Probing the Progression, Properties, and Progenies ofMagnetic ReconnectionThesis byYoung Dae YoonIn Partial Fulfillment of the Requirements for theDegree ofDoctor of PhilosophyCALIFORNIA INSTITUTE OF TECHNOLOGYPasadena, California2020Defended April 23rd, 2020ii© 2020Young Dae YoonORCID: 0000-0001-8394-2076All rights reserved except where otherwise notediiiACKNOWLEDGEMENTSImust begin by cherishingmy roots and thankingmy family. Myparents’ unwaveringsupport, commitment, and wisdom are the pillar and base upon which I am merelystanding. I am so proud of my younger brother for all he has accomplished despitethe sacrifices he has had to make because of me. I also thank every single one ofmy relatives for always rooting for me along the way. Without their love and faithamidst all the hardships, I would not have made it this far.To paraphrase Sir Isaac Newton, if I have seen any bit further, it is by standing onthe shoulders of Professor Paul Bellan. He is a brilliant experimentalist as well asan excellent theorist having both breath and depth of knowledge in physics. I admirehis accessibility as he always drops whatever he is doing at the time to pay attentionto what the students have to say. There is just one time slot when he is not available,and that is 30 minutes before his plasma physics class; I am still amazed by hisconstant effort in preparing for a class he has been teaching for decades. He is verypatient and understanding, but also enthusiastic and motivational. I sincerely feelextremely lucky to have chosen Paul as my advisor.I would like to thank my candidacy committee—Professors E. Sterl Phinney, GreggHallinan, and Kerry Vahala—and my thesis committee—Professors E. Sterl Phin-ney, Gregg Hallinan, and Marco Velli—for their helpful feedback and insightfulquestions.Ryan Marshall and Pakorn (Tung) Wongwaitayakornkul joined the group with mein 2014, and we will all be graduating at the same time. We have made so manyfond memories in class, in the lab, at conferences, and at bars. I thank Ryan forkeeping the office lively, offsetting my laziness, and having probably the most beerswith me at conferences. I thank Tung for all the fun lunch conversations; however,I do NOT thank him for his insane badminton and pool skills.I have benefited greatly from the senior members of the group. I thank Xiang Zhai,Vernon Chaplin, and Bao Ha for effectively transferring their abundant knowledgeand skills to an ignorant first-year student while finishing up their graduate studies.I deeply thank Magnus Haw for being the senior graduate student for the longesttime so that I could make use of his "senior aura" that magically fixes everything. Iam also greatly indebted to Gunsu Yun, who has helped me so much career-wise bytaking my matter into his own hands.ivI have greatly enjoyed working with and learning from the post-docs in our group:Kil-Byoung Chai, Amelia Greig, and Byonghoon Seo. In particular, I have greatlybenefited from our two Korean post-docs, Kil-Byoung Chai and Byonghoon Seo,whose work ethic, enthusiasm, and knowledge have shaped how I think and what Iknow. The numerous everyday conversations that we have had over a cup of coffee("the Korean Coffee Break," as Ryan calls it) have amounted to a big chunk of ideas.The lab will now be managed by Yi Zhou, Yang Zhang, and André Nicolov, fromwhom I have also learned a lot. Yi rightfully questions seemingly obvious state-ments; I tell myself to be more like Yi when I seem to be overlooking importantassumptions underlying a problem. Yang and André are also very passionate, intel-ligent, and diligent. I am happy that the lab is in good hands.I have never met an administrative hurdle during my time at Caltech, largely thanksto Connie Rodriguez, Christy Jenstad, and Jennifer Blankenship. I especially thankConnie for dealing with all the purchases and travel reports and for furnishing theoffice with leftover pizza from APh 110.I also thank Dave Felt for imparting his outstanding electrical engineering wisdomwhen I was building my RF plasma source.The Korean society at Caltech has been a big part of my graduate life. There areso many of you to name here, so I will try my best not to leave anyone out. Ithank those you who started out graduate life at Caltech with me: Young Joon Choi,Kibeom Kim, Sangjun Lee, Gyu Baek Rah, Jaeyun Moon, Serim Ryou, Sung WonAhn, Jong Hun Kang, and Hyunjun Cho. I also thank Hyeong Chan Jo, JeongminKim, Sanghyun Yi, Jihong Min, Juhyun Kim, Hyungjoo Row, Jinmo Koo, JieunShin for being good friends with me all these years. Special shout-out to thetravelling/drinking/dawdling crew Hee Jeong Ahn, Jonghun Lim, Hye Young Shin,Areum Kim, without whom I would have been a lot more productive.TheWednesday tennis club made tennis my favorite sport, although tennis may havebeen an excuse to gather people for drinks. I tremendously thank Peter Lee for hisleadership, without which the club would not function. I am not sure whether I willever have a chance to play tennis regularly again, so I will miss the club dearly.I cannot describe with words my gratitude to the Kim family—Mr. and Mrs. Kim,Wonchan Kim and Robin Kim, who made me feel at home even when I am on theother side of the globe from mine. Thanks for all the delicious food and fun times.vFinally, I would like to thank Ha-Rry for being there with me through all the upsand downs. You have done much more for me than you could ever imagine.viABSTRACTMagnetic reconnection is a plasma phenomenon in which opposing magnetic fieldsannihilate and release theirmagnetic energy into other forms of energy. In this thesis,various aspects of collisionless magnetic reconnection are studied analytically andnumerically, and an experimental diagnostic for magnetic fields in a plasma isdescribed.The progression of magnetic reconnection is first illustrated through the formulationof a framework that revolves around canonical vorticity flux, which is ideally aconserved quantity. The reconnection instability, electron acceleration, and whistlerwave generation are explained in an intuitive manner by analyzing the dynamicsof canonical vorticity flux tubes. The validity of the framework is then extendeddown to first principles by the inclusion of the electron canonical battery effect. Theimportance of this effect during reconnection determines the overall structure andevolution of the process.A crucial property of magnetic reconnection is its accompaniment by anomalousion heating much faster than conventional collisional heating. Stochastic heating isa mechanism in which, under a sufficiently strong electric field, particles undergochaotic motion in phase space and heat up dramatically. Using the previouslyestablished canonical vorticity framework, it is demonstrated that the Hall electricfields that develop during reconnection satisfy the stochastic ion heating criterionand that the ions involved indeed undergo chaotic motion. This mechanism is thenkinetically verified via exact analyses and particle simulations and is thus ultimatelyestablished as the main ion heating mechanism in magnetic reconnection.An important progeny of magnetic reconnection is whistler waves. These wavesinteract with energetic particles and scatter their pitch-angles, triggering losses ofmagnetic confinement. A previous study demonstrated via exact relativistic analysesthat if a particle undergoes a "two-valley" motion, it undergoes drastic changes inits pitch-angle. This analysis is extended to a relativistic thermal distribution ofparticles. The condition for two-valley motion is first derived; it is then shownthat a significant fraction of the particle distribution meets this condition and thusundergoes large pitch-angle scatterings. The scaling of this fraction with the waveamplitude suggests that relativistic microburst events may be explained by the two-valleymechanism. It is also found that the widely-used second-order trapping theoryviiis an inaccurate approximation of the theory presented.A new method of probing the magnetic field in a plasma is described and developedto some extent. It utilizes the two-photon Doppler-free laser-induced fluorescencetechnique, where two counter-propagating laser beams effectively cancel out theDoppler effect and excite electron populations. The fluorescence resulting fromthe subsequent de-excitation is then measured, enabling the resolution of Zeemansplitting of the spectral lines from which the magnetic field information can beinferred. A high-power, repetitively-pulsed radio-frequency plasma source wasdeveloped as the subject of diagnosis, and preliminary results are presented.viiiPUBLISHED CONTENT AND CONTRIBUTIONS[1] Young Dae Yoon and Paul M. Bellan. A generalized two-fluid picture of non-driven collisionless reconnection and its relation to whistler waves. Physicsof Plasmas, 24(5):052114, May 2017. ISSN 1070-664X. doi: 10.1063/1.4982812. URL http://aip.scitation.org/doi/10.1063/1.4982812.Y.D.Y. performed the analytical calculations, built and conducted the numericalsimulations, and wrote the manuscript.[2] Young Dae Yoon and Paul M. Bellan. An intuitive two-fluid picture of spon-taneous 2D collisionless magnetic reconnection and whistler wave generation.Physics of Plasmas, 25(5):055704, May 2018. ISSN 1070-664X. doi: 10.1063/1.5016345. URL http://aip.scitation.org/doi/10.1063/1.5016345.Y.D.Y. performed the analytical calculations, built and conducted the numericalsimulations, and wrote the manuscript.[3] YoungDaeYoon and PaulM. Bellan. Fast IonHeating in Transient CollisionlessMagnetic Reconnection via an Intrinsic Stochastic Mechanism. The Astrophys-ical Journal Letters, 868(2):L31, Nov 2018. ISSN 2041-8213. doi: 10.3847/2041-8213/aaf0a3. URL http://stacks.iop.org/2041-8205/868/i=2/a=L31?key=crossref.559ad6f22dfdfcdfe3b6e580c7240456. Y.D.Y.performed the analytical calculations, built and conducted the numerical simu-lations, and wrote the manuscript.[4] Young Dae Yoon and Paul M Bellan. Kinetic Verification of the Stochas-tic Ion Heating Mechanism in Collisionless Magnetic Reconnection. TheAstrophysical Journal Letters, 887(2):L29, Dec 2019. ISSN 2041-8213.doi: 10.3847/2041-8213/ab5b0a. URL https://iopscience.iop.org/article/10.3847/2041-8213/ab5b0a. Y.D.Y. performed the analytical cal-culations, conducted the numerical simulations, and wrote the manuscript.[5] Young Dae Yoon and Paul M. Bellan. The electron canonical battery effect inmagnetic reconnection: Completion of the electron canonical vorticity frame-work. Physics of Plasmas, 26(10):100702, Oct 2019. ISSN 1070-664X. doi:10.1063/1.5122225. URL http://aip.scitation.org/doi/10.1063/1.5122225. Y.D.Y. performed the analytical calculations, built and conducted thenumerical simulations, and wrote the manuscript.[6] Young Dae Yoon and Paul M. Bellan. Non-diffusive pitch-angle scattering of adistribution of energetic particles by coherent whistler waves. Submitted, 2020.Y.D.Y. performed the analytical calculations, built and conducted the numericalsimulations, and wrote the manuscript.ixTABLE OF CONTENTSAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viPublished Content and Contributions . . . . . . . . . . . . . . . . . . . . . . viiiBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiChapter I: Introduction to Plasma Physics . . . . . . . . . . . . . . . . . . . 11.1 Single-Particle Theory . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Vlasov Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Two-Fluid Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Magneto-Hydrodynamics (MHD) . . . . . . . . . . . . . . . . . . . 51.5 Magnetic Reconnection . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Chapter II: Canonical Vorticity Framework of Magnetic Reconnection: anIntuitive Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Canonical Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Numerical Construction . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5 Intuitive Description . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Chapter III: Electron Canonical Battery Term: Completion of the CanonicalVorticity Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1 The Completed Canonical Vorticity Framework . . . . . . . . . . . 363.2 Fluid Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3 Particle-in-cell Simulation . . . . . . . . . . . . . . . . . . . . . . . 433.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Chapter IV: Stochastic Ion Heating in Magnetic Reconnection . . . . . . . . 524.1 Stochastic Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Relevant Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3 Numerical Simulation and Results . . . . . . . . . . . . . . . . . . . 584.4 Stochastic Heating Condition Analysis . . . . . . . . . . . . . . . . 584.5 Test Particle Simulation . . . . . . . . . . . . . . . . . . . . . . . . 614.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67xChapter V: Kinetic Verification of Stochastic Ion Heating in Magnetic Re-connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.1 Stochastic Heating in the Harris Equilibrium Plasma Sheath . . . . . 695.2 Particle-in-cell verification . . . . . . . . . . . . . . . . . . . . . . . 725.3 Confirmation of Stochastic Motion . . . . . . . . . . . . . . . . . . 765.4 Heavy Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79ChapterVI: Pitch-Angle Scattering ofEnergetic Particles byCoherentWhistlerWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.1 Two-Valley Motion Review . . . . . . . . . . . . . . . . . . . . . . 856.2 Two-Valley Motion Condition . . . . . . . . . . . . . . . . . . . . . 876.3 Distribution of ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.4 Fraction of Particles Undergoing Two-Valley Motion . . . . . . . . . 916.5 Numerical Verification . . . . . . . . . . . . . . . . . . . . . . . . . 936.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.7 Comparison to Second-order Trapping Theory . . . . . . . . . . . . 986.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Chapter VII: Magnetic Field Diagnostic using Two-Photon Doppler-freeLaser-Induced Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . 1037.1 Sources of Spectral Broadening . . . . . . . . . . . . . . . . . . . . 1047.2 Two-Photon Doppler-Free Laser-Induced Fluorescence . . . . . . . . 1077.3 Repetitively Pulsed, High-Power, Inductively-CoupledRadio-FrequencyPlasma Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.5 One-Photon LIF Results . . . . . . . . . . . . . . . . . . . . . . . . 1167.6 Two-Photon LIF Preliminary Results . . . . . . . . . . . . . . . . . 1207.7 Suggestions for Further Work . . . . . . . . . . . . . . . . . . . . . 1227.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Chapter VIII: Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.1 Progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.3 Progenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1268.4 Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Appendix A: Whistler Wave Analytical Solution . . . . . . . . . . . . . . . . 149Appendix B: Algorithm Test using Whistler Waves . . . . . . . . . . . . . . 157Appendix C: Canonical Helicity Density and the Lagrangian Density inElectron-Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . 159Appendix D: Implicit Particle Integrator . . . . . . . . . . . . . . . . . . . . 162Appendix E: Harris Equilibrium Plasma Sheath Derivation . . . . . . . . . . 164Appendix F: One-Dimensional and Two-Dimensional Maxwell-Jüttner Dis-tributions and the Distribution of ξ . . . . . . . . . . . . . . . . . . . . . 167F.1 Derivation of fρ⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167F.2 Derivation of fρz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168F.3 Derivation of fξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169xiF.4 Derivation of non-relativistic fρ⊥ . . . . . . . . . . . . . . . . . . . 170xiiLIST OF ILLUSTRATIONSNumber Page1.1 Incremental change in surface area due to a displacement Uδt of aline segment dl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Frozen-in magnetic field lines within deforming magnetic flux tubes. 91.3 A graphical description of a torsional Alfvén wave. . . . . . . . . . . 101.4 (a) Non-reconnecting magnetic field lines (black) and stagnatingflows (green). (b) Reconnecting magnetic field lines and plasmainflow/outflow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Fast reconnection geometry. . . . . . . . . . . . . . . . . . . . . . . 141.6 Thesis outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 The background and perturbation magnetic fields and their sum asthe initial condition for magnetic reconnection. . . . . . . . . . . . . 242.2 Time evolution of the magnetic field during collisionless reconnec-tion. The colors and blue lines are the out-of-plane and in-planemagnetic fields, respectively, and the red arrows are the electron flowvectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 In-plane (blue lines) and out-of-plane (colors) components of (a)B and (b) Qe. Corresponding 3D views are shown in (c) and (d),respectively. The color of the lines represent the height in z. The redarrow in (c) is the direction of the out-of-plane electron velocity uez. 262.4 Temporal evolution of a Qe flux tube. The insets are side-on andoverhead views of the flux tube. The red arrow represents the di-rection and location of electron flow ue. The color of the flux tuberepresents the change of |Qe |. . . . . . . . . . . . . . . . . . . . . . 272.5 Time evolution of Qez, Bz, and (∇ × ue)z at (x, y) = (1, 2)de. They-axis is in natural logarithmic scale. . . . . . . . . . . . . . . . . . 292.6 (a) A Qe flux tube and the spatial variation of the magnitudes of uez(red arrows). (b) The the direction of uez in a frame traveling in the−z-direction with the center portion of the flux tube. . . . . . . . . . 32xiii2.7 Temporal evolution of one of the reconnected magnetic field lines(e.g. red line in Figure 2.2). The bottom figure shows the same fieldlines when the initial field line is straighted out. The A, B, and Csin both figures correspond to identical locations along the field line.The colors represent the height in z. . . . . . . . . . . . . . . . . . . 343.1 The staggered spatial grid used for the fluid simulation. . . . . . . . . 403.2 The staggered time grid and the time advancement algorithm for thefluid simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3 In-plane Qe (red), in-plane B (black) and uez (color) for varying µvalues and isotropic pressure at t = (a) 380, (b) 450, (c) 560. . . . . . 453.4 Same as Fig. 3.3 with pressure anisotropy and varying µ values att = (a) 400, (b) 500, (c) 630. . . . . . . . . . . . . . . . . . . . . . . 463.5 Same as 3.3 from the particle-in-cell simulation at t = 300. . . . . . . 473.6 (a) The y-component of the convective term ŷ · ∇ × (ue ×Qe) (color)and the anisotropic contribution to the canonical battery term −ŷ ·∇×(∇ · pe,aniso/ne)(contour) for the simulation corresponding to Fig.3.4b. (b) The sum of ŷ · ∇ × (ue ×Qe) and −ŷ · ∇ ×(∇ · pe,aniso/ne),which is equal to ∂Qey/∂t. The red arrows represent the direction ofQey. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.7 (a-c) Various quantities involved in the calculation of −ŷ · ∇ ×(∇ · pe,aniso/ne)' Bz∂ [B · ∇σ] /∂x for the simulation correspond-ing to Fig. 3.4b. (d) The y-component of the convective termŷ · ∇ × (ue ×Qe) and the anisotropic contribution to the canonicalbattery term −ŷ · ∇ ×(∇ · pe,aniso/ne), and (e) their sum. . . . . . . . 494.1 (a) Reconnected in-plane B field lines (white) and connected in-plane Qe field lines (red) in the reconnection geometry. The effectivepotential (color) was calculated from the cold version of Eq. 4.19.The inflows and outflows are respectively in the±x and±y directions.The x-axis and the y-axis have different scales to show the fieldlines more evidently. (b) Comparison of B2y/2 with −∫Exdx and−∫uezBydx along the black dotted line in (a). (c) Comparison of(u2ey + u2e)/2 with −∫Eydy along the magenta line in (a). . . . . . . 57xiv4.2 (a) Ten coherent ion trajectories undergoing Speiser-like orbits, and(b) their stochastic counterparts. (c) Positions in y phase space of all6000 ions at t = 580 |ωce |−1, and (d) their stochastic counterparts.(e), (f) Zoom-ins of (c) and (d), respectively. (g) Distribution of |∆ Ûy |2for the coherent case (blue) and the stochastic case (red). (h) Time-dependent 3D separation distances between two selected particleswith the minimum initial separation distance, 0.015de, under thestationary electric field (blue line; particles are red dots in (b)) andthe growing electric field (red line; particles are red dots in (f)). . . . 624.3 Positions in y phase space of ions that started with a thermal velocityof 0.01 de |ωce | in (a) the coherent case and (b) the stochastic case.Distribution of |∆ Ûy |2 in (c) the coherent case and (d) the stochasticcase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.1 (a) Contours of x̄sh(λ̄, Bg). Inset plots x̄sh as a function of Bg forthree λ̄ values. (b) Same as (a), but for x̄sh/λ̄. . . . . . . . . . . . . . 715.2 B (black lines) and Ex (color) for (a) bg = 0, (b) 0.1, (c) 0.3, and (d)0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3 lnα for bg = 0, 0.1, 0.3, and 0.5, respectively. . . . . . . . . . . . . . 765.4 Ti for bg = 0, 0.1, 0.3, and 0.5, respectively. . . . . . . . . . . . . . . 775.5 Ti⊥ for bg = 0, 0.1, 0.3, and 0.5, respectively. . . . . . . . . . . . . . 785.6 Ti‖ for bg = 0, 0.1, 0.3, and 0.5, respectively. . . . . . . . . . . . . . 795.7 (a) Two test ion trajectories (black solid and dashed lines) and Ex(color) under bg = 0. (b) The time-dependent spatial separationbetween the ions in (a). (c) µE×B (blue) and lnα (red) along theparticle trajectory represented by the black solid line in (a). Thered dashed line represents the stochastic heating criterion, abovewhich stochastic heating is expected. (d)-(f) are the same as (a)-(c),respectively, except for bg = 0.3. (g)-(i) are the same as (d)-(f),except for mi/me = 500. . . . . . . . . . . . . . . . . . . . . . . . . 815.8 (a) lnα, (b) Ti, (c) Ti⊥ for ions with mi/me = 500 under bg = 0.3. . . 82xv6.1 (a) An example of a two-valley ψ (ξ) for which b = −0.008 < 0.(b) An example of a one-valley ψ (ξ) for which b = 0.031 ≥ 0. (c)The time-dependent pitch-angle of the particle undergoing two-valleymotion, and (d) that of the particle undergoing one-valley motion.The wave parameters were κ = 0.01, α = 0.25, and n(α) = 18 fromEq. 6.27. (e), (f) The approximated pseudo-potentials χ obtainedby keeping only the term involving sκ′ in Eq. 6.7 for the respectiveparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.2 fξ for different (a) θ, (b) α and (c) n values. The default values areθ = 0.1, α = 0.25, and n = 10. The black dashed line is the resonantcondition ξ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.3 ptv as a function of α and θ for different κ values. . . . . . . . . . . . 936.4 (a) Regions of initial momentum space (dark green) that satisfy theunapproximated two-valley criteria (Eqs. 6.13 and b < 0) for φ =π/4. (b) Regions of this space that satisfy the approximated criterion(Eq. 6.16) for φ = π/4. (c) Pitch-angle range (in degrees) within asingle particle trajectory for a range of initial particle momenta forφ = π/4. (d-f) are the same as (a-c) except for φ = −π/4. Bluelines represent the resonance condition (Eq. 6.1; ξ = 0). The waveparameters were α = 0.25, κ = 0.005, and n = 18 from Eq. 6.27. . . 946.5 Pitch-angle range of 10,000 particles whose initial momenta wererandomly sampled from Eq. 6.18 for different κ values. Red pointsrepresent particles that meet the two-valley criterion (Eq. 6.16), andthe text inside represents the percentage of red particles. The redhorizontal lines represents the median ∆θ◦pitch of the red particles . . 956.6 Pitch-angle change per wave period of the respective simulations inFig. 6.5. The red horizontal lines respectively represent the medianvalue of the pitch-angle change per wave period of the red particles. . 966.7 Same as Fig. 6.4, but for κ = 0.02. . . . . . . . . . . . . . . . . . . . 977.1 A diagram of a laser-induced fluorescence scheme. . . . . . . . . . . 1077.2 Two-photon interaction of an atom in the lab frame and in the atomframe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.3 Two-photon laser-induced fluorescence scheme for Argon II. Dis-played wavelengths are vacuum values. . . . . . . . . . . . . . . . . 109xvi7.4 Example of two-photon LIF in Rubidium. (a) The original Doppler-broadened spectrum. (b)Result after applying twophotons. (c),(d),(e),(f)Zoom-ins of respective peaks, showing hyperfine splittings. Repro-duced from Jacques et al. [77]. © European Physical Society. Re-produced by permission of IOP Publishing. All rights reserved. . . . 1107.5 Two-photon LIF observation of Zeeman splitting of a Rubidium hy-perfine line. Reproduced from Jacques et al. [77]. © EuropeanPhysical Society. Reproduced by permission of IOP Publishing. Allrights reserved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.6 A schematic diagram of the RF plasma source. . . . . . . . . . . . . 1127.7 The resultant plasma discharge. . . . . . . . . . . . . . . . . . . . . 1147.8 Experimental setup for two-photon laser-induced fluorescence. Lstands for lens, M for mirror, BS for beam-splitter, AOM for acousto-optic modulator, OI for optical isolator, and PMT for photo-multipliertube. The purple color is the end-on view of the plasma. . . . . . . . 1157.9 One-photon LIF results for 1, 5, and 20 mTorr. The laser wavelengthis scanned around the 394.5 nm line in Fig. 7.3 and the plots showthe subsequent 442.7 nm fluorescence as a function of the laser wave-length. Red lines are fitted Voigt profiles. γ is the Lorentzian width,σ is the Gaussian width, Ti is the ion temperature calculated from σ,and λ0 is the center wavelength. . . . . . . . . . . . . . . . . . . . . 1187.10 The expected locations of the Zeeman splitted transition lines involv-ing the 1 → 3 transition in Fig. 7.3 for |B| = 50 G. The red line isthe original un-splitted line. . . . . . . . . . . . . . . . . . . . . . . 1207.11 Two-photon LIF results of two separate scan instances at 1 mTorr.The red line is the theoretical center transition wavelength, and thegrey area is the expected Zeeman splitting for |B| = 50 G. . . . . . . 121A.1 Analytical solution for Bφ in Eq. A.73 of an arbitrary magnitude.The red vertical line at z = 0 represents the ring current source. Thenumerical integration of Eq. A.73 is dubious at r = 0 (black dashedline). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155A.2 The fast Fourier transform of the result in Fig. A.1. The red linesrepresent the dispersion relation in Eq. A.80 for Ω = 0.35. . . . . . . 156B.1 Bφ at t = 80|ωce |−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 157B.2 The fast Fourier transform of the results in Fig. B.1. The red linesrepresent the dispersion relation in Eq. A.80 for Ω = 0.35. . . . . . . 158xviiLIST OF TABLESNumber Page2.1 Dimensional quantities and their normalization parameters in Chapter 2. 222.2 Comparison of simulated γs and calculated γc for different lengthparameters of the current sheet . . . . . . . . . . . . . . . . . . . . . 303.1 Dimensional quantities and their normalization parameters in Chapter 3. 375.1 Dimensional quantities and their normalization parameters in Chapter 5. 727.1 Plasma parameters of regions where magnetic reconnection occursor is thought to occur. L is the global length scale of the plasma,n is the density, and de is the electron skin depth. Unless specifiedotherwise, di is the ion skin depth assuming that the ion species ishydrogen. Referenced from Refs. [29, 74–76, 109, 127] . . . . . . . 1037.2 Comparison of different broadening effects in two types of plasmas:the CaltechMHD-driven Jet Experiment [29, 105, 109], and a pulsed,inductively-coupled, radio-frequency plasma [32]. Stark broadeningis calculated by using the experimentallymeasured values inKonjevićet al. [83] and using the scaling ∝ ne for Argon [173]. Zeemansplitting is calculated by assuming m j and g j are of order 1, andB = 0.1 T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071C h a p t e r 1INTRODUCTION TO PLASMA PHYSICSA plasma is often dubbed the "fourth state of matter" and is a mixture of ions,electrons, and possibly neutral atoms. Because the constituents of a plasmamutuallyinteract with one another, a plasma tends to develop collective behavior. Plasmaphysics is the study of this behavior in various circumstances. The majority ofthe observable matter in the Universe is in the plasma state, examples of whichinclude stars, planetary nebulae, the Earth’s magnetosphere, astrophysical accretiondisks and jets, and the interstellar medium. Naturally occurring plasmas on theEarth include lightning and flames, while man-made plasmas are found in fusion-grade experiments, research experiments, semiconductor manufacturing, and so on.In contrast to conventional gas or liquid dynamics which are dominated by inter-particle binary collisions, plasma dynamics are dominated by long-range electricand magnetic fields. The coupling of these fields with fluid effects leads to complexplasma behavior.Plasma dynamics can be described by four different models, which are— in order ofdecreasing fundamentality—single-particle theory, Vlasov theory, two-fluid theory,andmagneto-hydrodynamics (MHD). Single-particle theory is themost fundamentaland follows individual dynamics of all mutually-interacting particles in the plasmaand their interaction with electromagnetic fields. Vlasov theory constructs velocitydistribution functions from a statistically large number of particles and follows thetemporal evolution of these functions. Two-fluid theory treats the plasma as a systemof finite-pressure fluids, one for each species. MHD treats the plasma as a singleconducting fluid with finite pressure. The most suitable model to use depends onthe given situation; the specified plasma regime and the underlying assumptionsdetermine which toolkit is the most appropriate in solving the particular problem.All four plasma models will be used in pertinent places throughout the thesis. In thischapter, the fundamentals of each model are first presented as a proper introductionand background. Then, an introduction to magnetic reconnection — the mainsubject of this thesis — will be presented.21.1 Single-Particle TheoryThe equation of motion for a particle with charge q and mass m acted on by electricand magnetic fields E and B is given by the Lorentz equationmdvdt= qE + qv × B, (1.1)where v is the particle velocity. This equation will now be solved iteratively to obtaina hierarchy of slow and fast motions. The derivations will be more detailed thanusually necessary in order to highlight the underlying assumptions, the violation ofwhich will be important in Chapter 4. E and B are assumed to evolve adiabatically,i.e., slowly change in space and time.Cyclotron motionDividing Eq. 1.1 by q |B| yields1ωcdvdt=E|B| + v ×B|B| , (1.2)where ωc = q |B|/m is the cyclotron frequency. Assuming E = 0, we have1ωcdvcdt= vc ×B|B| , (1.3)which yields the cyclotron motion with velocity vc and frequency ωc.E × B driftLet us now define vE to be a steady drift on top of vc so that |dvE/dt | |dvc/dt |.Writing v = vc + vE and using |dvE/dt | |dvc/dt |, Equation 1.2 now becomes1ωcdvcdt=E|B| + (vc + vE ) ×B|B| . (1.4)Using Eq. 1.3, this becomesE + vE × B = 0, (1.5)and the solution isvE =E⊥ × BB2, (1.6)where E⊥ is the electric field perpendicular to B. This steady drift is called theE × B drift.3Polarization driftNext, let us define vp to be the next-order correction to vE so that |dvp/dt | |dvE/dt |. After writing v = vc + vE + vp and averaging out the cyclotron motion,Eq. 1.2 becomes1ωcd(vE + vp)dt=E|B| +(vE + vp)× B|B| . (1.7)Using the assumption |dvp/dt | |dvE/dt | and Eq. 1.5, we have1ωcdvEdt= vp ×B|B| , (1.8)for which the solution isvp =mqB2dE⊥dt. (1.9)This drift is called the polarization drift.µ conservationThemotion of the particle after averaging out the cyclotron motion is called guiding-center motion. In the frame of reference moving with the guiding-center velocityvgc = vE + vp, the magnetic momentµ =mv2c2B(1.10)is a conserved quantity.Breakdown of HierarchyThe drift solutions presented here rely on a delicate set of assumptions which, ifviolated, lead to non-adiabatic particle behavior. In fact, it will be shown in detailin Chapter 4 that if the drift hierarchy breaks down, chaotic behavior arises. In thiscase, a fast, strong heating of associated particles occurs.1.2 Vlasov TheoryLet us now consider the dynamics of a statistically large number of particles. Theinstantaneous particle density in phase space is called the distribution function andis denoted f (x, v, t). Conservation of the number of particles signifies that the rateof change of f must equal the flux of f in and out of the phase space volume.Consideration of this fact yields the Vlasov equation [11]:∂ f∂t+ v · ∂ f∂x +∂∂v · (a f ) = 0, (1.11)4where a (x, v, t) is the particle acceleration.Charged particles are subject to the Lorentz force, i.e.,aL =qm(E + v × B) . (1.12)Although aL is a function of v, aL commutes with the velocity derivative because(v × B)i = v j Bk − vk Bi does not depend on vi. Therefore, Eq. 1.11 becomes∂ f∂t+ v · ∂ f∂x +qm(E + v × B) · ∂ f∂v = 0, (1.13)which, together with Maxwell’s equations, is called the Vlasov-Maxwell system ofequations.1.3 Two-Fluid TheoryBecause x and v are independent quantities in phase space, Eq. 1.11 can be expressedas∂ f∂t+∂∂x · (v f ) +∂∂v · (a f ) = 0. (1.14)By taking moments of Eq. 1.14, fluid equations can now be obtained.Continuity EquationSince f is the instantaneous density in phase-space, taking its zeroth moment yieldsthe instantaneous spatial density, i.e.,n (x, t) =∫f d3v. (1.15)Taking the first moment yields the mean fluid velocity:u (x, t) =∫v f d3vn (x, t) . (1.16)Also, by Gauss’s Law, ∫∂∂v · (a f ) d3v =∫a f dsv, (1.17)where sv is the surface vector at v → ∞. Because the distribution function mustvanish at v→∞ (i.e., f (v→∞) = 0), this is equal to zero.Integrating Eq. 1.14 in three-dimensional velocity space, i.e., taking the zerothmoment and recalling that x, v, and t are independent variables thus yield thecontinuity equation∂nσ∂t+∂∂xσ· (nσuσ) = 0, (1.18)where the subscript σ represents each species, e.g., σ = i, e respectively for ionsand electrons.5Fluid Equation of MotionLet us now write velocity v is the sum of its mean value and its random part so thatv = u + v′ and define the pressure tensor↔p = m∫v′v′ f d3v′. (1.19)Here d3v = d3v′ and∫v′ f d3v′ = 0 because the mean value of the random part iszero. Then, taking the first moment of Eq. 1.13 and integrating the third term byparts yield the fluid equation of motionnσmσDuσDt= nσqσ (E + uσ × B) − ∇ ·↔pσ, (1.20)whereDDt=∂∂t+ uσ · ∇ (1.21)is the convective derivative.Equations 1.18 and 1.20 for ions and electrons constitute the system of two-fluidequations.1.4 Magneto-Hydrodynamics (MHD)Equation 1.20 yields two equations involving ui and ue for ions and electrons,respectively. Magneto-hydrodynamics (MHD) involves two new variables whichare linear combinations of ui and ue. These are the current densityJ =∑σnσqσuσ, (1.22)and the center-of-mass velocityU = 1ρ∑σmσnσuσ, (1.23)whereρ =∑σmσnσ (1.24)is the mass density. Writing particle velocity v as a sum of the center-of-massvelocity and a random part, i.e., v = U + v′, the MHD pressure tensor is defined as↔PMHD =∑σmσ∫v′v′ fσd3v′. (1.25)6Multiplying Eq. 1.13 by mσ, taking its zeroth and first moment and summing overall species, we obtain respectively the MHD continuity equation∂ρ∂t+ ∇ · (ρU) = 0, (1.26)and the MHD equation of motionρ(∂U∂t+ U · ∇U)= J × B − ∇ ·↔PMHD. (1.27)These equations, together with Maxwell’s equations,∇ · E = ρcε0, (1.28)∇ · B = 0, (1.29)∇ × E = −∂B∂t, (1.30)∇ × B = µ0J, (1.31)constitute the set of MHD equations. Note that the pre-Maxwell form of Ampère’slawwas used becauseMHDusually deals with phenomenawith characteristic veloc-ities much slower than the light speed, so the displacement current can be ignored.Unless specified otherwise, this form of Ampère’s law will be used throughout thethesis.It can be seen from Eq. 1.27 that MHD plasma motion is determined by two forces:the magnetic force J × B and the hydrodynamic force ∇ ·↔PMHD. Since Ampère’slaw gives J = ∇ × B/µ0, we can define plasma beta asβ =PB2/2µ0∼ ∇ ·↔PJ × B . (1.32)As long as the scale length of the two forces are comparable, β measures the ratioof the hydrodynamic force to the magnetic force. Low-β plasmas are thereforedominated by magnetic forces and vice versa.Generalized Ohm’s LawDividing the electron fluid equation by neqe = −ne and rearranging giveE + ue × B = −meeDueDt− ∇ ·↔pene. (1.33)Since ue = ui − J/nee and ui ' U due to the much heavier ion mass, Eq. 1.33becomes the generalized Ohm’s lawE + U × B = J × Bne− meeDueDt− ∇ ·↔pene+ ηJ, (1.34)7Figure 1.1: Incremental change in surface area due to a displacement Uδt of a linesegment dl.where ηJ is introduced to represent resistive effects due to collisions. The terms onthe right-hand side are respectively called the Hall term, the electron inertia term,the pressure term, and the resistive term. The roles these terms play in plasmadynamics will become apparent in Chapters 2 and 3.Ideal Ohm’s Law and Frozen-in FluxIf the right-hand side of Eq. 1.34 is neglected, then we obtain the ideal Ohm’s lawE + U × B = 0. (1.35)In regimes where this equation is valid — typically involving very slow time scalesand very large spatial scales — the plasma acts as a perfectly conducting fluid andis thus said to be ideal. Taking the curl of Eq. 1.35 and using Faraday’s law∇ × E = −∂B/∂t yield the plasma induction equation∂B∂t= ∇ × (U × B) . (1.36)The induction equation signifies an important aspect of ideal MHD, namely thatmagnetic flux through any surface moving with the plasma is conserved. To provethis property, we define the magnetic flux through a moving surface S(t) boundedby C(t) asΦB =∫SB · ds. (1.37)The rate of change of ΦB is determined by that of B and that of S. The former issimply ∂B/∂t. The latter can be found by noting that the incremental change insurface area due to a displacement Uδt of a line segment dl of the bounding contour8C is Uδt × dl (illustrated in Fig. 1.1). Therefore, the rate of change of S is U × dl,anddΦBdt=∫S∂B∂t· ds +∫CB · U × dl, (1.38)=∫S∂B∂t· ds +∫CB × U · dl, (1.39)=∫S[∂B∂t+ ∇ × (B × U)]· ds. (1.40)It can be seen that if Eq. 1.36 is true, then dΦB/dt = 0. This property of idealMHD is called the frozen-in flux phenomenon.Magnetic Flux TubesThe concept of frozen-in flux in ideal MHD equivalently means that magneticfields lines are frozen into the plasma. By connecting a continuous sequence ofsurfaces, a tube-like structure can be constructed. If magnetic field lines penetratethis structure, it is called a magnetic flux tube. Figure 1.2 shows a sketch of twodeforming magnetic flux tubes. The frozen-in field lines move with the tubes asthey deform.It should be noted thatmagnetic flux tubes are flux tubes associatedwith themagneticfield B, and that these are not the only kind of flux tubes. In fact, it will be shownin Chapter 2 that we can also define flux tubes associated with a quantity called thecanonical vorticity Q and that these Q flux tubes have similar but separate dynamicsfrom those of magnetic flux tubes.Magnetic HelicityMagnetic helicity is an important concept in MHD and is defined asK =∫VA · Bd3r, (1.41)where A is the vector potential associated with the magnetic field B and V is theplasma volume. This quantity measures the linkage and twist of magnetic flux tubes.AlthoughA is undefinedwith respect to a gauge, K turns out to be gauge-independentif magnetic field lines do not penetrate the boundary of the volume.Magnetic helicity is a conserved quantity in ideal MHD. To prove this, consider the9Figure 1.2: Frozen-in magnetic field lines within deforming magnetic flux tubes.time evolution of the helicity density κ = A · B∂κ∂t=∂A∂t· B + A · ∂B∂t, (1.42)= − (E + ∇φ) · B − A · (∇ × E) , (1.43)= −2E · B − ∇ · (φB + E × A) . (1.44)Dotting ideal Ohm’s law (Eq. 1.35) with B shows that E · B = 0. Integrating Eq.1.44 over the entire volume, we obtain the helicity conservation equation∂K∂t+∫ds · (φB + E × A) = 0, (1.45)where∫ds is the closed surface integral over the enclosed volume. Assuming thatthe magnetic field and the plasma flow do not penetrate the volume, i.e., B and Uare tangential to the boundary at the boundary, E = −U ×B is purely perpendicularto the boundary (i.e., E ‖ ds, so ds · (E × A) = 0). Thus, the surface integral in Eq.1.45 vanishes andK = const., (1.46)10Figure 1.3: A graphical description of a torsional Alfvén wave.so the total magnetic helicity is conserved for an ideal plasma.Alfvén WaveThe normal mode of ideal MHD is called the Alfvén wave [3], and there are twomodes of this wave: the compressional mode and the shear mode. The formerinvolves compressions and rarefactions of the magnetic field and the latter involvestransverse perturbations to the magnetic field. Both modes travel with the AflvénvelocityvA =B√µ0ρ, (1.47)where B is the magnetic field strength and ρ is the plasma mass density.A particular mode of Alfvén waves is called a "torsional Alfvén wave," which isa class of shear Alfvén waves. A graphical description is shown in Fig. 1.3. Anazimuthal source (red circle) twists the magnetic field lines (black lines) withina magnetic flux tube, and this twist propagates away from the source as torsionalAlfvénwaves. This twistmechanismwill be an analogy to the generationmechanismof another wave named "whistler waves" in Section 2.5.1.5 Magnetic ReconnectionMagnetic reconnection is a plasma process in which opposing magnetic field linescome together, annihilate, reconnect, and release their stored magnetic potential11(a)BU(b)BUFigure 1.4: (a) Non-reconnecting magnetic field lines (black) and stagnating flows(green). (b) Reconnecting magnetic field lines and plasma inflow/outflow.energy as other forms of energy, such as kinetic, thermal, and/or wave energies[165]. This process is observed in a variety of space and astrophysical situations— solar flares, coronal mass ejections, the Earth’s magnetopause and magnetotail,etc [40, 53, 86, 124]. It is also frequent in laboratory plasmas, such as tokamaks,spheromaks, and reversed-field pinches [13, 29, 109, 161, 163]. Magnetic recon-nection is almost always accompanied by strong ion and electron acceleration andheating, as evidenced by ion jets [85], abnormally high ion temperatures [74], andextreme-ultraviolet and X-ray emissions [29, 105]. Since reconnection acts to lowerthe magnetic energy in a plasma while nearly preserving magnetic helicity, it is animportant part of relaxation processes that lead to equilibrium plasma states withminimum magnetic energy [147].In ideal MHD, magnetic field lines are frozen into the plasma. Therefore, opposingfield lines cannot annihilate or break because two counter-propagating plasma flowsstagnate at some point in the middle. This can be seen graphically in Fig. 1.4a,where opposing magnetic field lines (black) pile up at the center due to stagnatingplasma inflows (green). Thus, magnetic reconnection is forbidden in ideal MHDdue to the ideal Ohm’s law (Eq. 1.35) and can only happen if the frozen-in condition12is violated.Sweet-Parker ReconnectionOne means of breaking the frozen-in condition is finite resistivity. If only theresistive term in Eq. 1.34 is kept, we have the resistive Ohm’s lawE + U × B = ηJ. (1.48)The mechanism by which the resistive term allows for reconnection becomes clearafter taking the curl of the above equation, using Faraday’s law to eliminate theelectric field, and rearranging:∂B∂t= ∇ × (U × B) + ηµ0∇2B, (1.49)whereAmpère’s lawwas used so∇×J = µ−10 ∇×(∇ × B) = µ−10 ∇ (∇ · B)−µ−10 ∇2B =−µ−10 ∇2B. The first term on the right-hand side portrays the frozen-in condition. Thesecond term, on the other hand, is a diffusion-like term with the diffusion coefficientη/µ0. Therefore, finite resistivity allows for magnetic field diffusion across plasmaand thus magnetic reconnection as shown in Fig. 1.4b.Using this picture, Sweet and Parker [119] made quantitative predictions of thereconnection rate, i.e., how fast the magnetic field lines reconnect. However, thetheory predicted reconnection rates thatwere far slower thanwhat had been observed.For example, magnetic field diffusion predicted the time scale of solar coronaleruptions to be months to years, whereas the actual observed time scale is minutesto hours. As enticing as this theory is, it does not accurately describe what is actuallyhappening in most physical situations.Fast Collisionless ReconnectionSince resistivity is not a fast enough source for magnetic reconnection, one mustlook to other terms in Eq. 1.34,E + U × B = J × Bne− meeDueDt, (1.50)where only the Hall term and the electron inertia term are kept, and a low-β plasmais assumed so the pressure term is dropped. Let us now investigate the conditionsfor which the non-ideal terms on the right-hand side become important.First, the Hall term will be compared with the U × B term. Assuming U is of theorder of the Alfvén velocity vA = B/√µ0ρ ' B/√µ0nmi and using J ∼ B/µ0L13where L is the length scale of the magnetic field, the following comparison applies:U × B : J × Bne, (1.51)B2√µ0nmi:B2µ0neL, (1.52)L :√miµ0ne2, (1.53)L : di, (1.54)where di = c/√ne2/miε0 = c/ωpi is the ion collisionless skin depth. Therefore, ifthe length scale of the magnetic field becomes comparable to the ion skin depth, i.e.,L ∼ di, the Hall term becomes important. Note that if L di, then the Hall termdominates, so the ions can be assumed to be stationary because |ui | ' |U| |J/ne|.Second, the electron inertia term will be compared with the Hall term. Sinceelectrons are much lighter than ions, they carry most of the current, so ue '−J/ne ∼ B/µ0neL. Considering the time-dependent part of the inertia term,writing ∂/∂t ∼ ω,J × Bne:mee∂ue∂t, (1.55)B2µ0neL:meeωBµ0neL, (1.56)ωce : ω, (1.57)where ωce = eB/me is the electron cyclotron frequency. Now considering thetime-independent part of the inertia term,J × Bne:meeue · ∇ue, (1.58)B2µ0neL:meeB2µ20n2e2L3, (1.59)L2 :meµ0ne2, (1.60)L2 : d2e, (1.61)where de = c/√ne2/meε0 = c/ωpe is the electron collisionless skin depth. Thus,the electron inertia term becomes as important as the Hall term if ω ∼ ωce and/orL ∼ de. Reconnection phenomena mostly involve ω ωce, so we concludethat the Hall term is important at di length scales and the electron inertia termis important at de length scales. Thus, at length scales comparable to or smaller14Figure 1.5: Fast reconnection geometry.than di, fast, collisionless magnetic reconnection consistent with observations mayoccur. Henceforth, the term "magnetic reconnection" will be used interchangeablywith "fast reconnection" and/or "collisionless reconnection."A graphical description of fast reconnection is shown in Fig. 1.5. The inset showsthe set of coordinates that will be used throughout the analysis. The initial opposingmagnetic field lines are in the y-direction and the reconnected fields are in the x-direction. The particle inflow is mainly in the x-direction and the outflow is mainlyin the y-direction. The z-direction represents the out-of-plane components. Sincethe Hall term signifies the difference between ion and electron flows and becomesimportant at L ∼ di, ions "unfreeze" from the magnetic field below these scales andflow outwards (green arrows). Electrons, on the other hand, are still frozen to themagnetic field down to L ∼ de. Only below these scales can the magnetic fieldsreconnect and the electrons flow outwards (red arrows). The particular shape ofthe electron flow creates quadrupole out-of-plane magnetic fields (plus and minus),which are characteristic of fast reconnection. The origin of these fields as well asother facets of reconnection will be explained more intuitively in Chapter 2.The question of how the macroscopic system couples to the microscopic scalesrequired for collisionless reconnection naturally arises. There are various ways thiscould be achieved, and some examples include the kink-induced Rayleigh-Taylorinstability [109], the sausage-to-kink instability [136], ideal tearing instability [148],and plasmoid/fractal instabilities [99, 138].Figure 1.5 represents two directly opposing magnetic fields. If a background out-of-plane magnetic field exists, this is called the guide field. Effects due to the guidefield as well as finite electron pressure will be discussed in Chapter 3.15Magnetic ReconnectionProgression Properties ProgeniesChapter 2Canonical Vorticity FrameworkChapter 3Completion ofthe FrameworkChapter 4StochasticIon HeatingMechanismChapter 5Kinetic VerificationChapter 6RelativisticWhistler-Wave-ParticleInteractionChapter 7Magnetic FieldLaserDiagnosticProbeFigure 1.6: Thesis outline.1.6 Thesis OutlineFigure 1.6 shows the outline of this thesis. The progression ofmagnetic reconnectionwill be studied in Chapters 2 and 3. In Chapter 2, the canonical vorticity frameworkwill be constructed and collisionless magnetic reconnection will be described usingthis framework in an intuitive manner. In Chapter 3, the framework will be extendedto kinetic regimes by the introduction of the canonical battery term.A crucial property of magnetic reconnection, namely its accompaniment by anoma-lous, extreme ion heating, will be studied in Chapters 4 and 5. In Chapter 4, thecanonical vorticity framework will be used to establish the existence and importanceof the stochastic ion heating mechanism in collisionless reconnection. In Chapter5, this mechanism will be analytically and numerically verified to be the main ionheating mechanism up to moderate guide fields.16An important progeny of collisionless reconnection is whistler waves, and theirgeneration mechanism will be discussed in Chapter 2. In Chapter 6, the interactionbetween whistler waves and relativistic particles will be studied in detail. In partic-ular, how coherent waves significantly scatter the pitch-angle of relativistic particleswill be illustrated.A new method of probing magnetic fields is described in Chapter 7. This methoduses two laser beams to excite the electron population in a plasma while suppressingthermal broadening effects in order to resolve the magnetic field information.17C h a p t e r 2CANONICAL VORTICITY FRAMEWORK OF MAGNETICRECONNECTION: AN INTUITIVE DESCRIPTIONAs shown in Section 1.5, collisionless magnetic reconnection is an extremely com-plicated process involving different length scales and multiple spatial dimensions.In this Chapter, a relatively intuitive description of collisionless reconnection willbe given. This is achieved by considering the dynamics of a quantity called thecanonical vorticity, which is a combination of the particle flow and the magneticfield, instead of focusing solely on the magnetic field. Unlike the Sweet-Parkerdescription in Chapter 1 where a steady magnetic diffusion is the main process,collisionless reconnection will be shown to be an exponentially growing instability.2.1 Canonical VorticityThe quantity "canonical vorticity" is defined asQσ = mσ∇ × uσ + qσB, (2.1)= mσwσ + qσB, (2.2)where uσ is the flow velocity of each species σ and wσ = ∇ × uσ is the vorticity ofeach species. By writing B = ∇ × A,Qσ = ∇ × (mσuσ + qσA) , (2.3)= ∇ × Pσ, (2.4)where Pσ = mσuσ + qσA is the canonical momentum of each species. Therefore,canonical vorticity Qσ is the curl of the species canonical momentum Pσ, hence thename. Qσ is a combination of the inertial term which involves finite species mass,and the magnetic term which involves the magnetic field.Frozen-in Property of Canonical VorticityConsider the two-fluid equation (Eq. 1.20)mσ∂uσ∂t+ mσuσ · ∇uσ = qσ (E + uσ × B) −∇ · ↔pσnσ, (2.5)where the convective derivative was decomposed into partial derivatives. Thepressure term will now be dropped by simply assuming low β, or by assuming both18isotropic (∇ · ↔p = ∇p) and barotropic (∇p × ∇n = 0) pressure when intending totake the curl of this equation so that∇ × ∇ ·↔pσnσ= ∇ × ∇pσnσ=1n2σ∇pσ × ∇nσ = 0. (2.6)Taking the curl of Eq. 2.5 and using Faraday’s law to eliminate E,mσ∂∇ × uσ∂t+ mσ∇ × (uσ · ∇uσ) = −qσ∂B∂t+ qσ∇ × (uσ × B) . (2.7)Using the identities ∇(V2/2)= V · ∇V +V × ∇ ×V and ∇ × ∇S = 0 for any vectorV and scalar S,mσ∂∇ × uσ∂t− mσ∇ × (uσ × ∇ × uσ) = −qσ∂B∂t+ qσ∇ × (uσ × B) . (2.8)Writing ∇ × uσ = wσ and rearranging,∂ (mσwσ + qσB)∂t= ∇ × (uσ × [mσwσ + qσB]) , (2.9)or∂Qσ∂t= ∇ × (uσ ×Qσ) , (2.10)which is the canonical induction equation. Equation 2.10 is an extremely importantresult in that it has the same form as the plasma induction equation (Eq. 1.36).Because the plasma induction equation means that the magnetic field B is frozeninto the plasma flow U, Eq. 2.10 means that the canonical vorticity Qσ is frozeninto the species flow uσ. It also equivalently means that the flux associated withQσ,ΦQ =∫SQσ · ds, (2.11)is a conserved quantity, and that flux tubes associated with Qσ can therefore bedefined. Each species has its own flux tubes that convect with the respective speciesflow. The conservation of Qσ flux is a combination of Kelvin’s circulation theorem[149] in hydrodynamics which states that∫S ∇ × uσ · ds =∮C uσ · dl = const., andthe plasma induction equation which states that magnetic flux is conserved.Because ΦQ, not ΦB, is the conserved quantity, B is free to annihilate, reconnect,and be converted to vorticity wσ while preserving ΦQ. In other words, collisionlessmagnetic reconnection is allowed because the topological constraint on B in MHDis relieved in the two-fluid regime.The frozen-in property of Qσ and the interchangeability between wσ and B withinQσ are the crux of the intuitive description of collisionless reconnection that willbe given in this chapter.192.2 AssumptionsA couple of assumptions will now be made in order to readily relate the concept ofcanonical vorticity to collisionless reconnection.Stationary Ion AssumptionThe region of interest in this chapter is the region where L < di (green box in Fig.1.5). In this region, the Hall term dominates over the U×B term, so the ions can beassumed to be stationary; detailed ion dynamics will be presented in Chapter 4. Thestationary ion assumption implies that the current is carried solely by the electrons,i.e.,ue = −Jnee= −∇ × Bµ0nee. (2.12)The electron canonical vorticity then becomesQe = me∇ × ue + qeB, (2.13)= − meµ0nee∇ × (∇ × B) − eB, (2.14)= e(− meµ0nee2∇ × (∇ × B) − B), (2.15)= e(d2e∇2B − B), (2.16)wheremeµ0nee2=1ε0µ0meε0nee2=c2ω2pe= d2e (2.17)and ∇ × (∇ × B) = ∇ (∇ · B) − ∇2B = −∇2B were used. Writing ∇ ∼ 1/L, theLaplacian term which arises from finite electron inertia scales as d2e/L2. Therefore,at length scales much larger than the electron skin depth, i.e., L de, the magneticterm dominates and Qe is topologically indistinguishable from B. At L de, theinertial term dominates so Qe ' mewe = ed2e∇2B. At L ∼ de, both the inertial andmagnetic terms are important.Incompressibility AssumptionSince ions are assumed to be stationary, a local change in electron density willresult in a local change in the charge density. This violates a fundamental plasmaassumption that plasmas are quasi-neutral, i.e., charge-neutral over scale lengthsmuch larger than the Debye length λD. Now, λD/de scales as vTe/c where vTe isthe electron thermal velocity, so vTe c means that λD de. Since de is thesmallest length scale in our analysis of collisionless reconnection, incompressibility20is a valid assumption as long as the electron thermal velocity is much smaller thanthe speed of light. Thus, a time-independent, spatially uniform electron density willbe used throughout the analysis.Mathematically, incompressibility means that∇ · ue = 0. (2.18)This can be seen by rearranging the electron continuity equation,∂ne∂t+ ∇ · (neue) = 0, (2.19)∂ne∂t+ ue · ∇ne + ne (∇ · ue) = 0. (2.20)Therefore, Dne/Dt = 0 if and only if ∇ · ue = 0.Assuming incompressibility is equivalent to assuming that the displacement currentis negligible. This fact is illustrated by taking the divergence of the Ampère’s lawincluding the displacement current assuming stationary ions:∇ · ∇ × B = −∇ · (µ0neeue) + µ0ε0∂∇ · E∂t, (2.21)0 = −∇ · (neue) −∂ne∂t, (2.22)where Gauss’s law was used so ε0∇ · E = (ni − ne) e and ∂ni/∂t = 0 because ionsare stationary. This result clearly shows that the time-dependence of ne is equivalentto finite displacement current.This set of assumptions and the ensuing system of equations have historically beencalled electron-magnetohydrodynamics (EMHD) [25]. Studies of magnetic recon-nection in similar formulations have been done before [17, 25, 27], albeit not in thethree-dimensional, intuitive manner that will be presented here.2.3 Numerical ConstructionA numerical simulation was developed in order to verify the ensuing analysis andgraphically represent the results. The algorithm that solves Eq. 2.10 will now bepresented. The code was written in Python 3.6.NormalizationIn order to facilitate the scaling of the simulation to real-life situations, relevantequations must be made dimensionless, i.e., normalized with respect to a set of21dimensional quantities. For example, dividing Eq. 2.12 by de |ωce | where |ωce | =eB0/me and B0 is a reference magnetic field,uede |ωce |= − me∇ × Bµ0nee2B0de, (2.23)= −de∇ ×BB0. (2.24)Normalizing each quantity, i.e., ūe = ue/de |ωce |, B̄ = B/B0, and ∇̄ = de∇, we havethe normalized Ampère’s law,∇̄ × B̄ = −ūe. (2.25)Likewise, length is normalized to de, time to |ωce |−1, and magnetic field to B0.Normalizing electron canonical vorticity Qe by eB0,Q̄e = ∇̄ × ūe − B̄, (2.26)= ∇̄2B̄ − B̄. (2.27)The normalized electron canonical induction equation ismee2B20∂Qe∂t=mee2B20∇ × (ue ×Qe) (2.28)∂Q̄e∂ t̄= de∇ ×(uede |ωce |× Q̄e)(2.29)∂Q̄e∂ t̄= ∇̄ ×(ūe × Q̄e), (2.30)where t̄ = eB0t/me = |ωce | t. Using Eqs. 2.25 and 2.27, Eq. 2.30 can be writtenentirely in terms of B̄:∂∂ t̄(∇̄2B̄ − B̄)= −∇̄ ×( [∇̄ × B̄]×[∇̄2B̄ − B̄] ). (2.31)Therefore, the entire system can be expressed solely as a function of the magneticfield.From this point on the bars will be dropped to reduce notational clutter. Onlynormalized quantities will be used unless specified otherwise. A summary of thenormalizations used in this chapter is illustrated in Table 2.1.Discretization2.5D Cartesian coordinates are used. This means that any quantity σ at any giventime t is defined on a 2D grid as σti, j , where i and j are the x and y coordinates, and22Quantity Normalizationt |ωce |−1x deue de |ωce |B B0Qe eB0Table 2.1: Dimensional quantities and their normalization parameters in Chapter 2.any vector V has three components, namely Vx , Vy, and Vz. Also, the z coordinateis an ignorable coordinate so that ∂/∂z → 0. The central differentiation scheme isused for the spatial derivative in the x-direction, i.e.,∂σ∂x→σti+1, j − σti−1, j2∆x(2.32)for any quantity q and similarly for the y-direction. The second spatial derivative isexpressed as∂2σ∂x2→σti+1, j + 2σti, j − σti−1, j∆x2. (2.33)The explicit forward differentiation scheme is used for the time derivative, i.e.,∂σ∂t= s→σt+1i, j − σti, j∆t= sti, j (2.34)for any source term s.AlgorithmThe following algorithm is used to solve Eq. 2.30 and equivalently Eq. 2.31.1. Given B̄t , calculate ūte and Q̄te using Eqs. 2.25 and 2.27, respectively.2. Obtain Q̄t+1e using Eq. 2.30.3. Obtain B̄t+1 by solving Eq. 2.27 iteratively using the relaxation method [84] .4. Repeat.The relaxationmethod in step 3 can be explained as follows. Consider the discretizedform of Eq. 2.27 at any given time:Qi, j =Bi+1, j + 2Bi, j − Bi−1, j∆x2+Bi, j+1 + 2Bi, j − Bi, j+1∆y2− Bi, j . (2.35)23This equation can be iteratively solved for B̄i, j :(Bi, j)n+1 =(2∆x2+2∆y2− 1)−1 (Qi, j −(Bi+1, j)n −(Bi−1, j)n∆x2−(Bi, j+1)n −(Bi, j−1)n∆y2),(2.36)where n is each iterative step. The iteration is halted when the difference of B at anytwo consecutive iterative steps reaches below a specified value of tolerance. Thismethod does not always yield a converging, stable solution; convergence tests mustbe first conducted in order to avoid spurious numerical solutions.Algorithm TestThe normal mode of Eq. 2.31 is a plasma wave called a "whistler wave." Therefore,the algorithm should reproduce whistler waves, and relevant tests were conductedto validate the code. In Appendix A, an analytical solution for whistler waves ispresented in detail. In Appendix B, tests of the algorithm against the prediction areconducted and confirm that the code produces whistler waves.Initial ConditionsThe magnetic field profile is the Harris magnetic shear profile [66] (see AppendixE), orBy = tanhxλ, (2.37)where λ is the shear length scale. This profile represents two regions of opposingmagnetic fields separated by a region of magnetic null.By Ampère’s law, this magnetic field is produced by the out-of-plane current profileuez = −Jz = −∂By∂x= −1λcosh−2xλ, (2.38)so the electrons are flowing in the −z-direction near x = 0. This configuration isfrequently called a "current sheet" because of this current profile.Reconnection is initiated by a 2D-localized vector potential perturbation of the formAz = ε exp(− x22λ2− y22σ2y), (2.39)where ε is the perturbation strength and is a small number, e.g., 10−5, and σy is theperturbation length scale in the y-direction. The perturbation magnetic fields from24Background Perturbation ResultzyxFigure 2.1: The background and perturbation magnetic fields and their sum as theinitial condition for magnetic reconnection.this vector potential areB̃x =∂Az∂y= −ε yσ2yexp(− x22λ2− y22σ2y), (2.40)B̃y = −∂Az∂x= εxλ2exp(− x22λ2− y22σ2y). (2.41)The background and perturbation magnetic fields and their sum are shown graphi-cally in Fig. 2.1. The perturbation adds to the background field in such a way thatfield lines close to the center (x = 0) are reconnected; those that are further awaycome closer together to the center at a localized point in y; i.e., the field lines bowtoward the center.This initial condition triggers the instability that is fast magnetic reconnection;analytical and intuitive explanations for why an instability occurs will be illustratedin Section 2.5.Typical grid dimensionswere x×y = 150×600 = 30de×120de. Neumann boundaryconditions were used, but the simulation was halted before any significant physicalchange reached the boundary.2.4 Simulation ResultsFigure 2.2 shows the time evolution of the magnetic field during collisionless recon-nection. As shown by the streamlines, the in-plane magnetic fields spontaneouslycome together at x = 0 and reconnect, accelerating electrons to high velocities (redarrows). The quadrupole out-of-plane magnetic fields, represented by the yellow25Figure 2.2: Time evolution of the magnetic field during collisionless reconnec-tion. The colors and blue lines are the out-of-plane and in-plane magnetic fields,respectively, and the red arrows are the electron flow vectors.and blue colors, also come together and grow in magnitude (note the different colorscales). These results — quadrupole Hall magnetic fields and the geometry of theelectron flow— are consistent with previous numerical and experimental studies ofHall-mediated collisionless reconnection [45, 54, 129, 166].Figures 2.3a and 2.3b show the in-plane (blue lines) and out-of-plane (colors)components of B and Qe, respectively. It can be clearly seen that while B field linesreconnect, Qe field lines do not reconnect and pile up near the center. The slightreconnection of Qe is due to the initial perturbation and finite numerical resistivitythat originates from unavoidable small numerical error.Figures 2.3c and 2.3d show three-dimensional views of B and Qe, respectively. Thisis done by stacking the 2D simulation results in the z-direction and plotting the3D vectors. It can be seen from Fig. 2.3c that the magnetic field lines generallymove in the direction of the out-of-plane electron flow uez (red arrow; recall thatthe magnetic field shear is created by the out-of-plane current profile in Eq. 2.38),but they do not exactly convect with uez. This fact is obvious in comparison toFig. 2.3d, where Qe field lines stay connected and exactly convect with uez in the−z-direction. This exact downward convection is an important process that will beexamined in the next Section.26Figure 2.3: In-plane (blue lines) and out-of-plane (colors) components of (a) B and(b) Qe. Corresponding 3D views are shown in (c) and (d), respectively. The colorof the lines represent the height in z. The red arrow in (c) is the direction of theout-of-plane electron velocity uez.2.5 Intuitive DescriptionInstability and Quadrupole FieldsUnder the assumptions presented in Section 2.2, collisionless reconnection is apurely growing instability with an exponential growth rate that depends on thesystem parameters. This fact was demonstrated in Bellan [12] via an extensiveanalytical calculation involving a pair of second-order differential equations thatform a fourth-order eigensystem. Here, an intuitive explanation of the exponentialgrowth using Qe flux tube dynamics is presented, followed by an analytical proofmuch simpler than that in Bellan [12]. The origin of the quadrupole magnetic fieldsin Fig. 2.3a will also be explained.A Qe flux tube is first defined by picking a Qe field line, .e.g., the red line in Fig.2.3d, and bunching up the field lines around it. The temporal evolution of this Qeflux tube is shown in Fig. 2.4. The small insets show the side-on and overheadviews of the flux tube. The progression of this flux tube is as follows:1. Initially due to the perturbation, a localized section of the flux tube bowstowards the center where electrons are flowing downwards (red arrow; Fig.27Figure 2.4: Temporal evolution of a Qe flux tube. The insets are side-on andoverhead views of the flux tube. The red arrow represents the direction and locationof electron flow ue. The color of the flux tube represents the change of |Qe |.2.4a).2. This section "feels" a stronger electron flow than other parts of the flux tubeand so convects downwards faster (Figs. 2.4b and 2.4c).3. This downward convection creates Qez fields and thus Bz fields that scale likeBz ∼ xy near the origin (colors in Fig. 2.3a). These are the quadrupole Hallfields that are generic to collisionless reconnection.4. Because Bz ∼ xy in the vicinity of the origin, uex = −∂Bz/∂y ∼ −x, sothe electron flow in the x-direction corresponds to inflow towards the center.Thus, more Qe field lines bow toward the center, and the process repeats toyield an instability.It is now clear why an instability occurs, but to prove exponential growth, ananalytical calculation must be done.Let us first explicitly prove that Qe does not reconnect by proving that Qex is zeroon the y-axis. Reconnection of Qe would require development of finite Qex on they-axis, so (Qex)x=0 = 0 means that Qe field lines stay connected. Expanding theright-hand side of Eq. 2.30,∂Qe∂t= Qe · ∇ue + ue (∇ ·Qe) − ue · ∇Qe −Qe (∇ · ue) , (2.42)= Qe · ∇ue − ue · ∇Qe, (2.43)28where ∇ · Qe = 0 by the definition of Qe (Eq. 2.26) and ∇ · ue = 0 by the incom-pressibility assumption. Expressing this equation using the convective derivative,DQeDt= Qe · ∇ue, (2.44)and its x-component isDQexDt= Qex∂uex∂x+Qey∂uex∂y, (2.45)where ∂/∂z → 0was used. We now examinewhat happens at x = 0 if (Qex)x=0,t=0 =0. The first term is obviously zero at t = 0. Because uex represents the electroninflow and so is zero everywhere on the y-axis, the second term is also zero. Thusat t = 0,DQexDt= 0. (2.46)Therefore, if (Qex)x=0 starts out with zero, it will remain zero. It has thus beenexplicitly established that Qe does not reconnect, and equivalently that Qex at ornear x = 0 is zero or extremely small.Next, consider the y-component of Eq. 2.44:DQeyDt= Qex∂uey∂x+Qey∂uey∂y. (2.47)Near x = 0, Qex ' 0, so(DQeyDt)x≈0=(Qey)x≈0(∂uey∂y)x≈0, (2.48)where x ≈ 0 means near but not exactly at x = 0. The solution is(Qey)x≈0,t =(Qey)x≈0,t=0[exp(∫ t0∂uey∂ydt)]x≈0, (2.49)which is a purely growing solution (∂uey/∂y > 0) because Bz has the profileBz ∼ γxy near x = 0 where γ is a positive value as can be seen from Fig. 2.2, and∂uey/∂y = ∂2Bz/∂x∂y ∼ γ > 0 using Eq. 2.25.This exponential growth propagates to other variables as well. Figure 2.5 shows thetime evolution of Qez, Bz, and (∇ × ue)z at (x, y) = (1, 2)de. The y-axis is in naturallogarithmic scale. Because the initial conditions are not an eigenfunction of thesystem, all relevant quantities relax to an eigenstate before t = 100 |ωce |−1. However,after t = 100 |ωce |−1, it can be seen that all relevant quantities exponentially growin time.29Figure 2.5: Time evolution of Qez, Bz, and (∇ × ue)z at (x, y) = (1, 2)de. The y-axisis in natural logarithmic scale.Thus, we can conclude that any slight gradient of electron velocity along Qe fieldlines is a sufficient condition for this instability that provides collisionless reconnec-tion.Bellan [12] predicted the growth rate in collisionless reconnection as (writing interms of the normalized variables used in this chapter)γ =√2101λσy. (2.50)Now, incompressibility dictates that the electron outflow uey satisfies ueyλ = const.,i.e., if the flow channel narrows, the outflow must be faster and vice-versa. Thus,the growth rate in Eq. 2.49 scales asγ '∂uey∂y'ueyσy∼ 1λσy, (2.51)which is in agreement with Eq. 2.50.Table 2.2 shows the comparison between the calculated growth rate in Bellan [12](γc) and the simulated growth rate γs obtained by linearly fitting through the linesin, e.g., Fig. 2.5. The agreement between the two rates is evident.30λ σy γs γc1 10 40.19 ± 0.40 43.402 10 17.56 ± 0.10 22.082 16 12.95 ± 0.50 13.805 16 5.21 ± 0.05 4.01Table 2.2: Comparison of simulated γs and calculated γc for different length pa-rameters of the current sheetElectron AccelerationQe flux tube dynamics also explains how collisionless reconnection triggers extremeelectron acceleration. As the flux tube convects downwardwith the electron flow, thetube lengthens and stretches like a rubber band as in Fig. 2.4c. Due to the assumedincompressibility, the total volume of the flux tube must remain the same, so anincrease in flux tube length must be accompanied by a decrease in its circumference,i.e., thinning. However, because Qe field lines are frozen-in, flux tube thinningmeans that the Qe lines bunch up together as in the top flux tube in Fig. 1.2. Thisbunching-up signifies that the magnitude of Qe increases, as can be seen from thecolors of the tube in Fig. 2.4. Because Qe ' ∇×ue at small length scales, this leadsto an increase of ue, i.e., electron acceleration.This process can be more simply understood by recalling that ΦQ is a conservedquantity, soΦQ =∫SQe · ds, (2.52)=∫S∇ × ue · ds −∫SB · ds, (2.53)=∮Cue · dl −∫SB · ds. (2.54)Magnetic reconnection involves the reduction of B, and flux tube thinning involvesthe reduction of S and C. Therefore, as B → 0 and C → 0, ue → ∞ in order toconserve ΦQ.Thus, extreme electron acceleration is due to a combination of interchange betweenthe magnetic field and the electron vorticity within Qe, and Qe amplification due toflux tube thinning.31Generation of Whistler WavesWhistler waves are right-handed circularly polarized electromagnetic plasma wavesin the frequency range ωci, ωpi ω ωce, ωpe with the dispersion relationc2k2ω2=ω2pe/ω2|ωce |ω cos θ − 1, (2.55)where θ is the wave propagation angle with respect to the background magnetic fieldB0 = B0 ẑ. In terms of normalized quantities used in this chapter, this dispersionrelation becomesk2 =1cos θω − 1, (2.56)where bars are dropped to reduce notational clutter. Whistler waves are frequentlyobserved in association with magnetic reconnection in both laboratory [29, 67] andspace [40] plasmas.Because the wave frequency is much larger than any ion-related frequency, ions canbe assumed to be stationary for a whistler wave. Incompressibility can then also beassumed due to quasi-neutrality. Thus, a whistler wave is the normal mode of Eq.2.31 upon which the present theory on reconnection is built upon.To prove this, let us linearize B so that it is expressed as a sum of the normalizedbackground field ẑ and the normalized perturbed field B1, i.e., B = ẑ+B1. Equation2.31 then becomes, to first order,∂∂t(∇2B1 − B1)= ∇ × ([∇ × B1] × ẑ) . (2.57)Assuming B1 ∼ exp (ik · x − iωt) so that ∂/∂t → −iω and ∇ → ik,iω(1 + k2)B1 = k × ([k × B1] × ẑ) , (2.58)= k × (kzB1 − kB1z) , (2.59)= kzk × B1. (2.60)Crossing both sides by k and using Eq. 2.60 to eliminate k × B,iω(1 + k2)k × B1 = kzk × (k × B) , (2.61)−ω2(1 + k2)2B1/kz = kz(k [k · B1] − k2B1), (2.62)ω2(1 + k2)2= k2z k2, (2.63)ω(1 + k2)= k2 cos θ, (2.64)32(a) (b)Figure 2.6: (a) A Qe flux tube and the spatial variation of the magnitudes of uez (redarrows). (b) The the direction of uez in a frame traveling in the −z-direction withthe center portion of the flux tube.where kz = k cos θ was used. Solving this equation in terms of k2 givesk2 =11cos θ − 1, (2.65)which is Eq. 2.56. Therefore, the whistler wave is the normal mode of Eq. 2.31.The wave source for magnetic reconnection is the central electron current, whichinduces a uez shear along a Qe flux tube. In Fig. 2.6a , the magnitude differencebetween uez at two different locations is apparent (red arrows). Now, in a frametraveling in the −z-direction with the center portion of the flux tube, uez at the twolocations have different directions, as can be seen from the orange arrows in Fig.2.6b. Because Qe field lines convect with ue, this relative shear induces a Qe fluxtube twist, which acts as a source for whistler waves. This is analogous to howtorsional Alfvén waves are excited (Fig. 1.3).To demonstrate this mathematically, the normalized electron canonical helicity den-sity κQ = Pe · Qe is first defined in analogy to magnetic helicity density κ = A · B.κQ evolves in time as follows (proof in Appendix C):DκQDt= Qe · ∇(u2e2− pe − ue · A), (2.66)= Qe · ∇L, (2.67)where pe is scalar electron pressure assuming pressure isotropy, and L = u2e2 −pe − ue · A is the Lagrangian density of the system. This is the canonical helicity33conservation equation[13, 172] for this specific regime. The central electron flowis a source of ∇L along the Q field lines, so it is in turn a source of canonicalhelicity density. Whistler waves therefore represent a dynamic winding/unwindingof Qe field lines just as torsional Alfvèn waves do of B lines. Therefore, whistlerwave generation and propagation in collisionless reconnection can be seen as thecentral electron flow curling up the Qe flux tubes and whistler waves propagatingthis helicity density in the ±y-directions.Now we will focus on how the whistler wave manifests itself in B fields. Figure 2.2shows that there is an additional region of out-of-plane magnetic fields that developsnear the outflow regions (e.g. black box in Fig. 2.2c). These fields, which havebeen previously observed by other numerical studies as well [45, 54] but generallynot discussed in three-dimensional detail, suggest wave-like behavior, so it is worthexamining how a typical reconnected B field line (e.g. red line in Fig. 2.2b) evolvestaking these fields into account.Figure 2.7 shows the 3D time evolution of a typical reconnected B field line. Wecan regard the initial field line as the background field, and the final field line asthe background field plus the perturbations to this field. If the initial field line isstretched so that it mimics a uniform background field (as in the bottom subfigure),we can see that the spatial polarization of the final field corresponds to the areaof strong uez (red rectangle) acting as the source of two temporally right-handedcircularly polarized whistler waves propagating away from this source in the ±ydirections.2.6 SummaryA quantity called "canonical vorticity" Qσ frozen into each species’ flow, and theQσ flux of each species is therefore conserved. By examining the dynamics ofQe flux tubes, various aspects of collisionless magnetic reconnection have beenexplained. First, the reason why collisionless reconnection is an exponentiallygrowing instability has been explained through the interaction of a perturbed Qe fluxtube with the central out-of-plane electron flow. Second, the electron accelerationmechanism has been explained through the lengthening and thinning of a Qe fluxtube in conjunction with conversion from the magnetic field to electron vorticitywhile preservingQe flux. Third, the generation of whistler waves has been explainedby the twisting of a Qe flux tube by the out-of-plane electron flow shear and thesubsequent propagation of this twist.34Figure 2.7: Temporal evolution of one of the reconnected magnetic field lines (e.g.red line in Figure 2.2). The bottom figure shows the same field lines when the initialfield line is straighted out. The A, B, and Cs in both figures correspond to identicallocations along the field line. The colors represent the height in z.35C h a p t e r 3ELECTRON CANONICAL BATTERY TERM: COMPLETION OFTHE CANONICAL VORTICITY FRAMEWORKIn Chapter 2, the pressure term was dropped by either assuming low-β or isotropicand barotropic electron pressure. This allowed the expression of the electron equa-tion of motion in the form of an ideal induction equation (Eq. 2.10). In thischapter, the effects of a finite pressure tensor will be investigated. By includingthe pressure tensor, the validity of the canonical vorticity framework extends to theVlasov frame of reference (Section 1.2). It will be shown that the pressure ten-sor acts as a battery term that generates additional canonical vorticity, and that itscompetition with the convective term determines the structure and stability of theelectron-diffusion-region (EDR; red box in Fig. 1.5).The EDR structure during reconnection, which is strongly correlated with the shapeof the electron out-of-plane flow, has long been a subject of controversy. It wasinitially presumed that stable electron-scale current layers could not exist becausevarious instabilities would break up these layers [37, 43, 44]. However, spacecraftobservations showed that, in fact, the EDR has a highly elongated stable struc-ture [121]. The stability of this elongated structure was interpreted via numericalsimulations as resulting from the divergence of the pressure tensor [79, 121, 137].However, this interpretation assumed a zero initial out-of-plane B field (guide field)and later numerical simulations showed that a small guide field alters the structurecompletely [57]. Another study showed that different regimes of the EDR existdepending on a magnetization parameter [89].In the completed canonical vorticity framework, these disparities are not only easilyresolved but also unified. In fact, the examination of canonical vorticity dynam-ics is essential for the correct interpretation of the physical origin of the currentstructure. Different kinetic effects contribute to the canonical battery term, whichthen competes with the convective term to determine the progression of the elec-tron canonical vorticity. This progression in turn determines the overall structure,evolution, and stability of the EDR. This framework provides a simpler and cleareralternative to the traditional approach where each of the multitude of terms in theelectron momentum equation is examined separately.363.1 The Completed Canonical Vorticity FrameworkAssumptionThe sole assumption is that the ions are stationary, which is valid for L < dicorresponding to the green box in Fig. 1.5. While the theory to be presentedis similar in many respects to classical electron-magnetohydrodynamics (EMHD;Chapter 2) [25], the minimal restriction of this stationary-ion assumption causesthe theory to differ substantively from EHMD because the assumption permits finitedisplacement current and the full electron pressure tensor pe = me∫v′ev′e f (ve) d3vewhere v′e is the random part of ve. Because new quantities will be introduced in thisChapter, unnormalized quantities will first be used and will be normalized later.The Generalized Canonical Induction EquationThe electron equation of motion iseE + eue × B = −meDueDt− ∇ ·↔pene. (3.1)Recall from Eq. 2.2 that the electron canonical vorticity is defined asQe = me∇ × ue − eB = mewe − eB, (3.2)where we = ∇× ue is the electron fluid vorticity. Then, Eq. 3.1 can be expressed aseE = ue ×Qe − me∂ue∂t− me∇(u2e2)− ∇ ·↔pene. (3.3)Taking the curl of Eq. 3.3 and using Faraday’s law ∇ × E = −∂B/∂t yields thegeneralized canonical induction equation∂Qe∂t= ∇ × (ue ×Qe) − ∇ ×(∇ · ↔pene). (3.4)It should be noted that Eq. 3.4 is also valid for any plasma species.The electron canonical vorticity dynamics governing electron physics is thus reducedto just the two terms on the right hand side of Eq. 3.4. The ∇ × (ue ×Qe) term isa convective term which prescribes that Qe flux is frozen into ue, and thus providesan intuitive understanding of the temporal development of Qe as demonstrated inChapter 2. Since the convective term describes freezing-in of the vorticity flux tothe flow, −∇ ×(∇ · ↔pe/ne)is the only term in Eq. 3.4 that enables diffusion of Qe37Quantity Normalizationt |ωce |−1x deue de |ωce |B B0Qe eB0E de |ωce |B0↔pe B20/µ0ne n0Table 3.1: Dimensional quantities and their normalization parameters in Chapter 3.across ue or vice-versa. This term will be called the "electron canonical battery"term because in the limit of isotropic pressure, i.e.↔pe = pe↔I = neTe↔I ,− ∇ ×(∇pene)=∇ne × ∇Tene(3.5)is the Biermann battery term [14], and because −∇ ×(∇ · ↔pe/ne)generates Qe asindicated by Eq. 3.4. Since Eq. 3.1 results from the collisionless Vlasov equationwithout approximations, Eq. 3.4 is kinetically exact.3.2 Fluid SimulationIn order to illustrate how the competition between the convective term and the canon-ical battery term governs magnetic reconnection, a fluid simulation was developedthat includes compressibility, displacement current, and kinetic effects. Because ofthe added complexity, the relatively simple algorithm used in Section 2.3 cannot beused. A new algorithm was devised, whose details will now be illustrated.Normalization of Relevant EquationsQuantities are normalized as follows: B to the background field strength B0, lengthto de, time to |ωce |−1,↔pe to B20/µ0, density ne to the background density n0, andE to vAeB0 = de |ωce | B0. A summary of the normalizations used in this chapter isillustrated in Table 3.1.38Dividing the equation of motion by ede |ωce | B0, we haveEde |ωce | B0+uede |ωce |× BB0= − meeB0de |ωce |DueDt− ∇ ·↔pede |ωce | B0nee, (3.6)Ē + ūe × B̄ = −DūeDt̄− ∇ ·↔pede |ωce | B0nee, (3.7)Ē + ūe × B̄ = −DūeDt̄− me∇̄ ·↔ped2e B20nee2, (3.8)Ē + ūe × B̄ = −DūeDt̄− µ0∇̄ ·↔pen̄eB20, (3.9)Ē + ūe × B̄ = −DūeDt̄− ∇̄ ·↔̄pen̄e, (3.10)where the barred quantities are dimensionless, andmed2e e2=meω2pec2e2=men0e2meε0c2e2= n0µ0 (3.11)was used.Faraday’s law is normalized as follows:de∇ ×Ede |ωce | B0= − 1|ωce | B0∂B∂t, (3.12)∇̄ × Ē = −∂B̄∂ t̄. (3.13)Ampère’s law, including the displacement current, is normalized as follows:deB0∇ × B = − deB0µ0neeue +deB01c2∂E∂t, (3.14)∇̄ × B̄ = −deω2pe|ωce | c2n̄eue +|ωce |ω2pe∂∂tEde |ωce | B0, (3.15)∇̄ × B̄ = −n̄eūe +ω2ceω2pe∂Ē∂t, (3.16)where j = −neeue due to the stationary ion assumption.The continuity equation is normalized as follows:1n0 |ωce |∂ne∂t+ de∇ ·(nen0uede |ωce |)= 0, (3.17)∂n̄e∂ t̄+ ∇̄ · (n̄eūe) = 0. (3.18)Unless specified otherwise, the bars will be dropped to reduce notational clutter,and only normalized quantities and dimensionless equations will be used throughoutthis chapter.39Expression in terms of jIn order to facilitate the application of the algorithm to be presented, the relevantequations must first be expressed in terms of the current density j = −neue (recallthat ions are assumed to be stationary). It is obvious how Faraday’s Law, Ampère’slaw, and the electron continuity equation are expressed in terms of j = −neue, buthow the equation of motion (Eq. 3.10) is expressed is not and so will be derivedhere.Multiplying Eq. 3.10 by ne,neE + neue × B = −ne∂ue∂t− neue · ∇ue − ∇ ·↔pe, (3.19)neE − j × B = −ne∂ue∂t− ue∂ne∂t− ue∇ · (neue) + j · ∇ue − ∇ ·↔pe, (3.20)neE − j × B =∂j∂t− ue∇ · (neue) + j · ∇ue − ∇ ·↔pe, (3.21)neE − j × B =∂j∂t− jne∇ · j − j · ∇ jne− ∇ · ↔pe, (3.22)neE − j × B =∂j∂t− ∇ ·(jjne)− ∇ · ↔pe. (3.23)Therefore, the set of relevant dimensionless equations are∂B∂t= −∇ × E,∂E∂t=ω2peω2ce(∇ × B − j) ,∂ne∂t= ∇ · j,∂j∂t= neE − j × B + ∇ ·(jjne)+ ∇ · pe. (3.24)The set of Eqs. 3.24 solves Eq. 3.4 exactly and includes finite displacement currentand non-uniform density.Numerical MethodThe numerical method is a two-dimensional finite-domain time-difference (FDTD)method [167]. The spatial grid is a 2D staggered grid, or the 2D Yee grid [167], asshown in Fig. 3.1; the x and y components are defined on the edges, the z componentis defined at the center, and scalars are defined at the corners. For example, thez-component of a curl is computed as, for any vector V and its curl W = ∇ × V,(Wz)i, j =(Vy)i+1/2, j − (Vy)i−1/2, j∆x−(Vx)i, j+1/2 − (Vy)i, j−1/2∆y. (3.25)40x Scalarzyi i+1/2i-1/2j+1/2jj-1/2Figure 3.1: The staggered spatial grid used for the fluid simulation.This grid allows for a second-order accuracy (∼ ∆x2) in spatial derivatives, andsatisfies the divergence-free condition of B to floating point precision.The time grid is also staggered as shown in Fig. 3.2; B and j are defined on integertime steps, and E, ne, and↔pe are defined on half-integer time steps. The timederivatives of quantities on integer steps are defined on half-integer steps, and thoseof quantities on half-integer steps are defined on integer steps.The time advancement algorithm is shown in Fig. 3.2 and can be described asfollows.1. Advance B in time using Faraday’s law.2. Calculate↔pe using a closure of choice relating↔pe to ne and B. The average ofBt and Bt+1 is used to obtain Bt+1/2.3. j is advanced in time using an iterative procedure, which will be explainedshortly.4. Using the new values of B and j at t + 1, advance E using Ampère’s law.5. Similarly, advance ne using the continuity equation.6. Repeat.In step 3, ∂j/∂t depends on j, but the former is defined on half-integer time stepswhile the latter is defined on integer time steps. Therefore, an iterative procedure is41Figure 3.2: The staggered time grid and the time advancement algorithm for thefluid simulation.used to solve the equation. Discretizing the equation for j,jt+1 − jt∆t= neE −jt+1 + jt2× B + ∇ ·( [jt+1 + jt] [jt+1 + jt]4ne)+ ∇ · ↔pe, (3.26)where quantities without superscripts are all defined at half-integer time steps, andBt+1/2 =(Bt + Bt+1)/2. This is then iteratively solved for jt+1, i.e.,jt+1n+1 − jt∆t= neE −jt+1n + jt2× B + ∇ ·( [jt+1n + jt] [jt+1n + jt]4ne)+ ∇ · ↔pe, (3.27)where the subscript n denotes the iteration number, and jt+10 = jt . For the recon-nection problem, it took 5 to 10 iterations before the solution converged to withinfloating point precision.Kinetic EffectsTwo different kinetic effects are considered, namely pressure anisotropy and electronviscosity. Pressure anisotropy is modeled using the following closure which isdescribed in Egedal et al. [49] and Le et al. [88]:p̃e‖ = ñe22 + α+πñ3e6B̃22α2α + 1, (3.28)p̃e⊥ = ñe11 + α+ ñeB̃αα + 1. (3.29)42In Eqs. 3.28 and 3.29 the tilde represents normalization to far upstream values(e.g., ñe = ne/n0) and the parameter α = ñ3e/B̃2 acts as a switch between having anisothermal (small α) or a double adiabatic (large α) equation of state. Equations 3.28and 3.29 are approximations of second order moments of an electron distributionfunction derived from an analysis of electron trapping by a combination of parallelelectric fields and magnetic mirrors. The situation where α 1 represents trappedelectrons because magnetic mirrors trap electrons at regions of low B and thesetrapped electrons increase the local electron density. The trapped electrons conservetheir first and second adiabatic invariants and thus obey Chew-Goldberg-Low [35]closure. On the other hand, α 1 represents untrapped electrons that providean isothermal closure. This closure given by Eqs. 3.28 and 3.29, valid in theregime vTe vA, enables fluid models to exhibit kinetic effects missing from anisotropic pressure closure by allowing for pressure anisotropy to have both spatialand temporal dependence as given by the local instantaneous value of α [111].Equations 3.28 and 3.29 contribute to the pressure tensor as↔pe,aniso = pe⊥↔I +(pe‖ − pe⊥)B2BB, (3.30)= pe⊥↔I + σBB. (3.31)The effect of electron viscosity is expressed as↔pe,vis = −µ∇ue, (3.32)where µ is the dynamic viscosity. Because conventional viscosity is in many casesnegligible, µ represents an effective viscosity that includes, for example, turbulentviscosity [23] and/or hyper-resistivity [143]. The total pressure tensor is then thesum of the partial pressures; i.e., pe = pe,aniso + pe,vis.Initial ConditionsThe simulation was initiated by imposing a magnetic perturbation in the form of Eq.2.39 on a periodic force-free equilibrium used in Drake [42] and Ohia et al. [111],given byBy ' tanh( xλ)− tanh(x − xmax/2λ)− 1, (3.33)Bz =√1 + B2g − B2y (x), (3.34)where xmax is the size of the domain, Bg is the out-of-plane field far from thereconnection region, and λ is the half-thickness of the shear and λ xmax . Equation433.33 characterizes two separate current sheets, but only one of the sheets is examined;the second current sheet just facilitates the application of the periodic boundaryconditions used in the simulation. Equation 3.34 characterizes a guide field thatrenders B2 = B2y + B2z uniform everywhere. Uniform initial pressure and density arealso imposed, so the equilibrium system is initially force-free. The system is solvedin 2.5D where ∂/∂z = 0, and the grid size is 512×1024 with 25 grid points per de inthe x direction and 10 per de in the y direction. Fixed parameters areωce/ωpe = 1/2,Bg = 0.4, and λ = 2. This extremely thin current sheet not only models the smallscale of the EDR, but is also highly relevant in space plasma phenomena such aselectron-only reconnection in the turbulent magnetosheath [122].3.3 Particle-in-cell SimulationSimulating plasmas by following every single particle is practically impossible withthe contemporary computing power due to the typically astronomical number ofparticles involved. A particle-in-cell (PIC) simulation is a practical alternativeand follows the trajectories of a much smaller number of quasi-particles that eachrepresents a collection of real particles. These quasi-particles have an artificially-imposed shape function S that represents how each quasi-particle motion affectsvalues on the numerical grid. More specifically, the distribution function of eachspecies isfσ(x, p, t) =N∑p=1wpS(x − xp)δ(p − pp), (3.35)where p ∈ [1, N] is the quasi-particle number, wp is the statistical weight of theparticle, and xp and pp are respectively the position and momentum of each particle.This distribution function can be used to calculate ρ and j using the followingrelations:ρ(x, t) =∑σqσ∫fσ(x, p, t)d3p, (3.36)j(x, t) =∑σqσ∫v fσ(x, p, t)d3p. (3.37)These are used to calculate E and B, which are in turn used to advance the quasi-particles in time using the relativistic equation of motion for each quasi-particle:dxdt=pγmσ, (3.38)dpdt= qσ(E + pγmσ× B), (3.39)44where γ =√1 + p2/m2σc2 is the Lorentz factor of the quasi-particle. The newpositions and momenta can be used to calculate fσ again, and the process can berepeated. A PIC simulation is usually considered to have the capability to resolvephysical phenomena down to first-principles, although extreme caution is requiredwhen equating these simulations to real-life situations because inevitable limitationsexist such as numerical dissipation and error.In order to verify the results from the fluid simulation, the open-source PIC codeSMILEI [41] was used to simulate reconnection with the same parameters as thefluid simulation. The realistic ion to electron mass ratio, mi/me = 1836, was usedwith ∼ 2 × 108 ions and the same number of electrons. The results from the fluidsimulation are mainly presented because of the ability to include, exclude, or controlparticular physics, and because of the clarity of presentation.3.4 ResultsViscosity DependenceWe first examine how electron viscosity affects the evolution of Qe. First recallfrom Chapter 2 and Eq. 2.27 that Qe ' −B for ∇−1 ∼ L 1 whereas Qe ' we forL 1. For regions where L ∼ 1 such as the EDR, both we and B are important.Now, electron viscosity contributes to the canonical battery term as− ∇ ×(∇ · [−µ∇ue]ne)= ∇ ×(µ∇2uene)' µ∇2wene, (3.40)assuming density variations are not too large. If L 1 so that Qe ' we and thus∇2we ' ∇2Qe, Eq. 2.10 becomes∂Qe∂t= ∇ × (ue ×Qe) +µne∇2Qe, (3.41)which has the same form as the resistive-MHD induction equation (Eq. 1.49),∂B∂t= ∇ × (U × B) + ηµ0∇2B. (3.42)Thus at electron scales, electron viscosity allows Qe to reconnect, similar to howresistivity allows B to reconnect.Figure 3.3 shows, for three different values of viscosity, the out-of-plane electronflow uez (color), in-plane B (black lines), and in-plane Qe (red lines) for a situationwith isotropic pressure (initially βe = 0.3). Note that the orientation of the plotsis rotated by 90◦ in comparison to those in Chapter 2 to conform to the popular45−4−2024x/de(a) µ = 0−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.10.0−4−2024x/de(b) µ = 10−6−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.10.0−40 −30 −20 −10 0 10 20 30 40y/de−4−2024x/de(c) µ = 10−5−0.7−0.6−0.5−0.4−0.3−0.2−0.10.00.1Figure 3.3: In-plane Qe (red), in-plane B (black) and uez (color) for varying µ valuesand isotropic pressure at t = (a) 380, (b) 450, (c) 560.orientation in the literature. The shape of the EDR corresponds to the shape of thepurple color. Different times are chosen for each viscosity value because viscositychanges howmuch time is required for the EDR to display its characteristic structure.For µ = 0 (Fig. 3.3a) the system is ideal, so Qe lines remain connected and pile upnear the x = 0 line in contrast to B lines which reconnect, as seen in Chapter 2. Forfinite µ (Figs. 3.3b and 3.3c), Qe lines reconnect as well.It is apparent from Fig.3.3 that the out-of-plane electron current structure (i.e., colorcontours) is well manifested by in-plane Qe but not by B — a feature that is animportant advantage of using Qe over B at electron scales. Fine structures (i.e.small L) of ue are not manifested by B because |B| ∼ |j| L ∼ |ue | L by Ampére’slaw, whereas they are well manifested by Qe because |Qe | ∼ |ue | /L.Another important feature is that the local increase of Qey shear corresponds tothe local increase of uez. To illustrate this point we consider the implications46−4−2024x/de(a) Anisotropic, µ = 0−0.96−0.84−0.72−0.60−0.48−0.36−0.24−0.120.00−4−2024x/de(b) Anisotropic, µ = 10−6−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.10.0−40 −30 −20 −10 0 10 20 30 40y/de−4−2024x/de(c) Anisotropic, µ = 10−5−0.875−0.750−0.625−0.500−0.375−0.250−0.1250.0000.125Figure 3.4: Same as Fig. 3.3 with pressure anisotropy and varying µ values at t =(a) 400, (b) 500, (c) 630.of an assumed hypothetical toy scenario where Qey ∼ A (t) x exp(−x2)whereA (t) increases in time. This profile represents locally sheared Qey. Using uez =−∫weydx ' −∫Qeydx for L 1 gives uez ∼ A (t) exp(−x2), corresponding to alocal increase in uez.Anisotropy DependenceWe next examine how pressure anisotropy affects Qe. In order to bring the systemto an anisotropy-driven state faster, an initial pressure anisotropy with βe‖ = 0.6and βe⊥ = 0.1 was imposed. The results for different µ are shown in Figs. 3.4a-c. In comparison to the isotropic case it is seen that pressure anisotropy greatlydistorts the in-plane Qe lines so that they pile up in the upper-left and lower-rightquadrants; this corresponds to the anisotropic regime in Ohia et al. [111]. Again,the out-of-plane current structure is correlated with Qe rather than with B; uez isenhanced at locations where Qey is sheared. The distortion of Qe field lines and the47−20 −10 0 10 20y/de−202x/deParticle-in-cell−0.375−0.300−0.225−0.150−0.0750.0000.0750.1500.225Figure 3.5: Same as 3.3 from the particle-in-cell simulation at t = 300.−101x/de0.0150.030-0.015-0.030 0.0000.0000.000(a) ŷ · ∇ × (ue ×Qe) (color), −ŷ · ∇ × (∇ · pe,aniso/ne) (contour)−0.0350.0000.035−30 −20 −10 0 10 20 30y/de−101x/de(b) ∂Qey/∂t = ŷ · ∇ × (ue ×Qe)− ŷ · ∇ × (∇ · pe,aniso/ne)−0.0350.0000.035Figure 3.6: (a) The y-component of the convective term ŷ ·∇×(ue ×Qe) (color) andthe anisotropic contribution to the canonical battery term −ŷ · ∇ ×(∇ · pe,aniso/ne)(contour) for the simulation corresponding to Fig. 3.4b. (b) The sum of ŷ · ∇ ×(ue ×Qe) and −ŷ · ∇ ×(∇ · pe,aniso/ne), which is equal to ∂Qey/∂t. The red arrowsrepresent the direction of Qey.corresponding elongation and tilt of the out-of-plane current are reproduced by thePIC simulation, as shown in Fig. 3.5.The anisotropic contribution to the electron canonical battery term explains theorigin of the distortion of Qe and equivalently of the elongation of uez. Figure3.6a shows the convective term ŷ · ∇ × (ue ×Qe) (color) and the battery term−ŷ · ∇ ×(∇ · pe,aniso/ne)(contour). Figure 3.6b shows the sum of the two termswhich is equal to ∂Qey/∂t, and the red arrows show the direction of Qey. It can beseen that −ŷ · ∇ ×(∇ · pe,aniso/ne)adds to ŷ · ∇ × (ue ×Qe) and increases the spatial48extent of ∂Qey/∂t; this in turn elongates the structure of uez.To further understand the distortion of Qe due to anisotropy, we examine theanisotropic contribution to the canonical battery term, which is equal to−∇ ×(∇ · pe,aniso/ne)' −∇ × (∇ · [σBB])= −∇ × (B [B · ∇σ] + σB · ∇B) . (3.43)Since an increase of Qey shear corresponds to an increase in uez, we consider they-component of the canonical induction equation (Eq. 3.4) and thus of Eq. 3.43which is∂∂x(B[B · ∇σ] + σB · ∇B)z =∂∂x(Bz[B · ∇σ] + σB · ∇Bz) . (3.44)Here, σ depends on |B| and Bz is of the same order of magnitude as By. Therefore,since By has the shortest spatial scale of the components of B, it follows that|∇σ | /σ |∇Bz/| Bz, so∂∂x(Bz[B · ∇σ] + σB · ∇Bz) '∂(Bz[B · ∇σ])∂x' Bz∂B · ∇σ∂x. (3.45)Therefore, we have− ŷ · ∇ ×(∇ · pe,anisone)' Bz∂B · ∇σ∂x(3.46)Figure 3.7 shows various quantities involved in the calculation of Bz∂ [B · ∇σ] /∂x.The quadrupole out-of-plane Hall fields (plus and minus signs in Fig. 1.5) add andsubtract to the background guide field Bz prescribed by Eq. 3.34. This, togetherwith the reduction of |B| due to reconnection, generates a region of low |B| (Fig.3.7a) with a pronounced tilt. Because p‖ ∼ n3e/B2 and p⊥ ∼ neB, this |B| tiltgenerates a tilted region of finite σ (Fig. 3.7b; color). The quantity B · ∇σ,which is the variation of σ along the in-plane B (Fig. 3.7b; lines), is shown inFig. 3.7c (color), and its gradient ∇ (B · ∇σ) ' x̂∂ [B · ∇σ] /∂x is represented bythe black arrows. The resultant −ŷ · ∇ ×(∇ · pe,aniso/ne)' Bz∂ [B · ∇σ] /∂x isshown in Fig. 3.7d (contour). The convective term ŷ · ∇ × (ue ×Qe) (Fig. 3.7d;color) enhances Qey shear (Fig. 3.7d; red arrows) as expected from Eq. 2.49,and −ŷ · ∇ ×(∇ · pe,aniso/ne)further enhances this shear in spatial extent in the ydirection. The resultant ∂Qey/∂t = ŷ · ∇ × (ue ×Qe) − ŷ · ∇ ×(∇ · pe,aniso/ne)is shown in Fig. 3.7e; this increase in Qey shear is the origin of the elongatedout-of-plane flow uez.49−101x/de(a) |B|0.8820.9120.9420.9721.0021.0321.0621.0921.1221.152−101x/de(b) σ (color), B (lines)0.180.210.240.270.300.330.360.390.420.45−101x/de(c) B · ∇σ (color), ∇ (B · ∇σ) (arrows)−0.0090−0.0075−0.0060−0.0045−0.0030−0.00150.00000.00150.0030−101x/de0.000-0.015-0.0300.015 0.030(d) ŷ · ∇ × (ue ×Qe) (color), −ŷ · ∇ × (∇ · pe,aniso/ne) (contour)−0.0350.0000.035−30 −20 −10 0 10 20 30y/de−101x/de(e) ŷ · ∇ × (ue ×Qe)− ŷ · ∇ × (∇ · pe,aniso/ne)−0.0350.0000.035Figure 3.7: (a-c) Various quantities involved in the calculation of −ŷ · ∇ ×(∇ · pe,aniso/ne)' Bz∂ [B · ∇σ] /∂x for the simulation corresponding to Fig. 3.4b.(d) The y-component of the convective term ŷ · ∇ × (ue ×Qe) and the anisotropiccontribution to the canonical battery term −ŷ · ∇ ×(∇ · pe,aniso/ne), and (e) theirsum.503.5 DiscussionThe examination of Qe dynamics and the electron canonical battery term is notonly advantageous but in fact essential for the correct interpretation of a given EDRstructure. For example, Ohia et al. [111] observed that imposing isotropic pressuredeformed the out-of-plane current structure to be less elongated, which seeminglycontradicts Fig. 3.3a. However, in Ohia et al. [111], an effective electron viscosity of1.5×10−5 with the same normalized units as this study was imposed, approximatelycorresponding to Fig. 3.3c. Thus, it is not isotropic pressure that leads to an out-of-plane current localized at the origin; instead, the current localization results fromthe reconnection of Qe by electron viscosity.Another such example is the origin of elongated EDR structures. The elongatedstructure in Fig. 3.3a results from the pile-up of Qe field lines due to the lackof significant electron viscosity, whereas the elongated structures in Fig. 3.4 resultfrom pressure anisotropy. Therefore, in order to give correct physical interpretations,the scrutinization of the electron canonical battery term and the ensuing canonicalvorticity dynamics is vital.In Ohia et al. [111], it was found that the plasma approaches the firehose condition,which is the condition for the firehose instability [131], along the elongated currentlayer. This phenomenon can be simply explained by a direct comparison betweenthe convective term and the electron canonical battery term. On scales L > de, theconvective term goes like ∇ × (ue ×Qe) ' ∇ × (ue × B) ' −∇ × ([∇ × B] × B) ∼B2/L2, and the anisotropic contribution to the canonical battery term goes like∇ × (∇ · [σBB]) ∼ σB2/L2. It follows that σ ∼ 1 or(pe‖ − pe⊥)/B2 ∼ 1 in orderfor the two terms to be commensurate; thus, the parallel and perpendicular electronpressures approach the firehose criterion.Equation 3.41 stipulates the condition for the instability of the EDR. Because Qe 'we for L 1, Eq. 3.41 is equivalent to the electron fluid vorticity equation∂we∂t= ∇ × (ue × we) +µne∇2we. (3.47)It then naturally follows from fluid dynamics that turbulent flow develops at asufficiently high Reynolds number Re ∼ ne/µ. This unstable regime correspondsto the regimes in Drake et al. [44] and Del Sarto et al. [37]. All fluid runs and PICruns eventually become unstable to turbulence, in accordance with Drake et al. [44]and Del Sarto et al. [37] in that electron-scale current sheets are unstable to variousinstabilities.51Equation 2.10 also clarifies the distinction between reconnection models where themagnetic field is broken only by electron inertia [27] and those where it is brokenalso by pressure tensor effects such as electron viscosity [69]. If only the electroninertia term is included, then the system is entirely described by the convective termin Eq. 2.10, whereas the electron viscosity is manifested by the electron canonicalbattery term. A possible scenario is that electron inertia breaks the magnetic fieldsduring earlier times, but the EDR is subject to instabilities and turbulence due tothe high Reynolds number at latter times. These instabilities increase the effectiveviscosity, which makes the canonical battery term break both the magnetic field andthe canonical vorticity field. Whether a particular reconnection event is affected bymostly the convective term or by both the convective term and the canonical batteryterm depends on the parameters and the time-scale of the system.The advantage of the canonical vorticity framework is now clear. The convectiveterm in Eq. 2.10 signifies the convection of Qe with ue, which intuitively explainsreconnection electron physics for zero beta (no pressure), as in Chapter 2. One canthen study how a particular kinetic effect, such as electron pressure anisotropy, vis-cosity, or distribution function foliation [97], influences reconnection by examininghow that effect is manifested in the canonical battery term and then competes withthe convective term. This is not only much simpler than examining multiple termsin the generalized Ohm’s law (Eq. 1.34), but also exhaustive because Eq. 2.10 is adirect consequence of the collisionless Vlasov equation.3.6 SummaryIn summary, the electron canonical battery term completes the canonical vorticityframework of the electron physics inmagnetic reconnection. As demonstration of thepower of this approach, the growth, saturation, stability, andmorphology of the EDRhave been reinterpreted, expanded, and unified within this framework. In particular,the framework illustrates how the changes in the electron fluid closure (Eqs. 3.28and 3.29) affect the current structure through the electron canonical battery term.The distinction between electron-inertia-driven and pressure-tensor-driven modelsof reconnection is also clearly demonstrated. The simple yet complete nature of thisframework makes it an appealing alternative to the traditional magnetic-field-basedapproach to magnetic reconnection.52C h a p t e r 4STOCHASTIC ION HEATING IN MAGNETIC RECONNECTIONMagnetic reconnection is ultimately a process in whichmagnetic energy is convertedto particle energies. Although magnetic reconnection has been the focus of manystudies for decades, the mechanisms underlying this energy conversion from themagnetic field to particles are still poorly understood. In particular, anomalousion heating much faster than conventional collisional heating has frequently beenobserved in various astrophysical and laboratory reconnection events [29, 50, 65,74, 96, 117, 118, 124, 141, 168], and resolving the fundamental cause of this heatingremains a crucial objective of reconnection research. The resolution of this problemis an important key to tackling critical conundrums such as the coronal heatingproblem [59], i.e., why the solar atmosphere is over a million degrees while thesolar surface is only about a few thousand degrees.Five features of ion heating are observed in reconnection simulations, experiments,and observations: (i) correlation with in-plane Hall electric fields [7, 120, 141, 168],(ii) temperature anisotropy (Ti⊥ , Ti‖) [47, 70, 95, 103], (iii) non-conservation ofthe ion magnetic moment µ [46, 47, 82], (iv) preferential heating of ions withhigher mass-to-charge ratios mi/qi [51, 106, 128], and (v) reduction of the heatingamount under a finite guide field (background out-of-plane magnetic field) [46].Since anomalous ion heating occurs over a broad range of plasma parameters, theheating mechanism is expected to involve physical quantities that are intrinsic to thereconnection process. So far, numerous mechanisms have been proposed to explainion heating; examples include pick-up [46, 47], turbulent interaction [107, 138],interactions with fluctuating electric fields [51], remagnetization [168], and shocksor viscous damping [118]. However, the proposed mechanisms are still underserious debate; there is no general consensus among the community on what thefundamental ion heating mechanism is.In this chapter, it is shown that stochastic ion heating is intrinsic to transient collision-lessmagnetic reconnection because stochastic ion heatingwill always occur at lengthscales smaller than the ion skin depth di = c/ωpi when there is a perturbation of aHarris-type magnetic equilibrium [16, 66]. It is shown via analytic methods supple-mented by numerical calculations that the in-plane electric fields near the magnetic53separatrices (lines that separate regions of reconnected and non-reconnected fields)satisfy the condition for stochastic ion heating. Thus, the correlation of anomalousheating with the in-plane Hall electric fields will be established; the other four fea-tures of ion heating in reconnection will be explicitly demonstrated in Chapter 5.The stochastic nature of ions will also be demonstrated via single-particle simula-tions. Comparisons to laboratory experiments support the analytical calculationspresented here. Hall electric fields have been attributed to ion energization seen inprevious numerical [7, 46, 125], observational [162] and experimental [141, 168]reconnection studies, but the specific mechanism linking the fields to the heatinghas not been identified.4.1 Stochastic HeatingStochastic heating occurs when a particle in a magnetic field is subject to a perpen-dicular potential gradient so strong that it causes a breakdown of the guiding-centerapproximation [10, 11, 108, 142]. The particle then experiences a large potentialdifference within an orbital period and its motion becomes chaotic. When thishappens to a large number of particles, the velocity distribution function broadensand this broadening is observationally equivalent to heating. Stochastic heatinghas been proposed to be an ion heating source in the solar wind via Alfvén waveinteractions [22, 30, 31, 142, 156–159] and has been observed in laboratory exper-iments [29, 108, 133] . The threshold in Coulomb gauge for ion stochastic heatingis [10, 108, 142],α =miqi |B|2∇2⊥φ = miqi |B|2|∇⊥ · E⊥ | > 1, (4.1)where φ is the electrostatic potential and ∇⊥ is the gradient perpendicular to B.Breakdown of the Guiding-Center ApproximationIn order to understand this mechanism, consider particle guiding-center motionunder the electromagnetic fieldsE = E ŷ sin (ky − ωt) , (4.2)B = Bẑ. (4.3)The E × B drift isvE =E × BB2= x̂EBsin (ky − ωt) , (4.4)54and the polarization drift isvp =mqB2dE⊥dt, (4.5)= ŷmEqB2ddtsin (ky − ωt) , (4.6)= ŷmEqB2(kdydt− ω)cos (ky − ωt) . (4.7)Here, dy/dt = vy is the velocity in the y-direction. Since vE is in the x-directionand vp is in the y-direction, ŷdy/dt = vp. Solving Eq. 4.7 in terms of vp givesvp = −ŷωmEqB2cos (ky − ωt)1 − α cos (ky − ωt), (4.8)whereα =mkEqB2(4.9)is the stochastic heating condition in Eq. 4.1 assuming that the particle is an ionand |∇⊥ · E⊥ | ' kE . It can clearly be seen from Eq. 4.8 that when α > 1,the denominator vanishes when cos (ky − ωt) = α−1. However, when deriving thepolarization drift in Section 1.1, it was explicitly assumed that |dvE/dt | dvp/dt.Because vE is finite as in Eq. 4.4, this assumption breaks down if vp → ∞ alongwith the guiding-center approximation. The drift equations in Section 1.1 thereforecannot be used, and the magnetic moment µ is not conserved.Chaotic MotionIt will now be shown that the satisfaction of Eq. 4.1 leads to chaotic particle motion.According to the concept of Lyapunov exponents [100], chaotic behavior entailsthe spatial divergence of two particles that have an infinitesimal initial separationin phase space. Now consider the motion of the spatial difference of two particles,δr = r1− r2 and its time derivatives. As above, the electric field is in the y-directionand the magnetic field is in the z-direction. The Lorentz force equation yieldsδ Üx = qBmδ Ûy, (4.10)δ Üy = qm∂E∂yδy − qBmδ Ûx, (4.11)δ Üz = 0. (4.12)55Differentiating the second equation in time and substituting the first equation in,d3δyd3t=qm∂E∂ydδydt− q2B2m2dδydt, (4.13)d3δyd3t=qm(∂E∂y− qB2m)dδydt, (4.14)which yields an exponentially growing solution for dδy/dt whenmqB2∂E∂y= α > 1. (4.15)Therefore, satisfaction of Eq. 4.1 signifies that particles undergo chaotic motion.Stochastic heating does not require the fields to be incoherent — it is a mechanismthat causes particle trajectories to become chaotic in coherent fields. Thus, stochasticheating should not be confused with mechanisms where the fields themselves arestochastic. It should be noted, however, that there has been extensive work on ionacceleration via stochastic interaction with magnetic islands [48, 61, 72, 90, 91,174, 175]. The present work shows that stochastic ion heating occurs at a singlereconnection region, but it will also work in the context of stochastic fields withmultiple reconnection regions if the condition in Eq. 4.1 remains satisfied.4.2 Relevant EquationsWe begin the analysis by noting that the electric field acting on the ions can bedetermined from the two-fluid electron equation of motion. Using this equation toprescribe the electric field is equivalent to invoking the generalized Ohm’s law withthe Hall, electron inertia, and electron pressure terms all included. The normaliza-tion of quantities is the same as in Chapter 3 and thus Table 3.1; the magnitude ofthe magnetic field far from the reconnection region is denoted B0 and this defines areference electron cyclotron frequency ωce = qeB0/me. On normalizing length bythe electron collisionless skin depth de = c/ωpe, time by |ωce |−1, magnetic field byB0, density by the background density n0, electric field by de |ωce | B0, and pressureby B20/µ0, the two-fluid electron equation of motion can be expressed asĒ + ūe × B̄ = −DūeDt̄− ∇̄p̄en̄e, (4.16)where barred quantities are dimensionless. The pressure has been assumed tobe scalar. To reduce clutter, the bars will be dropped from now on. Under thisnormalization, Eq. 4.1 becomes1εB2∇2⊥φ > 1, (4.17)56where ε = me/mi is the electron-to-ion mass ratio.Recall from Chapter 2 that assuming negligible pressure term by positing either coldelectrons (low βe) or barotropic pressure, and invoking Faraday’s law yield the idealelectron canonical induction equation:∂Qe∂t= ∇ × (ue ×Qe) , (4.18)whereQe ≡ ∇×ue−B is the electron canonical vorticity. An important consequenceof Eq. 4.18 was that Qe field lines remain connected whereas B field lines mayreconnect.It should be noted that as seen in Chapter 3, the canonical battery term can beimportant and will add an additional term to Eq. 4.18, allowing for the reconnectionof Qe. However, this effect will be disregarded in the present analysis becauseomitting the canonical battery term keeps the analysis simple and enables focusingon more dominant effects. Also for simplicity, a zero guide-field will be assumed.It will be shown in Chapter 5 that this stochastic heating mechanism is still valid inthe kinetic regime that includes the canonical battery term and under the presenceof moderate guide fields.Recall that by decomposing the convective derivative using ue · ∇ue = ∇u2e/2 −ue × ∇ × ue, Equation 4.16 may be expressed in terms of Qe rather than B, i.e.,E = −∂ue∂t+ ue ×Qe − ∇(u2e2)− ∇pene. (4.19)The connected nature of Qe will be now used to simplify Eq. 4.19 in a way thatenables checking whether the condition given by Eq. 4.1 for stochastic ion heatingis satisfied in transient reconnection situations. The cold electron assumption willbe used so that the hydrodynamic pressure term in Eq. 4.19 may be ignored;justification for this assumption will be given later.In Chapter 2, it was shown that the nominal growth rate of this system near the elec-tron diffusion region (EDR) can be expressed in terms of gradients of componentsof the electron flow. It is seen from the definition of Qe that ∇ ·Qe = 0 and it is alsoseen that ∇ ·ue ∼ ∇ · J ∼ ∇ ·∇×B = 0. Using these properties and Qex (x = 0) = 0,expansion of the y-component of Eq. 4.18 gives an exponentially growing solutionfor Qey with a growth rate∂∂t'∂uey∂y' −∂uex∂x. (4.20)57Figure 4.1: (a) Reconnected in-plane B field lines (white) and connected in-planeQe field lines (red) in the reconnection geometry. The effective potential (color)was calculated from the cold version of Eq. 4.19. The inflows and outflows arerespectively in the ±x and ±y directions. The x-axis and the y-axis have differentscales to show the field lines more evidently. (b) Comparison of B2y/2with−∫Exdxand −∫uezBydx along the black dotted line in (a). (c) Comparison of(u2ey + u2e)/2with −∫Eydy along the magenta line in (a).The growth rate is always positive since ∂uey/∂y ' ∂2Bz/∂x∂y and the quadrupoleout-of-plane Hall magnetic field has the dependence Bz ∼ xy in the EDR. Thisgrowth rate will be used to rewrite the ∂ue/∂t term in Eq. 4.19.584.3 Numerical Simulation and ResultsTo validate the predictions of the model presented here, the EMHD simulation thatwas developed in Chapter 2.2 was used. Again, a perturbation to the backgroundHarris-type magnetic fieldBy = tanhxλ(4.21)initialized the simulation.Figure 4.1a shows the numerically calculated reconnected in-plane B field lines(white) and the connected in-plane Qe field lines (red). The distinction between thereconnection of B and the non-reconnection of Qe is clearly evident. Specifically,Bx(x = 0) is finite but Qex(x = 0) = 0, so B reconnects but Qe remains connected.The electric field is calculated numerically using the values of ue and Qe on theright-hand side of Eq. 4.19, and the associated electrostatic potential (color) isplotted for reference. The potential is saddle-shaped with contours approximatelyfollowing the B field lines in agreement with previous simulations [46, 79, 125] andlaboratory experiments [168].4.4 Stochastic Heating Condition AnalysisInflowWe first consider the inflow (i.e., flow on y = 0, black dashed line in Fig. 4.1a.The B field is in the y-direction here, so the perpendicular electric field relevantto stochastic heating is Ex . Although Ez is also perpendicular to By, we need notconsider Ez for the purposes of analyzing stochastic heating because ∂/∂z = 0 in2D, so Ez is irrelevant to Eq. 4.1. Using the cold electron assumption (to be justifiedlater), the x-component of Eq. 4.19 becomesEx = −∂uex∂t+ ueyQez − uezQey −∂∂x(u2e2). (4.22)On invoking Eq. 4.20, the first term on the right-hand side of Eq. 4.22 is−∂uex/∂t 'uex∂uex/∂x = ∂(u2ex/2)/∂x. The second term disappears because uey (y = 0) =Qez (y = 0) = 0 due to the antisymmetry of the outflow and the quadrupole natureof the out-of-plane Qe fields as seen in Chapter 2. On expanding the third termusing Qey = −∂uez/∂x − By and the fourth term using u2e = u2ex + u2ez at y = 0, Eq.4.22 simplifies toEx = uezBy (4.23)' − ∂∂x(B2y2). (4.24)59In the last line, we have used ue ' −∇ × B so uez = ∂Bx/∂y − ∂By/∂x and∂By/∂x >> ∂Bx/∂y in the EDR; these assumptions are true because By > Bx andalso ∂/∂x >> ∂/∂y. Therefore, B2y/2 acts as an effective electrostatic potentialin the x-direction (this is equivalent to Eq. 3 in Li and Horiuchi [93]). Thisconclusion regarding the effective potential is shown graphically in Fig. 4.1b, where−∫Exdx and −∫uezBydx are compared with B2y/2, calculated from the simulation.Outside the EDR, ∂By/∂x −→ 0, so the approximation fails, but for the purposesof analyzing the slope of the potential valley into which the ions fall (ion inflowdirections represented by the green arrows) and thus for the purpose of investigatingthe existence of stochastic heating, B2y/2 remains a good approximation for theelectrostatic potential. The integration constant was set so that the three functionscoincide at x = 0 in Fig. 4.1b.Equation 4.17, the condition for ion stochasticity in the inflow direction, can thus bewritten as1εB2y ∂2∂x2(B2y2) > 1. (4.25)Using By(x) ' B0x/λ for x < λ and d2e/ε = d2i , Eq. 4.25 in dimensioned quantitiesbecomes x2 < d2i . A sufficient condition to satisfy x2 < d2i while x < λ isλ2 < d2i . (4.26)Therefore, at sub-ion-skin-depth length scales, at which collisionless reconnectionoccurs, the condition for stochastic heating in the inflow direction is satisfied.The cold electron assumption in the inflow direction and thus the validity of Eq.4.24 is justified by the fact that numerical [93] and experimental [168] studiesshow that the magnetic force on the electron fluid is balanced by the electric fieldcorresponding to Eq. 4.23 and thus Eq. 4.24.OutflowWe now consider the outflow (flow along the x = 0 line, magenta dashed line inFig. 4.1a). Because the magnitude of B is very small on this line, the cold electronassumption in principle cannot be used. However, it will be shown later thatincluding the electron pressure term has little effect on the stochasticity condition,so for now electrons will be assumed cold. Along the outflow (i.e., on the y-axis,x = 0), the magnetic field is in the x-direction, so the perpendicular electric fieldrelevant to stochastic heating is Ey. The magnetic field component Bx at x = 0 has60small amplitude and its being finite is what constitutes the reconnection. Ez againplays no role in Eq. 4.1 because ∂/∂z = 0. The y-component of Eq. 4.19 with thecold electron assumption invoked isEy = −∂uey∂t+ uezQex − uexQez −∂∂y(u2e2). (4.27)Using Eq. 4.20, the first term on the right-hand side of Eq. 4.27 is −∂uey/∂t '−uey∂uey/∂y = −∂(u2ey/2)/∂y. The second termdisappears becauseQex (x = 0) =0 since Qe does not reconnect. The third term disappears because of the antisym-metry of uex and the quadrupole nature of Qez. Equation 4.27 now simplifies toEy ' −∂∂y(u2ey + u2e2). (4.28)Thus,(u2ey + u2e)/2 acts as an effective potential in the y-direction. A comparisonbetween −∫Eydx and(u2ey + u2e)/2 along x = 0 shows that the latter is a goodapproximation, as shown in Figure 4.1c. The potential forms a hill off of whichthe ions fall downwards (ion outflow directions represented by green arrows). Theintegration constant was set in Fig. 4.1c so that the two functions coincide at y = 0.The normalized stochasticity condition (Eq. 4.17) in the outflow direction is1εB2x ∂2∂y2(u2ey + u2e2) > 1. (4.29)Here the connected nature of Qe gives Qex (x = 0) = ∂uez/∂y−Bx ' uez/Ly−Bx =0, so using ∂/∂y ∼ 1/Ly = Bx/uez on Eq. 4.29 gives1ε(u2ey + u2e2u2ez)> 1. (4.30)However,(u2ey + u2e)/u2ez ≥ 1 since u2e = u2ey + u2ez, so Eq. 4.30 reduces tomime> 1, (4.31)which is of course satisfied. Consequently, the stochasticity condition in the outflowdirection is always satisfied by a wide margin.The cold electron assumption in the outflowdirectionwill nowbe justified. Recallingthat p̄e = nekBTe/(B20/µ0) gives p̄e = n̄ev̄2Te/2 where v̄2Te = (2kBTe/me) /(de |ωce |)2.Dropping bars, the pressure term in Eq. 4.19 becomes ∇pene = ∇(v2Te2)+v2Te2∇nene'61∇(v2Te2). The v2Te2∇neneterm was dropped by invoking the quasi-neutrality condition;because the normalized Poisson’s equation gives ∇ · E = − c2v2Ae(ne − n0) wherev2Ae = B2/µ0mene is the electron Alfvén velocity, taking the gradient of both sidesand requiring ∇ · E to be small by quasi-neutrality yields ∇ne << v2Ae/c2. Usingthis new pressure term in Eq. 4.19 and applying the same reasoning used to deriveEq. 4.28, Ey approximates toEy ' −∂∂y(u2ey + u2e + v2Te2). (4.32)Equation 4.32 implies that if there is electron heating in the exhaust region, thepotential becomes flatter and thus less conducive to stochastic heating. This is not aproblem in guide-field reconnection in which electron heating is localized near theEDR [146]. For zero guide-field reconnection, however, if the exhaust region (finitey) is hotter than the EDR (y ≈ 0) [166], the potential is a well centered around y = 0which is not conducive to stochastic heating. Nevertheless, temperature effects donot negate the stochastic condition being satisfied for the following reasons.Consider the worst case scenario where the outflow becomes completely thermal-ized, i.e., u2ey + v2Te = const. In this case, only u2e is left in Eq. 4.32 so Eq. 4.30becomes 1ε(u2e2u2ez)> 1, which is always satisfied because u2e ≥ u2ez.Furthermore, previous studies are consistent with the cold electron assumption.Several numerical studies [79, 125] have shown that |eΦ/Te |—which correspondsto ∼ u2e/v2Te — increases to be 10 ∼ 20 inside the saddle potential, which meansthat the electron flow is cold and laminar. A previous experimental study [168]measured the plasma potential including electron temperature effects and showedthat the potential is a hill in the y-direction with the peak at y = 0, confirming theshape given in Fig. 4.1(c).The fact that Eqs. 4.26 and 4.31 do not depend on the magnetic field signifiesthat stochastic heating is intrinsic to collisionless reconnection. φ in Eq. 4.17has a dependence on B in such a way that this dependence cancels out B in thedenominator. Thus, Eq. 4.17 is always met given that the assumptions in the presentanalysis are met as well.4.5 Test Particle SimulationIn order to verify ion stochastic heating, 6000 test ion particle trajectories weresimulated using an implicit particle integrator [15]. The details of the algorithm62Figure 4.2: (a) Ten coherent ion trajectories undergoing Speiser-like orbits, and(b) their stochastic counterparts. (c) Positions in y phase space of all 6000 ionsat t = 580 |ωce |−1, and (d) their stochastic counterparts. (e), (f) Zoom-ins of (c)and (d), respectively. (g) Distribution of |∆ Ûy |2 for the coherent case (blue) andthe stochastic case (red). (h) Time-dependent 3D separation distances between twoselected particles with the minimum initial separation distance, 0.015de, under thestationary electric field (blue line; particles are red dots in (b)) and the growingelectric field (red line; particles are red dots in (f)).63are described in Appendix D. Trajectories were calculated for both stationary andgrowing electric fields. These two cases should respectively give ordered andstochastic ion motions. The particles were started at rest from a rectangular grid(x, y) = (0, 0) – (4.5, 0.4)de, spaced by (∆x,∆y) = (0.015, 0.02)de. Such a smallspatial separation was chosen because stochasticity is defined by particles’ havingdisparate trajectories with slightly different initial conditions. The results presentedin Fig. 4.2 are based on a mass ratio mi/me = 10. While this mass ratio is small,the purpose of the simulation is to reveal the distinction between coherent andstochastic situations. Simulations with higher mass ratios (20, 50, 100) have alsobeen conducted and exhibit similar qualitative behavior; mass and time scalings aresubject to future investigations.The growing electric field was the time-dependent numerically calculated field, andthe stationary electric field was chosen to have the value of the time-dependentfield at t = 5 |ωce |−1. A small value for this reference time was chosen becausethe particles were started at rest from different positions. This means that they allhave different initial total energies, so a small potential was chosen to keep the ionsinitially cold in order to explicitly show the thermal energy gain due to stochasticheating. Coherence was not destroyed by choosing the electric field at a later time.The focus will be on the outflow direction since Eq. 4.31 is met by a large marginso the stochastic evolution in this direction is more evident.Particle Trajectories and Phase MixingFigure 4.2a shows ten selected ion trajectories (red) in the stationary electric fieldcase, and Figure 4.2b shows the stochastic counterpart. The effective potential at(a) t = 5 |ωce |−1 and (b) 580 |ωce |−1 is plotted (color) for reference. The ions haveSpeiser-type (bouncing) orbits, consistent with previous numerical studies [7, 46].In the stationary electric field case (i.e., Fig. 4.2a, the trajectories show a coherentmotion. In the growing electric field case (i.e., Fig. 4.2b), the trajectories clearlydiverge and undergo phase mixing, the signature of stochastic motion. Stochasticityis further demonstrated by the central trajectory (white) which starts at x = 0 inboth situations and which, in principle, should have no motion in the x-direction.While the central trajectory in Fig. 4.2a (stationary electric field) is a straight,stable outflow, the trajectory in Fig. 4.2b (growing electric field) becomes unstableindicating stochasticity, where the seed for the instability is the ever-present smallnumerical error.64Ion Phase SpaceExamination of the ion phase space indicates chaotic motion. Figure 4.2c shows thepositions in y phase space of all ions at t = 580 |ωce |−1 for the stationary electricfield case, and Fig. 4.2d shows the counterpart in the growing electric field case.Figures 4.2e and 4.2f show zoomed-in versions of Figs. 4.2c and 4.2d, respectively.The left plots (stationary electric field) clearly show a periodic order, whereas theright plots (growing electric field) exhibit unpredictability and chaos.The two red dots in Fig. 4.2c (which look like one dot due to their proximity)represent the phase space positions of two selected particles that started with theminimal separation of 0.015de for the stationary electric field case, and those inFig. 4.2d represent their stochastic counterpart (growing electric field). The time-dependent 3D separation distance is plotted in Fig. 4.2h as blue (ordered) and red(stochastic). It can be seen in both phase and physical space that the two particlesin the stochastic case separate rapidly from one another, a behavior which is thecharacteristic of chaotic motion and in accordance with the concept of Lyapunovexponents [100].Effective HeatingThe stochasticity results in an effective heating. Figure 4.2g shows for ordered (blue)and stochastic (red) ions the distribution of the squared deviation of the ion outflowvelocity |∆ Ûy |2, a proxy for thermal energy. Significant heating is clearly seen for thestochastic case. The distribution is non-Maxwellian, however, because collisionsare not included.Initial TemperatureIn reality, ions have a finite initial temperature. Another simulation was conductedwhere the ions start with a thermal velocity of 0.01 de |ωce |, and the results areshown in Fig. 4.3. The phase space as shown in Fig. 4.3b is much broader than thatin Fig. 4.3a. The separation between the two red dots are again much larger in Fig.4.3b than in Fig. 4.3a. The thermal spread can be seen explicitly in Fig. 4.3d incontrast to Fig. 4.3c. Also, in comparison with Figs. 4.2c and 4.2d, it can be seenthat an initial temperature makes the ions spatially travel further and faster.It should be said that the purpose of this test particle simulation is to explicitly showthe contrast between coherent and stochastic ion motion, and the results presentedhere reflect the initial stage of stochastic evolution. A temporal limitation exists dueto the time scale separation between the ions and electrons, so the simulation must65Figure 4.3: Positions in y phase space of ions that started with a thermal velocityof 0.01 de |ωce | in (a) the coherent case and (b) the stochastic case. Distribution of|∆ Ûy |2 in (c) the coherent case and (d) the stochastic case.be halted before the ions are exhausted from the reconnection region. This alsomeans that the ions have yet to gain all the available energy from this mechanism.4.6 DiscussionHeating RateAn order-of-magnitude estimate for the heating rate and strength in a reconnectionsituation will be given as follows. A proxy for the exponential heating rate from thismechanism in a reconnection situation is its Lyapunov exponentγE =√qiEmia, (4.33)66where a is the length scale of the electric field E [10]. Here,√qiEmia=qi B0miB⊥B0√miEqi B2⊥a(4.34)and noting that the quantity inside the square root is equivalent α of the stochasticheating criterion (Eq. 4.1), γ ≈ ωci B⊥B0diLxfor the inflow and ≈ ωci B⊥B0√mimefor theoutflow. Assuming B⊥ . B0 and Lx . di, the theoretical heating rate γ is in therange ωci < γ <√ωciωce. This is much faster than the MHD time scale governingthe macrophysics of the solar environment.As noted in Sanders et al. [133], the maximally achieved ion thermal velocitydue to stochastic heating is vT = ωcia(α + 1.9), where α is defined in Eq. 4.1and thus satisfies α > 1. Using a ' di, the upper bound on the local beta isβlocal = vT/vA = α + 1.9, at a time scale faster than the ion cyclotron frequency.Since the reconnected magnetic field magnitude is less than the that of upstreamfield, the global beta may still stay sub-unity. Amore detailed analysis on the heatingamount is subject to future investigations.Heating ProfileOur results are consistent with a recent particle-in-cell calculation of ion temperatureprofile in collisionless reconnection. Haggerty et al. [63, Figure 2] shows a sharpion temperature increase across the magnetic separatrix where the potential dropoccurs, and a further increase in the ion outflow jet. This is congruous with the factthat the inflow condition (Eq. 4.26) is marginally satisfied and the outflow condition(Eq. 4.31) is satisfied by a wide margin; therefore, the temperature increase is moredramatic along the outflow.Comparisons to ExperimentsThe claim that Eq. 4.1 is satisfied in collisionless reconnection is also supportedby experimental observations. In the Magnetic Reconnection eXperiment [166],the measured in-plane electric field was 700 V/m with a 1 cm length scale. Usingthe upstream magnetic field B0 = 0.1 kG, Eq. 4.1 becomes 14.6 >> 1 and so thecondition for ion stochastic heating is satisfied by a wide margin. The conditionwas also met in the spontaneous collisionless reconnection event in the CaltechMHD-driven jet experiment [29].The calculated heating rate also agrees with experiments. Defining γobs to be the in-verse e-folding time scale of the ion temperature based on experimental observations,67γobs ≈ 105 s−1 for MRX [166] and γobs ≈ 106 s−1 for the Caltech jet experiment[29]. Calculation of γ range for both experiments gives γ ≈ 1–30 × 106 s−1 for theformer and γ ≈ 1–100 × 106 s−1 for the latter. Therefore, stochastic ion heating isfast enough to account for the observations in both experiments.4.7 SummaryIt has been shown that ion stochastic heating, which had already been established asan important heating mechanism in the solar wind, atmosphere, and flares, is alsoinherent to collisionless magnetic reconnection. The connected nature of electroncanonical vorticity has been exploited to show that the electric fields near the mag-netic separatrix satsify the stochastic heating criterion. The approximations for theelectric field were verified by an electron fluid simulation, and ion stochasticity wasdemonstrated by test particle simulations. Comparisons to laboratory reconnectionexperiments also support the existence of this mechanism.68C h a p t e r 5KINETIC VERIFICATION OF STOCHASTIC ION HEATING INMAGNETIC RECONNECTIONIn Chapter 4, it was demonstrated via a two-fluid, zero-beta analysis of the general-ized Ohm’s law that stochastic heating is intrinsic to anti-parallel (zero guide field)collisionless reconnection. Although the demonstration provided useful insightsinto the fundamental ion heating mechanism in reconnection, a fully self-consistentkinetic verification of this mechanism was not demonstrated. Also, four of the fivefeatures of ion heating in reconnection, namely temperature anisotropy (Ti⊥ , Ti‖),µ violation, preferential heating of higherm/q particles, and reduction of the heatingamount in the presence of a guide field, were not explicitly illustrated in terms ofstochastic heating.In this chapter, it is unambiguously shown via exact kinetic analyses and particle-in-cell simulations that stochastic heating [10, 108, 133, 142] is the main ion heatingmechanism in a generic collisionless reconnection process up to moderate guidefields and exhibits all five features of anomalous ion heating.Recall from Chapter 4 that the criterion for stochastic ion heating is given by a singledimensionless parameter α (x, t) whereα (x, t) = miqi |B|2∇2⊥φ > 1; (5.1)here ∇⊥ is the gradient perpendicular to the local magnetic field B = B (x, t), andφ = φ (x, t) is the local electrostatic potential. Equation 5.1 provides a spatio-temporal prediction for stochastic heating. Stochastic heating involves the break-down of the guiding-center approximation and causes associated non-conservationof the magnetic moment µ. Since µ involves v⊥, this mechanism preferentiallyincreases T⊥. In addition, according to Eq. 5.1, ions with higher mi/qi more easilysatisfy the α > 1 criterion and so are expected to be more readily heated. Thus,stochastic heating is an enticing candidate for the anomalous ion heating mechanismin magnetic reconnection.This chapter fills the gap between the prediction of stochastic ion heating inmagneticreconnection in Chapter 4 and a self-consistent verification by confirming the exis-69tence and inherent features of stochastic ion heating through fully kinetic analysesand simulations.5.1 Stochastic Heating in the Harris Equilibrium Plasma SheathWe begin with an analysis of the existence of stochastic heating in the Harrisequilibrium sheath [66], which is often used as the initial condition for reconnectionanalyses and simulations. Although the Harris-type hyperbolic tangent magneticfield profile has been used in previous chapters, the exact Harris equilibrium is morecomplicated and will now be briefly derived.Harris Equilibrium Plasma SheathThe Harris equilibrium plasma sheath is a kinetically exact specific solution fora steady-state velocity distribution function that simultaneously satisfies Poisson’sequation, Ampère’s law, and the Vlasov equation (Eq. 1.13). A sheath refers tothe region of plasma sandwiched between regions of oppositely-directed magneticfields. Using the specific distribution function, φ and A can be calculated, and fromthese the equilibrium electric and magnetic field can be derived. For a more detailedderivation, see Appendix E.The equilibrium is initially solved in a frame where ions and electrons are movingat equal and opposite velocities in the z direction, i.e., Viz = −Vez = V . Assuminguniform T = Ti = Te,φ = 0 (5.2)in this frame, soE = 0. (5.3)The equilibrium magnetic field is determined to have the profile [66]By (x) = 2√µ0n0kBT tanhxλ, (5.4)whereλ =√2cVλD (5.5)=√2cV1√λ−2Di + λ−2De(5.6)=cV√ε0kBTn0e2(5.7)70is the sheath half-thickness and n0 is the peak density of the sheath. The densityprofile isni(x) = ne(x) = n0 cosh−2xλ. (5.8)Lorentz tranforming to a frame where Viz = 0, which is approximately the center-of-mass frame (i.e., the lab frame due to mi me), a lab-frame electric field withthe following profile develops [66]:Ex (x) = −V By (x) = −2V√µ0n0kBT tanh( xλ), (5.9)where we have assumed V c, so the Lorentz factor γ = 1. The magnetic fieldstays the same after the transformation.In this lab frame, Viz = 0 and Vez = −2V , so Jz (x) = 2n (x) eV . Thus, Eq.5.9 is of the same order of magnitude as the Hall component of the electric fieldEx,Hall (x) = −Jz (x) By/n (x) e = −2V By (x) in the lab frame. In the vicinityof the reconnection region, the in-plane electric field is mainly balanced by theHall component [93, 171], so whether Eq. 5.9 satisfies Eq. 5.1 is important fordetermining the existence of stochastic heating.Stochastic Ion Heating ConditionTomake the analysismore general, a guide field Bz = bgBy (x →∞) is now includedas well where bg is the ratio of the guide field to the asymptotic value of By. Sinceonly Ex (x), By (x), and Bz are involved, Eq. 5.1 becomesmiqi[B2y (x) + B2z] ∂Ex (x)∂x > 1, (5.10)which, after inserting the Harris profiles (Eqs. 5.4 and 5.9), becomes2λ2d2icosh2( xλ) [tanh2( xλ)+ b2g]< 1, (5.11)where di is the ion skin depth. Denoting x̄ = x/di and λ̄ = λ/di, the solution to Eq.5.11 is x̄ < x̄sh(λ̄, bg)wherex̄sh =λ̄2lnλ̄2(1 − b2g)+ 1 +√(1 − 2λ̄2b2g) (2λ̄2 + 1)λ̄2(1 + b2g) . (5.12)A more detailed derivation of this solution is in Appendix E. Figure 5.1a showscontours of x̄sh(λ̄, bg), and the inset plots x̄sh as a function of bg for λ̄ = 0.5, 1, 2.710.0 0.5 1.0 1.5 2.0 2.5 3.0λ0.000.250.500.751.001.251.501.752.00b g0.1000.1000.2000.2000.3000.4000.5000.600(a) xsh0.0 0.5 1.0bg0.00.20.40.6x shλ=0.5λ=1λ=20.0 0.5 1.0 1.5 2.0 2.5 3.0λ0.000.250.500.751.001.251.501.752.00b g0.2500.5001.0001.5002.000(b) xsh/λ0.0 0.5 1.0bg0.000.250.500.751.00x sh/ λλ=0.5λ=1λ=2Figure 5.1: (a) Contours of x̄sh(λ̄, Bg). Inset plots x̄sh as a function of Bg for threeλ̄ values. (b) Same as (a), but for x̄sh/λ̄.Figure 5.1b shows similar plots, but for x̄sh/λ̄, the fractional extent of the sheath thatis subject to stochastic ion heating. For bg = 0 and x̄ λ̄ so that sinh(x̄/λ̄)' x̄/λ̄,Eq. 5.11 simplifies to x̄ < 1/√2, which shows that stochastic ion heating is intrinsicto a Harris sheath. Figure 5.1b shows that the fractional heating extent increasesas the sheath becomes thinner. However, Figs. 5.1a and 5.1b show that for each λ̄there exists a maximum guide field value above which stochastic ion heating doesnot occur (red dashed lines). This corresponds to the argument of the square root inEq. 5.12 becoming negative when bg >(√2λ̄)−1, so no real solution for x̄sh exists.For λ̄ = 1, the maximum bg is 1/√2 ' 0.7.This analytical result regarding the Harris equilibrium does not directly apply toan evolving reconnection sheet since the Harris sheath does not by itself involvereconnection. However, because the Harris electric field (Eq. 5.9) is comparable tothe Hall electric field, and the reconnecting magnetic field profile is closely relatedto the Harris profile [164], one can extrapolate a general trend that the fractionalextent of stochastic ion heating decreases as the guide field or the sheath widthincreases.72Quantity Normalizationt ω−1cix diu diωci = vAiB B0E diωci B0 = vAi B0n n0↔p B20/µ0T miv2Aiµ miv2Ai/B0Table 5.1: Dimensional quantities and their normalization parameters in Chapter 5.5.2 Particle-in-cell verificationSetupIn order to test the prediction that stochastic ion heating is intrinsic to collisionlessreconnection, fully kinetic particle-in-cell simulations were conducted using theSMILEI code [41]. The simulations were conducted in 2D, i.e., ∂/∂z → 0. Lengthswere normalized to di, time to ωci, velocity to vAi = diωci, B to the upstream B0,density to the peak density n0, T to miv2Ai, and the magnetic moment µ to miv2Ai/B0.A summary of the normalizations used in the simulation in this Chapter is detailedin Table 5.1.Double periodic Harris sheaths were employed as initial conditions, i.e.,By = B0[tanh(x̄λ̄)− tanh(x̄ − x̄max/2λ̄)− 1](5.13)andn = n0[cosh−2(x̄λ̄)+ cosh−2(x̄ − x̄max/2λ̄)+ nb], (5.14)where x̄max is the simulation box size in the x direction, nb = 0.2 the relativeupstream density, λ̄ = 1 the half-thickness, and T = Ti = Te the temperature. Onlyone of the two sheaths is examined; the double sheath facilitates the application ofperiodic boundary conditions. We work in the lab-frame, so an initial Harris-typeelectric field in the form of Eq. 5.9 is employed. The ion-to-electron mass ratio wasmi/me = 100 and the grid size was 1024 × 2048 = 40.96di × 81.92di. The numberof particles per cell ranged from 100 to 600 depending on the local density, andc/vA = 20. Although x̄sh depends on both λ̄ and bg, changing λ̄ requires significantalterations in simulation parameters such as the domain, and the current sheet tends73to be more unstable for smaller λ̄. Thus, in order to facilitate the analysis, bg waschosen to be the variable parameter.From now on, the bars will be dropped and only normalized quantities will be usedunless specified otherwise.Calculation of Ti⊥ and Ti‖Pressure and temperature are not variables that are intrinsic to the particle-in-cellalgorithm (Section 3.3) and therefore must be post-processed. The mean ion flowvelocity is first calculated asui =∫vi fid3vi, (5.15)where fi is calculated from Eq. 3.35.Now, the ion pressure tensor is defined as↔pi =∫v′iv′i fid3vi, (5.16)where v′i = vi − ui is the random part of the ion velocity. Consider the followingintegral: ∫vivi fid3vi =∫(ui + v′i)(ui + v′i) fid3vi, (5.17)=∫(uiui + uiv′i + v′iui + v′iv′i) fid3vi, (5.18)= uiui∫fid3vi +∫v′iv′i fid3vi (5.19)= uiui +↔pi, (5.20)where∫v′i fid3vi = 0 by the definition of v′i which is the randompart of vi. Therefore,↔pi =∫vivi fid3vi − uiui, (5.21)where the right-hand side can be calculated from the variables explicitly used in thePIC algorithm.Now, in terms of pi⊥ and pi‖ ,↔pi = pi⊥↔I + (pi‖ − pi⊥)B̂B̂, (5.22)74where B̂ is the local unit vector in the direction of the local magnetic field B. It iseasy to see thatpi‖ = B̂ ·↔pi · B̂, (5.23)=B · ↔pi · BB2. (5.24)Since pi‖ is now known, pi⊥ can be calculated by simply using the fact that the traceof a tensor does not change under coordinate transformations, i.e.,Tr↔pi = 2pi⊥ + pi‖, (5.25)orpi⊥ =Tr↔pi − pi‖2. (5.26)Ti‖ and Ti⊥ can then be found simply by dividing pi‖ and pi⊥ by ni =∫fid3vi,respectively.Hall Electric FieldsFigure 5.2 shows B (black lines) and Ex (color) for bg = (a) 0, (b) 0.1, (c) 0.3, and(d) 0.5. Times are chosen so that significant reconnection has taken place but theperiodicity of the domain has not yet affected the local system. It is well-known thatthe Hall term in the generalized Ohm’s law is important in collisionless reconnection[16]. Thus, strong in-plane Hall electric fields Ex ∼ (J × B)x (color) develop; thesehave also been seen in previous studies [7, 47, 93, 125, 162, 171]. Although a finiteguide field alters the structure, Ex is generally pointing toward x = 0, meaning thations are falling down a potential valley along the inflow. A Hall Ey (not plotted)also develops but is smaller in spatial extent and magnitude.Comparison of the Stochastic Heating Criterion to the Ion Temperature Profileα as defined by Eq. 5.1 can be calculated from the electric and magnetic fieldinformation. Figure 5.3 shows lnα; regions where lnα is above zero (red andyellow) represent the spatial extent of predicted stochastic ion heating. It is seenthat the spatial extent decreases as the guide field increases, and at bg = 0.5 (Fig.5.3d), the stochasticity onset criterion is not satisfied anywhere.Figure 5.4 shows the spatial profile of the ion temperature Ti =(2Ti⊥ + Ti‖)/3.Figures 5.4a and 5.4b show strong heating around the regions predicted by Figs.5.3a and 5.3b. Figure 5.4c also shows heating, but its amount is much reduced, as75−505x/d i(a) bg=0−101−505x/d i(b) bg=0.1−101−505x/d i(c) bg=0.3−101−15 −10 −5 0 5 10 15y/di−505x/d i(d) bg=0.5−101Figure 5.2: B (black lines) and Ex (color) for (a) bg = 0, (b) 0.1, (c) 0.3, and (d) 0.5.predicted from there being only sparse red regions in Fig. 5.3c. Figure 5.4d showsthat there is no strong heating at bg = 0.5, in agreement with the prediction of Fig.5.3d. As previously mentioned, this heating amount reduction with increasing guidefield is an observed feature of ion heating in reconnection.Temperature AnisoptropyFigures 5.5 and 5.6 show Ti⊥ and Ti‖ , respectively. It can clearly be seen that the twotemperature profiles are vastly different, confirming the temperature anisotropy thatis an observed feature of ion heating in reconnection. Also, Ti⊥ is more consistentwith the prediction in Fig. 5.3 thanTi‖ . AlthoughTi⊥ is generally lower inmagnitudethan Ti‖ , the spatial profile of Ti resembles that of Ti⊥ more because Ti⊥ contributesto Ti twice as much as Ti‖ does (recall that Ti = (2Ti⊥ + Ti‖)/3).76−505x/d i(a) bg=0−2−1012−505x/d i(b) bg=0.1−2−1012−505x/d i(c) bg=0.3−2−1012−15 −10 −5 0 5 10 15y/di−505x/d i(d) bg=0.5−2−1012Figure 5.3: lnα for bg = 0, 0.1, 0.3, and 0.5, respectively.5.3 Confirmation of Stochastic MotionThe correlation between perpendicular heating and the stochastic criterion involvesthe non-conservation of the ion magnetic moment µ = miv2c/2B, where vc is thecyclotron velocity that must be calculated in the frame moving with the guiding-center velocity [11, pages 96-98]. However, the guiding-center under zero guidefield is ill-defined for outflowing ions because they are not magnetized. Therefore,we define a pseudo-moment µE×B = mi |vi⊥ − vE×B |2 /2B, whose non-conservationmeasures the departure from guiding center motion and thus the breakdown ofthe drift hierarchy. Stochastic heating involves this very breakdown [10], so thecorrelation between µE×B and α indicates the existence of stochastic heating.Another indication of stochastic heating is the Lyapunov exponent [100], whichdescribes the rate of separation of two initially infinitesimally close particles. Asseen in Chapter 4, a positive Lyapunov exponent is characteristic of chaotic behavior.77−505x/d i(e) bg=00.000.250.500.751.00−505x/d i(f) bg=0.10.000.250.500.751.00−505x/d i(g) bg=0.30.000.250.500.751.00−15 −10 −5 0 5 10 15y/di−505x/d i(h) bg=0.50.000.250.500.751.00Figure 5.4: Ti for bg = 0, 0.1, 0.3, and 0.5, respectively.µE×B violationTo verify stochastic heating at a single-particle level, test-particle simulations wereconducted with the algorithm in Appendix D. Figure 5.7a shows two typical ionoutflow trajectories (black solid and dashed line) with an initial spatial separationof 0.04di for bg = 0. Both ions undergo oscillatory motion in the x-directionunder the Hall electric field (color) [7, 47, 162, 168, 171]. Figure 5.7b shows thetime-dependent spatial ion separation, which exhibits divergence and thus a positiveLyapunov exponent. Figure 5.7c shows µE×B (blue) and lnα (red) along the particletrajectory represented by the black solid line in Fig. 5.7a. The red dashed linerepresents the stochastic heating criterion, above which µE×B violation is expected.The locations where the stochastic criterion is satisfied coincides with kicks in µE×B.This µE×B is an observed feature of ion heating in reconnection.The correlation between α and chaotic behavior becomes evenmore obvious in Figs.5.7d, 5.7e, and 5.7f, which are respectively the same as Figures 5.7a, 5.7b, and 5.7c78−505x/d i(a) bg=00.00.51.01.5−505x/d i(b) bg=0.10.00.51.01.5−505x/d i(c) bg=0.30.00.51.01.5−15 −10 −5 0 5 10 15y/di−505x/d i(d) bg=0.50.00.51.01.5Figure 5.5: Ti⊥ for bg = 0, 0.1, 0.3, and 0.5, respectively.except for bg = 0.3 (recall from Fig. 5.4c that there is limited stochastic heating inthis case). In Fig. 5.7e, it is apparent that the spatial separation of the ions doesnot diverge, illustrating non-stochastic behavior. In Fig. 5.7f, lnα rarely goes abovezero, and µE×B is relatively conserved.5.4 Heavy IonsFinally, Equation 5.1 predicts that ions with larger mi/qi more easily satisfy thestochastic heating criterion— an observed feature of ion heating in reconnection. Asimulation containing a mix of heavy ions (mi/me = 500) of density 0.1n and lightions (mi/me = 100) of density 0.9n for bg = 0.3 was run. Figure 5.4a shows lnα forthe heavy ions, which, because of their larger mi/qi, satisfy the stochastic heatingcriterion across a broader range compared to the light ions in Fig. 5.3c. Figure5.4b shows Ti for the heavy ions, which, in comparison to the light ions in Fig.5.4c, undergo stronger stochastic heating. The spatial profile of Ti agrees with Fig.79−505x/d i(a) bg=00.00.51.01.5−505x/d i(b) bg=0.10.00.51.01.5−505x/d i(c) bg=0.30.00.51.01.5−15 −10 −5 0 5 10 15y/di−505x/d i(d) bg=0.50.00.51.01.5Figure 5.6: Ti‖ for bg = 0, 0.1, 0.3, and 0.5, respectively.5.4a, but there is additional heating (blue arrows) that is not due to the stochasticmechanism and which will be subject to future investigations. Figure 5.4c showsthat Ti⊥ agrees well with the prediction from Fig. 5.4a, hinting that the additionalheating comes from Ti‖ .Single-particle trajectories of the heavy ions confirm that stochastic heating is occur-ring. Figure 5.7g is the same as Figure 5.7d, but for the heavy ions. The separationdivergence (Fig. 5.7h) and the kicks in µE×B in correlation with α (Fig. 5.7i) verifychaotic behavior. At tωci > 20, lnα is positive only sporadically, so the kicks aresmaller, whereas at tωci < 20, lnα stays positive for longer times, so the kicks arebigger.5.5 SummaryThe study presented in this chapter provides strong confirmation that stochasticheating is the fast ion heating mechanism in collisionless magnetic reconnection80up to moderate guide fields. The heating is consistent with calculations based onthe properties of the in-plane Hall electric fields that intrinsically develop duringreconnection. Ion temperature profiles were found to agree with the predictions.The stochastic nature of ions was confirmed through the examination of Lyapunovexponents and the correlation between µE×B violation and the stochastic heatingcriterion. Heavier ions were found to be heated more strongly because they moreeasily satisfy the criterion.81Figure 5.7: (a) Two test ion trajectories (black solid and dashed lines) and Ex (color)under bg = 0. (b) The time-dependent spatial separation between the ions in (a). (c)µE×B (blue) and lnα (red) along the particle trajectory represented by the black solidline in (a). The red dashed line represents the stochastic heating criterion, abovewhich stochastic heating is expected. (d)-(f) are the same as (a)-(c), respectively,except for bg = 0.3. (g)-(i) are the same as (d)-(f), except for mi/me = 500.82−505x/d i(a) ln(α) @ mi/me=500−2−1012−505x/d i(b) Ti @ mi/me=5000.00.51.0−15 −10 −5 0 5 10 15y/di−505x/d i(c) Ti⟂ @ mi/me=5000.00.51.01.5Figure 5.8: (a) lnα, (b) Ti, (c) Ti⊥ for ions with mi/me = 500 under bg = 0.3.83C h a p t e r 6PITCH-ANGLE SCATTERING OF ENERGETIC PARTICLES BYCOHERENT WHISTLER WAVESIn Chapter 2, it was seen that whistler waves are generated as a result of collisionlessmagnetic reconnection. Because reconnection is ubiquitous in magnetized plasmaenvironments, whistler waves are also ubiquitous in the Earth’s magnetosphere[26, 62, 132, 151], Jupiter’s magnetosphere [92, 135, 152], and Saturn’s magne-tosphere [1, 8, 73]. These waves are also important in the solar wind [36, 155],fast magnetic reconnection [12, 29, 67, 104, 169, 170], and helicon plasma sources[20, 33]. In particular, the interaction between energetic charged particles andmagnetospheric whistler waves is important since the interaction can change thepitch-angle of the particles, potentially scattering them into the loss cone of a mag-netic mirror configuration such as the Earth’s dipole magnetic field. Because theescaped energetic particles can cause pulsating auroras at the Earth’s poles andenergetic particles in general can damage spacecraft, this interaction has been thefocus of many studies for decades [2, 5, 6, 68, 71, 80, 81, 101, 114, 145, 154].Relativistic wave-particle resonance has been known to be an important element ofparticle energization and pitch-angle scattering. Resonant interaction arises whenω − kvz =Ωγ. (6.1)Here, ω is the wave frequency, k is the wavenumber parallel to the backgroundmagnetic field B0 which is oriented in the z direction, vz is the parallel particlevelocity, and Ω = qB0/m is the cyclotron frequency of the particle with charge qand mass m. Also, γ =(1 − v2/c2)−1/2 is the particle Lorentz factor where v is theparticle speed and c is the speed of light. Kennel and Petschek [80] first quantifiedthe scattering mechanism by which incoherent whistler waves lead to velocity spacediffusion, and numerous studies have further developed this mechanism [2, 81,101, 153]. However, recent spacecraft measurements indicate that the observedchorus bursts are, in fact, extremely coherent and that these waves, especially large-amplitude ones (δB/B0 ∼ 0.01 where δB is the wave magnetic field), are directlylinked to electron energization, loss, and microbursts [4, 24, 28, 55, 153, 154]. Thislinkage suggests that a non-diffusive process could be governing what is observed.84There has thus been a continuing and substantial theoretical effort to investigate thedynamics of energetic particles under coherent whistler waves. Bortnik et al. [19]numerically investigated ad hoc the coherent interaction between large-amplitudewhistler waves and relativistic particles. Lakhina et al. [87] showed via calculationsof pitch-angle diffusion coefficients that coherent chorus subelements can causerapid pitch angle scattering, although Lakhina et al. [87] used diffusion coefficientscalculated from incoherent whistler waves [80] and used non-relativistic equations ofmotionwhereas the actual wave-particle interaction involves relativistic particles (10keV toMeV [24, 154]). Bellan [9] presented an exact analytical calculation involvinga relativistic particle in a right-handed circularly polarized electromagnetic wave.This calculation showed that a certain class of particles undergo quick, drastic pitch-angle scattering depending on whether the individual particle’s initial conditionsmeet a certain criterion, which will be discussed in the next section. Also notethat other studies have investigated this single-particle problem via various methods[21, 56, 126, 130]. However, an analysis of the importance of this mechanism fora distribution of particles has not yet been done. To demonstrate importance, onemust show that a significant fraction of the particles in the distribution experiencesthis drastic scattering. If this can be demonstrated, then the particle interaction withcoherent whistler waves will be a dominant pitch-angle scattering mechanism.In this chapter, the analysis presented in Bellan [9] is extended to the relativisticgeneralization of a thermal distribution of particles; the generalization is prescribedby the Maxwell-Jüttner distribution [78]. It is found that for parameters relevant tomagnetospheric chorus, coherent right-handed circularly polarized waves propagat-ing parallel to the background magnetic field trigger large, non-diffusive pitch-anglescatterings for a significant fraction (1% − 5%) of the energetic particles. The scal-ing of this fraction with the wave amplitude may also explain the association ofrelativistic microbursts to large-amplitude chorus [24]. A new condition for largepitch-angle scattering is also presented; this condition is a certain range relatedto Eq. 6.1, but may or may not include exact resonance depending on the particleinitial conditions. Test-particle simulations corroborate the predictions made by thisanalysis. It is also demonstrated that the widely-used second-order trapping theory[110, 113, 115, 116, 144] is a simplified approximation of the theory presented inthis chapter and that this simplified approximation effectively misses critical detailsof the wave-particle interaction. The present study illustrates that coherent whistlerwaves are an important cause of non-diffusive pitch-angle scattering and providesan accurate condition for this scattering.856.1 Two-Valley Motion ReviewLet us begin with a brief review of the large pitch-angle scattering mechanism pre-sented in Bellan [9]. A thorough comprehension of this single-particle mechanismis essential for understanding the ensuing analysis presented here. It is assumed thatthe wave is right-handed circularly polarized and travels parallel to a uniform back-ground magnetic field, so the total magnetic field can be expressed as B = B0 ẑ + B̃whereB̃ = κB0 [x̂ sin (kz − ωt) + ŷ cos (kz − ωt)] . (6.2)Here κ is thewave amplitude relative to the background B0. Faraday’s lawdeterminesthe wave electric field to beẼ = −ωkẑ × B̃ = ωkB̃ [x̂ cos (kz − ωt) − ŷ sin (kz − ωt)] . (6.3)The relativistic Lorentz force equation determines the motion of a charged particle:ddt(γβ) = qm(Ẽc+ β × B), (6.4)where β = v/c and γ =(1 − β2)−1/2.In Bellan [9], a left-handed circularly polarized wave was used although the studywas intended for right-handed waves. However, the result therein is unaffected bythis apparent error because the sign of the particle charge was unspecified. Althoughit was not explicitly stated, the analysis was carried out assuming that the chargeis positive, e.g., for positrons or ions. If the charge is assumed to be negative,the same wave-particle interaction arises when the wave is assumed to have aright-handed polarization. Therefore, the theory in Bellan [9] describes wave-particle interactions between positively charged particles and left-handed waves,and equivalently between negatively charged particles and right-handed waves— orelectrons and right-handed whistler waves. This equivalence can also be seen usingcharge-parity-time symmetry, which is a fundamental law of any Lorentz-invariantsystem [58]; making the changes z → −z and t → −t in Eqs. 6.2 and 6.3 changesthe sense of rotation of the wave, and the relevant physics must be equivalent whenthe change q→ −q is made.In this chapter, the analysis in Bellan [9] with the left-handed wave and positivelycharged particles will be used for two reasons. First, the analysis can then be keptgeneral for any particle with any sign of charge. Second, the derivation of a separatetheory for negatively charged particles will merely be a matter of some sign changes86and is not worth the additional complexity in understanding the core points of thischapter.In Bellan [9], a "frequency mismatch" parameterξ = 1 + αγ (nβz − 1) (6.5)was defined, where α = ω/Ω is the normalized frequency, βz = vz/c is the nor-malized parallel velocity, and n = ck/ω is the refractive index. Equation 6.1 issatisfied when ξ = 0, so ξ is a measure of the departure from resonance. An exactrearrangement of Eq. 6.4 leads to an equation of motion for a particle moving inξ-space [9]:1Ω′d2ξdt′2= −∂ψ∂ξ, (6.6)whereψ (ξ) = 18ξ4 +(κ′2 −ξ202− sκ′ sin φ0)ξ22− κ′2ξ (6.7)is the pseudo-potential for ξ-spacemotion. Here the primed quantities are calculatedin the wave frame, i.e., a frame moving with a velocity ẑω/k. The subscript 0 refersto the value at the initial time t = t′ = 0 and there are two parameters, namely sand φ0. The parameter s is defined as s = αnβ⊥0γ0/γT where γT =(1 − n−2)−1/2is the Lorentz factor of the wave. The parameter φ0 is defined as the initial angularorientation of the perpendicular velocity in the x − y plane, i.e., the angle betweenβ⊥0 and Ẽ(t = 0, z = 0). The shape of the pseudo-potential is entirely determinedby the initial conditions of the particle with respect to the wave as prescribed by ξ0,s, and φ0. Note that s is an initial condition of the particle because α and n are fixedparameters in the present analysis.Multiplying Eq. 6.6 by dξ/dt′ and integrating with respect to t′ yields the particlepseudo-energy,W =12Ω′2(dξdt′)2+ ψ (ξ) , (6.8)which is a constant of the motion. For certain initial conditions, ψ (ξ) consists oftwo valleys separated by a hill in between. If the initial particle pseudo-energyis sufficiently large to go over the hill between the two valleys, then the particleundergoes two-valley motion in ξ-space. This motion involves large changes in ξand thus in βz, β⊥ and the pitch-angle θpitch = tan−1 β⊥/βz.876.2 Two-Valley Motion ConditionLet us now derive the conditions for two-valley motion for a given particle. Theconditions consist of two parts: ψ(ξ) must first be two-valleyed, and the particlemust have sufficient pseudo-energy to overcome the hill between the two-valleys.The initial particle kinetic pseudo-energy can be expressed as [9]12Ω′2(dξdt′)2t ′=t=0=12s2κ′2 cos2 φ0, (6.9)so the total pseudo-energy isW =12s2κ′2 cos2 φ0 −ξ408+ξ202κ′ (κ′ − s sin φ0) − κ′2ξ0. (6.10)We write Eq. 6.7 as ψ (ξ) = ξ4/8 + bξ2/2 − κ′2ξ where b = κ′2 − ξ20/2 − sκ′ sin φ0.Then dψ/dξ = ξ3/2 + bξ − κ′2, so one extremum is at small ξ ' κ′2/b and twoextrema are at large ξ ' ±√−2b. Since d2ψ/dξ2 = 3ξ2/2 + b, for b < 0 the largeextrema are local minima (two valleys) and the small extremum is a local maximum(a hill). For b ≥ 0, the large extrema are undefined, so there is a minimum atξ ' κ′2/b. Figure 6.1a shows an example of a two-valley ψ (ξ) for which b < 0, andFig. 6.1b shows a one-valley ψ (ξ) for which b ≥ 0.We now make the assumptionκ′ s, (6.11)which will be shown in Section 6.4 to be appropriate for relevant magnetosphericsituations. Then, b ' −ξ20/2 − sκ′ sin φ0 is negative forξ20 ≥ −2sκ′ sin φ0. (6.12)All particles having sin φ0 > 0 satisfy this equation because ξ20 is non-negative.Particles having sin φ0 ≤ 0 satisfy Eq. 6.12 only if they are in a certain distanceaway from exact resonance (ξ = 0).Now, inserting ξ = κ′2/b in Eq. 6.7, we have the height of the hill to be ψmax '−κ′4/(2b). Therefore, the particle has enough pseudo-energy to cross over the hillif12s2κ′2 cos2 φ0 −ξ408+ξ202κ′ (κ′ − s sin φ0) ≥ κ′2ξ0 −κ′42b. (6.13)We now assume and justify later that the terms on the right-hand side of Eq. 6.13 aremuch smaller than those on the left-hand side. Using Eq. 6.11, Eq. 6.13 becomes12s2κ′2 cos2 φ0 −ξ408−ξ202κ′s sin φ0 ≥ 0, (6.14)88−0.2 −0.1 0.0 0.1 0.2ξ−0.50.00.51.0ψ1e−4b=-0.008(a) Two-valley ψ(ξ)−0.2 −0.1 0.0 0.1 0.2ξ0123 1e−4b=0.031(b) One-valley ψ(ξ)0 200 400 600t/Ω−1100110120θ pitch(∘) Two-valley θ pitch(t)0 200 400 600t/Ω−1100110120(d) One-valley θ pitch(t)−0.2 −0.1 0.0 0.1 0.2ξ0.00.51.0χ1e−4 (e) χ(ξ)−0.2 −0.1 0.0 0.1 0.2ξ01231e−4 (f) χ(ξ)Figure 6.1: (a) An example of a two-valley ψ (ξ) for which b = −0.008 < 0. (b)An example of a one-valley ψ (ξ) for which b = 0.031 ≥ 0. (c) The time-dependentpitch-angle of the particle undergoing two-valley motion, and (d) that of the particleundergoing one-valley motion. The wave parameters were κ = 0.01, α = 0.25, andn(α) = 18 from Eq. 6.27. (e), (f) The approximated pseudo-potentials χ obtainedby keeping only the term involving sκ′ in Eq. 6.7 for the respective particles.89whose solution isξ20 ≤ 2sκ′ (1 − sin φ0) . (6.15)Now we derive the conditions for which the assumptions regarding Eq. 6.13 arevalid. This is done by using the solution (i.e., Eq. 6.15) obtained under theassumptions and deriving the conditions for which the terms on the right-hand sideof Eq. 6.13 are indeed small compared to those on the left-hand side. Using Eq.6.15 as an equality, it is seen that each term on the left-hand side of Eq. 6.13 isO(s2κ′2)except for the κ′2ξ20/2 term which is ignored by Eq. 6.11. On the right-hand side, κ′2ξ0 = O(√sκ′5)so it can be ignored if κ′ s3. Examining the secondterm, κ′4/2b = O(κ′3/s)because b = O (sκ′), so it can be ignored if κ′ s. Sinceκ′ 1 for linear waves, κ′ s3 and κ′ s are both true for s ≥ 1, and κ′ s3 isa stronger statement than κ′ s if s < 1. Therefore, for κ′ s3—which will laterbe demonstrated to be valid for relevant magnetospheric parameters—the followinggives the condition for which a particle undergoes two-valley motion and thus alarge pitch-angle scattering:− 2sκ′ sin φ0 ≤ ξ20 ≤ 2sκ′ (1 − sin φ0) . (6.16)Equation 6.16 is one of the main results of this chapter. For φ0 ≥ 0, Eq. 6.16becomes Eq. 6.15 and specifies a certain range around ξ0 = 0. However, for φ0 < 0that statistically represents half of the particle population, Eq. 6.16 does not includeξ0 = 0, which means that particles further away from exact resonance undergo two-valley motion and thus large pitch-angle scattering. Therefore, Eq. 6.16 specifiesthe exact range of the initial distance from resonance that leads to two-valley motion.Figure 6.1c shows the time-dependent pitch-angle θ◦pitch(t) of the particle that hasenough pseudo-energy to undergo two-valley motion in the two-valley pseudo-potential in Fig. 6.1a. Figure 6.1d shows θ◦pitch(t) of the particle moving in theone-valley pseudo-potential. The particle in Fig. 6.1c experiences a much largerchange in pitch-angle than that in Fig. 6.1d. The rate of change of the pitch-anglein Fig. 6.1c is also very large; the wave period is TwaveΩ = 2π/α ' 25, so thepitch-angle changes by ∼ 15◦ in tΩ ' 40 or in about one to two wave periods.6.3 Distribution of ξThe initial particle distribution in ξ-space will now be derived. The subscript zerowill henceforth be dropped because only the initial conditions are being examined.The thermal distribution is assumed to be the Maxwell-Jüttner distribution [78]90with an isotropic temperature. This distribution is a considerable simplification,and repercussions of this simplification and possible remedies will be discussedin Section 6.6. This is the relativistic generalization of the Maxwell-Boltzmanndistribution and can be expressed in terms of the Lorentz factor γ asfγ =γ2√1 − 1/γ2θK2 (1/θ)exp(−γθ), (6.17)where θ = kBT/mc2 is the normalized temperature and Kn is the modified Besselfunction of the second kind of order n. Using γ =√1 + p2/m2c2 =√1 + ρ2 whereρ = p/mc is the normalized particle momentum, Eq. 6.17 can be expressed asfρ =14πθK2 (1/θ)exp(−√1 + ρ2θ). (6.18)Integrating Eq. 6.18 in ρz and over all angles gives fρ⊥:fρ⊥ =ρ⊥√1 + ρ2⊥θK2 (1/θ)K1©«√1 + ρ2⊥θª®®¬ . (6.19)Note that fρ is defined in 3D ρ-space so that∫fρd3ρ = 1, whereas fρ⊥ is definedin 1D ρ⊥-space so that∫fρ⊥dρ⊥ = 1. Integrating Eq. 6.18 in ρx and ρy gives fρz :fρz =θ2K2 (1/θ)©«1 +√1 + ρ2zθª®®¬ exp©«−√1 + ρ2zθª®®¬ , (6.20)where∫fρz dρz = 1. The details of the derivations of fρ⊥ and fρz are given inAppendices F.1 and F.2, respectively.Now, noting that γβ = γv/c = p/mc = ρ, the mismatch parameter (Eq. 6.5) canbe expressed asξ = 1 + α (nρz − γ) . (6.21)The probability distribution of having a specific ξ is obtained by multiplying theprobability distribution of having a certain γ by that of having the corresponding ρzwhich yields the specified ξ, and then integrating over all γ (full derivation given inAppendix F.3). The solution isfξ =∫ ∞1γ2√1 − 1/γ22αnK22 (1/θ)©«1 +√1 + ρ2z (γ, ξ)θª®®¬ exp©«−γ +√1 + ρ2z (γ, ξ)θª®®¬ dγ,(6.22)91−4 −2 0 2 4ξ0.00.51.01.5f ξ(a) fξ for varying θθ=0.01θ=0.04θ=0.16θ=0.64−4 −2 0 2 4ξ0.00.51.01.52.0(b) fξ for varying αα=0.05α=0.1α=0.2α=0.4−4 −2 0 2 4ξ0.000.250.500.751.00(c) fξ for varying nn=4n=8n=16n=32Figure 6.2: fξ for different (a) θ, (b) α and (c) n values. The default values areθ = 0.1, α = 0.25, and n = 10. The black dashed line is the resonant conditionξ = 0.where ρz (γ, ξ) = [(ξ − 1) /α + γ] /n is a rearrangement of Eq. 6.21 and∫fξdξ = 1.Given θ, α and n, Eq. 6.22 is an integral solution for fξ .Figure 6.2 shows fξ for different (a) θ, (b) α and (c) n values. The default valuesare θ = 0.1, α = 0.25, and n = 10, where α = 0.25 and θ = 0.1 are relevantvalues for the dayside outer magnetosphere [153], and n = 18 ∼ 10 from thewhistler dispersion relation (Eq. 6.27). The black dashed vertical line representsthe resonant condition ξ = 0 (or equivalently Eq. 6.1). As θ, α, and n increasefrom zero, fξ broadens and more particles are resonant. After a certain threshold,however, too much broadening leads to the decrease of the local magnitude offξ (ξ = 0) and reduces the number of resonant particles. Increasing α significantlychanges the mean value of ξ as well, raising this threshold higher.6.4 Fraction of Particles Undergoing Two-Valley MotionBefore calculating the fraction of particles undergoing two-valleymotion, the proba-bility distribution of the limits of integration (Eq. 6.16) must first be derived. Again,the subscript zero will be dropped. Since s = αnβ⊥γ/γT = α√n2 − 1ρ⊥, the rele-vant distribution is that of ρ⊥ and sin φ. Equation 6.19 prescribes fρ⊥ , and assumingthat φ is isotropic, the probability distribution of Φ = sin φ is the Arcsine(-1,1)distribution,fΦ =1π√1 − Φ2, (6.23)for Φ ∈ (−1, 1).We now have all the ingredients to calculate the fraction of particles that undergotwo-valley motion in ξ-space and thus experience large pitch-angle scattering. Thisfraction can be found by calculating the probability that both Eqs. 6.12 and 6.1592(i.e., Eq. 6.16) are satisfied. In the case Φ > 0 when Eq. 6.12 is always met,after defining a numerical factor a = 2ακ′√n2 − 1 so that 2sκ′ sin φ = aρ⊥Φ and2sκ′ (1 − sin φ) = aρ⊥ (1 − Φ), the probability of two-valley motion isp+ =∫ ∞ρ⊥=0∫ 1Φ=0fρ⊥ fΦ∫ √aρ⊥(1−Φ)ξ=−√aρ⊥(1−Φ)fξdξdΦdρ⊥. (6.24)In the opposite case where Φ ≤ 0, the probability of two-valley motion is,p− =∫ ∞ρ⊥=0∫ 0Φ=−1fρ⊥ fΦ(∫ −√−aρ⊥Φ−√aρ⊥(1−Φ)fξdξ +∫ √aρ⊥(1−Φ)√−aρ⊥Φfξdξ)dΦdρ⊥. (6.25)The total fraction of particles undergoing two-valley motion is then ptv = p+ + p−.There are four degrees of freedom when calculating ptv: θ, α, n and κ. However,one degree of freedom can be eliminated by linking α and n through the whistlerwave dispersion relation, which, for parallel propagation and Ωp/Ω 1 where Ωpis the electron plasma frequency, isc2k2ω2=Ω2p/ω2|Ω| /ω − 1 . (6.26)In terms of the dimensionless variables used in this chapter, this becomesn =Ωp/Ω√α (1 − α), (6.27)which can be used to express n (α) if Ωp/Ω is specified. Using parameters inTsurutani et al. [153] (ne ' 10 cm−3,B0 ' 125 nT), we obtain Ωp/Ω ' 8; this valuewill be used throughout the rest of the analysis.Let us now calculate ptv for the parameters in the range 0.0001 ≤ κ ≤ 0.01,0.1 ≤ α ≤ 0.8 and 0.01 ≤ θ ≤ 10 (corresponding to electron thermal energies from5.11 keV to 5.11 MeV). Since the parameter range is determined, the conditions forwhich the assumption κ′ s3 that was used to derive Eq. 6.16 is true can now bedetermined. Because n 1, κ′ = κ/γT = κ√1 − 1/n2 ' κ and s = α√n2 − 1ρ⊥ 'αnρ⊥. From Eq. 6.27 it follows that αn =(Ωp/Ω) √α/(1 − α). We now comparethe largest value of κ to the lowest value of s3, which involves the smallest valuesof α and θ. For θ 1, the most likely ρ⊥ is√θ (see Appendix F.4). Thus, thecondition κ′ s3 can be expressed asκ (ΩpΩ√αθ1 − α)3, (6.28)930.25 0.50 0.75α10−210−1100101θ(a) ptv for κ=0.00010.00000.00080.00160.00240.00320.00400.00480.00560.00640.25 0.50 0.75α10−210−1100101(b) ptv for κ=0.00100.00000.00250.00500.00750.01000.01250.01500.01750.02000.25 0.50 0.75α10−210−1100101(c) ptv for κ=0.01000.00000.00800.01600.02400.03200.04000.04800.05600.0640Figure 6.3: ptv as a function of α and θ for different κ values.orθ κ2/3(Ωp/Ω)2 (α/(1 − α)) . (6.29)Inserting α = 0.1 and κ = 0.01 shows that κ′ s3 is valid if θ 0.0066. Thus,0.01 ≤ θ ≤ 10 is consistent with κ′ s3.Figure 6.3 shows contours of ptv as a function of α and θ for different κ values.For κ ≥ 0.001, which is typical for magnetospheric chorus [102, 153], a significantfraction (1%−5%) of particles undergo two-valleymotion and thus large pitch-anglescattering. However, ptv decreases at high θ (θ & 1), and this phenomenon is relatedto the decrease of the local magnitude of fξ (ξ = 0) if there is too much broadeningof fξ , as shown in Fig. 6.2. Figure 6.3 also shows that ptv (α, θ) has more or less thesame shape across a wide range of κ but its magnitude is proportional to√κ. Thisis because the limits of the ξ integrals in Eqs. 6.24 and 6.25 scale as√a ∼√κ, so ifthe integration range is sufficiently small so that the integrand may be approximatedby a linear function, it follows that ptv ∝√κ.6.5 Numerical VerificationThe analytical predictions presented so far will now be verified via numerical sim-ulations. A computer code was written which solves Eq. 6.4 and dx/dt = cβ usingthe fully implicit Runge-Kutta method of the Radau IIA family of order 5 [64] in thescipy.integrate.solve_ivp package in Python 3.7. This particular method was usedbecause it yielded the smallest numerical error out of the available methods in thePython package, measured by the drift of the average value of the pitch-angle overtime which should be zero in principle (see, e.g., Fig. 6.1c where the particle’spitch-angle oscillates around a stationary average value). The error was quantifiedby using the statistics of the 10,000 particles in Fig. 6.5c. The Radau method witha time step ∆t = 0.2 yielded a median value for the pitch-angle drift of 0.07◦ with a94−0.4 −0.2ρz0.00.51.01.5ρ⟂(a) Two-Valley Predictions (ϕ= π/4)0.00.20.40.60.81.0−0.4 −0.2ρz0.00.51.01.5(b) App oximate P edictions (ϕ= π/4)0.00.20.40.60.81.0(0.4 (0.2ρz0.00.51.01.5(c) Δθ)pitch(ρ⟂ Δ ρz) (ϕ= π/4)102030(0.4 (0.2ρz0.00.51.01.5ρ⟂(d) Two-Valley Predictions (ϕ=−π/4)0.00.20.40.60.81.0−0.4 −0.2ρz0.00.51.01.5(e) App oximate P edictions (ϕ=(π/4)0.00.20.40.60.81.0(0.4 (0.2ρz0.00.51.01.5(f) Δθ)pitch(ρ⟂ Δ ρz) (ϕ=(π/4)102030Figure 6.4: (a) Regions of initial momentum space (dark green) that satisfy theunapproximated two-valley criteria (Eqs. 6.13 and b < 0) for φ = π/4. (b) Regionsof this space that satisfy the approximated criterion (Eq. 6.16) for φ = π/4. (c)Pitch-angle range (in degrees) within a single particle trajectory for a range of initialparticle momenta for φ = π/4. (d-f) are the same as (a-c) except for φ = −π/4.Blue lines represent the resonance condition (Eq. 6.1; ξ = 0). The wave parameterswere α = 0.25, κ = 0.005, and n = 18 from Eq. 6.27.standard deviation of 0.14◦, which is far smaller than the pitch-angle change of a vastmajority of the particles. The electromagnetic fields were prescribed by Eqs. 6.2and 6.3, which is a simplified model of a whistler wave. The code was parallelizedwith the multiprocessing package.It will first be verified that particles which satisfy Eq. 6.16 and thus undergo two-valley motion experience large pitch-angle scattering. 2,500 particle trajectorieswere numerically integrated, and the initial particle momenta were scanned in therange ρ⊥ ∈ [0, 2], ρz ∈ [−0.5, 0], and φ = π/4,−π/4. The wave amplitude wasκ = 0.005, and the wave frequency was α = 0.25, which gives n = 18 using Eq.6.27.Figure 6.4a shows the regions of initial momentum space (dark green) that satisfy theunapproximated two-valley criteria (Eqs. 6.13 and b < 0) for φ = π/4. Figure 6.4bshows regions of this space that satisfy the approximated criterion (Eq. 6.16). Theregions are virtually identical except for ρ⊥ . 0.1 because for sufficiently large ρ⊥,the κ s3 approximation holds. Figures 6.4a and 6.4b are effectively predictionsof large pitch-angle scattering. The colors in Fig. 6.4c show the pitch-angle rangethat a particle undergoes for each point in (ρ⊥, ρz) space; this pitch-angle range is950 2000 4000 6000 8000 10000Particle Number0123Δθpitch ( ∘Red = 0.43Δ(a∘ κ= 0.00010 2000 4000 6000 8000 10000Particle Number0.02.55.07.510.012.5Δθpitch ( ∘Red = 1.52Δ(b∘ κ= 0.00100 2000 4000 6000 8000 10000Particle Number010203040Δθpitch ( ∘Red = 4.29Δ(c∘ κ= 0.0100Figure 6.5: Pitch-angle range of 10,000 particles whose initial momenta wererandomly sampled from Eq. 6.18 for different κ values. Red points representparticles that meet the two-valley criterion (Eq. 6.16), and the text inside representsthe percentage of red particles. The red horizontal lines represents the median∆θ◦pitch of the red particlesdefined by the absolute difference between the maximum andminimum pitch-anglesalong the particle trajectory. For example, the particle in Fig. 6.1c has a pitch-anglerange of ∼ 15◦, and that in Fig. 6.1d has a pitch-angle range of ∼ 3◦. Figures 6.4d-fare the same as Figs. 6.4a-c except for φ = −π/4. It can be clearly seen that ifa particle’s initial momentum satisfies the two-valley criteria, it undergoes a largepitch-angle scattering.The blue curves in Fig. 6.4 represent the resonance condition (Eq. 6.1; ξ = 0). Thecurve is found by solving ξ = 1 + α(nρz − γ) = 1 + α(nρz −√1 + ρ2⊥ + ρ2z) = 0 forρ⊥ (ρz) and restricting the domain of ρz to be consistent with γ = α−1 + nρz ≥ 1.In Figs. 6.4d-f, the blue lines do not pass through regions of two-valley motion andlarge scattering. This fact is consistent with Eq. 6.16 which qualitatively states thatfor φ < 0, the condition for two-valley motion and large scattering does not includeξ = 0.Next, the analytical prediction for ptv will be verified via the Monte-Carlo method.The trajectories of 10,000 particles whose initial momenta were randomly sampledfrom Eq. 6.18 were respectively integrated for κ = 0.0001, 0.001, and 0.01. Otherparameters were α = 0.25, n = 18, and θ = 0.1.Figure 6.5 shows the pitch-angle range (in degrees) of the randomly sampled parti-cles for different κ values. Red points represent particles that meet the two-valleycriterion (Eq. 6.16), and the text inside represents the percentage of red particles.Figure 6.3 shows that for α = 0.25 and θ = 0.1, the predicted percentage rangesare 0.4 − 0.48%, 1.25 − 1.5%, and 4.00 − 4.80% for κ = 0.0001, 0.001, and 0.01,respectively, which approximately agree with the results in Fig. 6.5. Red points960 2000 4000 6000 8000 10000Particle Number0.000.050.100.15Δθpitch ( ∘ per Twave Red = 0.43Δ(a∘ κ= 0.00010 2000 4000 6000 8000 10000Particle Number0.00.51.01.5Δθpitch ( ∘ per Twave Red = 1.52Δ(b∘ κ= 0.00100 2000 4000 6000 8000 10000Particle Number051015Δθpitch ( ∘ per Twave Red = 4.29Δ(c∘ κ= 0.0100Figure 6.6: Pitch-angle change per wave period of the respective simulations inFig. 6.5. The red horizontal lines respectively represent the median value of thepitch-angle change per wave period of the red particles.generally experience significantly larger pitch-angle scattering than other particles,as can be seen from the median value of the red points (red horizontal lines). How-ever, it can be seen that there are blue points that also experience large scattering;examining the pseudo-potential ψ (ξ) for these points shows that these particles havepseudo-energies that are just short of overcoming the two-valley hill, so they "al-most" undergo two-valley motion and experience substantial pitch-angle scattering.Therefore, we conclude that ptv is a lower-bound for the fraction of particles withlarge pitch-angle scattering.Even if two-valley motion were to cause large pitch-angle scattering, the mechanismwould not be significant if this scattering could not occur within a short enough time.Thus, it is necessary to show that the coherent wave lasts sufficiently long for two-valleymotion to occur. Figure 6.6 shows the pitch-angle change within a single waveperiod for the respective simulations in Fig. 6.5. Tsurutani et al. [153] observedin the outer magnetosphere coherent chorus elements with amplitudes κ ' 0.0016that are 0.1 ∼ 0.5 s long with a frequency of ∼ 700Hz. These elements consistedof subelements or packets lasting 5 ∼ 10 ms, corresponding to about 3.5 to 7 waveperiods. κ ' 0.0016 approximately corresponds to Fig. 6.6b, which shows thatred particles can reach their median pitch-angle range (∼ 5◦ from Fig. 6.5b) in fivewave periods on average. For κ = 0.01 (Fig. 6.6b), this rate is even faster as the redparticles can reach their median pitch-angle range of ∼ 15◦ (Fig. 6.5c) in about twowave periods.6.6 DiscussionThe results presented here may help explain the the association of large-amplitudewhistler waves to relativistic microbursts (∼ 1MeV) [24] andmay explain the lack of97−0.4 −0.2ρz0.00.51.01.5ρ⟂(a) Two-Valley Predictions (ϕ= π/4)0.00.20.40.60.81.0−0.4 −0.2ρz0.00.51.01.5(b) App oximate P edictions (ϕ= π/4)0.00.20.40.60.81.0(0.4 (0.2ρz0.00.51.01.5(c) Δθ)pitch(ρ⟂ Δ ρz) (ϕ= π/4)2040(0.4 (0.2ρz0.00.51.01.5ρ⟂(d) Two-Valley Predictions (ϕ=−π/4)0.00.20.40.60.81.0−0.4 −0.2ρz0.00.51.01.5(e) App oximate P edictions (ϕ=(π/4)0.00.20.40.60.81.0(0.4 (0.2ρz0.00.51.01.5(f) Δθ)pitch(ρ⟂ Δ ρz) (ϕ=(π/4)10203040Figure 6.7: Same as Fig. 6.4, but for κ = 0.02.such energetic microbursts in small-amplitude chorus [154]. Particle energization isnot a subject of this Chapter and thuswill not be discussed; it will be assumed that theparticles are first energized by somemechanism that yields a relativistic distribution,and then the ensuing pitch-angle dynamics are studied in order to concentrate onone topic. It should be noted, however, that energization and pitch-angle scatteringmay occur simultaneously.In Fig. 6.3, for small amplitudes (0.0001 ≤ κ ≤ 0.001), only up to 0.5% of particlesin a distribution with a temperature of ∼ 1 MeV (corresponding to θ ' 2) interactwith the wave, whereas for large amplitudes (κ ' 0.01), ∼2% of such particles do.This is because the range of the two-valley condition in Eq. 6.16 scales with thewave amplitude κ; i.e., as the wave amplitude increases, more particles, includingenergetic particles, satisfy the two-valley condition.The interaction of large-amplitude waves with relativistic particles is further ex-plained in Fig. 6.7, which is the same as Fig. 6.4 but for a larger wave amplitude(κ = 0.02). It can clearly be seen that the predictions of large scattering in Fig. 6.7are much broader in phase space than those in Fig. 6.4. This is important becausein Fig. 6.4, relativistic particles with ρ & 1 must have large initial pitch-angles tointeract with the wave since the two-valley condition is a narrow range related to theexact resonance condition, and thus these particles must undergo extremely largepitch-angle scatterings in order to jump into the loss cone. However, in Fig. 6.7, therange for two-valleymotion ismuch increased, allowing for relativistic particles with98smaller initial pitch-angles to interact with the wave. The deviation of the two-valleycondition from the exact resonance condition is because the range in Eq. 6.16 scaleswith κ. Furthermore, the pitch-angle range itself is significantly increased in Fig.6.7. Therefore, a larger wave amplitude allows for relativistic particles with lowerinitial pitch-angles to interact with the wave, while simultaneously increasing theamount of pitch-angle scattering; these two effects lead to more relativistic particlesbeing pitch-angle scattered into the loss-cone.There are a few limitations to the present analysis that may be subject to futurework. First, the Maxwell-Jüttner distribution is a simplification and should not beconsidered as a distribution representing the entire electron population. The actualdistribution is a sum of these Maxwellians or other functions such as the kappadistribution [123]. If the actual distribution can be expressed as a weighted sumof Maxwelll-Jüttner distributions, then the total fraction of particles that undergotwo-valley motion is the sum of the partial fractions for each distribution. On theother hand, if the actual distribution is another sufficiently simple function, then ananalysis similar to that in Sections 6.3 and 6.4 may be conducted by replacing Eq.6.17 by the actual distribution.Second, the particle temperature is assumed for simplicity to be isotropic, whereasobservations indicate that electron temperature in the magnetosphere in general isanisotropic and electron distribution functions can be more complex than simpleanisotropic distributions [94]. The transition to an anisotropic Maxwell-Jüttnerdistribution is outlined in Livadiotis [98] and Treumann and Baumjohann [150].Third, the wave is assumed to have exact parallel propagation, whereas many in-stances of magnetospheric chorus involve oblique propagation [6, 134]. Also,chorus typically exhibits frequency and amplitude changes over a short time pe-riod [153], but the model presented here is based on a plane wave with a fixedfrequency and wavenumber (Eqs. 6.2 and 6.3). However, including obliquity andvariable frequency makes the analysis considerably more complicated and so wouldbe inappropriate for an inaugural analysis.6.7 Comparison to Second-order Trapping TheoryA popular theory describing wave-particle interactions is the second-order trappingeffect presented in, e.g., Sudan and Ott [144], Nunn [110] and Omura et al. [113].Omura et al. [115] and Omura et al. [116] present relativistic generalizations of thetheory. However, it will now be shown that this previous theory is an approximation99of the theory presented here; this approximation effectively misses the critical two-valley nature of the pseudo-potential.Omura et al. [113] use the following coupled equations for non-relativistic speeds:dζdt= k(vz − VR), (6.30)ddt(vz − VR) =ω2tk(sin ζ + S), (6.31)whereVR =ω −Ωk, (6.32)ζ is the angle between v⊥ and B̃, ωt =√kv⊥Ωκ is the trapping frequency, and Sis a parameter that is equal to zero when the background magnetic field is spatiallyuniform and ω is a constant. Therefore, setting S = 0, differentiating Eq. 6.31 intime, and using Eq. 6.30,d2dt2(vz − VR) =ω2tkcos ζdζdt, (6.33)= ω2t (vz − VR) cos ζ . (6.34)Letting γ = 1 in Eq. 6.5 and rearranging shows thatξ =kΩ(vz − VR) , (6.35)so Eq. 6.34 becomesd2ξdt2= ξω2t cos ζ, (6.36)= − ∂∂ξ(−ξ22ω2t cos ζ), (6.37)1Ω2d2ξdt2= −∂ χ(ξ)∂ξ, (6.38)where χ(ξ) = −ξ2ω2t cos ζ/2Ω2 is the pseudo-potential of this system.Now, let us examine the term involving sκ′ in Eq. 6.7 assuming γ0 = γT = 1;−sκ sin φ0ξ22= −αnκβ⊥0 sin φ0ξ22, (6.39)= −ωΩckωκv⊥0csin φ0ξ22, (6.40)= −ω2t0Ω2sin φ0ξ22, (6.41)= χ0(ξ), (6.42)100because ζ and φ are related by ζ = φ − π/2, so cos ζ = sin φ. χ0(ξ) is χ(ξ) exceptthat v⊥0 and φ0 are used instead of v⊥ and φ, and the relationship is similar for ωt0and ωt . Therefore, χ(ξ) results from keeping only the sκ′ term in ψ(ξ). This isimportant because χ(ξ) only describes either a trapping or a non-trapping potentialbut not a two-valley potential.Figure 6.1e and 6.1f plot the approximated pseudo-potentials χ(ξ) for the particlesin Fig. 6.1a and 6.1b, respectively. For both particles, χ(ξ) is clearly a one-valleypotential, whereas the unapproximated ψ(ξ) is two-valleyed for the particle in Fig.6.1a and thus it undergoes much larger pitch-angle scattering than the particle inFig. 6.1b . Therefore, if the theory in Omura et al. [113] were to be used, it would beimpossible to distinguish between the two particles which clearly have an extremelylarge difference in the amount of pitch-angle scattering.Another important problem with the second-order trapping theory is that the time-dependence of the variables is ambiguous at best. Omura et al. [113] imply that v⊥and thus ωt are time-dependent but then treat v⊥ as a constant when they state thatcombining Eqs. 6.30 and 6.31 gives a pendulum equation. Sudan and Ott [144]admit that v⊥ is time-dependent, but then argue that it can be treated as a constant,as specified in the sentence after their Eq. 10. In the present theory, however,the initial variables and the time-dependent ones are explicitly differentiated, so noapproximation regarding time-dependence needs to be made. This is an extremelyimportant point because this time-dependence of v⊥ gives the two-valley potentialwhereas treating it as a constant does not. This fact can be more explicitly illustratedby examining Eq. 26 in Bellan [9] which is an equation for the parallel velocity(recall that βz = vz/c and prime refers to the wave frame):1Ω′d2β′zdt′2= ξβ′⊥ ·B̃′⊥B0− β′zB̃′⊥B0·B̃′⊥B0. (6.43)The second-order trapping theory effectively drops the last term in Eq. 6.43 andignores the time-dependence of the first term on the right-hand side. This leads to1Ω′d2ξdt′2= αnγ′γTξβ′⊥ ·B̃′⊥B0, (6.44)which is equivalent to Eq. 6.38 if γ′ = γT = 1 is assumed. However, Eq. 35 ofBellan [9] states thatβ′⊥ ·B̃′⊥B0= β′⊥0 ·B̃′⊥0B0− γT2αnγ′(ξ2 − ξ20), (6.45)101which means that treating v⊥ as a constant effectively misses the ξ-dependence inEq. 6.45, which is the reason for the two-valley shape of the pseudo-potential.For example, neglecting the ξ20 term in Eq. 6.45 leads to erroneous conclusionsregarding the shape of the potential near ξ = 0. In Fig. 6.1e, χ(ξ) is a valleybecause −sκ′ sin φ0 is positive in this case. However, the correct pseudo-potentialψ(ξ) in Fig. 6.1a is a hill near ξ = 0 because −ξ20/2− sκ′ sin φ0 in Eq. 6.7 is negativein this case. Also, the ξ2 term in Eq. 6.45, which leads to the positive ξ4 term in Eq.6.7, prevents the pseudo-potential from diverging to −∞ as ξ → ±∞. This preventsthe particle ξ from veering off to infinity; this phenomenon is unphysically allowedif the approximated χ(ξ) is used and sin φ0 > 0. The term linear in ξ in Eq. 6.7which affects the asymmetry of the two-valleys is also neglected in χ(ξ). The factthat v⊥ is not constant can be explicitly seen in Fig. 5g of Bellan [9], where v⊥ of aparticle undergoing two-valley motion varies in time by over a factor of three.It should be noted, however, that for a non-uniform background field and/or time-dependent wave frequencies, S is finite in Eq. 6.31 and this may have an importantrole in the system additional to the effects described in the present chapter.6.8 SummaryThe interaction of a relativistically-consistent thermal distribution of particles witha coherent right-handed circularly polarized wave has been investigated. Departurefromwave-particle resonance for each particle is expressed by a frequencymismatchparameter ξ, where ξ = 0 represents perfect resonance. An exact rearrangement ofthe relativistic particle equation of motion shows that ξ follows pseudo-Hamiltoniandynamics with an associated pseudo-potential ψ (ξ). If ψ (ξ) has two-valleys sepa-rated by a hill, and the particle has enough pseudo-energy to overcome the hill, thenthe particle undergoes two-valley ξ-spacemotion that produces a large, non-diffusivepitch-angle scattering.An accurate condition for two-valley motion and thus for large pitch-angle scatteringhas been derived; this condition is related to but may or may not include the exactresonance condition (Eq. 6.1), and the range of this condition scales with thewave amplitude. Assuming that the particle distribution is Maxwell-Jüttner, whichis a relativistic generalization of the Maxwell-Boltzmann distribution, for typicalmagnetospheric parameters a significant fraction (1−5%) of the particles undergoestwo-valley motion. The pertinent analysis can potentially be used for the actual localelectron distribution, which may not be exactly Maxwellian. Numerical simulations102confirm the analytical results. The scaling of the fraction of interacting particleswith the wave amplitude may also explain the association of relativistic microburststo large-amplitude chorus. The present theory is more accurate and exact than thewidely-used second-order trapping theory as second-order trapping theory fails totake into account two-valley motion.103C h a p t e r 7MAGNETIC FIELD DIAGNOSTIC USING TWO-PHOTONDOPPLER-FREE LASER-INDUCED FLUORESCENCEAs can be seen in Chapters 2 through 5, fast magnetic reconnection involves ion skindepth length scales ∼ di as well as electron skin depth length scales ∼ de. Therefore,any experimental diagnostic that probes magnetic fields in fast reconnection mustbe able to resolve these scales. However, di and de in many situations are minutequantities compared to the global scale L of the plasma, as can be seen in Table7.1. This scale constraint as well as other limitations to existing magnetic fielddiagnostics impose the need for a new, alternative diagnostic.A widely-used magnetic field diagnostic is the Faraday pickup coil, which is simplya loop of an arbitrary number of turns made of a conducting wire. Because Faraday’slaw states that the voltage induced in the loop is proportional to the rate of change ofmagnetic flux, a Faraday pickup coil can change fluctuating magnetic fields into anelectrical signal. Because of the simplicity of their design and application, these coilsare frequently used to probe magnetic fields in a plasma [29, 67, 129]. However,when probing smaller length scales, the area through the loop must inevitablybecome smaller. This smaller area leads to smaller magnetic flux and thus a smallersignal-to-noise ratio, which limits the spatial dimensions of the coil. Anotherimportant limitation is that the coil, being a physical wire, essentially perturbs thePlasma Type L (m) n (m−3) di (m) de (m)Flaring Solar Loops 107 − 108 1017 1 10−2Solar Active Regions 107 − 108 1015 − 1016 1 − 10 10−2 − 10−1Fusion Plasmas 1 − 10 1021 10−2 10−4Earth’s Magnetotail 108 − 109 109 104 102Caltech MHD-driven Jet Ex-periment (Argon)10−1 1022 10−2 10−4Magnetic Reconnection Ex-periment (MRX)0.5 1020 10−1 10−3Table 7.1: Plasma parameters of regions where magnetic reconnection occurs or isthought to occur. L is the global length scale of the plasma, n is the density, and de isthe electron skin depth. Unless specified otherwise, di is the ion skin depth assumingthat the ion species is hydrogen. Referenced from Refs. [29, 74–76, 109, 127]104plasma; the effect of this perturbation obviously becomes larger as the region ofinterest becomes smaller. These constraints hinder the usage of pickup coils forinvestigating magnetic fields at small length scales.Another magnetic field diagnostic uses Faraday rotation, which is an effect wherethe plane of polarization of a linearly polarized light wave rotates when passingthrough a magnetic field in a medium. The amount of this rotation is proportionalto the strength of the magnetic field parallel to the light propagation direction, sothis method has also been widely used to probe magnetic fields [18, 140]. Thismethod induces little to no physical perturbation to the plasma and can probe smalllength scales because the size of the light source is arbitrary. However, becausethe light wave has to pass through a finite region of space for this effect to occur,only line-integrated data can be obtained, i.e., the magnetic field measurement isnot 3D-localized.In this chapter, a new magnetic field diagnostic based on the laser-induced fluores-cence (LIF) technique is presented. This diagnostic is essentially non-perturbing,3D-localized, and can probe arbitrary scale lengths limited only by the diameterof the laser source. The particular scheme suppresses Doppler broadening of thespectral lines, enabling the resolution of Zeeman splitting, which can then be used toinfer the magnetic field information. The theoretical background is first presented,followed by a description of a repetitively pulsed radio-frequency plasma sourcethat was constructed in order to test the diagnostic. Preliminary results are thenpresented.7.1 Sources of Spectral BroadeningSeveral textbook sources of spectral broadening will now be described; these effectscan be analyzed to yield various properties of the diagnosed plasma [38].Doppler BroadeningDoppler broadening of a spectral line arises due to the thermal motion of atoms. Itis first assumed that the distribution function f is a 1D Maxwellian in the directionof the laser beam, i.e,f (v)dv =√m2πkBTexp(− mv22kBT)dv, (7.1)where m is the atommass, v is the atom velocity in the direction of laser propagation,and T is the temperature. In the reference frame of an atom moving at v c where105c is the speed of light, the laser wavelength is Doppler shifted byλ = λ0(1 +vc), (7.2)where λ0 is laser wavelength in the lab frame, orv = c(λ − λ0λ0), (7.3)dv =cλ0dλ. (7.4)Inserting these into Eq. 7.1, we havef (λ)dλ =√mc22πkBTλ20exp(− mc22kBTλ20[λ − λ0]2), (7.5)which has the standard deviationσλ =√kBTmc2λ0. (7.6)This effect is called the Doppler broadening effect. Note that if the atoms are movingat a finite average velocity, a Doppler shift in the center wavelength occurs as well.Zeeman SplittingWhen an atom is subject to an external magnetic field, the interaction between thetwo effectively splits any energy level to multiple levels. One transition line betweentwo energy levels then splits into multiple transition lines. The energy perturbationis∆E = µBg jm j |B| , (7.7)where µB = e~/2me is the Bohr magneton, m j is the total angular momentum inthe z-direction, B is the external magnetic field, g j is the Landé g-factor which is afunction of the total angular momentum j, the orbital angular momentum l, and thespin angular momentum s. One energy level therefore splits into multiple levels,each corresponding to certain possible values of the quantum numbers. Since theamount of the splitting depends on the magnitude of B, this amount can be used toinfer the strength of the magnetic field.Natural BroadeningNatural broadening inevitably arises due to the Heisenberg uncertainty principle,which effectively relates the lifetime of a particular energy level to the uncertaintyof its energy value. The broadening is thus∆E ' ~τ, (7.8)106where τ is the lifetime of the energy level. The broadening profile is Lorentzian.Stark BroadeningStark broadening is analogous to the Zeeman effect in that the electric field insteadof the magnetic field perturbs the energy levels. Stark broadening is significant forhigh plasma densities. For hydrogen lines, Stark broadening scales as ∼ n2/3e wherene is the electron density, whereas for non-hydrogen lines it scales as ∼ ne [173].The broadening profile is Lorentzian.Instrumental BroadeningInstrumental broadening occurs due to limitations of the measurement devices.For example, imperfect focusing inside a spectrometer physically diffuses light,effectively leading to a broadened spectral measurement.Saturation BroadeningSaturation broadening, or power broadening, occurs when the laser intensity is sohigh that the population of the base level becomes equal to the excited level. Atthis saturated point, no further absorption of the laser by the plasma can occur. Aspectral absorption probability typically peaks at the center wavelength and is lowerat the wings of the profile. Therefore, if the laser intensity is above a threshold, theplasma can no longer absorb the photons at the center wavelength but can absorbthose further away from the center wavelength. This effectively leads to a broadenedline profile with the width [38]γs = w√1 +B12ρ(ω12)A12, (7.9)where w is the original profile width, A12 and B12 are respectively Einstein coef-ficients for spontaneous and induced transition between levels 1 and 2, and ρ(ω12)is the spectral energy density of the laser at the transition frequency ω12. Thebroadening profile is Lorentzian.Comparison of Broadening EffectsThe measured spectral profile is a convolution of all the broadening effects. De-pending on the plasma parameters, different effects contribute in different amounts.For example, Table 7.2 shows the contribution of different broadening effects onthe 394.539 nm (vacuum; 4D◦5/2 →4 D7/2) Ar II line for two different plasmas.The Caltech MHD-driven plasma jet [109] has a relatively high density and tem-107LaserFluorescenceBase LevelExcitation LevelDe-excitation LevelFigure 7.1: A diagram of a laser-induced fluorescence scheme.perature, so Stark and Doppler effects dominate over Zeeman splitting. The pulsed,inductively-coupled radio-frequency plasma source developed by Chaplin [32] has alower density and temperature, and so Doppler broadening is comparable to Zeemansplitting while Stark broadening is negligible. Natural broadening (not shown) isnegligible for both cases.7.2 Two-Photon Doppler-Free Laser-Induced FluorescenceLaser-Induced FluorescenceLaser-induced fluorescence, or LIF, is an active spectroscopy method in which alaser source excites the electron population at a certain energy level to a higherlevel [39]. These electrons then de-excite to other lower energy levels, emittingfluorescent signals that can be analyzed to obtain various information such as thetemperature, density, and magnetic field of the diagnosed plasma. Figure 7.1graphically illustrates the scheme.ne (m−3) Ti (eV) Stark (pm) Doppler (pm) Zeeman (pm)Caltech JetExperiment1022 2 2.2 3 0.7Inductively-CoupledPlasma1019 0.05 2.2 × 10−3 0.5 0.7Table 7.2: Comparison of different broadening effects in two types of plasmas:the Caltech MHD-driven Jet Experiment [29, 105, 109], and a pulsed, inductively-coupled, radio-frequency plasma [32]. Stark broadening is calculated by using theexperimentally measured values in Konjević et al. [83] and using the scaling ∝ nefor Argon [173]. Zeeman splitting is calculated by assuming m j and g j are of order1, and B = 0.1 T.108vLab FrameAtom Frame(1+v/c) (1-v/c)Figure 7.2: Two-photon interaction of an atom in the lab frame and in the atomframe.Because the absorption of the laser is dependent on the laser wavelength, the popula-tion of the excited level and thus the fluorescence intensity is wavelength-dependent.Therefore, the full spectrum can be obtained by scanning the laserwavelength aroundthe spectral line.The fluorescence intensity depends on the population of the excited level, which inturn depends on the population of the base level and the laser intensity. Therefore, ahighly populated base level needs to be chosen in order to maximize the fluorescencesignal. An obvious choice is the ground state of the atom, but the transition energybetween the ground state and the first excited state is typically too high for anysimple, conventional laser. Therefore, a metastable level is typically chosen. Bydefinition, there are no allowed first-order transitions to a lower energy level frommetastable levels, so these levels have relatively long lifetimes and so are highlypopulated.Two-Photon LIFAs in can be seen in Table 7.2, Doppler broadening can dominate over Zeemansplitting so that the latter is unresolvable. Therefore, in order to resolve Zeemansplitting and thus obtain the magnetic field information, Doppler broadening mustbe eliminated. Two-photon laser-induced fluorescence achieves this feat by usingtwo counter-propagating laser beams with equal wavelengths [39], as in Fig. 7.2.In the lab frame, the atom is moving with a velocity v and is subject to two counter-propagating photons of wavelength λ. Then, in the frame of the atom, one of thephotons is blue-shifted while the other is red-shifted. If both photons are absorbed1094D7/24D5/2o2P3/2394.539 nm393.430 nm393.984 nm442.724 nm4P3/22D5/2o410.499 nm123542'3d4p5s4p4sFigure 7.3: Two-photon laser-induced fluorescence scheme for Argon II. Displayedwavelengths are vacuum values.by the atom, then the absorbed energy is, assuming v c,E =cλ(1 + v/c) +cλ(1 − v/c), (7.10)' cλ(1 − vc+ 1 +vc), (7.11)=2cλ, (7.12)which does not depend on velocity. Therefore, Doppler broadening is cancelled outin this scheme.Figure 7.3 shows the proposed two-photon LIF scheme in Argon II. Two counter-propagating laser beams at 393.984 nm pumps the metastable 3d4D7/2 level (level1) to the 5s2P3/2 level (level 3) via a virtual level (level 2′). The intermediate level(level 2) is not necessary for two-photon transition but allows it to be achieved withrelatively low-power lasers; if the intermediate level does not exist, then high-powerlasers are necessary. This fact can be seen by examining the two-photon transitionprobability A13 [39]:A13 ∝γ13I1I2[ω13 − ω1 − ω2 − v · (k1 + k2)]2 + (γ13/2)× ↔D12 · ê1 · ↔D23 · ê2ω12 − ω1 − v · k1 +↔D12 · ê2 ·↔D23 · ê1ω12 − ω2 − v · k22, (7.13)where ωi, ki, êi and Ii are respectively the angular frequency, the wavevector, thepolarization vector, and the intensity of the i-th laser beam, and ωi j , γi j , and↔Di j110Figure 7.4: Example of two-photon LIF in Rubidium. (a) The original Doppler-broadened spectrum. (b) Result after applying two photons. (c),(d),(e),(f) Zoom-insof respective peaks, showing hyperfine splittings. Reproduced from Jacques et al.[77]. © European Physical Society. Reproduced by permission of IOP Publishing.All rights reserved.are the transition angular frequency, the natural linewidth, and the dipole matrixelement [38] between energy levels i and j. It can be seen that the second factorincreases as ω1 and ω2 get closer to ω12.Two-Photon LIF in RubidiumThere is a well-known two-photon LIF scheme using neutral Rubidium vapors[60, 77, 112]. In particular, Jacques et al. [77] provides a measurement of Zeemansplitting of the hyperfine structure of Rubidium.Figure 7.4 shows the result of applying two-photon LIF in Rubidium. The Doppler-broadened spectrum is significantly narrowed after application of two-photon LIF,and even hyperfine splittings are resolvable. A Zeeman splitting of the line in Fig.7.4f is shown in Fig. 7.5. The x-axes in Figs. 7.4 and 7.5 are in relative units, i.e.,how far away the laser frequency is from the center value.111Figure 7.5: Two-photon LIF observation of Zeeman splitting of a Rubidium hy-perfine line. Reproduced from Jacques et al. [77]. © European Physical Society.Reproduced by permission of IOP Publishing. All rights reserved.7.3 Repetitively Pulsed, High-Power, Inductively-Coupled Radio-FrequencyPlasma SourceA new plasma source was designed and built as the subject of diagnosis of theproposed two-photon LIF diagnostic. This source is based on the battery-poweredpulsed inductively-coupled radio-frequency (RF) plasma source previously devel-oped by Chaplin [32]. The prior source uses a battery rack to charge a capacitor toa high voltage, which is then discharged through a homemade RF generator circuitto deliver an electrical power of 2.7 kW to the load. This power creates a plasmaof a relatively high density (∼ 1019 m−3) which can last up to about a millisecond.However, this source can be pulsed only once every about 5 seconds, which corre-sponds to a 0.02% duty cycle. Also, it cannot last for more than about 20 hoursbefore the batteries need to be replaced. Therefore, it is not suitable for experimentsthat possibly require long averaging times such as LIF experiments.The new source uses wall power instead of a battery rack and so can be left onindefinitely. The circuit was designed so that the capacitor can be repetitivelypulsed at varying frequencies depending on the pulse length. The final design wassuch that the source was pulsed for 250 µs at 60 Hz, corresponding to a duty cycleof about 1.5%. Figure 7.6 shows a schematic diagram of the RF plasma source.Each of the components comprising the source will now be described in detail.112DRF130127.12MHz OscillatorI VRF OUT+5V+15V13.56MHzRF Driver CircuitI LimiterVARIACTransformer0-130VVoltagex2TransformerFuseBridgeRectifierDC Power SupplyAntenna+PlasmaOpticalGateOpticalConverterTimingCircuitIVProbe47uFTransformerFigure 7.6: A schematic diagram of the RF plasma source.RF GeneratorAcommercial power supply converts wall power to +5V and +15VDCoutputs. The+5 V output is used to power the crystal oscillator which provides the fundamentalfrequency of 27.12MHz. The +15 V output is used to power aMicrosemi DRF1301powerMOSFET hybrid chip containing two powerMOSFETs. A current limiter wasput in place to prevent the chip fromdrawing excessive current from the power supply.Without this current limiter, the power supply would heat up to extremely hightemperatures (> 100 C◦). The two power MOSFETs inside DRF1301 effectively actas periodic grounds that alternate 180◦ out-of-phase with each other at 13.56 MHz.Although not shown in Fig 7.6, the +15 V output is also fed through a DC step-downconverter to power a cooling fan at +12 V.Pulsed Power DriverA VARIAC (variable AC) transformer rated for 2 kW and 20A converts wall powerto AC voltages between 0-130 V. This voltage is then doubled through anothertransformer. A diode bridge rectifier then flips the negative values of the AC voltageto positive values (sinωt → |sinωt |). The output of the rectifier is used to charge a47 µF capacitor rated for 300 V, and because the rectifier does not allow backwardcurrent, it can only charge, not discharge, the capacitor. Therefore, the capacitoris charged to a voltage of√2 times the VARIAC transformer voltage — recall thatAC voltages are rms values — in about 4 ms, which is a quarter of the wall powerperiod.The gate for the circuit is made optical in order to isolate it from the high-powercomponents. When the gate is open, the main 47 µF capacitor is discharged through113one of the two MOSFETs in an alternating manner. This oscillating discharge ispicked up by an on-board transformer made out of a ferrite and wound-up wires.The output of this transformer is the final output of the high-power RF generator.IV ProbeAn IV probe measures the current and voltage outputs of the RF generator. Thecurrent was measured by an Ion Physics CM-10-M current monitor. The voltagewas measured by a 10x or a 1000x voltage probe, depending on the voltage level ofthe main capacitor.Because the current probe and the voltage probe use different measurement leads,an instrumental phase delay is inevitable. This delay, however, can be measured byreplacing the impedance matcher and the antenna by a resistor and then probing thecurrent and voltage profiles across it. Because the impedance of a resistor is purelyreal, there should theoretically be no phase delay between I and V. Therefore, anyphase delay across the resistor measured by the IV probe can be considered to beinstrumental and thus can be subtracted out from future measurements.Impedance MatcherIn order for maximal power to be transferred to the load, the load impedance mustbe equal to the the complex conjugate of the source impedance [32]. Because thesource impedance is unknown in the homemade RF generator, the load impedancemust be adjusted. Instead of expensive commercial impedance matchers, two binaryarrays of ceramic capacitors serve as variable parallel and serial capacitances — Cpand Cs — constituting the impedance matcher. The binary arrays could be adjustedby using copper jumpers to "activate" each of the ceramic capacitors with values of1 pF, 2 pF, 4 pF, and so on, providing nearly any capacitance value up to Cp = 2604pF and Cs = 886 pF with 1 pF resolution.Antenna and PlasmaAhalf-turn helical (HTH) [139] antenna design is used. Other types of antennas suchas Nagoya type III antennas [34] were tested as well, but the HTH antenna provedto be the most efficient in terms of plasma generation. The antenna wraps around aglass tube fed into a vacuum chamber and determines the spatial dimensions of theplasma. Its radius is about 1.27 cm, and its length is about 7.5 cm.114Figure 7.7: The resultant plasma discharge.Timing CircuitThe timing circuit consists of a trigger generator that determines the frequency ofthe pulse, which is fed into a pulse generator that determines the gate length of thepulse. The gate is then converted to an optical signal and fed into the RF driver.An example of the resultant plasma discharge is shown in Fig. 7.7.7.4 Experimental SetupFigure 7.8 shows the basic experimental setup for two-photon laser-induced fluores-cence.Laser Excitation SystemA tunable diode laser (Sacher Lasertechnik LYNX Littrow Cavity) is controlled bya laser controller, which is in turn controlled by a personal computer (PC). The laserwavelength is controlled by the laser current, temperature, and the piezo voltage,but the piezo voltage was mainly used to scan the wavelength during the mainmeasurements. The other parameters were varied to stabilize the laser during thescans. The diode laser produces a laser beam with a maximum power of 30 mWand a diameter of a few mm. The beam is fed through an acousto-optic modulator(AOM), which deflects the beam if an RF signal is applied. The deflected beam isfocused into a multimode optical fiber and is collimated at the output. Therefore, byproviding a TTL signal that turns the RF signal on and off at the lock-in frequencyflock-in via the lock-in amplifier, the output laser beam is modulated (turned on andoff) at flock-in.A 30:70 beamsplitter directs 30%of the output beam into awavemeter forwavelength115Diode Laserw/ OILock-in AmplifierAOMRFGeneratorTTL SlitOptical FiberL1M1CollimatorM2BSWavemeterL2FilterL3PMTPCw/ Labview ProgramLock-inSignalLaserControllerWavemeter Switch(For Two Lasers)Figure 7.8: Experimental setup for two-photon laser-induced fluorescence. L standsfor lens, M for mirror, BS for beam-splitter, AOM for acousto-optic modulator, OIfor optical isolator, and PMT for photo-multiplier tube. The purple color is theend-on view of the plasma.reference. The rest of the beam is directed through the plasma, bounces off of amirror (M2), and is redirected back into the plasma, providing counter-propagatingphotons.Fluorescence Collection SystemThe laser-induced fluorescence from a single point in the plasma passes througha lens (L2) and is made into a collimated beam, which is in turn passed througha band-pass optical filter of the desired wavelength. The beam is then focusedby another lens (L3) into a battery-operated photo-multiplier tube (PMT; HamatsuH10721-110). In order to maximize the detection of the signal, it is imperative touse a lens to collimate the fluorescence first and then pass it through the filter. Thisis because the wavelength band of the filter is usually rated at a 90◦ incident angleand thus is rated for a different wavelength band if the incident angle is different.Because the laser beam is modulated at flock-in, the signal is also modulated atflock-in, provided that the fluorescence rise and decay times are not long compared116to Tlock-in = 1/ flock-in. Thus, the PMT signal is fed into the input of the lock-inamplifier for averaging. The output of the lock-in amplifier is transferred to the PCvia a data acquisition system (National Instruments USB-6001).Wavelength Feedback and Scanning SystemA LabVIEW program was written that scans the laser wavelength around a centervalue at set intervals and collects lock-in data while locking the laser wavelengthat each set value. The wavelength locking is performed by feeding the real-timewavemeter measurement back to the laser piezo voltage via the PC. The user inputsthe center wavelength, the scan range, the number of data points, and the averagingtime, and these are translated into an array of wavelength values for which the datais to be taken. Then, the laser wavelength is sequentially locked at each wavelengthvalue for the duration of the averaging time, and the data from the lock-in is recordedat the end of each averaging time.For example, let us assume that the center wavelength is 394.538 nm, the scan rangeis±4 pm, the number of data points is 200, and the averaging time is 15 seconds. Thescan interval is thus 8/200 = 0.04 pm. So, the program first locks laser wavelengthat 394.534 nm for 15 s, records the value of the lock-in output, and then movesthe laser wavelength to 394.53404 nm. The process then continues until the laserwavelength reaches 394.542 nm.Two-Photon LIF with Different ColorsThe setup can be altered so that two separate counter-propagating laser beams arerespectively tuned to the transition wavelengths of the first (level 1 → 2) and thesecond (level 2→ 3) transitions. In this case, a second laser + AOM + fiber systemis placed at the location of mirror M2 and the laser beam is directed downwards.The beamsplitter directs 30% of the beam into another optical fiber. The two opticalfibers that respectively measure the wavelength of the first and the second laser arethen fed into a wavemeter switch — a device that allows a single wavemeter tomeasure multiple wavelengths simultaneously.7.5 One-Photon LIF ResultsIn the one-laser setup, the laser wavelength was first scanned around 394.539 nm,which is the transition wavelength of the first transition (level 1 → 2 in Fig. 7.3),and the fluorescence signal of the 442.6 nm line (level 2 → 4 in Fig. 7.3) wasmeasured for different background Argon pressures. This allowed for the testing117of the alignment and the overall setup because the expected signal is theoreticallyhigher for one-photon transition.Figure 7.9 shows the LIF profiles for background Argon pressures of 1, 5, and 20mTorr, respectively. Red lines are fitted Voigt profiles, which are a convolution ofGaussian and Lorentzian profiles; i.e.,V(λ;σ, γ) =∫ ∞−∞G(λ′;σ)L(λ − λ′, γ)dλ′, (7.14)whereG(λ;σ) = 1√2πσ2exp[−(λ − λ0)22σ2], (7.15)andL(λ; γ) = γπ[(λ − λ0)2 + γ2]. (7.16)Here, γ is the Lorentzian width, σ is the Gaussian width, and λ0 is the centerwavelength.For background pressures of 5 and 20 mTorr, the broadening is mostly Gaussian.Assuming that this Gaussian profile is entirely due to Doppler broadening, the iontemperatures are respectively 1230±186Kand 756±189K.Because the input poweris fixed, the more ions there are (i.e., higher density), the less power is distributed toindividual ions. Therefore, the temperature decreases as the background pressureincreases. However, it can be seen that the temperature at 1 mTorr is lower thanthat at 5 mTorr. This is because the ion-electron collision frequency becomes loweras the density decreases. Since the RF power is primarily transferred to the muchlighter electrons, if the number of ion-electron collisions decreases, then electronstransfer less energy to ions.The fact that the turnaround in the ion temperature is observed can actually be used toestimate the electron (and thus the ion) density. Ion-electron collisions will becomenegligible when the mean-free-path lmfp is comparable to the plasma length lplasma,which is about 5 cm. From the measured amount of Ar II, the electron temperatureis estimated to be around Te = 3 eV. The highest electron density at which collisionsare negligible can therefore be estimated to bene =1σlplasma, (7.17)=2πlplasma(2ε0Tee)2 1lnΛ, (7.18)= 3 × 1018 m−3, (7.19)118394.536 394.538 394.540 394.542−4−3−2−1Fluorescence (Arb.)Ti = 428±259 Kσ = 0.391±0.119 pmγ = 0.769±0.134 pmλ0 = 394.53890±0.00002 nm(a) 1 mTorr (Scan No. 75)394.536 394.538 394.540 394.542−4−3−2−1Fl orescence (Arb.)Ti = 1230±186 Kσ = 0.663±0.050 pmγ = 0.000±0.102 pmλ0 = 394.53891±0.00001 nm(b) 5 mTorr (Scan No. 77)394.536 394.538 394.540 394.542λ (nm)−4−3−2−1Fl orescence (Arb.)Ti = 756±189 Kσ = 0.520±0.065 pmγ = 0.099±0.128 pmλ0 = 394.53885±0.00001 nm(c) 20 mTorr (Scan No. 78)Figure 7.9: One-photon LIF results for 1, 5, and 20 mTorr. The laser wavelengthis scanned around the 394.5 nm line in Fig. 7.3 and the plots show the subsequent442.7 nm fluorescence as a function of the laser wavelength. Red lines are fittedVoigt profiles. γ is the Lorentzian width, σ is the Gaussian width, Ti is the iontemperature calculated from σ, and λ0 is the center wavelength.119where σ is the collision cross-section, lnΛ is a constant of order 10, and Te is inunits of eV. Above this density, collisions start becoming significant, so electronsstart imparting their energies to ions. Previously measured electron densities in asimilar setup yielded ∼ 1019 m−3 at ∼ 10 mTorr, with which the present estimateagrees well.The line broadening at 1 mTorr is dominated by a Lorentzian profile. Because theStark width and the natural width for the given parameters are both ∼ 2 fm andso are negligible, it can be concluded that the broadening comes from saturationbroadening. This makes sense because if the density is lower, there is a smallernumber of atoms that can be excited, so the laser intensity threshold for populationsaturation becomes lower.The measured saturation broadening can be used to calculate the laser bandwidth— the spectral uncertainty of the laser wavelength. The Einstein coefficients arerelated byB12 =λ30 A128πh, (7.20)where λ0 is the transition wavelength and h is Planck’s constant. The spectraldensity is related to the laser intensity byI =∫ρ(ν)cdν ' ρc∆ν, (7.21)where I is the laser intensity, ν is the laser frequency, and ∆ν is the laser bandwidth.Thus, we haveB12ρA12=λ30 I8πhc∆ν. (7.22)Using Eq. 7.9, we have∆ν =λ30 I8πhc1γ2s /w2 − 1. (7.23)In Fig. 7.9a, the total broadened of profile is γs =√γ2 + σ2 = 0.863 ± 0.173,and w ' σ because other sources of Lorentzian broadening, namely Stark andnatural broadening, are negligible. Using the values λ0 = 394.539 nm, I = 10mW/(π × 1 mm2) because the laser power at the output of the collimator is ∼ 10mW and the beam radius is ∼ 1 mm, we have ∆ν = 10.1 ± 9.7 MHz. Although thepropagated error is very large, themean value indicates a laserwavelength bandwidthof ∆λ = 5.25 fm, which is very small compared to other dominant effects such asDoppler broadening.120393.98358 393.98360 393.98362 393.98364 393.98366 393.98368λ (nm)Figure 7.10: The expected locations of the Zeeman splitted transition lines involvingthe 1→ 3 transition in Fig. 7.3 for |B| = 50G. The red line is the original un-splittedline.7.6 Two-Photon LIF Preliminary ResultsIn order to observe the two-photon transition, the laser wavelength was shifted tothe average of the transition wavelengths λ12 = 394.539 nm and λ23 = 393.430 nmin Fig. 7.3, which is 393.984 nm. The 442.7 nm optical filter that was used in one-photon LIF was replaced by a 410.5 nm filter in order to measure the fluorescencefrom level 3 to level 5 in Fig. 7.3. A permanent magnet was also placed near theplasma in order to provide a source of magnetic field. The presence of the magneticfield increased the ionization fraction of the plasma source. This could be seen bothvisibly as the color of the plasma turned bluer (a large fraction of Ar II spectrallines are around 400 nm) and measurably as the natural emission intensity of the410.499 nm line increased. The explanation is that the magnetic field traps theionizing electrons for a longer period of time compared to a situation where there isno magnetic field, so they impart more energy to the ions.The strength of themagnetic field at the location of the collection point wasmeasuredby a Hall effect sensor to be around 50 G. Figure 7.10 shows the expected locationsof the Zeeman splitted transition lines from level 1 to level 3 for |B| = 50 G. Thedistance between each line scales linearly with the strength of the magnetic field.Because the selection rule for one-photon Zeeman transition is mJ = 0,±1, that fortwo-photon Zeeman transition is mJ = 0,±1,±2.Figure 7.11 shows the two-photon LIF results of two separate scan instances at1 mTorr. The red line is the theoretical center transition wavelength, and the121393.982 393.984 393.9867.06.56.05.55.04.5Fluorescence (Arb.)3.54.04.55.0Scan No. 82 (blue) and 83 (green)Figure 7.11: Two-photon LIF results of two separate scan instances at 1 mTorr.The red line is the theoretical center transition wavelength, and the grey area is theexpected Zeeman splitting for |B| = 50 G.grey area is the expected Zeeman splitting for |B| = 50 G. Although saturationbroadening (∼0.8 pm) dominates over Zeeman splitting (∼0.04 pm) at this pressure,the former is in fact favorable when trying to find any signal. This is becausethe measured center wavelength may be shifted from the theoretical result due to,e.g., sources of systematic error such as the uncertainty of the wavemeter. Thisshift was actually observed in the one-photon LIF results (Fig. 7.9), where themeasured center wavelength (∼394.5389 nm) is different from the theoretical value(394.5387 nm). A simple analysis of the refractive index of the plasma shows thatthe predicted fractional wavelength shift due to finite plasma density is 7 × 10−10,which is negligible compared to other shift effects. Significant broadening allowsthe detection of a signal even when the laser wavelength is reasonably far from thecenter value.It can be seen from Fig. 7.11 the measured peak is shifted to the left, and that thereis broadening comparable to that in Fig. 7.9a. Line fitting the measurements toVoigt profiles yielded too big of an error to warrant any analysis. Nevertheless, thepeak near 393.9836 nm is evident from the two measurements.1227.7 Suggestions for Further WorkDue to the COVID-19 pandemic in the year 2020 and subsequent restricted accessto the laboratory, no further experiments could be done to improve upon the resultsthat have been presented so far. Therefore, some suggestions for further work willnow be given that may ultimately lead to the completion of this diagnostic.First, the center wavelength of the scan must first be shifted to that revealed by themeasurements in Fig. 7.11. This allows for the measurement of the complete lineprofile.Second, once it is confirmed that the peaks shown in Fig. 7.11 are indeed two-photon LIF signals with saturation broadening, the background gas pressure canthen be increased to reduce the broadening. Laser power may be reduced as well,but this will compromise the signal-to-noise ratio of the signal. The elimination ofthe saturation broadening will reveal sharp two-photon peaks as in Fig. 7.4.Third, optimization and tests of the signal should be conducted. The amount ofZeeman splitting can be altered by changing the proximity of the magnet, or simplyby using a different source of magnetic field such as solenoids. Two-color LIF canalso be conducted to increase the signal. The polarization of the laser beam mayalso be controlled to alter the selection rules.7.8 SummaryIn this chapter, a new laser magnetic field diagnostic was conceived and preliminaryresults were presented. The method uses the two-photon laser-induced fluorescence(LIF) technique, which allows for the elimination of Doppler broadening arisingfrom finite temperature. The riddance of the dominant broadening effect enablesthe resolution of Zeeman splitting, which can be used in turn to infer the magneticfield information.A repetitively-pulsed, inductively-coupled high-power RF plasma source was newlydeveloped as the subject of diagnosis. This source improved upon a previously builtbattery-powered source and is wall-powered.One-photon LIF results revealed various properties of the plasma as well as thelaser. The temperature of the plasma was measured via Doppler broadening to bearound 103 K, and this value was dependent on the background gas pressure. Thelaser bandwidth was measured via saturation broadening to be around 10 MHz.Finally, Preliminary two-photon LIF results were presented. Initial results indicate123that there is two-photon excitation around the expected wavelength. Althoughfurther developments were thwarted by the COVID-19 pandemic, suggestions forimprovements on this diagnostic were presented.124C h a p t e r 8SUMMARYThe progression, properties, and progenies of magnetic reconnection were studied,and a method of probing the magnetic field in a plasma was partially developed inthis thesis.8.1 ProgressionCanonical Vorticity FrameworkAn intuitive framework for explaining the progression of magnetic reconnection wasfirst presented. In order to describe various facets of the process, the frameworkutilizes a quantity called the electron canonical vorticityQe, which is perfectly frozeninto the electron flowwhile the magnetic fieldB is not frozen-in and thus reconnects.A careful analysis of the dynamics of a flux tube defined by Qe intuitively explainswhy magnetic reconnection is an instability, how electrons are accelerated, and howwhistler waves are generated. A two-fluid simulation was built to verify the analysis.When a slight perturbation is applied to a magnetic shear profile, a localized sectionof a Qe flux tube is subject to a stronger out-of-plane electron flow at the centerthan other parts of the tube. Because the flux tube convects with the electron flow,the localized section is pulled in the out-of-plane direction, in such a way thatgenerates more electron flow toward the center. The process then feeds back toitself, generating an instability.As the instability progresses and the flux tube is pulled in the out-of-plane direction, itis also lengthened and narrowed in order to conserve its volume, just like a stretchingrubber band. This decrease in the circumference of the flux tube, combined withmagnetic field cancellation due to magnetic reconnection, increases the electronflow velocity, i.e., accelerates electrons.Finally, the spatial gradient in the electron flow twists up the Qe, and this twist ispropagated as whistler waves. This is analogous to the generation mechanism oftorsional Alfvén waves.125Completion of the FrameworkBy including the "canonical battery" effect, the canonical vorticity framework ofmagnetic reconnection was completed down to first principles. Assuming that col-lisions are ignored, this battery effect is the only non-ideal term that relaxes thetopological constraint on Qe. Different kinetic effects such as electron viscosity andpressure anisotropy are manifested by the canonical battery term and significantlyaffect the reconnection process. The distinction between a Sweet-Parker-type dif-fusion process and a purely growing instability could be made by the significanceof the canonical battery term. Two-fluid and particle-in-cell simulations aided theanalysis.8.2 PropertiesStochastic Ion Heating in Magnetic ReconnectionOut of the various observed properties of magnetic reconnection, its accompani-ment by anomalous ion heating much faster than conventional heating has been alongstanding dilemma.Stochastic heating — a mechanism in which particles subject to sufficiently strongelectric fields undergo chaotic trajectories — is fast enough to account for theobserved heating rate. Using the established canonical vorticity framework, theHall electric fields that develop during reconnection could be calculated. It wasthen shown that these fields satisfy the stochastic ion heating criterion, and that theions involved indeed undergo chaotic trajectories. This results in stochastic motionin phase space, which is equivalent to heating. Single-particle simulations wereconducted to verify stochastic motion.Kinetic Verification of Stochastic Ion HeatingThe stochastic ion heating mechanism duringmagnetic reconnection was then kinet-ically verified through analytical calculations and particle-in-cell simulations. First,it was shown that stochastic ion heating is intrinsic to the Harris equilibrium currentsheet, which is profoundly related to magnetic reconnection. Then, it was veri-fied via particle-in-cell simulations that the Hall electric fields that develop duringmagnetic reconnection indeed satisfy the stochastic ion heating criterion even in thekinetic regime, and that the ion heating profile agreed well with the prediction fromthe criterion. The mechanism was shown to be valid up to moderate guide fields,and frequently observed features of ion heating were also observed to be consistentwith stochastic ion heating.126It was thus confirmed that stochastic ion heating is an intrinsic ion heating mecha-nism in magnetic reconnection.8.3 ProgeniesA very important progeny of magnetic reconnection is whistler waves, the genera-tion mechanism of which had already been discussed using the canonical vorticityframework. The interaction of whistler waves with relativistic particles, specificallyhow these waves scatter the pitch-angle of the particles, was examined.A previous study showed that exact rearrangements of the particle equation ofmotionunder a circularly polarized wave leads to a pseudo-Hamiltonian equation of motionof a frequency mismatch parameter ξ. If the pseudo-potential of this motion is two-valleyed, and the particle has enough pseudo-energy to undergo periodic motionin these two valleys, then the particle was shown to undergo drastic pitch-anglescattering.Here, the conditions were derived for which the particle undergoes two-valley mo-tion and thus undergoes drastic scattering. The conditions are a certain range relatedto the exact resonance condition but may or may not include this condition depend-ing on the initial conditions of the particle. It was then shown that if the particledistribution is assumed to be the Maxwell-Jüttner distribution, which the relativis-tic generalization of the Maxwell-Boltzmann distribution, a significant fraction ofthe particles undergo large scattering for typical magnetospheric parameters. Thescaling of this fraction with the wave amplitude suggests that relativistic microburstevents may be explained by the two-valley mechanism. The widely-used second-order trapping theory is shown to be an approximation of the theory presented hereand actually misses critical details of the wave-particle interaction.8.4 ProbeA new magnetic field diagnostic using two-photon Doppler-free laser-induced flu-orescence was described and developed to some extent. By using two counter-propagating laser beams, Doppler broadening of a spectral line can be cancelledout, enabling the resolution of Zeeman splitting. This splitting can then be used toextract the magnetic field information.A repetitively-pulsed, inductively-coupled high-power RF plasma source was newlydeveloped as the subject of diagnosis. One-photon LIF scans of this plasma showedthat the temperature of the plasma was around 103 K. The laser bandwidth was also127measured via saturation broadening to be around 10 MHz.Preliminary two-photon LIF results were presented. Initial results indicate thatthere is two-photon excitation around the expected wavelength. 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It will beassumed that all linear quantities have the dependence ∼ eik·x−iωt . The linearizeddimensionless equations are, under uniform B0 = ẑ,∇ × E = −∂B∂t, (A.1)∇ × B = −ue + Iδ (r − a) δ (z) e−iΩt φ̂, (A.2)∂ue∂t= −E − ue × ẑ, (A.3)where Iδ (r − a) δ (z) e−iΩt φ̂ is the ring current source with frequencyΩ, normalizedsource strength I, and radius a. Also, Due/Dt = ∂ue/∂t + ue · ∇ue ' ∂ue/∂tbecause ue is a linear quantity. Because the background magnetic field and the ringcurrent source has no dependence on φ, there is cylindrical symmetry; therefore,the wavevector k = kr r̂ + kz ẑ, i.e., kφ = 0, and k2 = k2r + k2z . Fourier analyzing inspace, the source term S (r) = δ (r − a) δ (z) turns intoS (k) =∫ ∞r=0∫ 2πφ=0∫ ∞z=−∞δ (r − a) δ (z) e−ikrr cos(kφ−φ)−ikz zrdrdφdz, (A.4)=∫ ∞r=0∫ 2πφ=0δ (r − a) e−ikrr cos φrdrdφ, (A.5)=∫ 2πφ=0ae−ikra cos φdφ, (A.6)= 2πaJ0 (kr a) , (A.7)where Jn is the Bessel function of the first kind of order n. Fourier analyzing thetime dependence of the source,∫ ∞−∞e−iΩt+iωt dt = 2πδ (Ω − ω) . (A.8)Thus, Fourier analyzing the relevant equations, i.e., ∇ → ik and ∂/∂t → −iω, givesk × E = ωB, (A.9)ik × B = −ue + I (2π)2 aJ0 (kr a) δ (Ω − ω) φ̂, (A.10)−iωue = −E − ue × ẑ. (A.11)150We will now define I (2π)2 aJ0 (kr a) δ (Ω − ω) = α to reduce clutter. Taking k× ofthe Eq. A.10:ik × (k × B) = −k × ue + αk × φ̂, (A.12)−ik2B = −k × ue + αkr ẑ − αkzr̂, (A.13)where ∇ · B = ik · B = 0 was used.Now taking k× of Eq. A.11 and using k · ue = 0 for incompressible electrons:−iωk × ue = −k × E − k × (ue × ẑ) , (A.14)−iωk × ue = −ωB − kzue + (k · ue) ẑ, (A.15)−iωk × ue = −ωB − kzue. (A.16)Using Eq. A.13 and Eq. A.10,−iω(ik2B + αkr ẑ − αkzr̂)= −ωB − kz(−ik × B + αφ̂), (A.17)ω(k2 + 1)B = ikzk × B − iαωkzr̂ − αkz φ̂ + iαωkr ẑ, (A.18)ω(k2 + 1)B = ikzk × B + C, (A.19)whereC = −iαωkzr̂ − αkz φ̂ + iαωkr ẑ. (A.20)Taking k× of Eq. A.19 and using Eq. A.19,ω(k2 + 1)k × B = ikzk × (k × B) + k × C, (A.21)ω(k2 + 1)k × B = −ikzk2B + k × C, (A.22)ω(k2 + 1)ikzk × B = k2z k2B + ikzk × C, (A.23)ω(k2 + 1) [ω(k2 + 1)B − C]= k2z k2B + ikzk × C, (A.24)[ω2(k2 + 1)2− k2z k2]B = ω(k2 + 1)C + ikzk × C, (A.25)B =ω(k2 + 1)C + ikzk × Cω2(k2 + 1)2 − k2z k2 . (A.26)Let us examine the φ direction. Since(k × C)φ = kzCr − krCz, (A.27)= −iαω(k2z + k2r), (A.28)= −iαωk2, (A.29)151it follows thatω(k2 + 1)C + ikzk × C = −αω(k2 + 1)kz + kzαωk2, (A.30)= αωkz, (A.31)= I (2π)2 aJ0 (kr a) δ (Ω − ω)ωkz . (A.32)The solution for Bφ is thusBφ (ω, k) =−I (2π)2 aδ (Ω − ω)ωJ0 (kr a) kzω2(k2 + 1)2 − k2z k2 . (A.33)The inverse transform in space givesBφ (ω, r) = −I (2π)2 aδ (Ω − ω)ω(12π)3(A.34)×∫ ∞kr=0∫ 2πkφ=0∫ ∞kz=−∞J0 (kr a) kzω2(k2 + 1)2 − k2z k2 eikrr cos(kφ−φ)eikz zkr dkr dkφdkz,(A.35)= Iaδ (Ω − ω)ωi ∂∂z∫ ∞kr=0∫ ∞kz=−∞J0 (kr a) J0 (krr) krω2(k2 + 1)2 − k2z k2 eikz zdkr dkz .(A.36)Let us now do a contour integral for the kz integral. Solving for the poles:ω2(k2 + 1)2− k2z k2 = 0, (A.37)ω2(k4 + 2k2 + 1)− k2z k2 = 0, (A.38)ω2(k2z + k2r)2+ 2ω2(k2z + k2r)+ ω2 − k2z(k2z + k2r)= 0, (A.39)ω2k4z + 2ω2k2z k2r + ω2k4r + 2ω2k2z + 2ω2k2r + ω2 − k4z − k2z k2r = 0, (A.40)(ω2 − 1)k4z +(2ω2k2r + 2ω2 − k2r)+ ω2k4r + 2ω2k2r + ω2 = 0, (A.41)(ω2 − 1)k4z +(2ω2[1 + k2r]− k2r)+ ω2(1 + k2r)2= 0, (A.42)whose solution for kz isk2z =−2ω2[1 + k2r]+ k2r ±√(2ω2[1 + k2r]− k2r)2 − 4 (ω2 − 1) ω2 (1 + k2r )22(ω2 − 1) .(A.43)152Here, the determinant isD =(2ω2[1 + k2r]− k2r)2− 4ω2(ω2 − 1) (1 + k2r)2, (A.44)= 4ω4(1 + k2r)2− 4ω2(1 + k2r)k2r + k4r − 4ω4(1 + k2r)2+ 4ω2(1 + k2r)2,(A.45)= −4ω2(1 + k2r)k2r + k4r + 4ω2(1 + k2r)2, (A.46)= −4ω2(1 + k2r) (k2r − 1 − k2r)+ k4r , (A.47)= 4ω2(1 + k2r)+ k4r . (A.48)Therefore, there exist 4 poles atkz = ±√√2ω2(1 + k2r)− k2r ±√4ω2(1 + k2r)+ k4r2(1 − ω2) . (A.49)To check the complexity of the poles, we need to check the following:2ω2(1 + k2r)− k2r <√4ω2(1 + k2r)+ k4r , (A.50)in which case there are at least two imaginary poles. The right-hand side is alwayspositive, so if the following is true,2ω2(1 + k2r)> k2r , (A.51)ω2 >k2r2(1 + k2r) , (A.52)then 2ω2(1 + k2r)− k2r is also positive. In this case, by squaring both sides Eq.A.50,4ω4(1 + k2r)2− 4ω2k2r(1 + k2r)+ k4r < 4ω2(1 + k2r)+ k4r , (A.53)4ω4(1 + k2r)− 4ω2k2r < 4ω2, (A.54)4ω4(1 + k2r)< 4ω2(1 + k2r), (A.55)ω2 < 1, (A.56)so for k2r2(1+k2r ) < ω2 < 1, two poles are real and two poles are imaginary.In the opposite case whereω2 0. The integral in Eq. A.36 now becomesBφ (ω, r) = Iaδ (Ω − ω)ωi (A.60)×∫ ∞kr=0dkr J0 (kr a) kr eikrr cos(kφ−φ) (A.61)× ∂∂z∫ ∞kz=−∞dkzeikz z(1 − ω2)(kz − A) (kz + A) (kz − iB) (kz + iB). (A.62)The(1 − ω2)term in the denominator exists because the coefficient of the quadraticterm of equation for kz is(1 − ω2). Doing the complex contour integral fromkz = −∞ to +∞, we pick up half the residues at A and −A, and the full residue atiB. With respect to the kz integral, the residues at A, −A, and iB are, respectively,Res (A) = eiAz(1 − ω2)(A + A) (A − iB) (A + iB), (A.63)=eiAz2A(1 − ω2) (A2 + B2) , (A.64)Res (−A) = e−iAz−2A(1 − ω2) (A2 + B2) , (A.65)Res (iB) = e−Bz(1 − ω2)(iB + A) (iB − A) 2iB, (A.66)=e−Bz−2iB(1 − ω2) (A2 + B2) . (A.67)154It can be inferred from Eq. A.49 thatA2 =2ω2(1 + k2r)− k2r +√4ω2(1 + k2r)+ k4r2(1 − ω2) , (A.68)B2 =−2ω2(1 + k2r)+ k2r +√4ω2(1 + k2r)+ k4r2(1 − ω2) , (A.69)A2 + B2 =√4ω2(1 + k2r)+ k4r1 − ω2. (A.70)Thus, the kz integral evaluates toπi√4ω2(1 + k2r)+ k4r(eiAz − e−iAz2A− e−BziB)=π√4ω2(1 + k2r)+ k4r(−sin AzA− e−BzB).(A.71)Doing the partial z derivative,π√4ω2(1 + k2r)+ k4r(− cos Az + e−Bz). (A.72)The solution for Bφ is nowBφ (ω, r) = −Iπaδ (Ω − ω)ωi∫ ∞kr=0J0 (kr a) J0 (krr) kr√4ω2(1 + k2r)+ k4r[cos (Az) − e−Bz]dkr .(A.73)Because A, B > 0, this is a sum of an oscillating solution and an evanescent solution.Figure A.1 shows the analytical solution for Bφ in Eq. A.73 for a source withω = 0.35, a = 1, and an arbitrary amplitude. The red vertical line represents thelocation and extent of the ring current source. The numerical integration was carriedout using the scipy.integrate.quad package in Python 3.7. The result of the integralat r = 0 (black horizontal dashed line) does not converge fast enough and is thusdubious. The cone-like structures emanating from the source are called "resonancecones" [52]. These arise from the frequency dependence of the plasma dielectrictensor; their angles have the following relation in dimensional quantities [52]:sin2 θ =Ω2(ω2pe + ω2ce −Ω2)ω2ceω2pe, (A.74)where θ is the cone angle, ωpe is the electron plasma frequency, ωce is the electroncyclotron frequency, and Ω is the frequency of the source. In the limit ωpe ωce,1550 5 10 15 20 25 30 35 40z/de−10−50510r/deBϕ (Arb.)−0.8−0.6−0.4−0.20.00.20.40.60.8Figure A.1: Analytical solution for Bφ in Eq. A.73 of an arbitrary magnitude.The red vertical line at z = 0 represents the ring current source. The numericalintegration of Eq. A.73 is dubious at r = 0 (black dashed line).which is equivalent to ignoring the displacement current (see Eq. 3.16), and Ω ωce for whistler waves, the relation becomessin2 θ =Ω2ω2ce, (A.75)which shows that θ = 0.35 = 21◦ for Ω/ωce = 0.35. The cone-angle in Fig. A.1 isapproximately θ = arctan(9/25) = 0.35, which agrees with the prediction.Reverting back to dimensionless quantities, the whistler dispersion relation in Eq.2.63 can be expressed as an equation for kz:Ω2 =k2z(k2z + k2r)(1 + k2r + k2z)2 , (A.76)Ω2(1 + k2r + k2z)2= k4z + k2z k2r , (A.77)Ω2(1 + k2r)2+ 2Ω2(1 + k2r)k2z +Ω2k4z = k4z + k2z k2r , (A.78)(1 −Ω2)k4z +[k2r − 2Ω2(1 + k2r)]k2z −Ω2(1 + k2r)2= 0. (A.79)The solution for kz isk2z =2Ω2(1 + k2r)− k2r +√k4r + 4Ω2(1 + k2r)2(1 −Ω2), (A.80)156−4 −3 −2 −1 0 1 2 3 4krde−2−1012k zd ekz vs. kr100200300400500Figure A.2: The fast Fourier transform of the result in Fig. A.1. The red linesrepresent the dispersion relation in Eq. A.80 for Ω = 0.35.which, for a particular value of Ω, yields a relation between kz and kr .Figure A.2 shows the fast Fourier transform of the result in Fig. A.1. The red linesrepresent the dispersion relation in Eq. A.80 for Ω = 0.35. It can be seen that theareas with high magnitude follow the red lines. The match is not perfect, of course,because the dispersion relation is based on plane waves.An analytical solution for a whistler wave excited by a delta function ring currentsource has thus been presented.157A p p e n d i x BALGORITHM TEST USING WHISTLER WAVESThe algorithm presented in Section 2.3 should yield whistler waves similar to thatin Appendix A. A 3D code that mimics the current source in Appendix A.73 wasconstructed.Because it is difficult to realize a delta function in a numerical simulation, the currentsource was approximated by a Gaussian of the formjφ = φ̂ε exp(−(r − a)2 + z22σ2)sinΩt, (B.1)where ε is the current amplitude, a is the ring radius, and σ is the ring thickness. ACartesian grid of dimensions (x, y, z) = (80, 80, 320) = (20, 20, 80)de was used, andthe time step was ∆t = 0.05. The source parameters were a = 1, σ = 0.5, ε = 0.1,and Ω = 0.35. Periodic boundary conditions were used.Figure B.1 shows Bφ from the numerical simulation at t = 80|ωce |−1. The generalspatial structure is similar to that in Fig. A.1, except for at r = 0. The cone angle isslightly wider (θ = 0.4), which may be due to the fact that here the current sourceis not a perfect delta function.0 5 10 15 20 25 30 35 40z/de−10−50510r/deBϕ−0.03−0.02−0.010.000.010.020.03Figure B.1: Bφ at t = 80|ωce |−1158−4 −3 −2 −1 0 1 2 3 4krde−2−1012k zd ekz vs. kr0510152025Figure B.2: The fast Fourier transform of the results in Fig. B.1. The red linesrepresent the dispersion relation in Eq. A.80 for Ω = 0.35.Figure B.2 shows the fast Fourier transform of the results in Fig. B.1. The red linesrepresent the dispersion relation in Eq. A.80 for Ω = 0.35. The regions of highmagnitude follow the red lines, arguably more closely than those in Fig. A.2.It has thus been confirmed that the algorithm and simulation described in Section2.3 produce whistler waves as expected.159A p p e n d i x CCANONICAL HELICITY DENSITY AND THE LAGRANGIANDENSITY IN ELECTRON-MAGNETOHYDRODYNAMICSRelevant normalizations are carried out according to Section 2.3. The normalizedelectron canonical momentum and canonical vorticity are defined as, respectively,Pe = ue − A, (C.1)Qe = ∇ × Pe, (C.2)= ∇ × ue − B. (C.3)The normalized electron equation of motion is∂ue∂t+ ue · ∇ue = −E − ue × B − ∇pe, (C.4)or, in terms of Qe,∂ue∂t+ ∇(u2e2)= −E + ue ×Qe − ∇pe, (C.5)where scalar electron pressure and uniform electron density were assumed. Also,because both ion and electron densities are assumed to be uniform, the electrostaticpotential φ = 0, so E = −∂A/∂t. Examining the time evolution of the normalizedelectron canonical helicity density κQ = Pe ·Qe,∂κQ∂t=∂Pe∂t·Qe + Pe ·∂Qe∂t, (C.6)=(∂ue∂t− ∂A∂t)·Qe + Pe · ∇ × (ue ×Qe) , (C.7)=(∂ue∂t+ E)·Qe + ∇ · ([ue ×Qe] × Pe) +:ue ×Qe ·Qe = 0(ue ×Qe) · (∇ × Pe), (C.8)=(−∇(u2e2)+:0ue ×Qe − ∇pe)·Qe + ∇ · (Qe [ue · Pe] − ue [Pe ·Qe]) ,(C.9)= Qe · ∇(−u2e2− pe + ue · Pe)− ue · ∇κQ, (C.10)where ∇ ·Qe = ∇ · ue = 0 has been used. Expressing this in terms of the convectivederivative and using ue · Pe = u2e − ue · A,DκQDt= Qe · ∇(u2e2− pe − ue · A). (C.11)160Now we prove that L = u2e2 − pe − ue · A is the Lagrangian density of the system.The Euler-Lagrange equation∇x L −DDt∇uL = 0, (C.12)where L (x, ue, t) =∫L (x, ue, t) dV is the Lagrangian, must give the equation ofmotion of the system (Eq. C.4). ∇x is the spatial gradient and ∇u is the gradient invelocity space. Here,∇x L =∫[−∇x pe − ∇x (ue · A)] dV, (C.13)=∫[−∇x pe − ue · ∇xA − ue × (∇x × A)] dV, (C.14)=∫[−∇x pe − ue · ∇xA − ue × B] dV, (C.15)since only pe and A are explicitly spatially dependent. Also,∇uL =∫∇uLdV, (C.16)=∫(ue − A) dV . (C.17)So,DDt∇uL =∫∂∂t(ue − A) dV +∫(ue − A) (ue · ds) , (C.18)=∫ (∂ue∂t− ∂A∂t)dV +∫∇x · [ue (ue − A)] dV, (C.19)=∫ (DueDt+ E − ue · ∇xA)dV . (C.20)Therefore,∇x L −DDt∇uL =∫ [ −∇x pe − ue · ∇xA − ue × B−DueDt − E + ue · ∇xA]dV, (C.21)=∫ [−∇x pe − ue × B −DueDt− E]dV, (C.22)= 0. (C.23)Setting the integrand to zero gives us the original normalized equation of motion(Eq. C.4):DueDt= −E − ue × B − ∇pe. (C.24)161Therefore, L is the Lagrangian density of the system andDκQDt= Qe · ∇L. (C.25)Also, integrating this equation over a volume and writing the total canonical helicityas KQ =∫V κQdV ,DKQDt=∫VQe · ∇LdV, (C.26)=∫V∇ · (QeL) dV, (C.27)=∮SLQe · ds. (C.28)Therefore, if Qe does not penetrate the volume at the boundary, KQ is a conservedquantity.162A p p e n d i x DIMPLICIT PARTICLE INTEGRATORAn algorithm [15] will now be illustrated that implicitly solves the particle equationof motionmdvdt= qE + qv × B. (D.1)Defining Ω = qB/m and Σ = qE/m,dvdt= Σ + v ×Ω. (D.2)Now, velocity is defined at integer time steps, and position is defined at half-integertime steps. The time derivative of the velocity is evaluated at the half-integer timestep, i.e.,vt+1 − vt∆t= f(vt+1 + vt2), (D.3)= Σt+1/2 +vt+1 + vt2×Ωt+1/2. (D.4)This equation may be expressed asvt+1 + A × vt+1 = C, (D.5)whereA =Ωt+1/2∆t2, (D.6)andC = vt + ∆t(Σt+1/2 +vt2×Ωt+1/2). (D.7)Now, dotting Eq. D.5 with A,vnew · A = C · A. (D.8)Crossing Eq. D.5 with A,vt+1 × A + (A × vt+1) × A = C × A, (D.9)vt+1 × A + vt+1 A2 − A (vt+1 · A) = C × A, (D.10)vt+1 − C + vt+1 A2 − A (C · A) = C × A, (D.11)vt+1 =C + A (C · A) + C × A1 + A2, (D.12)163which can be used to update a given vt .The new position is given byxt+3/2 = xt+1/2 + vt+1∆t. (D.13)164A p p e n d i x EHARRIS EQUILIBRIUM PLASMA SHEATH DERIVATIONThe Harris equilibrium is a specific stationary solution of the Vlasov equation,Gauss’s law, and Ampère’s law [66]. It is assumed that the spatial dependence isonly in the x-direction, i.e., only ∂/∂x is finite, and that A = Az ẑ. It then followsthat B = By ŷ and E = Ex x̂. Equations to solve are0 = vx∂ fσ∂x+qσmσ(E + vσ × B) ·∂ fσ∂vσ, (E.1)∂Ex∂x=1ε0∑σqσ∫fσd3vσ, (E.2)∂By∂x= µ0∑σqσ∫fσvzd3vσ . (E.3)In terms of the electrostatic and vector potentials φ and Az,0 = vx∂ fσ∂x+qσmσ(E + vσ × B) ·∂ fσ∂vσ, (E.4)∂2φ∂x2= − 1ε0∑σqσ∫fσd3vσ, (E.5)∂2 Az∂x2= −µ0∑σqσ∫fσvzd3vσ . (E.6)Now, because the system is independent of y and z coordinates, the canonicalmomenta py and pz are constants of the motion, as well as the total energy. In otherwords,W =12m(v2x + v2y + v2z)+ qφ, (E.7)py = mvy, (E.8)pz = mvz + qAz (E.9)are constants of motion for each species. Now consider the distribution functionfσ =(mσ2πkBTσ)3/2n0 exp[− mσ2kBTσ(v2x + v2y + (vz − Vσ)2 +2qmσ(φ − AzVσ))],(E.10)165where Tσ is the temperature of each species, n0 is the reference density, and Vσ isthe average velocity of each species in the z-direction. Becausev2x + v2y + (vz − Vσ)2 +2qσmσ(φ − AzVσ) =2Wmσ− 2pzmσ+ V2σ (E.11)is a function of constants of motion only, Eq. E.10 is a solution of Eq. E.4 byLiouville’s theorem. Thus we have∫fσd3v = n0 exp[− qσkBTσ(φ − AzVσ)], (E.12)∫vz fσd3v = n0Vσ exp[− qσkBTσ(φ − AzVσ)]. (E.13)Inserting these into Eqs. E.5 and E.6,∂2φ∂x2= −en0ε0[exp[− ekBTi(φ − AzVi)]− exp[ekBTe(φ − AzVe)] ], (E.14)∂2 Az∂x2= −µ0en0[Vi exp[− ekBTi(φ − AzVi)]− Ve exp[ekBTe(φ − AzVe)] ]. (E.15)It is now assumed thatVi = −Ve = V , which can be done by moving to a frame wherethe average velocities of ions and electrons are equal in magnitude and opposite.It is also assumed that Ti = Te = T for simplicity; the situation where Ti , Te isdescribed in Ref. [164]. Then,∂2φ∂x2= −en0ε0exp[ekBTAzV] (exp[− eφkBT]− exp[eφkBT] ), (E.16)∂2 Az∂x2= −µ0en0V exp[ekBTAzV] [exp[− eφkBT]+ exp[eφkBT] ]. (E.17)Since the first equation is trivially solved by φ = 0, the only equation to solve is∂2 Az∂x2= −2µ0en0V exp[ekBTAzV]. (E.18)The solution to this equation isAz = −2kBTeVln cosh(V x√2cλD), (E.19)whereλD =(1λ2Di+1λ2De)− 12=√ε0kBT2n0e2(E.20)is the Debye length.166Using Eq. E.19, the solution for the magnetic field isBy = −∂Az∂x=2kBTec√n0e2ε0kBTtanh(V x√2cλD), (E.21)= 2√µ0n0kBT tanh(V x√2cλD), (E.22)= B0 tanh(V x√2cλD). (E.23)The magnitude of the equilibrium magnetic field illustrates that the magnetic pres-sure is balance by the thermal pressure, i.e.,B202µ0= 2n0kBT = pi + pe, (E.24)where pi and pe are ion and electron pressures, respectively.Finally, inserting Eq. E.19 and φ = 0 into Eq. E.12 gives the equilibrium densityn = n0 cosh−2(V x√2cλD), (E.25)which shows that the density is peaked at x = 0 and goes to zero as x → ±∞. Thisis why the Harris equilibrium is also called the Harris "sheath."167A p p e n d i x FONE-DIMENSIONAL AND TWO-DIMENSIONALMAXWELL-JÜTTNER DISTRIBUTIONS AND THEDISTRIBUTION OF ξF.1 Derivation of fρ⊥In cylindrical coordinates, Eq. 6.18 is equivalent tofρd3ρ =14πθK2 (1/θ)exp(−√1 + ρ2θ)ρ⊥dρ⊥dφdρz . (F.1)Integrating in ρz gives∫ ∞ρz=−∞fρρ⊥dρ⊥dφdρz =∫ ∞ρz=−∞14πθK2 (1/θ)exp©«−√1 + ρ2⊥ + ρ2zθª®®¬ ρ⊥dρ⊥dφdρz(F.2)= ρ⊥dρ⊥dφ∫ ∞ρz=012πθK2 (1/θ)exp©«−√1 + ρ2⊥ + ρ2zθª®®¬ dρz,(F.3)where ρ2 = ρ2⊥ + ρ2z . Defininga2 =1 + ρ2⊥θ2(F.4)andt =ρzθ, (F.5)the ρz-integral in Eq. F.3 becomes12πK2 (1/θ)∫ ∞0exp(−√a2 + t2)dt . (F.6)Now we definet = a sinh z, (F.7)so√a2 + t2 = a√1 + sinh2 z = a cosh z and dt = a cosh zdz. Equation F.6 is nowa2πK2 (1/θ)∫ ∞0cosh z exp (−a cosh z) dz. (F.8)168The z-integral in Eq. F.8 evaluates toK1 (a)whereKn is themodifiedBessel functionof the second kind of order n [176, Section 8.432, 1.].Therefore, Eq. F.3 is now√1 + ρ2⊥2πθK2 (1/θ)K1©«√1 + ρ2⊥θª®®¬ ρ⊥dρ⊥dφ. (F.9)Integrating in φ yields the final result:fρ⊥dρ⊥ =ρ⊥√1 + ρ2⊥θK2 (1/θ)K1©«√1 + ρ2⊥θª®®¬ dρ⊥, (F.10)which is Eq. 6.19.F.2 Derivation of fρzIn cylindrical coordinates, Eq. 6.18 is equivalent tofρd3ρ =14πθK2 (1/θ)exp(−√1 + ρ2θ)ρ⊥dρ⊥dφdρz . (F.11)Integrating in all φ and ρ⊥, Eq. F.11 becomesfρz dρz =∫ ∞ρ⊥=012θK2 (1/θ)exp©«−√1 + ρ2⊥ + ρ2zθª®®¬ ρ⊥dρ⊥dρz . (F.12)Letting η2 = 1 + ρ2⊥ + ρ2z while keeping ρz constant so thatηdη = ρ⊥dρ⊥, (F.13)we havefρz dρz =∫ ∞η=√1+ρ2z12θK2 (1/θ)exp(−ηθ)ηdηdρz . (F.14)Using the integral formula [176, Section 3.351, 2.]∫ ∞uxne−µxdx = e−uµn∑k=0n!k!ukµn−k+1, (F.15)169where x = η, u =√1 + ρ2z , µ = 1/θ, and n = 1 in this case, we havefρz dρz =12θK2(1/θ)(θ2 + θ√1 + ρ2z)exp©«−√1 + ρ2zθª®®¬ , (F.16)=θ2K2(1/θ)©«1 +√1 + ρ2zθª®®¬ exp©«−√1 + ρ2zθª®®¬ , (F.17)which is Eq. 6.20.F.3 Derivation of fξξ is defined asξ = 1 + α (nρz − γ) = 1 + αζ, (F.18)where ζ = nρz − γ.fζ will first be derived. Defining R = nρz (so dρz = dR/n and ζ = R − γ), we havefρz (ρz)dρz = fρz (R/n)dRn(F.19)=θ2nK2 (1/θ)(1 +√1 + R2/n2θ)exp(−√1 + R2/n2θ)dR (F.20)= fR(R)dR. (F.21)Now, in order for ζ = R − γ to be true, the value of R has to equal ζ + γ for a givenvalue of γ. The probability distribution of this occurrence integrated over all valuesof γ gives fζ :fζ (ζ) =∫ ∞1fγ (γ) fR (ζ + γ) dγ (F.22)=∫ ∞1γ2√1 − 1/γ22nK22 (1/θ)©«1 +√1 + (ζ + γ)2 /n2θª®®¬ exp©«−γ +√1 + (ζ + γ)2 /n2θª®®¬ dγ.(F.23)Finally, rearranging Eq. F.18 yields ζ (ξ) = (ξ − 1) /α so that dζ = dξ/α. It follows170thatfζ (ζ)dζ = fζ ([ξ − 1]/α)dξα(F.24)=∫ ∞1γ2√1 − 1/γ22αnK22 (1/θ)©«1 +√1 + (ζ (ξ) + γ)2 /n2θª®®¬ exp©«−γ +√1 + (ζ (ξ) + γ)2 /n2θª®®¬ dγdξ(F.25)= fξ(ξ)dξ. (F.26)Writing ρz (γ, ξ) = (ζ (ξ) + γ) /n yields a more compact expression:fξ(ξ) =∫ ∞1γ2√1 − 1/γ22αnK22 (1/θ)©«1 +√1 + ρ2z (γ, ξ)θª®®¬ exp©«−γ +√1 + ρ2z (γ, ξ)θª®®¬ dγ,(F.27)which is Eq. 6.22.F.4 Derivation of non-relativistic fρ⊥From Eq. F.10,fρ⊥ =ρ⊥√1 + ρ2⊥θK2 (1/θ)K1©«√1 + ρ2⊥θª®®¬ . (F.28)For ρ⊥ 1, √1 + ρ2⊥ ' 1 +ρ2⊥2. (F.29)For θ 1, it is seen that [160, Section 7.23]K2 (1/θ) '√πθ2e−1/θ, (F.30)and for small θ 1 and ρ⊥ 1,K1©«√1 + ρ2⊥θª®®¬ ' K1(1θ+ρ2⊥2θ)(F.31)'√πθ2 + ρ2⊥exp(−1θ−ρ2⊥2θ), (F.32)171so to lowest order,fρ⊥ 'ρ⊥θ√1 +ρ2⊥2exp(−ρ2⊥2θ)(F.33)' ρ⊥θexp(−ρ2⊥2θ). (F.34)The most likely ρ⊥ value given this probability distribution function isρ⊥,ML =√θ. (F.35)

Magnetic Reconnection

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