnetic islands that have grown to nearly one-tenth of the current sheet length [100]. 69 4.2 Numerical solution with merging and the r asymmetry If the merging terms are kept, the resulting integro-di erential equation must be solved computationally. As discussed in ?3.3.3, S N is the only free parameter in Eq. (3.12), so any parameter regime can be explored by solely modifying S N . As such, we choose L = 4000di (the size of the magnetopause), " = 0:1 (corresponding to fast reconnection [154]), Kmrg = 1, and vary SN over three orders of magnitude from SN = 0:01 ci to SN = 10 ci. Fig. 4.2 shows the evolution of the distribution function in ( ; r) phase space for SN = 1 ci, or S N = 4000. The source function S( ; r) generates islands at the de to di scales, in the far bottom left corner. Growth due to quasi-steady reconnection increases both and r at a constant rate, as discussed in ?4.1. In ( ; r) phase space, this gives growth along a line of slope _ = _r = B0. Therefore, the no-merging solution described by Eq. (4.4) would remain localized along the line = B0r, shown as the dashed diagonal lines in Fig. 4.2. On the other hand, the merging terms break this symmetry. Island coalescence increases area but not ux, so the distribution in Fig. 4.2 curves away from the diagonal = B0r. The r asymmetry introduced by the merging terms is an important aspect of this model and will receive much attention for the duration of this thesis. The integration of Eq. (3.11) eventually reaches the steady state island distri- bution, F1( ; r), as seen by the nearly identical Fig. 4.2(c)-(d). The steady state distribution is shown for four values of SN in Fig. 4.3. Note that the r asymmetry due to merging appears in various degrees among the four solutions. In the S N = 40 70 Figure 4.2: The time evolution of F ( ; r) with S N = 4000 at (a) t = 1000 1 ci , (b) t = 2000 1ci , (c) t = 7000 1 ci , and (d) t = 17000 1 ci . The contours are smoothed and on a logarithmic, such that F decreases by a factor of 4 for each contour going outwards. 71 Figure 4.3: The steady state distribution function F1( ; r) for (a) S N = 40, (b) S N = 400, (c) S N = 4000, and (d) S N = 40000. The contours are logarithmic as in Fig. 4.2. 72 case, the solution barely deviates from the no-merging solution along the = B0r diagonal. In other words, island coalescence is not playing a signi cant role. As S N increases, the degree of asymmetry follows suit. This should not surprise, since SN governs the number of islands in the system N , and the merging terms are the only N2 terms in Eq. (3.11). Naturally, the more islands present in the current sheet, the more likely island coalescence will play a role. Because islands gain area in merging, the large S N solutions also attain larger r. This will be shown more formally in the following sections. 4.3 Moments of the evolution equation 4.3.1 Total island number N The moments of Eq. (3.11) are of interest because the resulting equations predict quantities of global relevance. For example, integrating Eq. (3.11) in and A will produce an equation for the total number of islands N , de ned in Eq. (3.5). With the simplifying assumption that the merging velocity v cA, the moment equation for N becomes soluble. The analytical solution for N can then be compared with the numerical solution in Fig. 4.2. Applying the operator R1 0 d R1 0 dA to the left-hand side of Eq. (3.11), the @f=@ and @f=@A terms vanish since f ! 0 as ;A! 0 or ;A!1. (More intu- itively, those terms are responsible for island growth, so by necessity they conserve island number.) On the right-hand side, the source term simply becomes SN , and if we assume that v cA, the merging integrals / N2=2. The factor of 12 derives 73 from the simple fact that two islands coalesce into one. The resulting equation for N is dN dt = SN cA L N cA 2L N2: (4.5) Eq. (4.5) is a Riccati equation and can be solved analytically. We separate variables and integrate both sides to get t t0 = Z dN SN cA L N cA 2L N2 = 2L cA Z dN (N + 1)2 (2S N + 1) = 2L cA (2S N + 1) 12 tanh 1 N + 1 (2S N + 1) 1 2 ! (4.6) where S N is as de ned in Eq. (3.13). Solving for N(t), we get N(t) = (2S N + 1) 1 2 tanh t t0 ts 1 (4.7) where ts = 2L cA (2S N + 1) 12 (4.8) is the time-scale for the system to reach a statistical steady state. To solve for the integration constant t0, we employ the initial condition N(0) = 0, for which we nd that t0 = ts tanh 1(2S N + 1): (4.9) Substituting this back into Eq. (4.7), after some algebra we get N(t) = (Nf + 2) tanh t ts Nf + 1 + tanh t ts Nf (4.10) where Nf = (2S N + 1) 1 2 1 (4.11) 74 Figure 4.4: The number of islands N vs. t in the numerical solution (solid line; S N = 4000, as in Fig. 4.2) and predicted by Eq. (4.10) (dashed line; ts 89 1 ci , Nf = 88). is the asymptotic number of islands in steady state when t ts. The form of N(t) predicted by Eq. (4.10) shown in Fig. (4.4) compares favorably to that of the numerical solution, validating both the numerically integrated solution and the assumption that v cA. 4.3.2 Total island area AT Another quantity of interest is the total area of all the islands in the system AT = Z 1 0 dA Z 1 0 d Af( ;A): (4.12) An equation for AT can also be obtained by taking a moment of Eq. (3.11). However, a simpler approach is to take advantage of the known fact that the merging terms conserve AT . Therefore, we instead take the second moment of Eq. (4.2), the evolu- tion equation for F ( ; r) without merging. By applying the operator R1 0 R1 0 dr r 2 75 to both sides, we get dAT dt + "cA Z 1 0 d Z 1 0 dr r2 @F @r = d2iSN cA L AT (4.13) where, as in the moment equation forN , the @F=@ term vanished because F (0; r) = F (1; r) = 0, and we have used the delta function source given by Eq. (4.3) with r0 = di. The second term can be integrated by parts: Z 1 0 d Z 1 0 dr r2 @F @r = Z 1 0 d r2F ( ; r) 1 r=0 Z 1 0 d Z 1 0 dr2 rF ( ; r) (4.14) where, again, the former term vanishes since F ( ; 0) = F ( ;1) = 0, and the second term is of course the rst moment of F : rT = Z 1 0 dA Z 1 0 d rF ( ; r): (4.15) Thus, Eq. (4.13) becomes dAT dt = d2iSN + 2 "cArT cA L AT : (4.16) At late time, when the system has reached a statistical steady state, dAT=dt = 0 and the source term is negligible. (Newly created islands are of negligible size compared to the existing islands.) Balancing the remaining two terms, we nd that AT = 2 "LrT : (4.17) One might be tempted here to take the rst moment of Eq. (4.2) and obtain closure for the moment equations by solving for rT . Unfortunately, this does not work because the merging terms cannot be neglected. Whereas simple rules exist for N and AT , rT does not simplify so easily in the merging terms. 76 4.4 Solutions at large A Of greatest interest in the distribution of islands are the largest, most easily observed islands. The behavior of these curves for the largest islands can be deduced from Eq. (3.11) for large A. As usual, simplifying the merging term requires some assumptions. We again assume that the merging velocity v cA, but also that the bulk of the islands are much smaller than A. Consequently, if we de ne Z 1 A dA f(A) Z 1 0 dA f(A) (4.18) where f(A) = Z 1 0 d f( ;A): (4.19) is the distribution of islands in A with the dependence integrated away. A similar assumption is made of , so that R 0 d 0 R1 0 d 0. The merging terms can then be written as @ f @t mrg KmrgcA L Z 1 0 dA0 f(A0) f(A A0) f(A) Z 1 0 dA0 f(A0) KmrgcA L Z 1 0 dA0 f(A0) f(A A0) f(A) : (4.20) where of course the source term does not contribute since it only generates small A islands. Again, since the majority of the islands are small compared to A, the contributions to R1 0 dA 0 f(A0) come primarily from A0 A. A Taylor expansion for f(A A0) about A gives: f(A A0) f(A) A0 @ f @A + :::: (4.21) 77 Then Eq. (4.20) simpli es to @ f @t mrg KmrgcA L Z 1 0 dA0A0 f(A0) @ f @A KmrgcA L AT @ f @A (4.22) With this expression for the merging terms, applying the R1 0 d operator to Eq. (3.11) gives an evolution equation for f at large A: @ f @t + @ @A _A f + KmrgcA L AT @ f @A = cA L f (4.23) The third term on the left side, which arises from the merging terms, describes how large islands grow in area by devouring smaller islands. The coe cient KmrgcAAT=L is the rate at which the total area AT of all the smaller islands is consumed. Of course, AT as de ned by Eq. (4.12) is clearly dependent upon f , but we will henceforth treat AT as a constant determined by Eq. (4.16) and the moment equations. By balancing the second and third terms on the left side we obtain a characteristic length scale above which growth via reconnection dominates growth via merging. Accordingly, we would expect F exp r r^ (4.24) with r^ = "L in that regime, in accordance with Eq. (4.4) from the solution without merging. Recalling that _A = 2 "cAr, we nd that the transition occurs at RT = KmrgAT 2 "L = KmrgrT (4.25) from Eq. (4.17). In reality, by the very de nition of rT , all islands in the system must have r < rT , so unless Kmrg 1,1 this regime will never be attained. Never- 1We will nd in ?5.2 that this is not the case. 78 theless, we will nd it constructive to think of the solution as steadily approaching exponential-like behavior as we consider larger and larger r. In particular, we will nd that Eq. (4.24) is close to valid for large r, with r^ = 1 d dr ln F1 (4.26) approaching "L as r !1. Transforming Eq. (4.23) into an equation for F and substituting in RT , we have @ F @t + "cA @ F @r + "cART 1 r @ F @r F r2 = cA L F (4.27) from which we can solve for the steady state solution F1(r) = Z 1 0 d F1( ; r) (4.28) since Eq. (4.27) is now just a rst-order di erential equation in r. The result is Z d F1 F1 = 1 "L Z dr "LRT r2 r(r +RT ) = 1 "L Z dr "L r + RT "L r +RT 1 (4.29) ln F1 F0 = ln r "L + RT "L 1 ln r +RT "L r "L (4.30) F1 = Cr r +RT RT ="L 1 exp r "L (4.31) where F0 and C are constants of integration. As previously suggested, the exponen- tial behavior of Eq. (4.31) will dominate the polynomial factors for su ciently large r. This self-consistently validates our assumption in Eq. (4.18). Fig. 4.5(a) shows the distributions of islands in radius F1(r) for each SN (the -integrated solutions from Fig. 4.3). The distribution function is shown on a log- 79 Figure 4.5: (a) The steady state distribution function in r, given by F1(r) as de ned in Eq. (4.28) for various SN . (b) The average magnetic eld strength B as a function of island radius r for various SN . 80 linear plot, so the nearly straight lines for large r are indicative of the exponential behavior in F1 predicted by Eq. (4.31). The characteristic in-plane magnetic eld B = =r will also be a ected by island merging because merging increases island size but not ux. The largest islands, which may have attained their sizes from multiple mergers, typically have a weaker in-plane B. For larger SN , where merging plays a more prominent role, this e ect is more pronounced. This is borne out in Fig. 4.3(a), in that the steady state solution for larger SN tilts farther away from the = B0r diagonal. Fig. 4.5(b) shows, for various SN , that the average magnetic eld strength as a function of island radius B(r) = R1 0 d r F1( ; r) R1 0 d F1( ; r) (4.32) is weaker for the largest islands, especially for larger SN . In order to maintain force balance with the outside plasma, a weakened in-plane magnetic eld should be accompanied by a compressed guide eld Bz. Enhanced core elds are routinely observed in FTEs [141] and in magnetotail plasmoids [165]. It is now evident that both island merging and quasi-steady reconnection con- tribute to the growth of islands into macroscale objects. In the following chapter, we will further explore the competition between these two distinct mechanisms for island growth using a Hall MHD simulation of a large current sheet. This simulation is the rst of its kind to investigate both mechanisms for island growth in a system of many islands. 81 Chapter 5 A Hall MHD simulation of many islands We shall use the Hall MHD code F3D to test the model described in Ch. 3 against a simulated 2-D current sheet of length L 1638:4di. It is worth noting, however, that the simulation is not a perfect replica of the systems described by our model; some of the dynamics included in the statistical model are missing from the simulation. Nonetheless, it is useful in validating those aspects of the model that it does describe accurately, in particular the two mechanisms for island growth: quasi-steady reconnection and merging. For the rst of these mechanisms, the growth of islands is governed by a nor- malized reconnection rate ". The characteristic radius r = p A= increases at the rate _r = "cA and the ux increases at the rate _ = "cAB0. In full particle-in-cell simulations of reconnection, after reaching a nonlinear stage, the normalized re- connection rate often plateaus at roughly " = 0:1 [154]. Although resistive MHD simulations do not show reconnection rates this fast, the inclusion of the Hall term has been shown to enable reconnection rates comparable to those including kinetic e ects explicitly [11]. Therefore, the Hall MHD simulations should describe accu- rately the island growth by quasi-steady reconnection in the system. With regards to the merging terms, although the incompressibility assumed by the model is not explicitly enforced by the code, in practice the density variations observed in Hall 82 MHD simulations of reconnection do not appear to be strong enough to signi cantly alter the merging rules described in ?3.2, namely that the areas add but the ux of the merged island only takes on that of the larger of the two original islands. This quantitative di erence in the two competing mechanisms for island growth will allow us to distinguish between them in the simulation. In particular, based on Eq. (4.25), Kmrg governs the relative importance of these two processes, so an important goal of these simulations is to t to a value for Kmrg consistent with the results of the simulation. Unfortunately, the Hall MHD code is not as e ective in modeling the source term S( ;A). PIC simulations have shown that in reconnecting current layers, islands form and are convected out, leaving behind a thin, elongated current sheet that is again unstable to island formation [46, 35, 63, 90, 93]. We therefore require S( ;A) to generate islands continuously and at a steady rate. This process does not occur naturally in a Hall MHD simulation of reconnection. MHD simulations generate magnetic islands spontaneously for su ciently large values of the Lundquist number, S & 3 104 [13, 8], but the onset of Hall reconnection during island formation sweeps away secondary islands, leaving a single large x-line [157]. Instead, we initialize the current sheets with a small perturbation to the magnetic eld. The perturbations (with wavelengths down to Lx=256) eventually grow into hundreds of islands. Although the simulation does not properly describe spontaneous secondary island formation, the islands that do form as a result of the initial perturbation are quite small, comparable in size with those generated by a tearing mode instability. Also, this Hall MHD simulation does not model the convective loss term of Eq. (3.11) 83 because it employs periodic boundaries along the out ow. Periodicity is necessary because the simulation does not model island formation correctly yet, and it is preferrable that the islands remain in the simulation domain for as long as possible since the goal of the simulations is to explore the growth and dynamics of large numbers of magnetic islands. 5.1 F3D computational details The 2D simulations were performed using the compressible two- uid code F3D, which solves the Hall MHD equations [155]: @n @t = r Ji (5.1) @Ji @t = r JiJi n + J B T n rn (5.2) @B0 @t = r E0 (5.3) E0 = 1 n J B0 1 n Ji B (5.4) B0 = (1 d2er 2)B (5.5) J = r B (5.6) where Ji is the ion ux, ue = (Ji J)=n the electron velocity, and de = c=!pe the electron skin depth. The mass ratio me=mi = 1=25, and the grid scale is 0:1di, enough to marginally resolve the electron skin depth de = 0:2di. The system size is Lx Ly = 1638:4di 204:8di with periodic boundary conditions. Magnetic elds and densities are normalized to their asymptotic values far from the current sheets, B0 and n0 respectively. (Quasineutrality is assumed, so that ni ne). Time is 84 normalized to the ion cyclotron period, t0 = 1 i = mic=eB0, length to the ion skin depth L0 = di = p mic2=4 n0e2, velocity to the Alfv en velocity v0 = cA = B0= p 4 n0mi, electric eld to E0 = cAB0=c, and temperature to T0 = mic2A. The electron charge and ion mass are both normalized to unity. The initial con guration consists of double Harris sheets of width w0 = di: Bx(y) = 8 >>>>< >>>>: B0 tanh y 34Ly w0 y < Ly 2 B0 tanh y 14Ly w0 y > Ly 2 (5.7) There is no initial guide eld (Bz = 0). The density at the center of the current sheets is 1:5n0 and falls to 1:0n0 to balance magnetic pressure. The initial con gura- tion also includes a magnetic perturbation with wave numbers up to k = (5=32)d 1i . This perturbation acts as the seed for magnetic islands. The maximum k pro- duces perturbations with wavelength Lx=256. This ensures that each current sheet will produce & 100 islands, su cient for a statistically signi cant analysis. The simulation includes neither viscosity nor resistivity explicitly, but does include a fourth-order di usion term for B in Faraday?s law, Eq. (5.3). The fourth-order dif- fusion coe cient is initially 2 10 5d4i ci but later, in order to prevent grid-scale instabilities, is increased to 5 10 5d4i ci after the islands have grown from the initial perturbation to a discernible size. The time step starts o as t = 0:04 1ci but is adjusted to t = 0:025 1ci at the time of the increase of the fourth-order di usion coe cient. 85 Figure 5.1: Snapshots of Jz in the Hall MHD simulation showing island formation, growth, and merging at (a) t = 420 1ci , (b) t = 520 1 ci , (c) t = 620 1 ci , and (d) t = 686 1ci . Values of Jz shown are capped at Jz = 0:6n0ecA. 5.2 Simulation results and analysis Fig. 5.1 shows the time-evolution of this system. The initial perturbations grow into magnetic islands that undergo the dynamics described above. The islands begin as very small perturbations that rst grow into discernible islands in Fig. 5.1(a). In Fig. 5.1(b) these islands have reached scales large enough so that islands on a single layer interact with one another. The full merger of several islands is evident by Fig. 5.1(c). The nal state of our simulation is shown in Fig. 5.1(d). Fig. 5.2 shows the positions of all the o-lines in the lower current sheet in time. The convergence of two or more o-lines indicates the coalescence of those islands. 86 Figure 5.2: Location of all o-lines in time. Horizontal dashed lines correspond to snapshots of Jz in Fig. 5.1. 5.2.1 Characterizing the ux and size of islands An automated algorithm goes through successive frames of the simulation and tracks the size and ux of each of the magnetic islands in a particular current sheet. The tracking algorithm calculates the ux function (x; y) de ned such that B(x; y) = z^ r (x; y) +Bz(x; y)z^ (5.8) Contours of the ux function uniquely de ne the in-plane magnetic eld. For a horizontal cut of the ux function cut(x) along the current sheet, the symmetry across the current sheet ensures that a local extremum in cut(x) corresponds to either a saddle point (x-line) or a local extremum (o-line) in (x; y). For example, using the lower current sheet in our simulations, the x-lines (o-lines) correspond to local minima (maxima) in cut(x). (Fig. 5.2 tracks the positions of the o-lines 87 according to this de nition.) A magnetic island can now be de ned by associating it with a particular x- line and a particular o-line. To do this, x-lines and o-lines are paired o by starting with the most highly embedded x-line, say, at xX;1, and after considering each of those o-lines located within its separatrices on both the left and the right, pairing it with the o-line xO;1 that minimizes j cut(xX;1) cut(xO;1). This is repeated for the second most highly embedded x-line, and so on, eventually matching xX;i with xO;i for i = 1; 2; :::; n for all n x-lines. Fig. 5.3(a) demonstrates this pairing scheme for a simple example. Note that de ning islands in this way allows for islands to be contained entirely within other islands. This allows for a more realistic description of the merging process. If the x-line between the two islands is pushing the two islands apart, as in Fig. 5.3(b), this de nition introduces a small error in that the lesser island (island A on the left, in green), despite being very nearly the same size as the dominant island (island B on the right, in red), has a border completely within the dominant island. These two islands will both grow at similar rates, and their di erence (signi ed by the region on the left surrounding island A but belonging to island B) becomes eventually negligible compared to their overall size. On the other hand, if the two islands are merging with one another as in Fig. 5.3(c), then the size of the dominant island (island A, in green) steadily increases as it consumes the lesser island (island B, in blue). Meanwhile, island B gradually shrinks until it is fully subsumed within island A. Once all of the x-lines and o-lines have been paired o , then the ux i and 88 Figure 5.3: (a) A simple cartoon example of how magnetic islands can be de ned by an x-line, o-line, and separatrix, all of the same color for a particular island. The corresponding cut(x) is shown below its current sheet, x-lines aligned with local minima and o-lines with local maxima. (b) A similar cartoon for two islands with a reconnecting x-line in between them pushing them apart. (c) Another example of two islands, now with a merging x-line in between. The lesser island shrinks as it is consumed by the dominant island. 89 area Ai of magnetic island i are simply given by: i = cut(xO;i) cut(xX;i) (5.9) Ai = Z Ri dA (5.10) where Ri is the region de ned by the separatrix of the x-line at xX;i for island i on the side that encompasses its o-line at xO;i, but subtracting the regions of other islands entirely contained within Ri. 5.2.2 Tracking islands in r phase space Using this technique for any given time slice, each island can be plotted in the r phase space as in Fig. 5.4. The selected times correspond to the snapshots in Fig. 5.1, with a green x corresponding to a particular magnetic island with ux and characteristic radius r. By following the trajectories of individual islands, we can see both mechanisms for island growth at play. During normal island growth (quasi-steady reconnection), an island?s trajectory through the r phase space in Fig. 5.4 is generally along a diagonal with slope B0 as reconnection injects ux into the island and consequently increases its area. Occasionally, a merger between two islands will push the dominant island o that diagonal into higher r, while not a ecting its ux . Meanwhile, the lesser island gravitates back towards the origin as it is consumed by the dominant one, losing both ux and area and eventually disappearing. The merging process therefore breaks the r symmetry and allows islands to move above the symmetry diagonal = B0r. The region below the symmetry diagonal in Fig. 5.4 is prohibited since no mechanism exists for increasing 90 Figure 5.4: The distribution of islands in r phase space in the Hall MHD simulation overlaid upon the the integrated numerical solution shown in red. Each island is marked by a green x { overlaid on top of a numerical solution to Eq. (3.11) at times (a) t = 420 1ci , (b) t = 520 1 ci , (c) t = 620 1 ci , and (d) t = 686 1 ci . The numerical solution in red shows the evolution of Eq. (3.11) on a logarithmic scale with the parameters L = 1638:4di, " = 0:055, SN = 1:5 ci, and Kmrg = 6. This gure is also available as a movie in the online auxiliary material in Fermo et al. (2011) [62]. 91 the ux without also proportionally increasing the scale size r. In other words, the model predicts an allowed region, < B0r (5.11) that is consistent with all of the islands in the simulation. 5.3 Matching to the numerical solution The results of the simulation allow us to study the two mechanisms for the growth of large islands by examining their trajectories in r space and also to benchmark Eq. (3.11). Using the system size L = 1638:4di, we empirically match the numerical solution of Eq. (3.11) to determine appropriate values for the reconnection rate " and the merging coe cient Kmrg in the simulation. The quasi- steady reconnection rate " governs the speed with which the distribution travels along the main diagonal = B0r. In particular, since growth due to reconnection is the only mechanism that allows for growth in , a simple empirical t for _ = "cAB0 yields ". A non-trivial numerical solution to Eq. (3.11) requires a source of island cre- ation. The rate of island creation in Eq. (3.11) is SN = R1 0 dr R1 0 d S( ; r). How- ever, since the Hall MHD simulations do not produce islands naturally, this method is not a reliable predictor for SN in actual space plasmas. (Future PIC simula- tions may produce a better model for predicting this quantity.) The choice of SN will instead re ect a value that generates a total number of islands N(t) = R1 0 dr R1 0 d f( ; r; t) consistent with the number of islands in the Hall MHD simu- 92 lation exceeding some threshold ux min 0:5B0di. Although most of the islands arise directly from the initial perturbation, rapid island growth / " proceeds only after an initial period of stagnation. Di erent islands reach this stage of growth at di erent times; only after starting its rapid growth phase do we count an island towards the total number of islands N . For most islands, this growth phase was in a rougly 60 1ci interval (between 370 1 ci and 430 1 ci in Fig. 5.2). We can also t the merging coe cient Kmrg, which controls the relative mag- nitudes of the merging and island growth terms and drives the r asymmetry. We adjust Kmrg until the asymmetry of the numerical solution accounts for all of the islands in the Hall MHD simulation. Choosing " = 0:055, SN = 1:5 ci, and Kmrg = 6 produces the numerical solution shown in red in Fig. 5.4, or alternatively in the online auxiliary material for Fermo et al. (2011) [62]. The fact that Kmrg is greater than unity emphasizes that merging plays an even more important role than was hypothesized by the original model described in ?3.3.1. The distributions of islands in the Hall MHD simulation (at the times of the panels shown in Fig. 5.1) are overlaid on top of the numerical solution of Eq. (3.11) in red in Fig. 5.4 as both evolve in time. The distribution of islands follows the numerical solution quite well. The number of islands tracked in the simulation still is not enough to interpolate this data to a smooth distribution function, so this is largely still just a qualitative comparison. Generating a smooth two-parameter distribution function from discrete data would require still many more islands, especially in the regions of our distribution that are of greatest interest (i.e., the largest islands). Nevertheless, the behavior of the 93 islands in our Hall MHD simulation shows that Eq. (3.11) e ectively describes island growth by quasi-steady reconnection (along the diagonal in r phase space) and coalescence (adding to the characteristic radius r but not its ux ). It is clear that island merging, and the subsequent asymmetry in r phase space described in ?4.2, plays an important role in the dynamics of this system. 94 Chapter 6 Island formation in a PIC simulation of guide eld reconnection In the previous chapter, we explored our model?s two mechanisms for island growth using a Hall MHD simulation. A better understanding of the generation mechanism for magnetic islands (and consequently of the source term SN in our model) is also highly desirable, and has been explored in other recent literature [46, 106, 144, 37]. The results of the present study point towards PIC simulations as the preferred method. Global simulations of the magnetopause, which often show at most a few FTEs at a time [135, 38, 121], might lack the resolution to see the small-scale FTEs generated at kinetic scales. To this end, we return to the simulations performed by Drake et al. discussed in Ch. 2 and shown in Fig. 2.13 [46]. In those simulations, reconnection with a guide eld led to elongated electron current sheets as described in ?2.3.2, within which secondary islands developed. In this chapter, we perform similar simulations at high resolutions for the purpose of investigating how these secondary islands rst form at scales of the electron skin depth. The 2D simulations were performed using the particle-in-cell code P3D, which evolves the electromagnetic eld using the full Maxwell equations and steps particles forward using the Lorentz force law [187]. 95 6.1 P3D computational details The basic equations in the code use a normalization based on a reference density n0 and magnetic eld B0 { for our reconnection simulations, the peak Harris sheet density (corresponding to about 100 particles per cell in our simulation) and the asymptotic upstream Bx. Given n0 and B0, we can normalize lengths to the ion skin depth di = c p mi=4 n0e2, time to the ion cyclotron period 1 ci = mic=eB0, velocities to the Alfv en speed cA = B0= p 4 min0, the electric eld to E0 = B0vA=c, and temperatures to T0 = miv2A. With these normalizations, the relevant equations for P3D are Maxwell?s equations @B @t = r E (6.1) @E @t = c2(r B J) (6.2) r B = 0 (6.3) r E = c2 ; (6.4) the equations of motions for ions dx dt = v (6.5) d( v) dt = E + v B; (6.6) and the equations of motions for electrons dx dt = v (6.7) d( v) dt = mi me (E + v B) (6.8) 96 where = e(ni ne), J = e(nivi neve), and = 1= p 1 v2=c2.1 In our simulation, e = 1, c = 15, and mi=me = 25. The equations of motion advance the particles relativistically in the Lorentz force law using the Boris algorithm [10]. Faraday?s law steps the magnetic eld and Amp ere?s law steps the electric eld using a trapezoidal leapfrog algorithm, for which spatial derivatives are taken to second order. It turns out that this method also ensures that r B = 0 remains true for all times, as long as it holds true in the initial conditions. Gauss? law, on the other hand, can be violated due to accumulated discretization errors, so a correction to the electric eld is regularly applied using a multigrid Poisson solver [179]. The simulation domain is Lx Ly = 102:4di 51:2di with periodic boundary conditions, using 8192 4096 cells, ensuring that we comfortably resolve the electron skin depth de = 0:2di. The time step is 0:0025 1ci . The initial con guration consists of double Harris sheets of width w0 = di: Bx(y) = 8 >>>>< >>>>: B0 tanh y 3Ly=4 w0 y < Ly=2 B0 tanh y Ly=4 w0 y > Ly=2 (6.9) and a uniform guide eld Bg = 2B0. A magnetic eld perturbation in each current sheet places x-lines at (Lx=4; 3Ly=4) and (3Lx=4; Ly=4), producing a single primary magnetic island on each current sheet. An ambient background density of 0:2n0 supplements the Harris sheet density necessary to maintain pressure balance. The initial distribution is a Maxwellian with Te = 0:4mic2A for the electrons and Ti = 1We distinguish between an individual particle?s velocity v and the bulk ion or electron ow in a particular cell vi or ve. 97 Figure 6.1: The out-of-plane current density Jz at t = 87 1 ci depicting the tilted electron current sheet. Compare with the simulation by Pritchett and Coroniti (2004) [133] in Fig. 2.11(a). 0:1mic2A for the ions. 6.2 A non-tearing mechanism for island generation Just as in the simulations of guide eld reconnection by Drake et al. (2006) [46], our simulation produced a tilted electron current sheet along the separatrices, as described in ?2.3.2 and depicted in Fig. 6.1. These current layers elongate and become unstable to the formation of secondary islands. In Drake et al. (2006), the creation of secondary islands is attributed to the tearing instability, citing that k 4:0d 1i and w 0:13di such that kw 0:5 < 1 as required by Eq. (2.20). This is in accord with most of the present literature based on MHD theory [64, 17, 106], 98 MHD simulations [144], 2-D PIC simulations without a guide eld [35, 63, 93], and most recently, in 3-D reconnection simulations [37]. However, we shall consider here whether a di erent mechanism can be responsible for the generation of magnetic islands. 6.2.1 Vortical ow in a secondary island The impetus for such a consideration is illustrated in Fig. 6.2, which shows one such secondary magnetic island that had originally formed in the electron current sheet. The structure appears to be a simple magnetic island in Fig. 6.2(a), but a closer inspection reveals a surprising feature: vortical electron ows around the island. In Fig. 6.2(b), we see similarities to Kelvin-Helmholtz vortices associated with shear ows [91, 183]. Is it possible that the structure in Fig. 6.2 started out not as a magnetic island, but as a Kelvin-Helmholtz vortex? To answer this question, it would be bene cial to make a case study out of a single magnetic island, following it from its birth in the electron current sheet until it is expelled from the x-line region into the primary magnetic island. The island shown in Fig. 6.2 was born very close to the primary x-line around 4 1ci prior to the time shown. For this reason, the island continued to grow for some time without convecting away from the x-line. However, having been born so close to the primary x-line, it would be di cult to distinguish a tearing mode responsible for the island?s birth from the primary x-line. A more interesting case study, then, is an island that was born along the tilted electron current sheet 99 Figure 6.2: (a) The out-of-plane current density Jz and contours of the magnetic eld near the x-line at t = 98 1ci . (b) A zoom-in of the region within the green box of (a), with arrows showing the electron ow ve. 100 slightly away from the x-line, where it might be easier to distinguish between a uid ow instability and the tearing instability. 6.2.2 Maturation of a de-scale vortex into a magnetic island In particular, we observe an island along the tilted electron current sheet that started out quite small in Fig. 6.3, at the scale of the electron skin depth de = 0:2di, and 5di away from the primary x-line. Note in Fig. 6.3(a) that the Jz enhancement within the green box, which we might be inclined to call an \island," is in fact not an island at all. At this scale, within the electron current layer, the structure is still decoupled from the magnetic eld. In fact, along the separatrix and away from the x-line, the current sheet is not close enough to the magnetic eld reversal region to trigger the tearing instability. The lack of any in ow of plasma to either side of the structure in Fig. 6.3(b) corroborates this story. Instead, we nd that (after transforming into the frame of the out ow of the primary reconnection x-line v = 2cA) the electrons at this very early stage exhibit vortical ows, more consistent with a Kelvin-Helmholtz vortex than with a magnetic island generated by the tearing mode. To verify the feasibility of such an explanation, Fig. 6.3(c) shows vex in the vicinity of this structure. Because it exists near the separatrix, there exists a sharp velocity gradient across the current sheet from vex 0 to vex 2cA. The velocity shear is shown clearly in a vertical cut of vex, Fig. 6.3(d), along with the out-of- plane current sheet. Unsurprisingly, the vortex forms in the vicinity of the strongest 101 Figure 6.3: (a) The out-of-plane current density Jz and contours of the magnetic eld near the x-line at t = 90 1ci . (b) A zoom-in of the region within the green box of (a), with arrows showing the electron ow in the frame of the out ow, ve + 2cA. (c) The reconnection electron out ow vex. (d) Vertical cuts of vex (solid curve) and Jz (dashed curve) through the center of the green box in (c), with dashed vertical lines denoting the top and bottom boundaries of the green box. 102 Figure 6.4: (a) The out-of-plane current density Jz and contours of the magnetic eld near the x-line at t = 91 1ci . (b) A zoom-in of the region within the green box of (a), with arrows showing the electron ow in the frame of the out ow ve + 2cA. velocity shear combined with the largest current density. As the vortex ows away from the x-line, it appears to grow, as seen in Fig. 6.4. This growth is due to plasma owing out from the x-line and piling up at the vortex. Although the vortex is itself owing outwards, plasma accelerated by Ek streams along the current sheet and crashes into the vortex in Fig. 6.4(a). Another source of growth is that by t = 91 1ci , a vortex that was born slightly upstream of the vortex observed in Fig. 6.3 has collided with it, and the two are now spiraling 103 Figure 6.5: (a) The out-of-plane current density Jz and contours of the magnetic eld near the x-line at t = 92 1ci . (b) A zoom-in of the region within the green box of (a), with arrows showing the electron ow in the frame of the out ow ve + 2cA. around one another in Fig. 6.4(b). It is also worth pointing out that the vortex has now grown larger than a de, and consequently is now beginning to recouple to the magnetic eld in Fig. 6.4(a). In Fig. 6.5(a), the recoupling to the magnetic eld has progressed even further so that the structure appears to be a true magnetic island. The coupling is still incomplete, however, as evidenced by the fact that the magnetic island is o -center from the Jz enhancement. Fig. 6.5(b) shows that the vortical ows have remained 104 intact, even though it is no longer a single coherent vortex. Unfortunately, we are not able to follow this particular structure much further because by now it has been ejected and is starting to interact with the primary island. Nonetheless, Fig. 6.2 showed that the ow vortices persisted for quite a long time in an island that had remained largely stationary. (Note that no frame transformation was necessary in Fig. 6.2(b) as was necessary for Figs. 6.3(b)-6.5(b).) 6.3 Implications for future work The evidence from this simulation suggests that a uid ow instability can be responsible for secondary island formation in component reconnection. The con- ditions under which this instability is dominant remain uncertain. To determine speci cally whether these are Kelvin-Helmholtz vortices, as they appear to be, or the result of a type of streaming instability such as an electron-electron two-stream instability [172] or the Buneman instability [18], further studies of these types of simulations are warranted. This distinction between islands generated by the is important for trying to understand the source term in our model. One might try to model the source term with linear MHD theories of plasmoid formation [106], but the results of this simulation and others [46, 35, 37] show that a complete picture of the dynamics of magnetic islands must start at scales as small as the electron skin depth de. If islands at these small scales are very numerous (corresponding to large S N and large N), then solutions to our evolution equation will be in regimes where merging plays 105 an important role, as was found in our Hall MHD simulation of Ch. 5. Arguably, a more important question is whether the large scale current sheets in nature reside within these regimes of our model. In the following chapter, we shall address this question using observational data of ux transfer events observed by Cluster. 106 Chapter 7 Cluster observations of ux transfer events As discussed in ?2.2.1, ux transfer events (FTEs) were rst identi ed by Russell and Elphic as ux ropes along the surface of the magnetopause, among the rst observational evidence of reconnection [141, 142]. We equate them here with the magnetic islands seen in 2D simulations. The behavior of simulated FTEs in a global hybrid MHD simulation conducted by Omidi and Sibeck is consistent with that of the islands in our model [121]. The FTEs form (generally near the subsolar region), grow, merge, and convect poleward until they are ejected into the cusps. We therefore treat the magnetopause as a very long single current sheet with a length (south to north) of L 30RE 4000di. In this chapter, based on this model of FTEs we shall compare observations by the four Cluster spacecraft to the predictions of our statistical model. 7.1 Dissecting the bipolar BN signature The distinct FTE signature is a magnetic eld component normal to the mag- netopause, BN , that exhibits a bipolar signature [141, 142]. Fig. 7.1 shows a sample BN pro le for a ux transfer event. This data was taken from the ESA Cluster Ac- tive Archive, and also appeared in a study by Wang et al. [184]. The FTE exhibits the characteristic bipolar signature in BN of Russell and Elphic [141]. This data 107 Figure 7.1: A sample ux transfer event observed by Cluster 4 at 0420 UT on 19 February 2001. The event occured at (5:8; 4:7; 9:0) RE in GSM coordinates. was obtained from the uxgate magnetotometer aboard Cluster spacecraft number 4 on a magnetopause crossing on 19 February 2001 at around 04:20 UT. 7.1.1 Extracting r and from the bipolar BN During a three-year dayside phase of Cluster (2001-2003), the four spacecraft made numerous magnetopause crossings during which more than a thousand ux transfer events (FTEs) were identi ed for a statistical study by Wang et al. [184]. From each BN pro le and the plasma ow velocity, one can derive the approximate size of the FTE and the ux it contains. If the bipolar BN signature is approximated by a sine wave, then we only need the time elapsed between peaks, tpk pk, the peak- to-peak magnitude of the BN oscillation, BN;pk pk, and the plasma ow speed vFTE. Fig. 7.1 shows how tpk pk and BN;pk pk were calculated for the 19 February 2001 108 event. From these, a characteristic radius for the FTE can be calculated as rmeas = vFTE tpk pk (7.1) and its magnetic ux can be approximated by integrating half of the sine wave: meas = Z tpk pk 0 dtvFTE BN;pk pk 2 sin t tpk pk = 1 rmeasBN;pk pk: (7.2) As an example, the particular FTE whose BN pro le is shown in Fig. 7.1 has vFTE = 186 km/s, tpk pk = 44 s, and BN;pk pk = 29 nT, which yields rmeas = 8214 km = 1:29RE and meas = 2:38 105 nT km. This approach shows that an estimate for the scale size and ux of magnetic islands can be obtained from observational data, subject to a few assumptions. Consequently, there exist a few caveats to this approach for which we should account. 7.1.2 Magnetic eld normalization In our model and in the simulations, the magnetic eld is normalized to the asymptotic magnetic eld upstream of the reconnection site B0, but the FTEs in our database span three years and many di erent solar wind and magnetospheric condi- tions. Therefore, in order to make a valid comparison with the model?s predictions, the magnetic eld BN;pk pk should be normalized to the reconnecting component of the tangential (to the magnetopause) magnetic eld surrounding the FTE, which we denote Bt;surr. The quantity Bt;surr obtained from Cluster, however, may include a guide eld component, for which we cannot account. The magnetic eld normal- ization might therefore be overestimated if guide eld reconnection is occurring in 109 the magnetopause. Determining the guide eld requires knowledge of the ux rope orientation, which is itself a very di cult observational challenge. Another concern is that the database of FTEs provides one of the Bt;surr values for either the magnetosheath or the magnetosphere side, depending on which side of the magnetopause the particular spacecraft was on when it encountered the FTE. The proper normalization requires Bt;surr for both sides. (Of the 1,098 events, 730 of them measured Bt;surr on the magnetosheath side, 368 on the magnetosphere side.) For reconnection with asymmetric magnetic elds, the reconnection rate scales with the harmonic mean of the two upstream magnetic elds [21]: B0 = 2BsheathBsphere Bsheath +Bsphere : (7.3) Since our database only provides either Bsheath or Bsphere, the proper normalization B0 may vary from the measured Bt;surr. After presenting the data we shall discuss how these normalization issues could potentially a ect our data. 7.1.3 Dependence on spacecraft trajectory Another limitation to consider is the uncertainty of the proximity of the space- craft?s trajectory to the center of the FTE. In principle, the method prescribed here should be most accurate for a trajectory that goes straight through the center of the FTE along the surface of the magnetopause, and more prone to error farther away from the center of the FTE. In actuality though, it is the FTE that passes over the spacecraft. Cluster 4 was traveling at an average speed of about 2:3 km/s when the 19 February 2001 FTE in Fig. 7.1 passed over the spacecraft, whereas the FTE ow 110 velocity was measured as 186 km/s in a direction e ectively along the surface of the magnetopause (within 9?). Consequently, the spacecraft trajectory through the FTE is essentially parallel to the magnetopause surface. In the context of an island in the Hall MHD simulation performed in Ch. 5, such a trajectory corresponds to a horizontal cut through a magnetic island. A typical island from the Hall MHD simulation performed in Ch. 5 is shown in Fig. 7.2(a). A typical spacecraft trajectory through the FTE at a distance 10di from the magnetopause surface (highlighted by a horizontal solid white line) would measure the bipolar BN signature shown in Fig. 7.2(b). For this particular trajec- tory, rmeas can be calculated from the peak-to-peak separation and meas from the peak-to-peak BN according to Eqs. (7.1)-(7.2). Performing this calculation for ev- ery horizontal cut across the magnetic island yields rmeas and meas as functions of y, the spatial separation between the trajectory and the current sheet. Both rmeas and meas as functions of y are shown in Fig. 7.2(c) and Fig. 7.2(d), respectively. The actual size rtrue and ux true (determined as in ?5.2) are displayed as dashed lines, for comparison. The measured values for both quantities are quite accurate when the trajectory passes through the island proper, even without going through the center. Far from the island, this method could overestimate the scale size and underestimate the ux, but in such cases where Cluster only passes by the outer edge of the FTE, the observed BN pro le will be weaker and generally more di cult to discern from the background noise. The signal-to-noise ratio is much lower in observational data such as Fig. 7.1 than in MHD simulations such as in Fig. 7.2(b). Therefore, such events are less likely to be recognized as FTEs by Cluster and should 111 Figure 7.2: (a) Jz for a sample magnetic island from the Hall MHD simulation. (See Fig. 5.1.) (b) The bipolar BN signature a spacecraft would measure from passing through this magnetic island traveling along the solid white horizontal line in (a), a distance 10di from the center of the main current sheet. To the right are (c) the scale size rmeas and (d) the magnetic ux meas that a spacecraft would measure by horizontally passing through the island a distance y from the current sheet. The dashed lines denote the actual scale size rtrue = (Atrue= )1=2 and magnetic ux true. 112 Figure 7.3: A distribution of the scale sizes of 1,098 ux transfer events detected by Cluster between 2001 and 2003. The dashed curve ts an exponential tail with r^ = 5277 km to the tail of the distribution (beyond its peak). not signi cantly skew the statistics. 7.2 A study of 1,098 FTEs 7.2.1 An exponential tail in r The evolution equation, Eq. (3.11), normalizes length scales to "L. Direct measurements of the reconnection rate and magnetopause length are not possible, so in order to normalize the data properly, we approximate the numerical steady- state solution to Eq. (3.11) for the largest islands with an exponential tail for large r according to Eq. (4.24). Fig. 7.3 shows that the distribution of FTEs in r agrees well with an exponential t for r^ = 5277 km. A numerical steady-state solution of the our evolution equation with S N = SNKmrgL=cA = 24000 yields an exponential tail with r^ = 2:64"L. (We now use Kmrg = 6 to match the results from ?5.3.) Equating r^ with our FTE histogram so 113 that r^ = 2:64"L = 5277 km, we estimate that "L 2000 km. Using L 30RE for the magnetopause, one can also indirectly estimate the average reconnection rate to be " 0:01. 7.2.2 Asymmetry in r phase space Fig. 7.4(a) shows most of the 1,098 ux transfer events accordingly normal- ized, r to "L = 2000 km and to Bt;surr"L. (Two events with or r outside of the box are excluded.) These data points are overlaid on the aforementioned steady-state numerical solution to Eq. (3.11) with normalized source amplitude S N = SNKmrgL=cA = 24000. Qualitatively, the distribution of FTEs compares fa- vorably with that predicted by the model, in that the distribution of islands appears to exhibit behavior consistent with island growth due to quasi-steady reconnection (along the symmetry diagonal) and merging (upward in phase space towards higher r). The large majority of the FTEs observed by Cluster fall into the region above the = Bt;surrr diagonal, as described by Eq. (5.11). The 52 outliers below the diagonal (in red) have meas > Bt;surrrmeas. To account for these outliers, Fig. 7.4(b) shows the average in-plane magnetic eld within the FTE meas=Bt;surrrmeas versus Bt;surr. In this plot, we discover that the 52 outliers were normalized to a comparatively small Bt;surr, almost all less than the mean Bt;surr of 23:4 nT. Furthermore, 42 out of the 52 outliers (or 81%) measured Bt;surr in the magnetosheath, where typically Bsheath < Bsphere. In ?7.1.2, we pointed out that the proper normalization for the magnetic eld should have 114 Figure 7.4: (a) The distribution of ux transfer events in r phase space, overlaid on a numerical steady state solution to Eq. (3.11) for appropriate parameters. Flux transfer events above the = B0r diagonal are in green, whereas those in the prohibited region below the diagonal are marked in red. (b) The (normalized) average in-plane magnetic eld within the FTE versus the absolute (non-normalized) Bt;surr. The horizontal dashed line corresponds to the diagonal dashed line of (a). Most of the outlier events have small Bt;surr. 115 accounted for Bt both in the magnetosheath and in the magnetosphere. When the measured Bt;surr is quite small, it is quite likely that Bt is larger on the other side of the magnetopause. If this is the case, then Eq. (7.3) dictates that B0 > Bt;surr by as much as a factor of two. Although the data necessary to make this correction are not available, the fact that all of the outliers have low Bt;surr suggests that the proper normalization could push many of those outliers back into the allowed region described by Eq. (5.11). On the other hand, if guide eld reconnection is occurring, the presence of a guide eld would have the opposite e ect, since Bt;surr would include the guide eld component as well as B0. Further observational work can be done to ascertain the guide eld and magnetic eld asymmetry in these events to obtain the corrent normalization for the magnetic elds, but obtaining this data remains a di cult challenge for observationalists. 7.3 Interpretation of the results Normalization issues notwithstanding, the strong asymmetry in r phase space indicates that, as in the Hall MHD simulations of Ch. 5, merging is a prominent factor in island dynamics. Yet for merging to play a signi cant role, very many islands must be present at any given time, most of them at small scales. How can we reconcile this statement with the precipitous dropo in FTEs smaller than roughly 4000 km in Fig. 7.3? We attribute this dropo not to a lack of FTEs . 4000 km but to the decreased likelihood of a spacecraft passing through them. Besides the indirect evidence for the preponderance of smaller FTEs on the 116 magnetopause, it should also be noted that 4000 km does not correspond to any physically signi cant scale for ux tube generation. Even assuming a simple linear tearing mode that produces FTEs typically with kw 1, where w di 50 km is the current sheet width as in Eq. (2.20), then islands can be generated by the tearing mode as small as 2 =k 2 w 300 km. The results of Ch. 6, simula- tions of collisionless reconnection [46, 35, 37], and observations of FTEs [178] and magnetotail plasmoids [51, 28], all point to the generation of these structures at small scales. Therefore, Fig. 7.3 probably does not paint the whole picture. A more likely scenario is that FTEs rst form at de or di scales, then grow to macroscales & 4000 km. These smaller FTEs are probably much more numerous than those which can actually be observed by the spacecraft. 117 Chapter 8 Conclusions Magnetic reconnection is a widespread phenomenon in laboratory, space, and astrophysical plasmas, responsible for the rapid conversion of magnetic energy into plasma ows. Magnetic islands { or their 3-D analogue, ux tubes { are the prod- uct of patchy (multiple x-line) reconnection and have been observed in the magne- topause, the magnetotail, and in coronal post- are loops. Recently, they have been the subject of much scrutiny because of the role they could play in enhanced re- connection rates, turbulent reconnection, and particle energization. Nonetheless, a uni ed picture of magnetic islands, from their birth at kinetic scales to their growth into observable macroscale objects, has been lacking. In this thesis, we have at- tempted to bridge that gap with a model that treats islands on a statistical level for very large current sheets. 8.1 Summary of the model and relevant simulations Because the current sheets in question are so large, we have proposed to model islands with a statistical distribution function f( ;A). We pattern the evolution of f in time after the behavior of magnetic islands observed in simulations and observations of magnetic islands: creation of islands at di or de scales, growth via quasi-steady reconnection, convection out of the current sheet, and coalescence of 118 islands. These four processes are formalized mathematically into Eq. (3.11), our evolution equation for f . We consider steady state solutions of Eq. (3.11) to corre- spond to the state of the current sheet when the system has itself reached a statistical steady state. Of the four kinds of behaviors considered, island coalescence had the most profound e ect on the steady state solutions of Eq. (3.11). While island growth by quasi-steady reconnection added to the ux and to its size proportionately, merging introduced a r asymmetry. The magnitude of that asymmetry corresponds to the relative signi cance of the merging terms, dictated by the total number of islands N , or equivalently, by the normalized rate of island creation S N . We nd that the rate of growth for the largest islands is governed both by the quasi-steady reconnection rate " and by the presence of smaller islands to be devoured by merging. These two competing mechanisms for island growth were the subject of study for our Hall MHD simulation of a large current sheet with many islands. The tra- jectories of islands in r phase space was consistent with island growth by both quasi-steady reconnection (adding both area and ux along a diagonal in r) and merging (adding only to the area). Consequently, the predicted r asymmetry was borne out in these simulations. We have also used the Hall MHD simulations to benchmark the model by comparing a numerical solution of Eq. (3.11) to the distribution of islands in the simulation. In particular, in tting a value to the merging coe cient Kmrg, which regulates the relative importance of island merging and quasi-steady growth by reconnection, the result that Kmrg 6 in our sim- ulations underscores the importance of merging in very large systems with many 119 islands. Whereas the Hall MHD simulation was capable of modeling the large simula- tion domains necessary to explore island growth, a PIC simulation was more suitable for investigating the small scales in which islands are created. Following upon the simulations of guide eld reconnection in Drake et al. (2006) [46], we found that secondary islands are created when the tilted electron current sheet becomes nar- row and elongated. Surprisingly, it was discovered that the islands did not appear to be generated by a tearing mode, but by a ow instability. The islands start o as a de-scale ow vortex disconnected from the magnetic eld in a region of high velocity shear. As out owing plasma piles up at these vortices, they grow and start to recouple to the magnetic eld. 8.2 Comparing statistical studies of FTEs and SADs Observations of 1,098 ux transfer events [184] from the four Cluster spacecraft are also consistent with the predictions of the model. The islands in the Hall MHD simulation and the FTEs observed by Cluster show good qualitative agreement with the predicted distributions of islands. More particularly, the preponderance of these islands reside in the allowed region predicted by Eq. (5.11). Just as in the Hall MHD simulations, this asymmetry is further evidence that island merging plays a signi cant role in the dynamics of the magnetopause. In this scenario, patchy reconnection generates many FTEs at small scales. This suggests that reconnection is quite bursty at the magnetopause. Further- 120 more, the bulk of these islands will be quite small, perhaps too small for spacecraft instrumentation to resolve or to distinguish from noise. Even a large, detailed sta- tistical study of FTEs such as that performed by Wang et al. [184] is likely to miss the preponderance of FTEs at small scales. Despite the dropo in FTEs smaller than roughly 4000 km in Fig. (7.3), there is little reason to believe that ux tubes should spontaneously form at that particular size. Most likely, FTEs are generated at much smaller scales, whether by the tearing mode or otherwise, and then grow into the macroscale objects observed by Cluster. We can also compare the results of our statistical study of magnetopause FTEs with another statistical study of magnetic islands observed in a very large current layer, namely, the work by McKenzie and Savage on supra-arcade down ows in post- are coronal current sheets [148]. (Recall the discussion of these structures in ?2.2.3, and in particular, Fig. 2.8.) As part of the work for her Ph.D. thesis, Sabrina Savage wrote a semiautomated algorithm to recognize these supra-arcade down ows in images of post-CME are arcades. By tracking them through successive movie frames, they characterized these ux tubes by their speeds, sizes, and magnetic ux and generated histograms for this data [112]. The right hand side of Fig. 8.1 shows histograms for the areas and uxes for SADs and SADLs observed by the SXT, XRT, and TRACE satellites [147, 148]. The resulting histogram could be described as consistent with exponential tails in r or as described by Eq. (4.24). However, this assertion comes with many caveats. The histogram was generated from a composite of many di erent ares of di erent sizes observed by three di erent instruments with di erent resolution 121 Figure 8.1: Histograms for the (a) areas and (b) uxes of SADs and SADLs observed by SXT (S), XRT (X), and TRACE (T), accompanied on the left by quartile plots for each instrument?s measurements of areas for SADs (left) and SADLs (right). Reprinted with permission from Savage and McKenzie (2011) [148]. ?2011 by the Institute of Physics. 122 capabilities at di erent viewing angles. The quartile plots on the left show the varying ranges observed by each instrument. Nonetheless, the plots are suggestive of the reasonableness of an exponential distribution. To address some of these concerns, Fig. 8.2 shows histograms of area and ux for SADs (excluding SADLs) observed by SXT [113]. The TRACE images have higher resolution than SXT and accordingly revealed smaller SADs (as seen in their respective quartile plots in Fig. 8.1(a)), but their TRACE data set only included 23 SADs, insu cient for a statistically signi cant analysis. In comparison, 122 SADs were detected in their SXT data. By tting to the SXT cumulative distribution functions, McKenzie and Savage concluded that the areas of SADs exhibit a log-normal distribution, while their uxes are consistent with either a log- normal distribution or an exponential. However, as with the lack of FTEs. 4000 km in Fig 7.3, it is possible that a selection bias is responsible for the lack of SAD areas . 30 Mm2 in Fig. 8.2. Near the resolution limitations of SXT, di erentiating these smaller SADs from the background becomes more challenging. In a system with di on the order of meters, it is highly unlikely that ux tubes are being generated spontaneously at (30 Mm2)1=2 5 Mm scales. Whether such a consideration would make either an exponential or a power-law distribution more likely remains to be determined. 123 Figure 8.2: (a) Histograms for the areas and uxes of SADs observed by SXT, shown with a tted log-normal distribution (dash-dotted) and for the latter an exponential distribution (dotted). (b) Cumulative distribution functions of SAD areas and uxes with corresponding theoretical CDFs. Reprinted with permission from McKenzie and Savage (2011) [113]. ?2011 by the Institute of Physics. 124 8.3 Future work The results of the simulations and the observations of magnetic islands in na- ture show good qualitative agreement with the model?s predicted distributions of islands. For example, the allowed region predicted by Eq. (5.11) and the corre- sponding r asymmetry was observed both in the Hall MHD simulation and in the distribution of FTEs observed by Cluster. This is further evidence that island merging plays a signi cant role in the dynamics of long current layers. In this sce- nario, bursty reconnection generates many islands, most of which are smaller than the resolving capabilities of spacecraft instrumentation. Nevertheless, continued work on this model would be pro table. The nature of 3-D reconnection, and in particular of ux tubes, is not yet well understood. Only very recent simulations have begun to explore the dynamics of ux tubes in 3-D reconnection simulations [37], so expanding this model into three dimensions would be very challenging. This thesis is just a rst step towards understanding 2-D magnetic islands. A Hall MHD simulation that generates islands spontaneously and convects those islands outwards towards an open boundary would allow for a more direct comparison with our model. Further studies of the guide eld PIC simulations would help to narrow down the nature of the ow instability responsi- ble for secondary islands. The conditions under which this instability or tearing is the dominant mechanism for seeding magnetic islands are still being explored. Ad- ditional development of observational techniques to determine the FTE guide eld and account for the magnetopause asymmetry using the Cluster data would increase 125 con dence in the magnetic eld normalizations. The expected launch of Magneto- spheric Multiscale Mission (MMS) will open doors to new microscale reconnection physics. Lastly, the statistical study by McKenzie and Savage [112, 147, 148, 113] shows great promise for validating this model with observations of SADs in coronal post- are loop arcades, especially if applied to images from the recently launched SDO mission. SDO images have higher resolution than the SXT images used to produce the distributions in Fig. 8.2. With more SDO images of coronal current sheets to complement the limited data from TRACE, one might nd enough SADs at a su ciently broad range of scales to compare with the predictions of this model. Despite the inherent challenges in describing systems with such a wide separation of scales, steady improvement in computational power, instrumentation capabilities, and observational techniques should enable a fuller and more complete theory of magnetic reconnection and magnetic islands. 126 Appendix A Solar are energy calculations The largest ares release as much as 3 1032 erg of energy [131]. We consider a large are (height L 105 km, volume V h3) under coronal conditions (particle density n 3 109 cm 3, temperature T 106 K, and magnetic eld B 100 G). The energy in the magnetic elds is EB B2 8 L3 4 1032 erg: (A.1) In constrast, the thermal energy available is just ET nTL 3 1029 erg (A.2) and the gravitational energy available is EG G (mHnL3)M R2 L 5 1029 erg (A.3) where G is the gravitational constant, mH the mass of hydrogen, M the solar mass, R the solar radius. Only the magnetic energy is adequate to explain the energy released by the largest solar ares. 127 Appendix B Area conservation in merging terms Consider the merging terms in Eq. (3.11): @f @t mrg = 1 L Z A 0 dA0f( ;A0) Z 0 d 0v( ;A0; 0; A A0)f( 0; A A0) 1 L f( ;A) Z 1 0 dA0 Z 1 0 d 0v( ;A; 0; A0)f( 0; A0): (B.1) To nd the rate of change in area from the merging terms, we multiply Eq. (B.1) by AdAd and integrate over all A and : dAmrg dt = 1 L Z 1 0 dAA Z 1 0 d Z A 0 dA0f( ;A0) Z 0 d 0v( ;A0; 0; A A0)f( 0; A A0) 1 L Z 1 0 dAA Z 1 0 d f( ;A) Z 1 0 dA0 Z 1 0 d 0v( ;A; 0; A0)f( 0; A0): (B.2) Note in the rst term that R1 0 dA R A 0 dA 0 is an integral over half of the A A0 plane, namely, the A0 < A half, and so can also be written as R1 0 dA 0 R1 A0 dA. Thus, dAmrg dt = 1 L Z 1 0 dA0 Z 1 A0 dA Z 1 0 d Z 0 d 0Av( ;A0; 0; A A0)f( ;A0)f( 0; A A0) 1 L Z 1 0 dA Z 1 0 dA0 Z 1 0 d Z 1 0 d 0Av( ;A; 0; A0)f( ;A)f( 0; A0)): (B.3) 128 Now, we de ne A00 = A A0 and substitute for A into the rst term: dAmrg dt = 1 L Z 1 0 dA0 Z 1 0 dA00 Z 1 0 d Z 0 d 0(A0 + A00)v( ;A0; 0; A00)f( ;A0)f( 0; A00) 1 L Z 1 0 dA Z 1 0 dA0 Z 1 0 d Z 1 0 d 0Av( ;A; 0; A0)f( ;A)f( 0; A0): (B.4) Note that A, A0, A00 are all simply dummy variables, so we may replace A0 ! A and A00 ! A0 in the rst term. In doing so, the A part of the (A+ A0) in the rst term will cancel part of the second term, namely the 0 2 [0; ] part. The remainder is dAmrg dt = 1 L Z 1 0 dA Z 1 0 dA0 Z 1 0 d Z 0 d 0A0v( ;A; 0; A0)f( ;A)f( 0; A0) 1 L Z 1 0 dA Z 1 0 dA0 Z 1 0 d Z 1 d 0Av( ;A; 0; A0)f( ;A)f( 0; A0) = 1 L Z 1 0 dA Z 1 0 dA0 Z 1 0 d Z 0 d 0A0v( ;A; 0; A0)f( ;A)f( 0; A0) 1 L Z 1 0 dA Z 1 0 dA0 Z 1 0 d 0 Z 0 0 d Av( ;A; 0; A0)f( ;A)f( 0; A0) (B.5) where, as we did with A and A0, we have swapped the order of integration in and 0 in the second term while keeping < 0. The merging velocity v is of course interchange symmetric, so dAmrg dt = 0: (B.6) 129 Bibliography [1] S.-I. Akasofu. The development of the auroral substorm. Planet. Space Sci., 12:273, 1964. [2] S.-I. Akasofu, S. Chapman, and D. Vankatesan. The main phase of great magnetic storms. J. Geophys. Res., 68:3345{3350, 1963. [3] H. Alfv en. Existence of electromagnetic-hydrodynamic waves. Nature, 150:405{406, 1942. [4] V. Angelopoulos, J. P. McFadden, D. Larson, C. W. Carlson, S. B. Mende, H. Frey, T. Phan, D. G. Sibeck, K.-H. Glassmeier, U. Auster, E. Donovan, I. R. Mann, I. J. Rae, C. T. Russell, A. Runov, X.-Z. Zhou, and L. Kepko. Tail reconnection triggering substorm onset. Science, 321(18):931, 2008. [5] V. Angelopoulos, D. G. Mitchell, D. J. Williams, R. W. McEntire, A. T. Y. Lui, R. B. Decker, E. C. Roelof, S. P. Christon, S. Kokubun, T. Yamamoto, W. J. Hughes, J. C. Samson, E. Friis-Christensen, and K. Hayashi. Growth and evolution of a plasmoid associated with a small, isolated substorm: IMP 8 and GEOTAIL measurements in the magnetotail. Geophys. Res. Lett., 22(22):3011{3014, 1995. [6] K. Beckwith, J. F. Hawley, and J. H. Krolik. The in uence of magnetic eld geometry on the evolution of black hole accretion ows: similar disks, drastically di erent jets. Ap. J., 678(2):1180, 2008. [7] A. Bhatnagar and W. Livingston. Fundamentals of solar astronomy. World Scienti c Publishing Co. Pte. Ltd., Singapore, 1949. [8] A. Bhattacharjee, Yi-Min Huang, H. Yang, and B. Rogers. Fast reconnection in high-Lundquist-number plasmas due to the plasmoid instability. Phys. Plasmas, 16:112102, 2009. [9] L. Biermann. Kometenschweife und solare korpuskularstrahlung. Z. Astro- phys., 29:274, 1951. [10] C. K. Birdsall and A. B. Langdon. Plasma Physics via Computer Simulations. McGraw-Hill, New York, 1991. [11] J. Birn, J. F. Drake, M. A. Shay, B. N. Rogers, R. E. Denton, M. Hesse, M. Kuznetsova, Z. W. Ma, A. Bhattacharjee, A. Otto, and P. L. Pritchett. Geospace Environmental Modeling (GEM) Magnetic Reconnection Challenge. J. Geophys. Res., 106:3715, 2001. [12] J. Birn, M. Hesse, and K. Schindler. MHD simulations of magnetotail dynam- ics. J. Geophys. Res., 101(A6):12939{12954, 1996. 130 [13] D. Biskamp. Magnetic reconnection via current sheets. Phys. Fluids, 29:1520, 1986. [14] G. S. Bisnovatyi-Kogan and S. I. Blinnikov. Disc accretion onto a black hole at subcritical luminosity. Astron. Astrophys., 59(1):111{125, 1977. [15] L. Boltzmann. Weitere Studien uber das W armegleichgewicht unter Gas- molek ulen. Wiener Ber., 66:275, 1872. [16] M. R. Brown and A. Martin. Spheromak experiment using separate guns for formation and sustainment. Fusion Tech., 30:300, 1996. [17] S. V. Bulanov, S. I. Syrovatskii, and J. Sakai. Stabilizing in uence of plasma ow in dissipative tearing instability. Pis?ma Zh. Eksp. Teor. Fiz., 28:177, 1978. [18] O. Buneman. Instability, turbulence, and conductivity in current-carrying plasma. Phys. Rev. Lett, 1:8{9, 1958. [19] R. C. Carrington. Description of a singular appearance seen in the sun on September 1, 1859. Mon. Not. R. Astron. Soc., 20:13{15, 1859. [20] P. Cassak. Catastrophic model for the onset of fast magnetic reconnection. PhD thesis, University of Maryland, 2006. [21] P. Cassak and M. A. Shay. Scaling of asymmetric magnetic reconnection: general theory and collisional simulations. Phys. Plasmas, page 102114, 2007. [22] P. A. Cassak, M. A. Shay, and J. F. Drake. Scaling of Sweet-Parker reconnec- tion with secondary islands. Phys. Plasmas, 16:120702, 2009. [23] S. Chapman. An outline of a theory of magnetic storms. In Proc. R. Soc. Lond. A, volume 95, pages 61{83, 1918. [24] S. Chapman. The motion of a neutral ionized stream in the earth?s magnetic eld. In Proc. Camb. Phil. Soc., volume 21, page 577, 1923. [25] S. Chapman and V. C. A. Ferraro. A new theory of magnetic storms. Nature, 126:129{130, 1930. [26] S. Chapman and V. C. A. Ferraro. A new theory of magnetic storms. Terr. Magn. Atmos. Electr., 36(3):171{186, 1931. [27] P. Charbonneau, S. W. McIntosh, H.-L. Liu, and T. J. Bogdan. Avalanche model for solar ares. Solar Phys., 203(2):321{353, 2001. [28] L.-J. Chen, N. Bessho, B. Lefebvre, H. Vaith, A. Fazakerley, A. Bhattachar- jee, P. A. Puhl-Quinn, A. Runov, Y. Khotyaintsev, A. Vaivads, E. Georgescu, and R. Torbert. Evidence of an extended electron current sheet and its neigh- boring magnetic island during magnetotail reconnection. J. Geophys. Res., 113:A12213, 2008. 131 [29] L.-J. Chen, A. Bhattacharjee, P. A. Puhl-Quinn, H. Yang, N. Bessho, S. Imada, S. Muhlbachler, P. W. Daly, B. Lefebvre, Y. Khotyaintsev, A. Vaivads, A. Fazakerley, and E. Georgescu. Observation of energetic elec- trons within magnetic islands. Nature Phys., 4:19{23, 2008. [30] A. Ciaravella and J. C. Raymond. The current sheet associated with the 2003 November 4 coronal mass ejection: density, temperature, thickness, and line width. Ap. J., 686:1372{1382, 2008. [31] G. C. Claridge. Coronium. R. Astron. Soc. Can., 31(8):337{346, 1937. [32] E. W. Cliver and L. Svalgaard. The 1859 solar-terrestrial disturbance and the current limits of extreme space weather activity. Sol. Phys., 224:407{422, 2004. [33] C. D. Cothran, M. Landreman, M. R. Brown, and W. H. Mattheaus. Gen- eralized Ohm?s law in a 3-D reconnection experiment. Geophys. Res. Lett., 32:L03105, 2005. [34] R. Dailey, C. A. Cattell, F. S. Mozer, and J. Berchem. Electric elds and convection velocities associated with ux transfer events. Geophys. Res. Lett., 12(12):843{846, 1985. [35] W. Daughton, H. Karimabadi, and J. Scudder. Fully kinetic simulations of un- driven magnetic reconnection with open boundary conditions. Phys. Plasmas, 13:072101, 2006. [36] W. Daughton, V. Roytershteyn, B. J. Albright, H. Karimabadi, L. Yin, and K. J. Bowers. Transition from collisional to kinetic regimes in large-scale reconnection layers. Phys. Rev. Lett., 103:065004, 2009. [37] W. Daughton, V. Roytershteyn, H. Karimabadi, L. Yin, B. J. Albright, B. Bergen, and K. J. Bowers. Role of electron physics in the development of turbulent magnetic reconnection in collisionless plasmas. Nature Phys., 7:539{542, 2011. [38] J. C. Dorelli and A. Bhattacharjee. On the generation and topology of ux transfer events. J. Geophys. Res., 114:A06213, 2009. [39] J. C. Dorelli and J. Birn. Electron magnetohydrodynamic simulations of mag- netic island coalescence. Phys. Plasmas, 8(9), 2001. [40] J. G. Doyle, P. B. Byrne, and G. H. J. van den Oord. Ultraviolet ares on II Pegasi. Astron. Astrophys., 24:153{161, 1989. [41] J. F. Drake, M. Opher, M. Swisdak, and J. N. Chamoun. A magnetic re- connection mechanism for the generation of anomalous cosmic rays. Ap. J., 709:963{974, 2010. 132 [42] J. F. Drake and M. A. Shay. Fundamentals of collisionless reconnection. In J. Birn and E. R. Priest, editors, Reconnection of Magnetic Fields: Magne- tohydrodynamics and Collisionless Theory and Observations. Cambridge Uni- versity Press, Cambridge, U.K., 2007. [43] J. F. Drake, M. A. Shay, and M. Swisdak. The hall elds and fast magnetic reconnection. Phys. Plasmas, 15:042306, 2008. [44] J. F. Drake, M. A. Shay, W. Thongthai, and M. Swisdak. Production of energetic electrons during magnetic reconnection. Phys. Rev. Lett., 94:095001{ 1{4, 2005. [45] J. F. Drake, M. Swisdak, H. Che, and M. A. Shay. Electron acceleration from contracting magnetic islands during reconnection. Nature, 443:553, 2006. [46] J. F. Drake, M. Swisdak, K. M. Schoe er, B. N. Rogers, and S. Kobayashi. Formation of secondary islands during magnetic reconnection. Geophys. Res. Lett., 33:L13105, 2006. [47] J. F. Drake, M. Swisdak, M. A. Shay, B. N. Rogers, A. Zeiler, and C. Cat- tell. Formation of electron holes and particle energization during magnetic reconnection. Science, 299:873, 2003. [48] J. W. Dungey. Conditions for the occurrence of electrical discharges in astro- physical systems. Phil. Mag., 44:725, 1953. [49] J. W. Dungey. Cosmic Electrodynamics. Cambridge University Press, Cam- bridge, Eng., 1958. [50] J. W. Dungey. Interplanetary magnetic eld and the auroral zones. Phys. Rev. Lett., 6, 1961. [51] J. P. Eastwood, T.-D. Phan, F. S. Mozer, M. A. Shay, M. Fujimoto, A. Retin, M. Hesse, A. Balogh, E. A. Lucek, and I. Dandouras. Multi-point observa- tions of the Hall electromagnetic eld and secondary island formation during magnetic reconnection. J. Geophys. Res., 112:A06235, 2007. [52] B. Edl en. An attempt to identify the emission lines in the spectrum of the solar corona. Ark. Mat. Astron. Fys., 28B(1):1{4, 1941. [53] B. Edl en. Die deutung der emissionlinien im spektrum der sonnenkorona. Z. Astrophys., 22:30{64, 1942. [54] J. Egedal and A. Fasoli. Single-particle dynamics in collisionless magnetic reconnection. Phys. Rev. Lett, 86:5047, 2001. [55] J. Egedal, A. Le^, Y. Zhu, W. Daughton, M. ieroset, T. Phan, R. P. Lin, and J. P. Eastwood. Cause of super-thermal electron heating during magnetotail reconnection. Geophys. Res. Lett., 37:L10102, 2010. 133 [56] R. C. Elphic, C. A. Cattell, K. Takahashi, S. J. Bame, and C. T. Russell. ISEE-1 and 2 observations of magnetic ux ropes in the magnetotail: FTE?s in the plasma sheet? Geophys. Res. Lett., 13(7):648{651, 1986. [57] A. G. Emslie, B. R. Dennis, G. D. Holman, and H. S. Hudson. Re nements to are energy estimates: a followup to \Energy partition in two solar are / CME events" by A. G. Emslie et al. J. Geophys. Res., 110:A11103, 2005. [58] D. H. Fair eld. Polar magnetic disturbances and the interplanetary magnetic eld. Space Res., 8:107{119, 1967. [59] D. H. Fair eld and L. G. Cahill. Transition region magnetic eld and polar magnetic disturbances. J. Geophys. Res., 71:155{169, 1966. [60] C. J. Farrugia, R. P. Rijnbeek, M. A. Saunders, D. J. Southwood, D. J. Rodgers, M. F. Smith, C. P. Chaloner, D. S. Hall, P. J. Christiansen, and L. J. C. Woolliscroft. A multi-instrument study of ux transfer event struc- ture. J. Geophys. Res., 93(A12):14465{14477, 1988. [61] R. C. Fear, S. E. Milan, A. N. Fazakerley, E. A. Lucek, S. W. H. Cowley, and I. Dandouras. The azimuthal extent of three ux transfer events. Ann. Geophys., 26:2353{2369, 2008. [62] R. L. Fermo, J. F. Drake, M. Swisdak, and K.-J. Hwang. Comparison of a statistical model for magnetic islands in large current layers with hall mhd simulations and cluster fte observations. J. Geophys. Res., 2011. In press; doi:10.1029/2010JA016271. [63] K. Fujimoto. Time evolution of the electron di usion region and the reconnec- tion rate in fully kinetic and large system. Phys. Plasmas, 13:072904, 2006. [64] H. P. Furth, J. Killeen, and M. N. Rosenbluth. Finite-resistivity instabilities of a sheet pinch. Phys. Fluids, 6(4):459, April 1963. [65] E. Gaddy, R. Decker, M. K. Lockwood, L. Roufberg, G. Knutzen, and D. Marsh. The Solar Probe Plus solar array development and design. In Photovoltaic Specialists Conference (PVSC), 2010 35th IEEE, pages 717{722, June 2010. [66] A. A. Galeev, R. Rosner, and G. S. Vaiana. Structured coronae of accretion disks. Ap. J., 229:318{326, 1979. [67] R. G. Giovanelli. Magnetic and electric phenomena in the sun?s atmosphere associated with sunspots. Mon. Not. R. Astron. Soc., 107:338, 1947. [68] J. T. Gosling, S. Erkisson, R. M. Skoug, D. J. McComas, and R. J. Forsyth. Petschek-type reconnection exhausts in the solar wind well beyond 1 AU: Ulysses. Ap. J., 644:613{621, 2006. 134 [69] J. T. Gosling, R. M. Skoug, and D. J. McComas. Direct evidence for magnetic reconnection in the solar wind near 1 AU. Geophys. Res. Lett., 110:A01107, 2005. [70] W. Grotrian. The question of the signi cance of the lines in the spectrum of the solar corona. Naturwiss., 27:214, 1939. [71] G. E. Hale. Some results and conclusions derived from a photographic study of the sun. Astron. Astrophys., 11:811{815, 1892. [72] G. E. Hale. On the probable existence of a magnetic eld in sun-spots. Ap. J., 28:315, 1908. [73] G. E. Hale. Solar vortices (contributions from the Mt. wilson solar observatory, no. 26). Ap. J., 28:100, 1908. [74] E. G. Harris. On a plasma sheath separating regions of oppositely directed magnetic eld. Nuovo Cim., 23(1):115{121, 1962. [75] H. Hasegawa, J. Wang, M. W. Dunlop, Z. Y. Pu, Q.-H. Zhang, B. Lavraud, M. G. G. T. Taylor, O. D. Constantinescu, J. Berchem, V. Angelopoulos, J. P. McFadden, H. U. Frey, E. V. Panov, M. Volwerk, and Y. V. Bogdanova. Evidence for a ux transfer event generated by multiple X-line reconnection at the magnetopause. Geophys. Res. Lett., 37:L16101, 2010. [76] M. Hesse and J. Birn. Plasmoid evolution in an extended magnetotail. J. Geophys. Res., 96(A4):5683{5696, 1991. [77] M. Hesse, Masha Kuznetsova, and M. Hoshino. The structure of the dissipa- tion region for component reconnection: Particle simulations. Geophys. Res. Lett., 29:1563, 2002. [78] R. Hodgson. On a curious appearance seen in the sun. Mon. Not. R. Astron. Soc., 20:15{16, 1859. [79] E. W. Hones, Jr. Transient phenomena in the magnetotail and their relation to substorms. Space Sci. Rev., 23:393, 1979. [80] E. W. Hones, Jr., D. N. Baker, S. J. Bame, W. C. Feldman, J. T. Gosling, D. J. McComas, R. D. Zwickl, J. A. Slavin, E. J. Smith, and B. T. Tsurutani. Structure of the magnetotail at 220re and its response to geomagnetic activity. Geophys. Res. Lett., 11(1):5{7, 1984. [81] F. Hoyle. Some recent researches in solar physics. Cambridge [Eng.] University Press, Cambridge, Eng., 1949. [82] Y.-M. Huang and A. Bhattacharjee. Scaling laws of resistive magnetohydrody- namic reconnection in the high-Lundquist-number, plasmoid-unstable regime. Phys. Plasmas, 17(6):062104, 2010. 135 [83] J. D. Huba, N. T. Gladd, and K. Papadopoulos. The lower-hybrid-drift in- stability as a source of anomalous resistivity for magnetic eld reconnection. Geophys. Res. Lett., 4:125, 1977. [84] W. J. Hughes. The magnetopause, magnetotail, and magnetic reconnection. In M. G. Kivelson and C. T. Russell, editors, Introduction to Space Physics. Cambridge University Press, New York, 1995. [85] K. Hurley, S. E. Boggs, D. M. Smith, R. C. Duncan, R. Lin, A. Zoglauer, S. Krucker, G. Hurford, H. Hudson, C. Wigger, W. Hajdas, C. Thompson, I. Mitrofanov, A. Sanin, W. Boynton, C. Fellows, A. von Kienlin, G. Lichti, A. Rau, and T. Cline. An exceptionally bright are from SGR 1806-20 and the origins of short-duration gamma-ray bursts. Nature, 434:1098{1103, 2005. [86] A. Ieda, S. Machida, T. Mukai, Y. Saito, T. Yamamoto, A. Nishida, T. Tere- sawa, and S. Kokubun. Statistical analysis of the plasmoid evolution with Geotail observations. J. Geophys. Res., 103(A3):4453{4465, 1998. [87] C. M. Jackman, N. Achilleos, E. J. Bunce, S. W. H. Cowley, M. K. Dougherty, G. H. Jones, S. E. Milan, and E. J. Smith. Interplanetary magnetic eld at 9 au during the declining phase of the solar cycle and its implications for saturn?s magnetospheric dynamics. J. Geophys. Res., 109:A11203, 2004. [88] J. R. Jokipii and B. Thomas. E ects of drift on the transport of cosmic rays IV. Modulation by a wavy interplanetary current sheet. Ap. J., pages 1115{1122, 1981. [89] B. B. Kadomtsev. On disruptive instability in tokamaks. Fiz. Plazmy, 1:710{ 715, 1975. [90] H. Karimabadi, W. Daughton, and J. Scudder. Multi-scale structure of the electron di usion region. Geophys. Res. Lett., 34:L13104, 2007. [91] W. T. Kelvin. Hydrokinetic solutions and observations. Phil. Mag., 4:362{377, 1871. [92] W. T. Kelvin. President?s address. In Proc. R. Soc., volume 52, pages 299{310, 1892. [93] A. Klimas, M. Hesse, and S. Zenitani. Particle-in-cell simulation of collisionless reconnection with open out ow boundaries. Phys. Plasmas, 15:082102, 2008. [94] Y.-K. Ko, J. C. Raymond, J. Lin, G. Lawrence, J. Li, and A. Fludra. Dynam- ical and physical properties of a post-coronal mass ejection current sheet. Ap. J., 594:1068{1084, 2003. [95] S. Krucker, H. S. Hudson, L. Glesener, S. M. White, S. Masuda, J.-P. Wuelser, and R. P. Lin. Measurements of the coronal acceleration of a solar are. Ap. J., 714:1108{1119, 2010. 136 [96] I. Langmuir. The interaction of electron and positive ion space charges in cathode sheaths. Phys. Rev., 39:954{989, 1929. [97] G. Lapenta. Self-feeding turbulent magnetic reconnection on macroscopic scales. Phys. Rev. Lett., 100:235001, 2008. [98] A. Lazarian and M. Opher. A model of acceleration of anomalous cosmic rays by reconnection in the heliosheath. Ap. J., 703(1):8{21, 2009. [99] L. C. Lee and Z. F. Fu. A theory of magnetic ux transfer at the earth?s magnetopause. Geophys. Res. Lett., 12:105, 1985. [100] J. Lin, Y.-K. Ko, L. Sui, J. C. Raymond, G. A. Stenborg, Y. Jiang, S. Zhao, and S. Mancuso. Direct observations of the magnetic reconnection site of an eruption on 2003 November 18. Ap. J., 622:1251{1264, 2005. [101] R. P. Lin and H. S. Hudson. 10-100keV electron acceleration and emission from solar ares. Sol. Phys., 17:412{435, 1971. [102] R. P. Lin and H. S. Hudson. Non-thermal processes in solar ares. Sol. Phys., 50(1):153{178, 1976. [103] F. A. Lindemann. Note on the theory of magnetic storms. Philos. Mag., 39:669{684, 1919. [104] M. G. Linton and D. W. Longcope. A model for patchy reconnection in three dimensions. Ap. J., 642:1177{1192, 2006. [105] G. D. Liveing and J. Dewar. Note on the unknown chromospheric substance of young. Proc. R. Soc. Lond., 28:475{477, 1878. [106] N. F. Loureiro, A. A. Schekochihin, and S.C. Cowley. Instability of current sheets and formation of plasmoid chains. Phys. Plasmas, 14:100703, 2007. [107] A. T. Y. Lui, P. H. Yoon, C. Mok, and C.-M. Ryu. Inverse cascade feature in current disruption. J. Geophys. Res., 113:A00C06, 2008. [108] H. B. Maris and E. O. Hulburt. A theory of auroras and magnetic storms. Phys. Rev., 33:412{431, 1929. [109] E. W. Maunder. Magnetic disturbances, 1882 to 1903, as recorded at the royal observatory, greenwich, and their association with sunspots. Mon. Not. R. Astron. Soc., 65:2{34, 1905. [110] H. J. McAndrews, C. J. Owen, M. F. Thomsen, B. Lavraud, A. J. Coates, M. K. Dougherty, and D. T. Young. Evidence for reconnection at Saturn?s magnetopause. J. Geophys. Res., 113:A04210, 2008. [111] D. E. McKenzie and H. S. Hudson. X-ray observations of motions and structure above a solar are arcade. Ap. J., 519:L93{L96, 2009. 137 [112] D. E. McKenzie and S. L. Savage. Quantitative examination of supra-arcade down ows in eruptive solar ares. Ap. J., 697:1569{2009, 2009. [113] D. E. McKenzie and S. L. Savage. Distribution functions of sizes and uxes determined from supra-arcade down ows. Ap. J., 735:L6, 2011. [114] F. S. Mozer, S. D. Bale, and T. D. Phan. Evidence of di usion regions at a subsolar magnetopause crossing. Phys. Rev. Lett., 89:015002, 2002. [115] T. Nagai, M. Fujimoto, Y. Saito, S. Machida, T. Teresawa, R. Nakamura, T. Yamamoto, T. Mukai, A. Nishida, and S. Kokubun. Structure and dynamics of magnetic reconnection for substorm onsets with Geotail observations. J. Geophys. Res., 103(A3):4419{4440, 1998. [116] T. Nagai, K. Takahash, H. Kawano, T. Yamamoto, S. Kokubun, and A. Nishida. Initial GEOTAIL survey of magnetic substorm signatures in the magnetotail. Geophys. Res. Lett., 21(25):2991{2994, 1994. [117] J. D. Nichols, S. W. H. Cowley, and D. J. McComas. Magnetopause recon- nection rate estimates for Jupiter?s magnetosphere based on interplanetary measurements at 5 au. Ann. Geophys., 24:393{406, 2006. [118] M. ieroset, R. P. Lin, T. D. Phan, D. E. Larson, and S. D. Bale. Evidence for electron acceleration up to 300keV in the magnetic reconnection di usion region in the earth?s magnetotail. Phys. Rev. Lett., 89:195001{1{4, 2002. [119] M. ieroset, T. D. Phan, M. Fujimoto, R. P. Lin, and R. P. Lepping. In situ detection of collisionless reconnection in the earth?s magnetotail. Nature, 412:417, 2001. [120] M. Oka, T.-D. Phan, S. Krucker, M. Fujimoto, and I. Shinohara. Electron acceleration by multi-island coalescence. Ap. J., 714:915{926, 2010. [121] N. Omidi and D. G. Sibeck. Flux transfer events in the cusp. Geophys. Res. Lett., 34:L04106, 2007. [122] M. Opher, J. F. Drake, M. Swisdak, K. M. Schoe er, J. D. Richardson, R. B. Decker, and G. Toth. Is the magnetic eld in the heliosheath laminar or a turbulent sheet of bubbles? Ap. J., 734(1):71, 2011. [123] E. N. Parker. Dynamics of the interplanetary gas and magnetic elds. Ap. J., 128:664, 1958. [124] E. N. Parker. The solar- are phenomenon and the theory of reconnection and annihilation of magnetic elds. Ap. J., 8:177, 1963. [125] E. N. Parker. Magnetic neutral sheets in evolving elds: II. Formation of the solar corona. Ap. J., 264:642{647, 1983. 138 [126] E. N. Parker. Nano ares and the solar x-ray corona. Ap. J., 330:474{479, 1988. [127] H. E. Petschek. Magnetic eld annihilation. In W. N. Ness, editor, AAS/NASA Symposium on the Physics of Solar Flares, page 425. NASA, Washington, DC, 1964. [128] T. D. Phan, J. T. Gosling, M. S. Davis, R. M. Skoug, M. ieroset, R. P. Lin, R. P. Lepping, D. J. McComas, C. W. Smith, H. Reme, and A. Balogh. A magnetic reconnection x-line extending more than 390 earth radii in the solar wind. Nature, 439:175, 2006. [129] T. D. Phan, G. Paschmann, C. Twitty, F. S. Mozer, J. T. Gosling, J. P. Eastwood, M. ieroset, H. Reme, and E. A. Lucek. Evidence for magnetic reconnection initiated in the magnetosheath. Geophys. Res. Lett., 34:L14104, 2007. [130] S. C. Prager, A. F. Almagri, S. Assadi, J. A. Beckstead, R. N. Dexter, D. J. Den Hartog, G. Chartas, S. A. Hokin, T. W. Lovell, T. D. Rempel, J. S. Sar , W. Shen, C. W. Spragins, and J. C. Sprott. First results from the Madison Symmetric Torus reversed eld pinch. Phys. Fluids B, 2:1367, 1990. [131] E. R. Priest and T. G. Forbes. The magnetic nature of solar ares. Astron. Astrophys. Rev., 10(4):313{377, 2002. [132] P. L. Pritchett. Geospace environmental modeling magnetic reconnection chal- lenge: simulations with a full particle electromagnetic code. J. Geophys. Res., 106:3783, 2001. [133] P. L. Pritchett and F. V. Coroniti. Three-dimensional collisionless magnetic reconnection in the presence of a guide eld. J. Geophys. Res., 109:A01220, 2004. [134] P. L. Pritchett and C. C. Wu. Coalescence of magnetic islands. Phys. Fluids, 22(11):2140{2146, 1979. [135] J. Raeder. Flux tranfer events: 1. Generation mechanism for nearly southward IMF. Ann. Geophys., 24:381, 2006. [136] Y. Ren, M. Yamada, S. Gerhardt, H. Ji, Russell Kulsrud, and A. Kuritsyn. Experimental veri cation of the Hall e ect during magnetic reconnection in a laboratory plasma. Phys. Rev. Lett., 95:005003, 2005. [137] R. P. Rijnbeek, S. W. H. Cowley, D. J. Southwood, and C. T. Russell. A survey of dayside ux transfer events observed by isee 1 and 2 magnetometers. J. Geophys. Res., 89(A2):786{800, 1984. 139 [138] C. R. Robinson and B. W. Bopp. A \helium are" on the active G5 dwarf Kappa Ceti. In J. L. Linsky and R. E. Stencel, editors, Cool Stars, Stellar Systems and the Sun, volume 291 of Lecture Notes in Physics, Berlin Springer Verlag, page 509, 1987. [139] B. B. Rossi and S. Olbert. Introduction to the Physics of Space. McGraw-Hill, New York, 1970. [140] A. Runov, R. Nakamura, W. Baumjohann, R. A. Treumann, T. L. Zhang, M. Volwerk, Z. V or os, A. Balogh, K.-H. Glabmeier, B. Klecker, H. R eme, and L. Kistler. Current sheet structure near magnetic X-line observed by Cluster. Geophys. Res. Lett., 30(11):1579, 2003. [141] C. T. Russell and R. C. Elphic. Initial isee magnetometer results: Magne- topause observations. Space Science Rev., 22:681, 1978. [142] C. T. Russell and R. C. Elphic. ISEE observations of ux transfer events at the dayside magnetopause. Geophys. Res. Lett., 6:33, 1979. [143] C. T. Russell, K. K. Khurana, D. E. Huddleston, and M. G. Kivelson. Lo- calized reconnection in the near Jovian magnetotail. Science, 280:1061{1064, 1998. [144] R. Samtaney, N. F. Loureiro, D. A. Uzdensky, A. A. Schekochihin, and S. C. Cowley. Formation of plasmoid chains in magnetic reconnection. Phys. Rev. Lett., 103:105004, 2009. [145] T. Sato and T. Hayashi. Externally driven magnetic reconnection and a pow- erful magnetic energy converter. Phys. Fluids, 22:1189, 1979. [146] M. A. Saunders, C. T. Russell, and N. Sckopke. Flux transfer events: scale size and interior structure. Geophys. Res. Lett., 11:131{134, 1984. [147] S. L. Savage. Using extreme ultra-violet and soft X-ray observations as probes of magnetic reconnection during solar ares. PhD thesis, Montana State Uni- versity, 2010. [148] S. L. Savage and D. E. McKenzie. Quantitative examination of a large sample of supra-arcade down ows in eruptive solar ares. Ap. J., 730:98, 2011. [149] S. L. Savage, D. E. McKenzie, K. K. Reeves, T. G. Forbes, and D. W. Long- cope. Reconnection out ows and current sheet observed with Hinode/XRT in the 2008 April 9 \Cartwheel CME" are. Ap. J., 722:329{342, 2010. [150] E. Schatzman. The heating of the solar corona and chromosphere. Ann. Astrophys., 12:203, 1949. [151] M. Scholer. Magnetic ux transfer at the magnetopause based on single x line bursty reconnection. Geophys. Res. Lett., 15:291, 1988. 140 [152] J. D. Scudder, F. S. Mozer, N. C. Maynard, and C. T. Russell. Fingerprints of collisionless reconnection at the separator, I, Ambipolar-Hall signatures. J. Geophys. Res., 107:1294, 2002. [153] M. A. Shay, J. F. Drake, B. N. Rogers, and R. E. Denton. The scaling of collisionless, magnetic reconnection for large systems. Geophys. Res. Lett., 26:2163, 1999. [154] M. A. Shay, J. F. Drake, and M. Swisdak. Two-scale structure of the electron dissipation region during collisionless magnetic reconnection. Phys. Rev. Lett., 99:155002, 2007. [155] M. A. Shay, J. F. Drake, M. Swisdak, and B. N. Rogers. The scaling of embedded collisionless reconnection. Phys. Plasmas, 11(5):2199, 2004. [156] N. R. Sheeley, H. P. Warren, and Y.-M. Wang. The origin of post are loops. Astrophys. J., 616:1224, 2004. [157] L. S. Shepherd and P. A. Cassak. Comparison of secondary islands in colli- sional reconnection to hall reconnection. Phys. Rev. Lett., 105:015004, 2010. [158] K. Shibata, S. Masuda, M. Shimojo, H. Hara, T. Yokoyama, S. Tsuneta, T. Kosugi, and Y. Ogawara. Hot-plasma ejections associated with compact- loop solar ares. Ap. J., 451:L83{L85, 1995. [159] K. Shibata and S. Tanuma. Plasmoid-induced-reconnection and fractal recon- nection. Earth, Planets and Space, 53:473{482, 2001. [160] A. Y. Shih, R. P. Lin, and D. M. Smith. RHESSI observations of the propor- tional acceleration of relativistic > 0:3 MeV electrons and > 30 MeV protons in solar ares. Ap. J., 698:L152{L157, 2009. [161] D. G. Sibeck, M. Kuznetsova, V. Angelopoulos, K.-H. Gla meier, and J. P. McFadden. Crater FTEs: Simulation results and THEMIS observations. Geo- phys. Res. Lett., 35:L17S06, 2008. [162] J. A. Slavin, M. H. Acu~na, B. J. Anderson, D. N. Baker, M. Benna, S. A. Boardsen, G. Gloeckler, R. E. Gold, G. C. Ho, H. Korth, S. M. Krimigis, R. L. McNutt, Jr., J. M. Raines, M. Sarantos, D. Schriver, S. C. Solomon, P. Tr avn cek, and T. H. Zurbuchen. MESSENGER observations of magnetic reconnection in Mercury?s magnetosphere. Science, 324:606{610, 2009. [163] J. A. Slavin, B. J. Anderson, D. N. Baker, M. Benna, S. A. Boardsen, G. Gloeckler, R. E. Gold, G. C. Ho, H. Korth, S. M. Krimigis, R. L. McNutt, Jr., L. R. Nittler, J. M. Raines, M. Sarantos, D. Schriver, S. C. Solomon, R. D. Starr, P. Tr avn cek, and T. H. Zurbuchen. MESSENGER observations of ex- treme loading and unloading of Mercury?s magnetic tail. Science, 329:665{668, 2010. 141 [164] J. A. Slavin, R. P. Lepping, J. Gjerloev, D. H. Fair eld, M. Hesse, C. J. Owen, M. B. Moldwin, T. Nagai, A. Ieda, and T. Mukai. Geotail observations of magnetic ux ropes in the plasma sheet. J. Geophys. Res., 108(A1):1015, 2003. [165] J. A. Slavin, C. J. Owen, M. M. Kuznetsova, and M. Hesse. ISEE 3 obser- vations of plasmoids with ux rope magnetic topologies. Geophys. Res. Lett., 22(15):2061{2064, 1995. [166] J. A. Slavin, M. F. Smith, E. L. Mazur, D. N. Baker, E. W. Hones, Jr., T. Iye- mori, and E. W. Greenstadt. ISEE 3 observations of traveling compression regions in the Earth?s magnetotail. J. Geophys. Res., 98(A9):15425{15446, 1993. [167] B. U. O. Sonnerup. Magnetopause structure during the magnetic storm of September 24, 1961. J. Geophys. Res., 76(28):6717{6735, 1971. [168] B. U. O. Sonnerup. Magnetic eld reconnection. In L. J. Lanzerotti, C. F. Kennel, and E. N. Parker, editors, Solar System Plasma Physics, volume 3, page 46. North Halland Pub., Amsterdam, 1979. [169] D. J. Southwood, C. J. Farrugia, and M. A. Saunders. What are ux transfer events? Planet. Space Sci., 36:503{508, 1988. [170] E. C. Stone, A. C. Cummings, F. B. McDonald, B. C. Heikkila, N. Lal, and W. R. Webber. Voyager 1 explores the termination shock region and the heliosheath beyond. Science, 309:2017{2020, 2005. [171] E. C. Stone, A. C. Cummings, F. B. McDonald, B. C. Heikkila, N. Lal, and W. R. Webber. An asymmetric solar wind termination shock. Nature, 454:71{ 74, 2008. [172] T. E. Stringer. Electrostatic instabilities in current-carrying and counter- streaming plasmas. J. Nucl. Energy, Part C Plasma Phys., 6:267{279, 1964. [173] L. Sui and G. D. Holman. Evidence for the formation of a large-scale current sheet in a solar are. Ap. J., 596:L251{L254, 2003. [174] P. A. Sweet. The neutral point theory of solar ares. In B. Lehnert, edi- tor, Electromagnetic Phenomena in Cosmical Physics, page 123. Cambridge University Press, New York, 1958. [175] M. Swisdak and J. F. Drake. Orientation of the reconnection x-line. Geophys. Res. Lett., 34:L11106, 2007. [176] M. Tanaka. Asymmetry and thermal e ects due to parallel motion of electrons in collisionless magnetic reconnection. Phys. Plasmas, 3:4010, 1996. 142 [177] S. Tanuma, T. Yokoyama, T. Kudoh, and K. Shibata. Two-dimensional mag- netohydrodynamic numerical simulations of magnetic reconnection triggered by a supernova shock in the interstellar medium: generation of X-ray gas in the galaxy. Ap. J., 551(1):312, 2001. [178] W.-L. Teh, S. Eriksson B. U. O Sonnerup, R. Ergun, V. Angelopoulos, K.-H. Glassmeier, J. P. McFadden, and J. W. Bonnell. THEMIS observations of a secondary magnetic island within the hall electromagnetic eld region at the magnetopause. Geophys. Res. Lett., 37:L21102, 2010. [179] U. Trottenberg, C. W. Oosterlee, and A. Sch uller. Multigrid. Academic Press, San Diego, CA, 2001. [180] S. Tsuneta. Moving plasmoid and formation of the neutral sheet in a solar are. Ap. J., 483:507{514, 1997. [181] D. A. Uzdensky, N. F. Loureiro, and A. A. Schekochihin. Fast magnetic recon- nection in the plasmoid-dominated regime. Phys. Rev. Lett., 105(23):235002, 2010. [182] S. von Goeler, W. Stodiek, and N. R. Sauto . Studies of internal disruptions and m = 1 oscillations in tokamak discharges with soft x-ray techniques. Phys. Rev. Lett., 1974. [183] H. von Helmholtz. Uber discontinuierliche Fl ussigkeits bewegungen. Monats. K onigl. Preuss. Akad. Wiss. Berlin, 23:215, 1868. [184] Y. L. Wang, R. C. Elphic, B. Lavraud, M. G. G. T. Taylor, J. Birn, J. Raeder, C. T. Russell, H. Kawano, Q.-G. Zong, H. Zhang, X. X. Zhang, and R. H. Friedel. Initial results of high-latitude magnetopause and low-latitude ank ux transfer events from 3 years of Cluster observations. J. Geophys. Res., 110:A11221, 2005. [185] Y. L. Wang, R. C. Elphic, B. Lavraud, M. G. G. T. Taylor, J. Birn, J. Raeder, C. T. Russell, H. Kawano, Q.-G. Zong, H. Zhang, X. X. Zhang, and R. H. Friedel. Initial results of high-latitude magnetopause and low-latitude ank ux transfer events from 3 years of Cluster observations. J. Geophys. Res., 110:A11221, 2005. [186] M. Yamada, H. Ji, S. Hsu, T. Carter, R. Kulsrud, N. Bretz, F. Jobes, Y. Ono, and F. Perkins. Study of driven magnetic reconnection in a laboratory plasma. Phys. Plasmas, 4:1936, 1997. [187] A. Zeiler, D. Biskamp, J. F. Drake, B. N. Rogers, M. A. Shay, and M. Scholer. Three-dimensional particle simulations of collisionless magnetic reconnection. J. Geophys. Res., 107:1230{1{9, 2002. 143 [188] H. Zhang, K. K. Khurana, M. G. Kivelson, V. Angelopoulos, Z. Y. Pu, Q.- G. Zong, J. Liu, and X.-Z. Zhou. Modeling a force-free ux transfer event probed by multiple Time History of Events and Macroscale Interactions during Substorms (THEMIS) spacecraft. J. Geophys. Res., 113:A00C05, 2008. [189] H. Zhang, M. G. Kivelson, K. K. Khurana, J. McFadden, R. J. Walker, V. An- gelopoulos, J. M. Weygand, T. Phan, D. Larson, K.-H. Glassmeier, and H. U. Auster. Evidence that crater ux transfer events are initial stages of typical ux transfer events. J. Geophys. Res., 115:A08229, 2010. [190] Q.-G. Zong, T. A. Fritz, Z. Y. Pu, S. Y. Fu, D. N. Baker, H. Zhang, A. T. Lui, I. Vogiatzis, K.-H. Glassmeier, A. Korth, P. W. Daly, A. Balogh, and H. Reme. Cluster observations of earthward owing plasmoid in the tail. Geophys. Res. Lett., 31:L18803, 2004. 144