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Probing the Progression, Properties, and Progenies of Magnetic Reconnection
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Yoon, Young Dae
(2020)
Probing the Progression, Properties, and Progenies of Magnetic Reconnection.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/hqwx-r892.
Abstract
Magnetic reconnection is a plasma phenomenon in which opposing magnetic fields annihilate and release their magnetic energy into other forms of energy. In this thesis, various aspects of collisionless magnetic reconnection are studied analytically and numerically, and an experimental diagnostic for magnetic fields in a plasma is described.
The progression of magnetic reconnection is first illustrated through the formulation of a framework that revolves around canonical vorticity flux, which is ideally a conserved quantity. The reconnection instability, electron acceleration, and whistler wave generation are explained in an intuitive manner by analyzing the dynamics of canonical vorticity flux tubes. The validity of the framework is then extended down to first principles by the inclusion of the electron canonical battery effect. The importance of this effect during reconnection determines the overall structure and evolution of the process.
A crucial property of magnetic reconnection is its accompaniment by anomalous ion heating much faster than conventional collisional heating. Stochastic heating is a mechanism in which, under a sufficiently strong electric field, particles undergo chaotic motion in phase space and heat up dramatically. Using the previously established canonical vorticity framework, it is demonstrated that the Hall electric fields that develop during reconnection satisfy the stochastic ion heating criterion and that the ions involved indeed undergo chaotic motion. This mechanism is then kinetically verified via exact analyses and particle simulations and is thus ultimately established as the main ion heating mechanism in magnetic reconnection.
An important progeny of magnetic reconnection is whistler waves. These waves interact with energetic particles and scatter their pitch-angles, triggering losses of magnetic confinement. A previous study demonstrated via exact relativistic analyses that if a particle undergoes a "two-valley" motion, it undergoes drastic changes in its pitch-angle. This analysis is extended to a relativistic thermal distribution of particles. The condition for two-valley motion is first derived; it is then shown that a significant fraction of the particle distribution meets this condition and thus undergoes large pitch-angle scatterings. The scaling of this fraction with the wave amplitude suggests that relativistic microburst events may be explained by the two-valley mechanism. It is also found that the widely-used second-order trapping theory is an inaccurate approximation of the theory presented.
A new method of probing the magnetic field in a plasma is described and developed to some extent. It utilizes the two-photon Doppler-free laser-induced fluorescence technique, where two counter-propagating laser beams effectively cancel out the Doppler effect and excite electron populations. The fluorescence resulting from the subsequent de-excitation is then measured, enabling the resolution of Zeeman splitting of the spectral lines from which the magnetic field information can be inferred. A high-power, repetitively-pulsed radio-frequency plasma source was developed as the subject of diagnosis, and preliminary results are presented.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Plasma Physics; Magnetic Reconnection; Canonical Vorticity; Stochastic Heating; Relativistic Particles; Whistler Waves; Laser Diagnostics
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Bellan, Paul Murray
Thesis Committee:
Phinney, E. Sterl (chair)
Hallinan, Gregg W.
Velli, Marco
Bellan, Paul Murray
Defense Date:
23 April 2020
Non-Caltech Author Email:
ydyoon93 (AT) gmail.com
Funders:
Funding Agency
Grant Number
Department of Energy (DOE)
DE-FG02-04ER54755
Air Force Office of Scientific Research (AFOSR)
FA9550-17-1-0023
NSF
1914599
NSF
1059519
Record Number:
CaltechTHESIS:04302020-151019178
Persistent URL:
DOI:
10.7907/hqwx-r892
Related URLs:
URL
URL Type
Description
DOI
Paper adapted for Chapter 2.
DOI
Paper adapted for Chapter 2.
DOI
Paper adapted for Chapter 3.
DOI
Paper adapted for Chapter 4.
DOI
Paper adapted for Chapter 5.
ORCID:
Author
ORCID
Yoon, Young Dae
0000-0001-8394-2076
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No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
13692
Collection:
CaltechTHESIS
Deposited By:
Young Dae Yoon
Deposited On:
11 May 2020 19:19
Last Modified:
18 May 2020 16:33
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Probing the Progression, Properties, and Progenies of
Magnetic Reconnection

Thesis by

Young Dae Yoon

In Partial Fulfillment of the Requirements for the
Degree of
Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2020
Defended April 23rd, 2020

Young Dae Yoon
ORCID: 0000-0001-8394-2076
All rights reserved except where otherwise noted

iii

ACKNOWLEDGEMENTS
I must begin by cherishing my roots and thanking my family. My parents’ unwavering
support, commitment, and wisdom are the pillar and base upon which I am merely
standing. I am so proud of my younger brother for all he has accomplished despite
the sacrifices he has had to make because of me. I also thank every single one of
my relatives for always rooting for me along the way. Without their love and faith
amidst 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 on
the shoulders of Professor Paul Bellan. He is a brilliant experimentalist as well as
an excellent theorist having both breath and depth of knowledge in physics. I admire
his accessibility as he always drops whatever he is doing at the time to pay attention
to 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 his
constant effort in preparing for a class he has been teaching for decades. He is very
patient and understanding, but also enthusiastic and motivational. I sincerely feel
extremely lucky to have chosen Paul as my advisor.
I would like to thank my candidacy committee—Professors E. Sterl Phinney, Gregg
Hallinan, and Kerry Vahala—and my thesis committee—Professors E. Sterl Phinney, Gregg Hallinan, and Marco Velli—for their helpful feedback and insightful
questions.
Ryan Marshall and Pakorn (Tung) Wongwaitayakornkul joined the group with me
in 2014, and we will all be graduating at the same time. We have made so many
fond memories in class, in the lab, at conferences, and at bars. I thank Ryan for
keeping the office lively, offsetting my laziness, and having probably the most beers
with 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 knowledge
and 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 longest
time so that I could make use of his "senior aura" that magically fixes everything. I
am also greatly indebted to Gunsu Yun, who has helped me so much career-wise by
taking my matter into his own hands.

iv
I 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 greatly
benefited 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 I
know. 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, from
whom I have also learned a lot. Yi rightfully questions seemingly obvious statements; I tell myself to be more like Yi when I seem to be overlooking important
assumptions underlying a problem. Yang and André are also very passionate, intelligent, 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 thanks
to Connie Rodriguez, Christy Jenstad, and Jennifer Blankenship. I especially thank
Connie for dealing with all the purchases and travel reports and for furnishing the
office with leftover pizza from APh 110.
I also thank Dave Felt for imparting his outstanding electrical engineering wisdom
when I was building my RF plasma source.
The Korean society at Caltech has been a big part of my graduate life. There are
so many of you to name here, so I will try my best not to leave anyone out. I
thank 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 Won
Ahn, Jong Hun Kang, and Hyunjun Cho. I also thank Hyeong Chan Jo, Jeongmin
Kim, Sanghyun Yi, Jihong Min, Juhyun Kim, Hyungjoo Row, Jinmo Koo, Jieun
Shin for being good friends with me all these years. Special shout-out to the
travelling/drinking/dawdling crew Hee Jeong Ahn, Jonghun Lim, Hye Young Shin,
Areum Kim, without whom I would have been a lot more productive.
The Wednesday tennis club made tennis my favorite sport, although tennis may have
been an excuse to gather people for drinks. I tremendously thank Peter Lee for his
leadership, without which the club would not function. I am not sure whether I will
ever 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 the
other side of the globe from mine. Thanks for all the delicious food and fun times.

Finally, I would like to thank Ha-Rry for being there with me through all the ups
and downs. You have done much more for me than you could ever imagine.

ABSTRACT
Magnetic reconnection is a plasma phenomenon in which opposing magnetic fields
annihilate and release their magnetic energy into other forms of energy. In this thesis,
various aspects of collisionless magnetic reconnection are studied analytically and
numerically, and an experimental diagnostic for magnetic fields in a plasma is
described.
The progression of magnetic reconnection is first illustrated through the formulation
of a framework that revolves around canonical vorticity flux, which is ideally a
conserved quantity. The reconnection instability, electron acceleration, and whistler
wave generation are explained in an intuitive manner by analyzing the dynamics
of canonical vorticity flux tubes. The validity of the framework is then extended
down to first principles by the inclusion of the electron canonical battery effect. The
importance of this effect during reconnection determines the overall structure and
evolution of the process.
A crucial property of magnetic reconnection is its accompaniment by anomalous
ion heating much faster than conventional collisional heating. Stochastic heating is
a mechanism in which, under a sufficiently strong electric field, particles undergo
chaotic motion in phase space and heat up dramatically. Using the previously
established canonical vorticity framework, it is demonstrated that the Hall electric
fields that develop during reconnection satisfy the stochastic ion heating criterion
and that the ions involved indeed undergo chaotic motion. This mechanism is then
kinetically verified via exact analyses and particle simulations and is thus ultimately
established as the main ion heating mechanism in magnetic reconnection.
An important progeny of magnetic reconnection is whistler waves. These waves
interact with energetic particles and scatter their pitch-angles, triggering losses of
magnetic confinement. A previous study demonstrated via exact relativistic analyses
that if a particle undergoes a "two-valley" motion, it undergoes drastic changes in
its pitch-angle. This analysis is extended to a relativistic thermal distribution of
particles. The condition for two-valley motion is first derived; it is then shown
that a significant fraction of the particle distribution meets this condition and thus
undergoes large pitch-angle scatterings. The scaling of this fraction with the wave
amplitude suggests that relativistic microburst events may be explained by the twovalley mechanism. It is also found that the widely-used second-order trapping theory

vii
is an inaccurate approximation of the theory presented.
A new method of probing the magnetic field in a plasma is described and developed
to some extent. It utilizes the two-photon Doppler-free laser-induced fluorescence
technique, where two counter-propagating laser beams effectively cancel out the
Doppler effect and excite electron populations. The fluorescence resulting from
the subsequent de-excitation is then measured, enabling the resolution of Zeeman
splitting of the spectral lines from which the magnetic field information can be
inferred. A high-power, repetitively-pulsed radio-frequency plasma source was
developed as the subject of diagnosis, and preliminary results are presented.

viii

PUBLISHED CONTENT AND CONTRIBUTIONS

[1] Young Dae Yoon and Paul M. Bellan. A generalized two-fluid picture of nondriven collisionless reconnection and its relation to whistler waves. Physics
of 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 numerical
simulations, and wrote the manuscript.
[2] Young Dae Yoon and Paul M. Bellan. An intuitive two-fluid picture of spontaneous 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 numerical
simulations, and wrote the manuscript.
[3] Young Dae Yoon and Paul M. Bellan. Fast Ion Heating in Transient Collisionless
Magnetic Reconnection via an Intrinsic Stochastic Mechanism. The Astrophysical 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 simulations, and wrote the manuscript.
[4] Young Dae Yoon and Paul M Bellan. Kinetic Verification of the Stochastic Ion Heating Mechanism in Collisionless Magnetic Reconnection. The
Astrophysical 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 calculations, conducted the numerical simulations, and wrote the manuscript.
[5] Young Dae Yoon and Paul M. Bellan. The electron canonical battery effect in
magnetic reconnection: Completion of the electron canonical vorticity framework. 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 the
numerical simulations, and wrote the manuscript.
[6] Young Dae Yoon and Paul M. Bellan. Non-diffusive pitch-angle scattering of a
distribution of energetic particles by coherent whistler waves. Submitted, 2020.
Y.D.Y. performed the analytical calculations, built and conducted the numerical
simulations, and wrote the manuscript.

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . viii
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Chapter I: Introduction to Plasma Physics . . . . . . . . . . . . . . . . . . . 1
1.1 Single-Particle Theory . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Vlasov Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Two-Fluid Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Magneto-Hydrodynamics (MHD) . . . . . . . . . . . . . . . . . . . 5
1.5 Magnetic Reconnection . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter II: Canonical Vorticity Framework of Magnetic Reconnection: an
Intuitive Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Canonical Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Numerical Construction . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Intuitive Description . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Chapter III: Electron Canonical Battery Term: Completion of the Canonical
Vorticity Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 The Completed Canonical Vorticity Framework . . . . . . . . . . . 36
3.2 Fluid Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Particle-in-cell Simulation . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Chapter IV: Stochastic Ion Heating in Magnetic Reconnection . . . . . . . . 52
4.1 Stochastic Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Relevant Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Numerical Simulation and Results . . . . . . . . . . . . . . . . . . . 58
4.4 Stochastic Heating Condition Analysis . . . . . . . . . . . . . . . . 58
4.5 Test Particle Simulation . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Chapter V: Kinetic Verification of Stochastic Ion Heating in Magnetic Reconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.1 Stochastic Heating in the Harris Equilibrium Plasma Sheath . . . . . 69
5.2 Particle-in-cell verification . . . . . . . . . . . . . . . . . . . . . . . 72
5.3 Confirmation of Stochastic Motion . . . . . . . . . . . . . . . . . . 76
5.4 Heavy Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Chapter VI: Pitch-Angle Scattering of Energetic Particles by Coherent Whistler
Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.1 Two-Valley Motion Review . . . . . . . . . . . . . . . . . . . . . . 85
6.2 Two-Valley Motion Condition . . . . . . . . . . . . . . . . . . . . . 87
6.3 Distribution of ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4 Fraction of Particles Undergoing Two-Valley Motion . . . . . . . . . 91
6.5 Numerical Verification . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.7 Comparison to Second-order Trapping Theory . . . . . . . . . . . . 98
6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Chapter VII: Magnetic Field Diagnostic using Two-Photon Doppler-free
Laser-Induced Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.1 Sources of Spectral Broadening . . . . . . . . . . . . . . . . . . . . 104
7.2 Two-Photon Doppler-Free Laser-Induced Fluorescence . . . . . . . . 107
7.3 Repetitively Pulsed, High-Power, Inductively-Coupled Radio-Frequency
Plasma Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.5 One-Photon LIF Results . . . . . . . . . . . . . . . . . . . . . . . . 116
7.6 Two-Photon LIF Preliminary Results . . . . . . . . . . . . . . . . . 120
7.7 Suggestions for Further Work . . . . . . . . . . . . . . . . . . . . . 122
7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Chapter VIII: Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.1 Progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.3 Progenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.4 Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Appendix A: Whistler Wave Analytical Solution . . . . . . . . . . . . . . . . 149
Appendix B: Algorithm Test using Whistler Waves . . . . . . . . . . . . . . 157
Appendix C: Canonical Helicity Density and the Lagrangian Density in
Electron-Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . 159
Appendix D: Implicit Particle Integrator . . . . . . . . . . . . . . . . . . . . 162
Appendix E: Harris Equilibrium Plasma Sheath Derivation . . . . . . . . . . 164
Appendix F: One-Dimensional and Two-Dimensional Maxwell-Jüttner Distributions and the Distribution of ξ . . . . . . . . . . . . . . . . . . . . . 167
F.1 Derivation of fρ⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
F.2 Derivation of fρz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
F.3 Derivation of fξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

xi
F.4 Derivation of non-relativistic fρ⊥ . . . . . . . . . . . . . . . . . . . 170

xii

LIST OF ILLUSTRATIONS

Number
Page
1.1 Incremental change in surface area due to a displacement Uδt of a
line segment dl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Frozen-in magnetic field lines within deforming magnetic flux tubes.
1.3 A graphical description of a torsional Alfvén wave. . . . . . . . . . . 10
1.4 (a) Non-reconnecting magnetic field lines (black) and stagnating
flows (green). (b) Reconnecting magnetic field lines and plasma
inflow/outflow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Fast reconnection geometry. . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Thesis outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 The background and perturbation magnetic fields and their sum as
the initial condition for magnetic reconnection. . . . . . . . . . . . . 24
2.2 Time evolution of the magnetic field during collisionless reconnection. 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
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 color of the lines represent the height in z. The red
arrow in (c) is the direction of the out-of-plane electron velocity uez .
26
2.4 Temporal evolution of a Qe flux tube. The insets are side-on and
overhead views of the flux tube. The red arrow represents the direction and location of electron flow ue . The color of the flux tube
represents the change of |Qe |. . . . . . . . . . . . . . . . . . . . . . 27
2.5 Time evolution of Q ez , Bz , and (∇ × ue )z at (x, y) = (1, 2)de . The
y-axis is in natural logarithmic scale. . . . . . . . . . . . . . . . . . 29
2.6 (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. . . . . . . . . . 32

xiii
2.7

3.1
3.2
3.3
3.4
3.5
3.6

3.7

4.1

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 initial field line is straighted out. The A, B, and Cs
in both figures correspond to identical locations along the field line.
The colors represent the height in z. . . . . . . . . . . . . . . . . . .
The staggered spatial grid used for the fluid simulation. . . . . . . . .
The staggered time grid and the time advancement algorithm for the
fluid simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In-plane Qe (red), in-plane B (black) and uez (color) for varying µ
values and isotropic pressure at t = (a) 380, (b) 450, (c) 560. . . . . .
Same as Fig. 3.3 with pressure anisotropy and varying µ values at
t = (a) 400, (b) 500, (c) 630. . . . . . . . . . . . . . . . . . . . . . .
Same as 3.3 from the particle-in-cell simulation at t = 300. . . . . . .
(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 ∂Q ey /∂t. The red arrows represent the direction of
Q ey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(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 anisotropic contribution to the canonical
battery term − ŷ · ∇ × ∇ · pe,aniso /ne , and (e) their sum. . . . . . . .
(a) Reconnected in-plane B field lines (white) and connected inplane Qe 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 are respectively in the ±x and ±y directions.
The x-axis and the y-axis have different scales to show the field
lines more evidently. (b) Comparison of By2 /2 with − E x dx and
− uez By dx along the black dotted line in (a). (c) Comparison of
uey + ue /2 with − E y dy along the magenta line in (a). . . . . . .

34
40
41
45
46
47

xiv
4.2

4.3

5.1
5.2
5.3
5.4
5.5
5.6
5.7

5.8

(a) Ten coherent ion trajectories undergoing Speiser-like orbits, and
(b) their stochastic counterparts. (c) Positions in y phase space of all
6000 ions at t = 580 |ωce | −1 , and (d) their stochastic counterparts.
(e), (f) Zoom-ins of (c) and (d), respectively. (g) Distribution of |∆ yÛ | 2
for the coherent case (blue) and the stochastic case (red). (h) Timedependent 3D separation distances between two selected particles
with the minimum initial separation distance, 0.015de , under the
stationary electric field (blue line; particles are red dots in (b)) and
the growing electric field (red line; particles are red dots in (f)). . . .
Positions in y phase space of ions that started with a thermal velocity
of 0.01 de |ωce | in (a) the coherent case and (b) the stochastic case.
Distribution of |∆ yÛ | 2 in (c) the coherent case and (d) the stochastic
case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(a) Contours of x̄sh λ̄, Bg . Inset plots x̄sh as a function of Bg for
three λ̄ values. (b) Same as (a), but for x̄sh /λ̄. . . . . . . . . . . . . .
B (black lines) and E x (color) for (a) bg = 0, (b) 0.1, (c) 0.3, and (d)
0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ln α for bg = 0, 0.1, 0.3, and 0.5, respectively. . . . . . . . . . . . . .
Ti for bg = 0, 0.1, 0.3, and 0.5, respectively. . . . . . . . . . . . . . .
Ti⊥ for bg = 0, 0.1, 0.3, and 0.5, respectively. . . . . . . . . . . . . .
Tik for bg = 0, 0.1, 0.3, and 0.5, respectively. . . . . . . . . . . . . .
(a) Two test ion trajectories (black solid and dashed lines) and E x
(color) under bg = 0. (b) The time-dependent spatial separation
between the ions in (a). (c) µE×B (blue) and ln α (red) along the
particle trajectory represented by the black solid line in (a). The
red dashed line represents the stochastic heating criterion, above
which 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. . . . . . . . . . . . . . . . . . . . . . . . .
(a) ln α, (b) Ti , (c) Ti⊥ for ions with mi /me = 500 under bg = 0.3. . .

65
71
75
76
77
78
79

81
82

xv
6.1

6.2

6.3
6.4

6.5

6.6

6.7
7.1
7.2
7.3

(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-valley
motion, and (d) that of the particle undergoing one-valley motion.
The wave parameters were κ = 0.01, α = 0.25, and n(α) = 18 from
Eq. 6.27. (e), (f) The approximated pseudo-potentials χ obtained
by keeping only the term involving sκ0 in Eq. 6.7 for the respective
particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
fξ for different (a) θ, (b) α and (c) n values. The default values are
θ = 0.1, α = 0.25, and n = 10. The black dashed line is the resonant
condition ξ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
ptv as a function of α and θ for different κ values. . . . . . . . . . . . 93
(a) Regions of initial momentum space (dark green) that satisfy the
unapproximated 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 a
single particle trajectory for a range of initial particle 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
parameters were α = 0.25, κ = 0.005, and n = 18 from Eq. 6.27. . . 94
Pitch-angle range of 10,000 particles whose initial momenta were
randomly sampled from Eq. 6.18 for different κ values. Red points
represent particles that meet the two-valley criterion (Eq. 6.16), and
the text inside represents the percentage of red particles. The red
horizontal lines represents the median ∆θ ◦pitch of the red particles . . 95
Pitch-angle change per wave period of the respective simulations in
Fig. 6.5. The red horizontal lines respectively represent the median
value of the pitch-angle change per wave period of the red particles. . 96
Same as Fig. 6.4, but for κ = 0.02. . . . . . . . . . . . . . . . . . . . 97
A diagram of a laser-induced fluorescence scheme. . . . . . . . . . . 107
Two-photon interaction of an atom in the lab frame and in the atom
frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Two-photon laser-induced fluorescence scheme for Argon II. Displayed wavelengths are vacuum values. . . . . . . . . . . . . . . . . 109

xvi
7.4

7.5

7.6
7.7
7.8

7.9

7.10

7.11

A.1

A.2
B.1
B.2

Example of two-photon LIF in Rubidium. (a) The original Dopplerbroadened spectrum. (b) Result after applying two photons. (c),(d),(e),(f)
Zoom-ins of respective peaks, showing hyperfine splittings. Reproduced from Jacques et al. [77]. © European Physical Society. Reproduced by permission of IOP Publishing. All rights reserved. . . . 110
Two-photon LIF observation of Zeeman splitting of a Rubidium hyperfine line. Reproduced from Jacques et al. [77]. © European
Physical Society. Reproduced by permission of IOP Publishing. All
rights reserved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
A schematic diagram of the RF plasma source. . . . . . . . . . . . . 112
The resultant plasma discharge. . . . . . . . . . . . . . . . . . . . . 114
Experimental setup for two-photon laser-induced fluorescence. L
stands for lens, M for mirror, BS for beam-splitter, AOM for acoustooptic modulator, OI for optical isolator, and PMT for photo-multiplier
tube. The purple color is the end-on view of the plasma. . . . . . . . 115
One-photon LIF results for 1, 5, and 20 mTorr. The laser wavelength
is scanned around the 394.5 nm line in Fig. 7.3 and the plots show
the subsequent 442.7 nm fluorescence as a function of the laser wavelength. 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. . . . . . . . . . . . . . . . . . . . . 118
The expected locations of the Zeeman splitted transition lines involving the 1 → 3 transition in Fig. 7.3 for |B| = 50 G. The red line is
the original un-splitted line. . . . . . . . . . . . . . . . . . . . . . . 120
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 the expected Zeeman splitting for |B| = 50 G. . . . . . . 121
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
numerical integration of Eq. A.73 is dubious at r = 0 (black dashed
line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
The fast Fourier transform of the result in Fig. A.1. The red lines
represent the dispersion relation in Eq. A.80 for Ω = 0.35. . . . . . . 156
Bφ at t = 80|ωce | −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
The fast Fourier transform of the results in Fig. B.1. The red lines
represent the dispersion relation in Eq. A.80 for Ω = 0.35. . . . . . . 158

xvii

LIST OF TABLES
Number
Page
2.1 Dimensional quantities and their normalization parameters in Chapter 2. 22
2.2 Comparison of simulated γs and calculated γc for different length
parameters of the current sheet . . . . . . . . . . . . . . . . . . . . . 30
3.1 Dimensional quantities and their normalization parameters in Chapter 3. 37
5.1 Dimensional quantities and their normalization parameters in Chapter 5. 72
7.1 Plasma parameters of regions where magnetic reconnection occurs
or 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 specified
otherwise, di is the ion skin depth assuming that the ion species is
hydrogen. Referenced from Refs. [29, 74–76, 109, 127] . . . . . . . 103
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 the experimentally measured values in Konjević
et al. [83] and using the scaling ∝ ne for Argon [173]. Zeeman
splitting is calculated by assuming m j and g j are of order 1, and
B = 0.1 T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Chapter 1

INTRODUCTION TO PLASMA PHYSICS
A 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 plasma mutually
interact with one another, a plasma tends to develop collective behavior. Plasma
physics is the study of this behavior in various circumstances. The majority of
the observable matter in the Universe is in the plasma state, examples of which
include stars, planetary nebulae, the Earth’s magnetosphere, astrophysical accretion
disks and jets, and the interstellar medium. Naturally occurring plasmas on the
Earth include lightning and flames, while man-made plasmas are found in fusiongrade experiments, research experiments, semiconductor manufacturing, and so on.
In contrast to conventional gas or liquid dynamics which are dominated by interparticle binary collisions, plasma dynamics are dominated by long-range electric
and magnetic fields. The coupling of these fields with fluid effects leads to complex
plasma behavior.
Plasma dynamics can be described by four different models, which are — in order of
decreasing fundamentality — single-particle theory, Vlasov theory, two-fluid theory,
and magneto-hydrodynamics (MHD). Single-particle theory is the most fundamental
and follows individual dynamics of all mutually-interacting particles in the plasma
and their interaction with electromagnetic fields. Vlasov theory constructs velocity
distribution functions from a statistically large number of particles and follows the
temporal evolution of these functions. Two-fluid theory treats the plasma as a system
of finite-pressure fluids, one for each species. MHD treats the plasma as a single
conducting fluid with finite pressure. The most suitable model to use depends on
the given situation; the specified plasma regime and the underlying assumptions
determine 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 this
chapter, the fundamentals of each model are first presented as a proper introduction
and background. Then, an introduction to magnetic reconnection — the main
subject of this thesis — will be presented.

1.1

Single-Particle Theory

The equation of motion for a particle with charge q and mass m acted on by electric
and magnetic fields E and B is given by the Lorentz equation

dv
= qE + qv × B,
dt

(1.1)

where v is the particle velocity. This equation will now be solved iteratively to obtain
a hierarchy of slow and fast motions. The derivations will be more detailed than
usually necessary in order to highlight the underlying assumptions, the violation of
which will be important in Chapter 4. E and B are assumed to evolve adiabatically,
i.e., slowly change in space and time.
Cyclotron motion
Dividing Eq. 1.1 by q|B| yields
1 dv
+v×
ωc dt
|B|
|B|

(1.2)

where ωc = q|B|/m is the cyclotron frequency. Assuming E = 0, we have
1 dvc
= vc ×
ωc dt
|B|

(1.3)

(1.4)

Using Eq. 1.3, this becomes
E + vE × B = 0,
and the solution is

(1.5)

E⊥ × B
(1.6)
B2
where E⊥ is the electric field perpendicular to B. This steady drift is called the
E × B drift.
vE =

Polarization drift
Next, let us define v p to be the next-order correction to vE so that |dv p /dt|

(1.8)

for which the solution is

m dE⊥
qB2 dt
This drift is called the polarization drift.
vp =

(1.9)

µ conservation
The motion of the particle after averaging out the cyclotron motion is called guidingcenter motion. In the frame of reference moving with the guiding-center velocity
vgc = vE + v p , the magnetic moment
µ=

mvc2
2B

(1.10)

is a conserved quantity.
Breakdown of Hierarchy
The drift solutions presented here rely on a delicate set of assumptions which, if
violated, lead to non-adiabatic particle behavior. In fact, it will be shown in detail
in Chapter 4 that if the drift hierarchy breaks down, chaotic behavior arises. In this
case, a fast, strong heating of associated particles occurs.
1.2

Vlasov Theory

Let us now consider the dynamics of a statistically large number of particles. The
instantaneous particle density in phase space is called the distribution function and
is denoted f (x, v, t). Conservation of the number of particles signifies that the rate
of 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
∂f
+v·
· (a f ) = 0,
∂t
∂x ∂v

(1.11)

where a (x, v, t) is the particle acceleration.
Charged particles are subject to the Lorentz force, i.e.,
a L = (E + v × B) .
(1.12)
Although a L is a function of v, a L 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
∂f
∂f
+v·
+ (E + v × B) ·
= 0,
(1.13)
∂t
∂x m
∂v
which, together with Maxwell’s equations, is called the Vlasov-Maxwell system of
equations.
1.3

Two-Fluid Theory

Because x and v are independent quantities in phase space, Eq. 1.11 can be expressed
as
∂f
· (v f ) +
· (a f ) = 0.
(1.14)
∂t
∂x
∂v
By taking moments of Eq. 1.14, fluid equations can now be obtained.
Continuity Equation
Since f is the instantaneous density in phase-space, taking its zeroth moment yields
the instantaneous spatial density, i.e.,
n (x, t) =
f d 3 v.
(1.15)
Taking the first moment yields the mean fluid velocity:
v f d3v
u (x, t) =
(1.16)
n (x, t)
Also, by Gauss’s Law,
· (a f ) d v =
a f dsv,
(1.17)
∂v
where sv is the surface vector at v → ∞. Because the distribution function must
vanish 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 zeroth
moment and recalling that x, v, and t are independent variables thus yield the
continuity equation
∂nσ
· (nσ uσ ) = 0,
(1.18)
∂t
∂xσ
where the subscript σ represents each species, e.g., σ = i, e respectively for ions
and electrons.

Fluid Equation of Motion
Let us now write velocity v is the sum of its mean value and its random part so that
v = u + v0 and define the pressure tensor
p = m v0v0 f d 3 v0 .
(1.19)
Here d 3 v = d 3 v0 and v0 f d 3 v0 = 0 because the mean value of the random part is
zero. Then, taking the first moment of Eq. 1.13 and integrating the third term by
parts yield the fluid equation of motion
nσ mσ

Duσ
= nσ qσ (E + uσ × B) − ∇ · pσ,
Dt

where

+ uσ · ∇
Dt ∂t

(1.20)

(1.21)

is the convective derivative.
Equations 1.18 and 1.20 for ions and electrons constitute the system of two-fluid
equations.
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 which
are linear combinations of ui and ue . These are the current density
J=
nσ qσ uσ,
(1.22)

and the center-of-mass velocity
U=

1Õ
mσ nσ u σ ,
ρ σ

(1.23)

where
ρ=

mσ nσ

(1.24)

is the mass density. Writing particle velocity v as a sum of the center-of-mass
velocity and a random part, i.e., v = U + v0, the MHD pressure tensor is defined as
PMHD =
mσ
v0v0 fσ d 3 v0 .
(1.25)

Multiplying Eq. 1.13 by mσ , taking its zeroth and first moment and summing over
all species, we obtain respectively the MHD continuity equation
∂ρ
+ ∇ · (ρU) = 0,
∂t
and the MHD equation of motion
∂U
+ U · ∇U = J × B − ∇ · PMHD .
∂t
These equations, together with Maxwell’s equations,
ρc
∇·E= ,
0

(1.26)

(1.27)

(1.28)

∇ · B = 0,

(1.29)

∂B
∂t
∇ × B = µ0 J,
∇×E=−

(1.30)
(1.31)

constitute the set of MHD equations. Note that the pre-Maxwell form of Ampère’s
law was used because MHD usually deals with phenomena with characteristic velocities 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 the
thesis.
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’s
law gives J = ∇ × B/µ0 , we can define plasma beta as

∇·P
β= 2
B /2µ0 J × B

(1.32)

As long as the scale length of the two forces are comparable, β measures the ratio
of the hydrodynamic force to the magnetic force. Low-β plasmas are therefore
dominated by magnetic forces and vice versa.
Generalized Ohm’s Law
Dividing the electron fluid equation by ne qe = −ne and rearranging give

me Due ∇ · pe
E + ue × B = −
(1.33)
e Dt
ne
Since ue = ui − J/ne e and ui ≃ U due to the much heavier ion mass, Eq. 1.33
becomes the generalized Ohm’s law

J × B me Due ∇ · pe
E+U×B=
+ ηJ,
ne
e Dt
ne

(1.34)

Figure 1.1: Incremental change in surface area due to a displacement Uδt of a line
segment dl.
where ηJ is introduced to represent resistive effects due to collisions. The terms on
the 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 plasma
dynamics will become apparent in Chapters 2 and 3.
Ideal Ohm’s Law and Frozen-in Flux
If the right-hand side of Eq. 1.34 is neglected, then we obtain the ideal Ohm’s law
E + U × B = 0.

(1.35)

In regimes where this equation is valid — typically involving very slow time scales
and very large spatial scales — the plasma acts as a perfectly conducting fluid and
is 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
= ∇ × (U × B) .
∂t

(1.36)

The induction equation signifies an important aspect of ideal MHD, namely that
magnetic flux through any surface moving with the plasma is conserved. To prove
this property, we define the magnetic flux through a moving surface S(t) bounded
by C(t) as
ΦB =

B · ds.

(1.37)

The rate of change of ΦB is determined by that of B and that of S. The former is
simply ∂B/∂t. The latter can be found by noting that the incremental change in
surface area due to a displacement Uδt of a line segment dl of the bounding contour

C is Uδt × dl (illustrated in Fig. 1.1). Therefore, the rate of change of S is U × dl,
and
dΦB
∂B
· ds +
B · U × dl,
(1.38)
dt
∂t
∂B
· ds +
B × U · dl,
(1.39)
S ∂t
∫
∂B
+ ∇ × (B × U) · ds.
(1.40)
S ∂t
It can be seen that if Eq. 1.36 is true, then dΦB /dt = 0. This property of ideal
MHD is called the frozen-in flux phenomenon.
Magnetic Flux Tubes
The concept of frozen-in flux in ideal MHD equivalently means that magnetic
fields lines are frozen into the plasma. By connecting a continuous sequence of
surfaces, a tube-like structure can be constructed. If magnetic field lines penetrate
this structure, it is called a magnetic flux tube. Figure 1.2 shows a sketch of two
deforming magnetic flux tubes. The frozen-in field lines move with the tubes as
they deform.
It should be noted that magnetic flux tubes are flux tubes associated with the magnetic
field B, and that these are not the only kind of flux tubes. In fact, it will be shown
in Chapter 2 that we can also define flux tubes associated with a quantity called the
canonical vorticity Q and that these Q flux tubes have similar but separate dynamics
from those of magnetic flux tubes.
Magnetic Helicity
Magnetic helicity is an important concept in MHD and is defined as
K=
A · Bd 3 r,

(1.41)

where A is the vector potential associated with the magnetic field B and V is the
plasma volume. This quantity measures the linkage and twist of magnetic flux tubes.
Although A is undefined with respect to a gauge, K turns out to be gauge-independent
if magnetic field lines do not penetrate the boundary of the volume.
Magnetic helicity is a conserved quantity in ideal MHD. To prove this, consider the

Figure 1.2: Frozen-in magnetic field lines within deforming magnetic flux tubes.
time evolution of the helicity density κ = A · B
∂B
∂κ ∂A
·B+A·
∂t
∂t
∂t
= − (E + ∇φ) · B − A · (∇ × E) ,

(1.43)

= −2E · B − ∇ · (φB + E × A) .

(1.44)

(1.42)

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
ds · (φB + E × A) = 0,
(1.45)
∂t
where ds is the closed surface integral over the enclosed volume. Assuming that
the magnetic field and the plasma flow do not penetrate the volume, i.e., B and U
are tangential to the boundary at the boundary, E = −U × B is purely perpendicular
to the boundary (i.e., E k ds, so ds · (E × A) = 0). Thus, the surface integral in Eq.
1.45 vanishes and
K = const.,
(1.46)

Figure 1.3: A graphical description of a torsional Alfvén wave.

so the total magnetic helicity is conserved for an ideal plasma.
Alfvén Wave
The normal mode of ideal MHD is called the Alfvén wave [3], and there are two
modes of this wave: the compressional mode and the shear mode. The former
involves compressions and rarefactions of the magnetic field and the latter involves
transverse perturbations to the magnetic field. Both modes travel with the Aflvén
velocity
(1.47)
vA = √
µ0 ρ
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 is
a class of shear Alfvén waves. A graphical description is shown in Fig. 1.3. An
azimuthal source (red circle) twists the magnetic field lines (black lines) within
a magnetic flux tube, and this twist propagates away from the source as torsional
Alfvén waves. This twist mechanism will be an analogy to the generation mechanism
of another wave named "whistler waves" in Section 2.5.
1.5

Magnetic Reconnection

Magnetic reconnection is a plasma process in which opposing magnetic field lines
come together, annihilate, reconnect, and release their stored magnetic potential

(a)

(b)

Figure 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 reconnection is almost always accompanied by strong ion and electron acceleration and
heating, as evidenced by ion jets [85], abnormally high ion temperatures [74], and
extreme-ultraviolet and X-ray emissions [29, 105]. Since reconnection acts to lower
the magnetic energy in a plasma while nearly preserving magnetic helicity, it is an
important part of relaxation processes that lead to equilibrium plasma states with
minimum magnetic energy [147].
In ideal MHD, magnetic field lines are frozen into the plasma. Therefore, opposing
field lines cannot annihilate or break because two counter-propagating plasma flows
stagnate 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 stagnating
plasma inflows (green). Thus, magnetic reconnection is forbidden in ideal MHD
due to the ideal Ohm’s law (Eq. 1.35) and can only happen if the frozen-in condition

12
is violated.
Sweet-Parker Reconnection
One means of breaking the frozen-in condition is finite resistivity. If only the
resistive term in Eq. 1.34 is kept, we have the resistive Ohm’s law
E + U × B = ηJ.

(1.48)

The mechanism by which the resistive term allows for reconnection becomes clear
after taking the curl of the above equation, using Faraday’s law to eliminate the
electric field, and rearranging:
∂B
= ∇ × (U × B) + ∇2 B,
∂t
µ0

(1.49)

−1
−1 2
where Ampère’s law was used so ∇×J = µ−1
0 ∇×(∇ × B) = µ0 ∇ (∇ · B)− µ0 ∇ B =
−µ−1
0 ∇ B. The first term on the right-hand side portrays the frozen-in condition. The
second term, on the other hand, is a diffusion-like term with the diffusion coefficient
η/µ0 . Therefore, finite resistivity allows for magnetic field diffusion across plasma
and thus magnetic reconnection as shown in Fig. 1.4b.

Using this picture, Sweet and Parker [119] made quantitative predictions of the
reconnection rate, i.e., how fast the magnetic field lines reconnect. However, the
theory predicted reconnection rates that were far slower than what had been observed.
For example, magnetic field diffusion predicted the time scale of solar coronal
eruptions to be months to years, whereas the actual observed time scale is minutes
to hours. As enticing as this theory is, it does not accurately describe what is actually
happening in most physical situations.
Fast Collisionless Reconnection
Since resistivity is not a fast enough source for magnetic reconnection, one must
look to other terms in Eq. 1.34,
E+U×B=

J × B me Due
ne
e Dt

(1.50)

where only the Hall term and the electron inertia term are kept, and a low-β plasma
is assumed so the pressure term is dropped. Let us now investigate the conditions
for 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 the
order of the Alfvén velocity v A = B/ µ0 ρ ≃ B/ µ0 nmi and using J ∼ B/µ0 L

13
where L is the length scale of the magnetic field, the following comparison applies:
J×B
ne
B2
B2
µ0 nmi µ0 neL
mi
L:
µ0 ne2
U×B:

L : di,

(1.51)
(1.52)
(1.53)
(1.54)

where di = c/ ne2 /mi 0 = c/ω pi is the ion collisionless skin depth. Therefore, if
the 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 term
dominates, 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. Since
electrons are much lighter than ions, they carry most of the current, so ue ≃
−J/ne ∼ B/µ0 neL. Considering the time-dependent part of the inertia term,
writing ∂/∂t ∼ ω,
J × B me ∂ue
ne
e ∂t
B2
me ωB
µ0 neL e µ0 neL
ωce : ω,

(1.55)
(1.56)
(1.57)

where ωce = eB/me is the electron cyclotron frequency. Now considering the
time-independent part of the inertia term,
J × B me
ue · ∇ue,
ne
B2
me
B2
µ0 neL e µ20 n2 e2 L 3
me
L2 :
µ0 ne2
L 2 : de2,

(1.58)
(1.59)
(1.60)
(1.61)

where de = c/ ne2 /me 0 = c/ω pe is the electron collisionless skin depth. Thus,
the electron inertia term becomes as important as the Hall term if ω ∼ ωce and/or
L ∼ de . Reconnection phenomena mostly involve ω
ωce , so we conclude
that the Hall term is important at di length scales and the electron inertia term
is important at de length scales. Thus, at length scales comparable to or smaller

Figure 1.5: Fast reconnection geometry.

than di , fast, collisionless magnetic reconnection consistent with observations may
occur. Henceforth, the term "magnetic reconnection" will be used interchangeably
with "fast reconnection" and/or "collisionless reconnection."
A graphical description of fast reconnection is shown in Fig. 1.5. The inset shows
the set of coordinates that will be used throughout the analysis. The initial opposing
magnetic field lines are in the y-direction and the reconnected fields are in the xdirection. The particle inflow is mainly in the x-direction and the outflow is mainly
in the y-direction. The z-direction represents the out-of-plane components. Since
the Hall term signifies the difference between ion and electron flows and becomes
important at L ∼ di , ions "unfreeze" from the magnetic field below these scales and
flow outwards (green arrows). Electrons, on the other hand, are still frozen to the
magnetic field down to L ∼ de . Only below these scales can the magnetic fields
reconnect and the electrons flow outwards (red arrows). The particular shape of
the 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 as
other facets of reconnection will be explained more intuitively in Chapter 2.
The question of how the macroscopic system couples to the microscopic scales
required for collisionless reconnection naturally arises. There are various ways this
could be achieved, and some examples include the kink-induced Rayleigh-Taylor
instability [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 outof-plane magnetic field exists, this is called the guide field. Effects due to the guide
field as well as finite electron pressure will be discussed in Chapter 3.

15
Magnetic
Reconnection

Progression

Properties

Progenies

Chapter 2

Chapter 4

Chapter 6

Canonical
Vorticity
Framework

Stochastic
Ion Heating
Mechanism

Chapter 3
Completion of
the Framework

Chapter 5
Kinetic
Verification

Relativistic
Whistler-WaveParticle
Interaction

Chapter 7

Magnetic Field
Laser
Diagnostic

Figure 1.6: Thesis outline.

1.6

Thesis Outline

Figure 1.6 shows the outline of this thesis. The progression of magnetic reconnection
will be studied in Chapters 2 and 3. In Chapter 2, the canonical vorticity framework
will be constructed and collisionless magnetic reconnection will be described using
this framework in an intuitive manner. In Chapter 3, the framework will be extended
to kinetic regimes by the introduction of the canonical battery term.
A crucial property of magnetic reconnection, namely its accompaniment by anomalous, extreme ion heating, will be studied in Chapters 4 and 5. In Chapter 4, the
canonical vorticity framework will be used to establish the existence and importance
of the stochastic ion heating mechanism in collisionless reconnection. In Chapter
5, this mechanism will be analytically and numerically verified to be the main ion
heating mechanism up to moderate guide fields.

16
An important progeny of collisionless reconnection is whistler waves, and their
generation mechanism will be discussed in Chapter 2. In Chapter 6, the interaction
between whistler waves and relativistic particles will be studied in detail. In particular, how coherent waves significantly scatter the pitch-angle of relativistic particles
will be illustrated.
A new method of probing magnetic fields is described in Chapter 7. This method
uses two laser beams to excite the electron population in a plasma while suppressing
thermal broadening effects in order to resolve the magnetic field information.

17
Chapter 2

CANONICAL VORTICITY FRAMEWORK OF MAGNETIC
RECONNECTION: AN INTUITIVE DESCRIPTION
As shown in Section 1.5, collisionless magnetic reconnection is an extremely complicated process involving different length scales and multiple spatial dimensions.
In this Chapter, a relatively intuitive description of collisionless reconnection will
be given. This is achieved by considering the dynamics of a quantity called the
canonical vorticity, which is a combination of the particle flow and the magnetic
field, instead of focusing solely on the magnetic field. Unlike the Sweet-Parker
description 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 Vorticity

The quantity "canonical vorticity" is defined as
Qσ = mσ ∇ × uσ + qσ B,

(2.1)

= mσ wσ + qσ B,

(2.2)

where uσ is the flow velocity of each species σ and wσ = ∇ × uσ is the vorticity of
each species. By writing B = ∇ × A,
Qσ = ∇ × (mσ uσ + qσ A) ,

(2.3)

= ∇ × Pσ,

(2.4)

where Pσ = mσ uσ + qσ A is the canonical momentum of each species. Therefore,
canonical vorticity Qσ is the curl of the species canonical momentum Pσ , hence the
name. 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 Vorticity
Consider the two-fluid equation (Eq. 1.20)

∇ · pσ
∂uσ
mσ
+ mσ uσ · ∇uσ = qσ (E + uσ × B) −
∂t
nσ

(2.5)

where the convective derivative was decomposed into partial derivatives. The
pressure term will now be dropped by simply assuming low β, or by assuming both

isotropic (∇ · p = ∇p) and barotropic (∇p × ∇n = 0) pressure when intending to
take the curl of this equation so that

∇ · pσ
∇pσ
∇×
=∇×
= 2 ∇pσ × ∇nσ = 0.
nσ
nσ
nσ

(2.6)

Taking the curl of Eq. 2.5 and using Faraday’s law to eliminate E,
∂∇ × uσ
∂B
+ mσ ∇ × (uσ · ∇uσ ) = −qσ
+ qσ ∇ × (uσ × B) .
(2.7)
∂t
∂t
Using the identities ∇ V 2 /2 = V · ∇V + V × ∇ × V and ∇ × ∇S = 0 for any vector
V and scalar S,
mσ

∂B
∂∇ × uσ
− mσ ∇ × (uσ × ∇ × uσ ) = −qσ
+ qσ ∇ × (uσ × B) .
∂t
∂t
Writing ∇ × uσ = wσ and rearranging,
mσ

∂ (mσ wσ + qσ B)
= ∇ × (uσ × [mσ wσ + qσ B]) ,
∂t

(2.8)

(2.9)

∂Qσ
= ∇ × (uσ × Qσ ) ,
(2.10)
∂t
which is the canonical induction equation. Equation 2.10 is an extremely important
result 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 frozen
into the plasma flow U, Eq. 2.10 means that the canonical vorticity Qσ is frozen
into the species flow uσ . It also equivalently means that the flux associated with
Qσ ,
ΦQ =

Qσ · ds,

(2.11)

is a conserved quantity, and that flux tubes associated with Qσ can therefore be
defined. Each species has its own flux tubes that convect with the respective species
flow. 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., and
the 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, collisionless
magnetic reconnection is allowed because the topological constraint on B in MHD
is relieved in the two-fluid regime.
The frozen-in property of Qσ and the interchangeability between wσ and B within
Qσ are the crux of the intuitive description of collisionless reconnection that will
be given in this chapter.

19
2.2

Assumptions

A couple of assumptions will now be made in order to readily relate the concept of
canonical vorticity to collisionless reconnection.
Stationary Ion Assumption
The 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 be
assumed to be stationary; detailed ion dynamics will be presented in Chapter 4. The
stationary ion assumption implies that the current is carried solely by the electrons,
i.e.,
∇×B
ue = −
=−
(2.12)
ne e
µ0 ne e
The electron canonical vorticity then becomes
Qe = me ∇ × ue + qe B,
me
=−
∇ × (∇ × B) − eB,
µ ne
0 e
me
=e −
∇ × (∇ × B) − B ,
µ0 ne e2
= e de2 ∇2 B − B ,
where

me
1 m e 0
c2
= de2
µ0 ne e
ω pe
0 0 ne e

(2.13)
(2.14)
(2.15)
(2.16)

(2.17)

and ∇ × (∇ × B) = ∇ (∇ · B) − ∇2 B = −∇2 B were used. Writing ∇ ∼ 1/L, the
Laplacian term which arises from finite electron inertia scales as de2 /L 2 . Therefore,
at length scales much larger than the electron skin depth, i.e., L
de , the magnetic
term dominates and Qe is topologically indistinguishable from B. At L
de , the
inertial term dominates so Qe ≃ me we = ede2 ∇2 B. At L ∼ de , both the inertial and
magnetic terms are important.
Incompressibility Assumption
Since ions are assumed to be stationary, a local change in electron density will
result in a local change in the charge density. This violates a fundamental plasma
assumption that plasmas are quasi-neutral, i.e., charge-neutral over scale lengths
much larger than the Debye length λD . Now, λD /de scales as vT e /c where vT e is
the electron thermal velocity, so vT e
c means that λD
de . Since de is the
smallest length scale in our analysis of collisionless reconnection, incompressibility

20
is a valid assumption as long as the electron thermal velocity is much smaller than
the speed of light. Thus, a time-independent, spatially uniform electron density will
be used throughout the analysis.
Mathematically, incompressibility means that
∇ · ue = 0.

(2.18)

This can be seen by rearranging the electron continuity equation,
∂ne
+ ∇ · (ne ue ) = 0,
∂t

∂ne
+ ue · ∇ne + ne (∇ · ue ) = 0.
∂t

(2.19)
(2.20)

Therefore, Dne /Dt = 0 if and only if ∇ · ue = 0.
Assuming incompressibility is equivalent to assuming that the displacement current
is negligible. This fact is illustrated by taking the divergence of the Ampère’s law
including the displacement current assuming stationary ions:
∇ · ∇ × B = −∇ · (µ0 ne eue ) + µ0 0
0 = −∇ · (ne ue ) −

∂ne
∂t

∂∇ · E
∂t

(2.21)
(2.22)

where Gauss’s law was used so 0 ∇ · E = (ni − ne ) e and ∂ni /∂t = 0 because ions
are stationary. This result clearly shows that the time-dependence of ne is equivalent
to finite displacement current.
This set of assumptions and the ensuing system of equations have historically been
called electron-magnetohydrodynamics (EMHD) [25]. Studies of magnetic reconnection in similar formulations have been done before [17, 25, 27], albeit not in the
three-dimensional, intuitive manner that will be presented here.
2.3

Numerical Construction

A numerical simulation was developed in order to verify the ensuing analysis and
graphically represent the results. The algorithm that solves Eq. 2.10 will now be
presented. The code was written in Python 3.6.
Normalization
In order to facilitate the scaling of the simulation to real-life situations, relevant
equations must be made dimensionless, i.e., normalized with respect to a set of

21
dimensional quantities. For example, dividing Eq. 2.12 by de |ωce | where |ωce | =
eB0 /me and B0 is a reference magnetic field,
ue
me ∇ × B
=−
de |ωce |
µ0 ne e2 B0 de
= −de ∇ × .
B0

(2.23)
(2.24)

Normalizing each quantity, i.e., ūe = ue /de |ωce |, B̄ = B/B0 , and ∇¯ = de ∇, we have
the 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)

= ∇¯ 2 B̄ − B̄.

(2.27)

The normalized electron canonical induction equation is
me
me ∂Qe
= 2 2 ∇ × (ue × Qe )
e B0 ∂t
e B0
∂ Q̄e
ue
= de ∇ ×
× Q̄e
∂ t¯
de |ωce |
∂ Q̄e
= ∇¯ × ūe × Q̄e ,
∂ t¯

(2.28)
(2.29)
(2.30)

where t¯ = eB0 t/me = |ωce | t. Using Eqs. 2.25 and 2.27, Eq. 2.30 can be written
entirely in terms of B̄:

∂ ¯2
∇ B̄ − B̄ = −∇¯ × ∇¯ × B̄ × ∇¯ 2 B̄ − B̄ .
(2.31)
∂ t¯
Therefore, the entire system can be expressed solely as a function of the magnetic
field.
From this point on the bars will be dropped to reduce notational clutter. Only
normalized quantities will be used unless specified otherwise. A summary of the
normalizations used in this chapter is illustrated in Table 2.1.
Discretization
2.5D Cartesian coordinates are used. This means that any quantity σ at any given
time t is defined on a 2D grid as σi,t j , where i and j are the x and y coordinates, and

22
Quantity
ue
Qe

Normalization
|ωce | −1
de
de |ωce |
B0
eB0

Table 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 coordinate
is an ignorable coordinate so that ∂/∂z → 0. The central differentiation scheme is
used for the spatial derivative in the x-direction, i.e.,
σi+1,
∂σ
j − σi−1, j
∂x
2∆x

(2.32)

for any quantity q and similarly for the y-direction. The second spatial derivative is
expressed as
σi+1,
∂2σ
j + 2σi, j − σi−1, j
(2.33)
∂ x2
∆x 2
The explicit forward differentiation scheme is used for the time derivative, i.e.,
σi,t+1
∂σ
j − σi, j
=s→
= si,t j
∂t
∆t

(2.34)

for any source term s.
Algorithm
The 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+1
e using Eq. 2.30.
3. Obtain B̄t+1 by solving Eq. 2.27 iteratively using the relaxation method [84] .
4. Repeat.
The relaxation method in step 3 can be explained as follows. Consider the discretized
form of Eq. 2.27 at any given time:
Qi, j =

Bi+1, j + 2Bi, j − Bi−1, j Bi, j+1 + 2Bi, j − Bi, j+1
− Bi, j .
∆x 2
∆y 2

(2.35)

23
This equation can be iteratively solved for B̄i, j :

+ 2 −1
Bi, j n+1 =
∆x
∆y

−1
Qi, j −

Bi+1, j n − Bi−1, j n
∆x 2

!
Bi, j+1 n − Bi, j−1 n
∆y 2

(2.36)
where n is each iterative step. The iteration is halted when the difference of B at any
two consecutive iterative steps reaches below a specified value of tolerance. This
method does not always yield a converging, stable solution; convergence tests must
be first conducted in order to avoid spurious numerical solutions.
Algorithm Test
The 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 conducted
to validate the code. In Appendix A, an analytical solution for whistler waves is
presented in detail. In Appendix B, tests of the algorithm against the prediction are
conducted and confirm that the code produces whistler waves.
Initial Conditions
The magnetic field profile is the Harris magnetic shear profile [66] (see Appendix
E), or
By = tanh ,
(2.37)
where λ is the shear length scale. This profile represents two regions of opposing
magnetic fields separated by a region of magnetic null.
By Ampère’s law, this magnetic field is produced by the out-of-plane current profile
uez = −Jz = −

∂By
= − cosh−2 ,
∂x

(2.38)

so the electrons are flowing in the −z-direction near x = 0. This configuration is
frequently called a "current sheet" because of this current profile.
Reconnection is initiated by a 2D-localized vector potential perturbation of the form
x2
y2
Az = exp − 2 −
(2.39)
2λ
2σy2
where is the perturbation strength and is a small number, e.g., 10−5 , and σy is the
perturbation length scale in the y-direction. The perturbation magnetic fields from

Background

Perturbation

Result

Figure 2.1: The background and perturbation magnetic fields and their sum as the
initial condition for magnetic reconnection.

this vector potential are
x2
y2
∂ Az
= − 2 exp − 2 −
B̃x =
∂y
2λ
σy
2σy2
∂ Az
x2
y2
B̃y = −
= 2 exp − 2 −
∂x
2λ
2σy2

(2.40)
(2.41)

The background and perturbation magnetic fields and their sum are shown graphically in Fig. 2.1. The perturbation adds to the background field in such a way that
field lines close to the center (x = 0) are reconnected; those that are further away
come closer together to the center at a localized point in y; i.e., the field lines bow
toward the center.
This initial condition triggers the instability that is fast magnetic reconnection;
analytical and intuitive explanations for why an instability occurs will be illustrated
in Section 2.5.
Typical grid dimensions were x×y = 150×600 = 30de ×120de . Neumann boundary
conditions were used, but the simulation was halted before any significant physical
change reached the boundary.
2.4

Simulation Results

Figure 2.2 shows the time evolution of the magnetic field during collisionless reconnection. As shown by the streamlines, the in-plane magnetic fields spontaneously
come together at x = 0 and reconnect, accelerating electrons to high velocities (red
arrows). The quadrupole out-of-plane magnetic fields, represented by the yellow

Figure 2.2: Time evolution of the magnetic field during collisionless reconnection. 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 color
scales). These results — quadrupole Hall magnetic fields and the geometry of the
electron flow — are consistent with previous numerical and experimental studies of
Hall-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 lines
reconnect, Qe field lines do not reconnect and pile up near the center. The slight
reconnection of Qe is due to the initial perturbation and finite numerical resistivity
that originates from unavoidable small numerical error.
Figures 2.3c and 2.3d show three-dimensional views of B and Qe , respectively. This
is done by stacking the 2D simulation results in the z-direction and plotting the
3D vectors. It can be seen from Fig. 2.3c that the magnetic field lines generally
move in the direction of the out-of-plane electron flow uez (red arrow; recall that
the 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 to
Fig. 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 be
examined in the next Section.

Figure 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 color
of the lines represent the height in z. The red arrow in (c) is the direction of the
out-of-plane electron velocity uez .
2.5

Intuitive Description

Instability and Quadrupole Fields
Under the assumptions presented in Section 2.2, collisionless reconnection is a
purely growing instability with an exponential growth rate that depends on the
system parameters. This fact was demonstrated in Bellan [12] via an extensive
analytical calculation involving a pair of second-order differential equations that
form a fourth-order eigensystem. Here, an intuitive explanation of the exponential
growth using Qe flux tube dynamics is presented, followed by an analytical proof
much simpler than that in Bellan [12]. The origin of the quadrupole magnetic fields
in 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 Qe
flux tube is shown in Fig. 2.4. The small insets show the side-on and overhead
views 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 bows
towards the center where electrons are flowing downwards (red arrow; Fig.

Figure 2.4: Temporal evolution of a Qe flux tube. The insets are side-on and
overhead views of the flux tube. The red arrow represents the direction and location
of 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 tube
and so convects downwards faster (Figs. 2.4b and 2.4c).
3. This downward convection creates Q ez fields and thus Bz fields that scale like
Bz ∼ x y near the origin (colors in Fig. 2.3a). These are the quadrupole Hall
fields that are generic to collisionless reconnection.
4. Because Bz ∼ x y in the vicinity of the origin, uex = −∂Bz /∂ y ∼ −x, so
the 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 to
yield an instability.
It is now clear why an instability occurs, but to prove exponential growth, an
analytical calculation must be done.
Let us first explicitly prove that Qe does not reconnect by proving that Q ex is zero
on the y-axis. Reconnection of Qe would require development of finite Q ex on the
y-axis, so (Q ex ) x=0 = 0 means that Qe field lines stay connected. Expanding the
right-hand side of Eq. 2.30,
∂Qe
= Qe · ∇ue + ue (∇ · Qe ) − ue · ∇Qe − Qe (∇ · ue ) ,
∂t
= Qe · ∇ue − ue · ∇Qe,

(2.42)
(2.43)

28
where ∇ · Qe = 0 by the definition of Qe (Eq. 2.26) and ∇ · ue = 0 by the incompressibility assumption. Expressing this equation using the convective derivative,
DQe
= Qe · ∇ue,
Dt

(2.44)

DQ ex
∂uex
∂uex
= Q ex
+ Q ey
Dt
∂x
∂y

(2.45)

and its x-component is

where ∂/∂z → 0 was used. We now examine what happens at x = 0 if (Q ex ) x=0,t=0 =
0. The first term is obviously zero at t = 0. Because uex represents the electron
inflow and so is zero everywhere on the y-axis, the second term is also zero. Thus
at t = 0,
DQ ex
= 0.
(2.46)
Dt
Therefore, if (Q ex ) x=0 starts out with zero, it will remain zero. It has thus been
explicitly established that Qe does not reconnect, and equivalently that Q ex at or
near x = 0 is zero or extremely small.
Next, consider the y-component of Eq. 2.44:
∂uey
∂uey
DQ ey
= Q ex
+ Q ey
Dt
∂x
∂y

(2.47)

DQ ey
∂uey
= Q ey x≈0
Dt x≈0
∂ y x≈0

(2.48)

Near x = 0, Q ex ≃ 0, so

where x ≈ 0 means near but not exactly at x = 0. The solution is
∫ t

∂uey
Q ey x≈0,t = Q ey x≈0,t=0 exp
dt
∂y
x≈0

(2.49)

which is a purely growing solution (∂uey /∂ y > 0) because Bz has the profile
Bz ∼ γx y near x = 0 where γ is a positive value as can be seen from Fig. 2.2, and
∂uey /∂ y = ∂ 2 Bz /∂ x∂ y ∼ γ > 0 using Eq. 2.25.
This exponential growth propagates to other variables as well. Figure 2.5 shows the
time evolution of Q ez , Bz , and (∇ × ue )z at (x, y) = (1, 2)de . The y-axis is in natural
logarithmic scale. Because the initial conditions are not an eigenfunction of the
system, 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 grow
in time.

Figure 2.5: Time evolution of Q ez , Bz , and (∇ × ue )z at (x, y) = (1, 2)de . The y-axis
is in natural logarithmic scale.

Thus, we can conclude that any slight gradient of electron velocity along Qe field
lines is a sufficient condition for this instability that provides collisionless reconnection.
Bellan [12] predicted the growth rate in collisionless reconnection as (writing in
terms of the normalized variables used in this chapter)
2 1
γ=
(2.50)
10 λσy
Now, incompressibility dictates that the electron outflow uey satisfies uey λ = 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 as
γ≃

∂uey uey
∂y
σy
λσy

(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 lines
in, e.g., Fig. 2.5. The agreement between the two rates is evident.

σy
10
10
16
16

γs
40.19 ± 0.40
17.56 ± 0.10
12.95 ± 0.50
5.21 ± 0.05

γc
43.40
22.08
13.80
4.01

Table 2.2: Comparison of simulated γs and calculated γc for different length parameters of the current sheet
Electron Acceleration
Qe flux tube dynamics also explains how collisionless reconnection triggers extreme
electron acceleration. As the flux tube convects downward with the electron flow, the
tube lengthens and stretches like a rubber band as in Fig. 2.4c. Due to the assumed
incompressibility, the total volume of the flux tube must remain the same, so an
increase 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 thinning
means that the Qe lines bunch up together as in the top flux tube in Fig. 1.2. This
bunching-up signifies that the magnitude of Qe increases, as can be seen from the
colors of the tube in Fig. 2.4. Because Qe ≃ ∇ × ue at small length scales, this leads
to an increase of ue , i.e., electron acceleration.
This process can be more simply understood by recalling that ΦQ is a conserved
quantity, so
ΦQ = Qe · ds,
(2.52)
= ∇ × ue · ds − B · ds,
(2.53)
ue · dl − B · ds.
(2.54)

Magnetic reconnection involves the reduction of B, and flux tube thinning involves
the reduction of S and C. Therefore, as B → 0 and C → 0, ue → ∞ in order to
conserve ΦQ .
Thus, extreme electron acceleration is due to a combination of interchange between
the magnetic field and the electron vorticity within Qe , and Qe amplification due to
flux tube thinning.

31
Generation of Whistler Waves
Whistler waves are right-handed circularly polarized electromagnetic plasma waves
in the frequency range ωci, ω pi
ωce, ω pe with the dispersion relation
ω2pe /ω2
c2 k 2
|ωce |
ω2
cos θ − 1

(2.55)

where θ is the wave propagation angle with respect to the background magnetic field
B0 = B0 ẑ. In terms of normalized quantities used in this chapter, this dispersion
relation becomes
(2.56)
k 2 = cos θ
where bars are dropped to reduce notational clutter. Whistler waves are frequently
observed in association with magnetic reconnection in both laboratory [29, 67] and
space [40] plasmas.
Because the wave frequency is much larger than any ion-related frequency, ions can
be assumed to be stationary for a whistler wave. Incompressibility can then also be
assumed 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 normalized
background field ẑ and the normalized perturbed field B1 , i.e., B = ẑ + B1 . Equation
2.31 then becomes, to first order,
∂ 2
∇ B1 − B1 = ∇ × ([∇ × B1 ] × ẑ) .
(2.57)
∂t
Assuming B1 ∼ exp (ik · x − iωt) so that ∂/∂t → −iω and ∇ → ik,
iω 1 + k 2 B1 = k × ([k × B1 ] × ẑ) ,

(2.58)

= k × (k z B1 − kB1z ) ,

(2.59)

= k z k × B1 .

(2.60)

Crossing both sides by k and using Eq. 2.60 to eliminate k × B,
iω 1 + k 2 k × B1 = k z k × (k × B) ,
2
−ω2 1 + k 2 B1 /k z = k z k [k · B1 ] − k 2 B1 ,
2
ω2 1 + k 2 = k z2 k 2,
ω 1 + k = k 2 cos θ,

(2.61)
(2.62)
(2.63)
(2.64)

32
(a)

(b)

Figure 2.6: (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.
where k z = k cos θ was used. Solving this equation in terms of k 2 gives
k2 =

cos 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, which
induces a uez shear along a Qe flux tube. In Fig. 2.6a , the magnitude difference
between uez at two different locations is apparent (red arrows). Now, in a frame
traveling in the −z-direction with the center portion of the flux tube, uez at the two
locations 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 flux
tube twist, which acts as a source for whistler waves. This is analogous to how
torsional Alfvén waves are excited (Fig. 1.3).
To demonstrate this mathematically, the normalized electron canonical helicity density 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):
2
DκQ
ue
= Qe · ∇
− pe − ue · A ,
(2.66)
Dt
= Qe · ∇L,

(2.67)
u2

where pe is scalar electron pressure assuming pressure isotropy, and L = 2e −
pe − ue · A is the Lagrangian density of the system. This is the canonical helicity

33
conservation equation[13, 172] for this specific regime. The central electron flow
is a source of ∇L along the Q field lines, so it is in turn a source of canonical
helicity density. Whistler waves therefore represent a dynamic winding/unwinding
of Qe field lines just as torsional Alfvèn waves do of B lines. Therefore, whistler
wave generation and propagation in collisionless reconnection can be seen as the
central electron flow curling up the Qe flux tubes and whistler waves propagating
this helicity density in the ±y-directions.
Now we will focus on how the whistler wave manifests itself in B fields. Figure 2.2
shows that there is an additional region of out-of-plane magnetic fields that develops
near the outflow regions (e.g. black box in Fig. 2.2c). These fields, which have
been previously observed by other numerical studies as well [45, 54] but generally
not discussed in three-dimensional detail, suggest wave-like behavior, so it is worth
examining how a typical reconnected B field line (e.g. red line in Fig. 2.2b) evolves
taking these fields into account.
Figure 2.7 shows the 3D time evolution of a typical reconnected B field line. We
can regard the initial field line as the background field, and the final field line as
the background field plus the perturbations to this field. If the initial field line is
stretched 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 area
of strong uez (red rectangle) acting as the source of two temporally right-handed
circularly polarized whistler waves propagating away from this source in the ±y
directions.
2.6

Summary

A quantity called "canonical vorticity" Qσ frozen into each species’ flow, and the
Qσ flux of each species is therefore conserved. By examining the dynamics of
Qe flux tubes, various aspects of collisionless magnetic reconnection have been
explained. First, the reason why collisionless reconnection is an exponentially
growing instability has been explained through the interaction of a perturbed Qe flux
tube with the central out-of-plane electron flow. Second, the electron acceleration
mechanism has been explained through the lengthening and thinning of a Qe flux
tube in conjunction with conversion from the magnetic field to electron vorticity
while preserving Qe flux. Third, the generation of whistler waves has been explained
by the twisting of a Qe flux tube by the out-of-plane electron flow shear and the
subsequent propagation of this twist.

Figure 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 initial
field line is straighted out. The A, B, and Cs in both figures correspond to identical
locations along the field line. The colors represent the height in z.

35
Chapter 3

ELECTRON CANONICAL BATTERY TERM: COMPLETION OF
THE CANONICAL VORTICITY FRAMEWORK
In Chapter 2, the pressure term was dropped by either assuming low-β or isotropic
and barotropic electron pressure. This allowed the expression of the electron equation of motion in the form of an ideal induction equation (Eq. 2.10). In this
chapter, the effects of a finite pressure tensor will be investigated. By including
the pressure tensor, the validity of the canonical vorticity framework extends to the
Vlasov frame of reference (Section 1.2). It will be shown that the pressure tensor acts as a battery term that generates additional canonical vorticity, and that its
competition with the convective term determines the structure and stability of the
electron-diffusion-region (EDR; red box in Fig. 1.5).
The EDR structure during reconnection, which is strongly correlated with the shape
of the electron out-of-plane flow, has long been a subject of controversy. It was
initially presumed that stable electron-scale current layers could not exist because
various instabilities would break up these layers [37, 43, 44]. However, spacecraft
observations showed that, in fact, the EDR has a highly elongated stable structure [121]. The stability of this elongated structure was interpreted via numerical
simulations 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 structure
completely [57]. Another study showed that different regimes of the EDR exist
depending on a magnetization parameter [89].
In the completed canonical vorticity framework, these disparities are not only easily
resolved but also unified. In fact, the examination of canonical vorticity dynamics is essential for the correct interpretation of the physical origin of the current
structure. Different kinetic effects contribute to the canonical battery term, which
then competes with the convective term to determine the progression of the electron canonical vorticity. This progression in turn determines the overall structure,
evolution, and stability of the EDR. This framework provides a simpler and clearer
alternative to the traditional approach where each of the multitude of terms in the
electron momentum equation is examined separately.

36
3.1

The Completed Canonical Vorticity Framework

Assumption
The sole assumption is that the ions are stationary, which is valid for L < di
corresponding to the green box in Fig. 1.5. While the theory to be presented
is similar in many respects to classical electron-magnetohydrodynamics (EMHD;
Chapter 2) [25], the minimal restriction of this stationary-ion assumption causes
the theory to differ substantively from EHMD because the assumption permits finite
displacement current and the full electron pressure tensor pe = me v0e v0e f (ve ) d 3 ve
where v0e is the random part of ve . Because new quantities will be introduced in this
Chapter, unnormalized quantities will first be used and will be normalized later.
The Generalized Canonical Induction Equation
The electron equation of motion is

Due ∇ · pe
eE + eue × B = −me
Dt
ne

(3.1)

Recall from Eq. 2.2 that the electron canonical vorticity is defined as
Qe = me ∇ × ue − eB = me we − eB,

(3.2)

where we = ∇ × ue is the electron fluid vorticity. Then, Eq. 3.1 can be expressed as
2
ue
∇ · pe
∂ue
eE = ue × Qe − me
− me ∇
∂t
ne

(3.3)

Taking the curl of Eq. 3.3 and using Faraday’s law ∇ × E = −∂B/∂t yields the
generalized canonical induction equation
∇ · pe
∂Qe
= ∇ × (ue × Qe ) − ∇ ×
(3.4)
∂t
ne
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 reduced
to just the two terms on the right hand side of Eq. 3.4. The ∇ × (ue × Qe ) term is
a convective term which prescribes that Qe flux is frozen into ue , and thus provides
an intuitive understanding of the temporal development of Qe as demonstrated in
Chapter 2. Since
term describes freezing-in of the vorticity flux to
the↔ convective
the flow, −∇ × ∇ · pe /ne is the only term in Eq. 3.4 that enables diffusion of Qe

37
Quantity
ue
Qe
pe
ne

Normalization
|ωce | −1
de
de |ωce |
B0
eB0
de |ωce |B0
B02 /µ0
n0

Table 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,
∇pe
∇ne × ∇Te
(3.5)
−∇×
ne
ne
↔
is the Biermann battery term [14], and because −∇ × ∇ · pe /ne generates Qe as
indicated by Eq. 3.4. Since Eq. 3.1 results from the collisionless Vlasov equation
without approximations, Eq. 3.4 is kinetically exact.
3.2

Fluid Simulation

In order to illustrate how the competition between the convective term and the canonical battery term governs magnetic reconnection, a fluid simulation was developed
that includes compressibility, displacement current, and kinetic effects. Because of
the added complexity, the relatively simple algorithm used in Section 2.3 cannot be
used. A new algorithm was devised, whose details will now be illustrated.
Normalization of Relevant Equations
Quantities are normalized as follows: B to the background field strength B0 , length
to de , time to |ωce | −1 , pe to B02 /µ0 , density ne to the background density n0 , and
E to v Ae B0 = de |ωce | B0 . A summary of the normalizations used in this chapter is
illustrated in Table 3.1.

38
Dividing the equation of motion by ede |ωce | B0 , we have

∇ · pe
ue
me
Due
=−
de |ωce | B0 de |ωce | B0
eB0 de |ωce | Dt
de |ωce | B0 ne e

(3.6)

∇ · pe
Dūe
Ē + ūe × B̄ = −
Dt
de |ωce | B0 ne e
me ∇¯ · pe
Dūe
Ē + ūe × B̄ = −
− 2 2 2,
Dt¯
de B0 ne e
Dūe µ0 ∇¯ · pe
Ē + ūe × B̄ = −
Dt¯
n̄e B02
Dūe ∇¯ · pe
Ē + ūe × B̄ = −
Dt¯
n̄e
where the barred quantities are dimensionless, and

me ω pe
me
me n0 e2
= n0 µ0
c 2 e2
me 0 c2 e2
de2 e2

(3.7)
(3.8)
(3.9)
(3.10)

(3.11)

was used.
Faraday’s law is normalized as follows:
∂B
de ∇ ×
=−
de |ωce | B0
|ωce | B0 ∂t
∂ B̄
∇¯ × Ē = − .
∂ t¯

(3.12)
(3.13)

Ampère’s law, including the displacement current, is normalized as follows:
de
de
de 1 ∂E
∇ × B = − µ0 ne eue +
(3.14)
B0
B0
B0 c2 ∂t
d ω2
¯∇ × B̄ = − e pe n̄e ue + |ωce | ∂
(3.15)
|ωce | c
ω pe ∂t de |ωce | B0
ω ∂ Ē
∇¯ × B̄ = −n̄e ūe + 2ce
ω pe ∂t

(3.16)

where j = −ne eue due to the stationary ion assumption.
The continuity equation is normalized as follows:
∂ne
ne u e
+ de ∇ ·
= 0,
(3.17)
n0 |ωce | ∂t
n0 de |ωce |
∂ n̄e ¯
+ ∇ · (n̄e ūe ) = 0.
(3.18)
∂ t¯
Unless specified otherwise, the bars will be dropped to reduce notational clutter,
and only normalized quantities and dimensionless equations will be used throughout
this chapter.

39
Expression in terms of j
In order to facilitate the application of the algorithm to be presented, the relevant
equations must first be expressed in terms of the current density j = −ne ue (recall
that ions are assumed to be stationary). It is obvious how Faraday’s Law, Ampère’s
law, and the electron continuity equation are expressed in terms of j = −ne ue , but
how the equation of motion (Eq. 3.10) is expressed is not and so will be derived
here.
Multiplying Eq. 3.10 by ne ,
∂ue
− ne ue · ∇ue − ∇ · pe,
∂t
∂ue
∂ne
ne E − j × B = −ne
− ue
− ue ∇ · (ne ue ) + j · ∇ue − ∇ · pe,
∂t
∂t
∂j
− ue ∇ · (ne ue ) + j · ∇ue − ∇ · pe,
ne E − j × B =
∂t
∂j
ne E − j × B =
− ∇ · j − j · ∇ − ∇ · pe,
∂t ne
ne

∂j
jj
ne E − j × B =
−∇·
− ∇ · pe .
∂t
ne

ne E + ne ue × B = −ne

(3.19)
(3.20)
(3.21)
(3.22)
(3.23)

Therefore, the set of relevant dimensionless equations are
∂B
= −∇ × E,
∂t
∂E ω pe
= 2 (∇ × B − j) ,
∂t
ωce
∂ne
= ∇ · j,
∂t

∂j
jj
= ne E − j × B + ∇ ·
+ ∇ · pe .
∂t
ne

(3.24)

The set of Eqs. 3.24 solves Eq. 3.4 exactly and includes finite displacement current
and non-uniform density.
Numerical Method
The 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], as
shown in Fig. 3.1; the x and y components are defined on the edges, the z component
is defined at the center, and scalars are defined at the corners. For example, the
z-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 (Vx )i, j+1/2 − (Vy )i, j−1/2
∆x
∆y

(3.25)

Scalar

j+1/2

j-1/2
i-1/2

i+1/2

Figure 3.1: The staggered spatial grid used for the fluid simulation.
This grid allows for a second-order accuracy (∼ ∆x 2 ) in spatial derivatives, and
satisfies 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 integer
time steps, and E, ne , and pe are defined on half-integer time steps. The time
derivatives of quantities on integer steps are defined on half-integer steps, and those
of quantities on half-integer steps are defined on integer steps.
The time advancement algorithm is shown in Fig. 3.2 and can be described as
follows.
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 of
Bt and Bt+1 is used to obtain Bt+1/2 .
3. j is advanced in time using an iterative procedure, which will be explained
shortly.
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 steps
while the latter is defined on integer time steps. Therefore, an iterative procedure is

Figure 3.2: The staggered time grid and the time advancement algorithm for the
fluid simulation.
used to solve the equation. Discretizing the equation for j,
t+1 t t+1 t !
j +j j +j
jt+1 + jt
jt+1 − jt
= ne E −
×B+∇·
+ ∇ · pe,
∆t
4ne

(3.26)

where quantities without superscripts are all defined at half-integer time steps, and
Bt+1/2 = Bt + Bt+1 /2. This is then iteratively solved for jt+1 , i.e.,
t+1 t t+1 t !
t+1 + jt
jt+1
jn + j jn + j
n+1
= ne E − n
×B+∇·
+ ∇ · pe, (3.27)
∆t
4ne
where the subscript n denotes the iteration number, and jt+1
= jt . For the recon0
nection problem, it took 5 to 10 iterations before the solution converged to within
floating point precision.
Kinetic Effects
Two different kinetic effects are considered, namely pressure anisotropy and electron
viscosity. Pressure anisotropy is modeled using the following closure which is
described in Egedal et al. [49] and Le et al. [88]:
π ñ3 2α
+ e2
2 + α 6 B̃ 2α + 1
p̃e⊥ = ñe
+ ñe B̃
1+α
α+1
p̃ek = ñe

(3.28)
(3.29)

42
In 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 an
isothermal (small α) or a double adiabatic (large α) equation of state. Equations 3.28
and 3.29 are approximations of second order moments of an electron distribution
function derived from an analysis of electron trapping by a combination of parallel
electric fields and magnetic mirrors. The situation where α
1 represents trapped
electrons because magnetic mirrors trap electrons at regions of low B and these
trapped electrons increase the local electron density. The trapped electrons conserve
their first and second adiabatic invariants and thus obey Chew-Goldberg-Low [35]
closure. On the other hand, α
1 represents untrapped electrons that provide
an isothermal closure. This closure given by Eqs. 3.28 and 3.29, valid in the
regime vT e
v A, enables fluid models to exhibit kinetic effects missing from an
isotropic pressure closure by allowing for pressure anisotropy to have both spatial
and temporal dependence as given by the local instantaneous value of α [111].
Equations 3.28 and 3.29 contribute to the pressure tensor as
pek − pe⊥
BB,
pe,aniso = pe⊥ I +
B2

(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 cases
negligible, µ represents an effective viscosity that includes, for example, turbulent
viscosity [23] and/or hyper-resistivity [143]. The total pressure tensor is then the
sum of the partial pressures; i.e., pe = pe,aniso + pe,vis .
Initial Conditions
The 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 by
x
x − xmax /2
− tanh
− 1,
(3.33)
By ≃ tanh
Bz = 1 + Bg2 − By2 (x),
(3.34)
where xmax is the size of the domain, Bg is the out-of-plane field far from the
reconnection region, and λ is the half-thickness of the shear and λ
xmax . Equation

43
3.33 characterizes two separate current sheets, but only one of the sheets is examined;
the second current sheet just facilitates the application of the periodic boundary
conditions used in the simulation. Equation 3.34 characterizes a guide field that
renders B2 = By2 + Bz2 uniform everywhere. Uniform initial pressure and density are
also imposed, so the equilibrium system is initially force-free. The system is solved
in 2.5D where ∂/∂z = 0, and the grid size is 512 × 1024 with 25 grid points per de in
the x direction and 10 per de in the y direction. Fixed parameters are ωce /ω pe = 1/2,
Bg = 0.4, and λ = 2. This extremely thin current sheet not only models the small
scale of the EDR, but is also highly relevant in space plasma phenomena such as
electron-only reconnection in the turbulent magnetosheath [122].
3.3

Particle-in-cell Simulation

Simulating plasmas by following every single particle is practically impossible with
the contemporary computing power due to the typically astronomical number of
particles involved. A particle-in-cell (PIC) simulation is a practical alternative
and follows the trajectories of a much smaller number of quasi-particles that each
represents a collection of real particles. These quasi-particles have an artificiallyimposed shape function S that represents how each quasi-particle motion affects
values on the numerical grid. More specifically, the distribution function of each
species is
fσ (x, p, t) =
w p S(x − x p )δ(p − p p ),
(3.35)
p=1

where p ∈ [1, N] is the quasi-particle number, w p is the statistical weight of the
particle, and x p and p p are respectively the position and momentum of each particle.
This distribution function can be used to calculate ρ and j using the following
relations:
Õ ∫
ρ(x, t) =
qσ
fσ (x, p, t)d 3 p,
(3.36)

j(x, t) =

qσ

v fσ (x, p, t)d 3 p.

(3.37)

These are used to calculate E and B, which are in turn used to advance the quasiparticles in time using the relativistic equation of motion for each quasi-particle:
dx
dt
γmσ
dp
= qσ E +
×B ,
dt
γmσ

(3.38)
(3.39)

44
where γ = 1 + p2 /mσ2 c2 is the Lorentz factor of the quasi-particle. The new
positions and momenta can be used to calculate fσ again, and the process can be
repeated. A PIC simulation is usually considered to have the capability to resolve
physical phenomena down to first-principles, although extreme caution is required
when equating these simulations to real-life situations because inevitable limitations
exist such as numerical dissipation and error.
In order to verify the results from the fluid simulation, the open-source PIC code
SMILEI [41] was used to simulate reconnection with the same parameters as the
fluid simulation. The realistic ion to electron mass ratio, mi /me = 1836, was used
with ∼ 2 × 108 ions and the same number of electrons. The results from the fluid
simulation are mainly presented because of the ability to include, exclude, or control
particular physics, and because of the clarity of presentation.
3.4

Results

Viscosity Dependence
We first examine how electron viscosity affects the evolution of Qe . First recall
from Chapter 2 and Eq. 2.27 that Qe ≃ −B for ∇−1 ∼ L
1 whereas Qe ≃ we for
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
2
µ∇ ue
µ∇2 we
∇ · [−µ∇ue ]
=∇×
(3.40)
−∇×
ne
ne
ne
assuming density variations are not too large. If L
1 so that Qe ≃ we and thus
∇2 we ≃ ∇2 Qe , Eq. 2.10 becomes
∂Qe
= ∇ × (ue × Qe ) + ∇2 Qe,
∂t
ne

(3.41)

which has the same form as the resistive-MHD induction equation (Eq. 1.49),
∂B
= ∇ × (U × B) + ∇2 B.
∂t
µ0

(3.42)

Thus at electron scales, electron viscosity allows Qe to reconnect, similar to how
resistivity allows B to reconnect.
Figure 3.3 shows, for three different values of viscosity, the out-of-plane electron
flow uez (color), in-plane B (black lines), and in-plane Qe (red lines) for a situation
with isotropic pressure (initially βe = 0.3). Note that the orientation of the plots
is rotated by 90◦ in comparison to those in Chapter 2 to conform to the popular

45
(a) µ = 0

0.0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
−0.8
−0.9

x/de

−2
−4

(b) µ = 10−6

0.0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
−0.8

x/de

−2
−4

x/de

−2
−4

−40

−30

−20

−10

y/de

0.1
0.0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7

Figure 3.3: In-plane Qe (red), in-plane B (black) and uez (color) for varying µ values
and isotropic pressure at t = (a) 380, (b) 450, (c) 560.
orientation in the literature. The shape of the EDR corresponds to the shape of the
purple color. Different times are chosen for each viscosity value because viscosity
changes how much 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 up
near the x = 0 line in contrast to B lines which reconnect, as seen in Chapter 2. For
finite 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., color
contours) is well manifested by in-plane Qe but not by B — a feature that is an
important 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’s
law, whereas they are well manifested by Qe because |Qe | ∼ |ue | /L.
Another important feature is that the local increase of Q ey shear corresponds to
the local increase of uez . To illustrate this point we consider the implications

46
(a) Anisotropic, µ = 0

0.00
−0.12
−0.24
−0.36
−0.48
−0.60
−0.72
−0.84
−0.96

x/de

−2
−4

(b) Anisotropic, µ = 10−6
0.0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
−0.8
−0.9

x/de

−2
−4

x/de

−2
−4

−40

−30

−20

−10

y/de

0.125
0.000
−0.125
−0.250
−0.375
−0.500
−0.625
−0.750
−0.875

Figure 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 Q ey ∼ A (t) x exp −x 2 where
A (t) increases in time. This profile represents locally sheared Q ey . Using uez =
− wey dx ≃ − Q ey dx for L
1 gives uez ∼ A (t) exp −x 2 , corresponding to a
local increase in uez .
Anisotropy Dependence
We next examine how pressure anisotropy affects Qe . In order to bring the system
to an anisotropy-driven state faster, an initial pressure anisotropy with βek = 0.6
and βe⊥ = 0.1 was imposed. The results for different µ are shown in Figs. 3.4ac. In comparison to the isotropic case it is seen that pressure anisotropy greatly
distorts the in-plane Qe lines so that they pile up in the upper-left and lower-right
quadrants; 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 is
enhanced at locations where Q ey is sheared. The distortion of Qe field lines and the

47
Particle-in-cell

x/de

−2
−20

−10

y/de

0.225
0.150
0.075
0.000
−0.075
−0.150
−0.225
−0.300
−0.375

Figure 3.5: Same as 3.3 from the particle-in-cell simulation at t = 300.
(a) ŷ · ∇ × (ue × Qe ) (color), −ŷ · ∇ × (∇ · pe,aniso/ne ) (contour)

x/de

0.035

0.015
0.030

0.000

-0.
030

0.000
0.000

-0.0
15

−1

0.00

(b) ∂Qey /∂t = ŷ · ∇ × (ue × Qe ) − ŷ · ∇ × (∇ · pe,aniso /ne )

−0.035
0.035

x/de

0.000

−1
−30

−20

−10

y/de

−0.035

Figure 3.6: (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 ∂Q ey /∂t. The red arrows
represent the direction of Q ey .

corresponding elongation and tilt of the out-of-plane current are reproduced by the
PIC simulation, as shown in Fig. 3.5.
The anisotropic contribution to the electron canonical battery term explains the
origin of the distortion of Qe and equivalently of the elongation of uez . Figure
3.6a 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 terms
which is equal to ∂Q ey /∂t, and the red arrows show the direction of Q ey . It can be
seen that − ŷ · ∇ × ∇ · pe,aniso /ne adds to ŷ · ∇ × (ue × Qe ) and increases the spatial

48
extent of ∂Q ey /∂t; this in turn elongates the structure of uez .
To further understand the distortion of Qe due to anisotropy, we examine the
anisotropic 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 Q ey shear corresponds to an increase in uez , we consider the
y-component of the canonical induction equation (Eq. 3.4) and thus of Eq. 3.43
which is
(B[B · ∇σ] + σB · ∇B)z =
(Bz [B · ∇σ] + σB · ∇Bz ) .
∂x
∂x

(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
∂(Bz [B · ∇σ])
∂B · ∇σ
≃ Bz
(Bz [B · ∇σ] + σB · ∇Bz ) ≃
∂x
∂x
∂x

(3.45)

Therefore, we have
∇ · pe,aniso
∂B · ∇σ
− ŷ · ∇ ×
≃ Bz
ne
∂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 and
subtract to the background guide field Bz prescribed by Eq. 3.34. This, together
with the reduction of |B| due to reconnection, generates a region of low |B| (Fig.
3.7a) with a pronounced tilt. Because p k ∼ n3e /B2 and p⊥ ∼ ne B, this |B| tilt
generates 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 in
Fig. 3.7c (color), and its gradient ∇ (B · ∇σ) ≃ x̂∂ [B · ∇σ] /∂ x is represented by
the black arrows. The resultant − ŷ · ∇ × ∇ · pe,aniso /ne ≃ Bz ∂ [B · ∇σ] /∂ x is
shown in Fig. 3.7d (contour). The convective term ŷ · ∇ × (ue × Qe ) (Fig. 3.7d;
color) enhances Q ey 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 y
direction. The resultant ∂Q ey /∂t = ŷ · ∇ × (ue × Qe ) − ŷ · ∇ × ∇ · pe,aniso /ne
is shown in Fig. 3.7e; this increase in Q ey shear is the origin of the elongated
out-of-plane flow uez .

(a) |B|

1.152
1.122
1.092
1.062
1.032
1.002
0.972
0.942
0.912
0.882

x/de

−1

(b) σ (color), B (lines)

0.45
0.42
0.39
0.36
0.33
0.30
0.27
0.24
0.21
0.18

x/de

−1

x/de

−1

(d) ŷ · ∇ × (ue × Qe ) (color), −ŷ · ∇ × (∇ · pe,aniso/ne ) (contour)

0.0030
0.0015
0.0000
−0.0015
−0.0030
−0.0045
−0.0060
−0.0075
−0.0090
0.035

x/de

0.01

0.03

0.000

-0.0
30 -0.01

−1

0.000

−0.035

(e) ŷ · ∇ × (ue × Qe ) − ŷ · ∇ × (∇ · pe,aniso /ne )

0.035

x/de

0.000

−1
−30

−20

−10

y/de

−0.035

Figure 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 anisotropic
contribution to the canonical battery term − ŷ · ∇ × ∇ · pe,aniso /ne , and (e) their
sum.

50
3.5

Discussion

The examination of Qe dynamics and the electron canonical battery term is not
only advantageous but in fact essential for the correct interpretation of a given EDR
structure. For example, Ohia et al. [111] observed that imposing isotropic pressure
deformed the out-of-plane current structure to be less elongated, which seemingly
contradicts Fig. 3.3a. However, in Ohia et al. [111], an effective electron viscosity of
1.5 × 10−5 with the same normalized units as this study was imposed, approximately
corresponding to Fig. 3.3c. Thus, it is not isotropic pressure that leads to an out-ofplane current localized at the origin; instead, the current localization results from
the reconnection of Qe by electron viscosity.
Another such example is the origin of elongated EDR structures. The elongated
structure in Fig. 3.3a results from the pile-up of Qe field lines due to the lack
of significant electron viscosity, whereas the elongated structures in Fig. 3.4 result
from pressure anisotropy. Therefore, in order to give correct physical interpretations,
the scrutinization of the electron canonical battery term and the ensuing canonical
vorticity 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 current
layer. This phenomenon can be simply explained by a direct comparison between
the convective term and the electron canonical battery term. On scales L > de , the
convective term goes like ∇ × (ue × Qe ) ≃ ∇ × (ue × B) ≃ −∇ × ([∇ × B] × B) ∼
B2 /L 2 , and the anisotropic contribution to the canonical battery term goes like
∇ × (∇ · [σBB]) ∼ σB2 /L 2 . It follows that σ ∼ 1 or pek − pe⊥ /B2 ∼ 1 in order
for the two terms to be commensurate; thus, the parallel and perpendicular electron
pressures 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
= ∇ × (ue × we ) + ∇2 we .
∂t
ne

(3.47)

It then naturally follows from fluid dynamics that turbulent flow develops at a
sufficiently high Reynolds number Re ∼ ne /µ. This unstable regime corresponds
to the regimes in Drake et al. [44] and Del Sarto et al. [37]. All fluid runs and PIC
runs 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 various
instabilities.

51
Equation 2.10 also clarifies the distinction between reconnection models where the
magnetic field is broken only by electron inertia [27] and those where it is broken
also by pressure tensor effects such as electron viscosity [69]. If only the electron
inertia term is included, then the system is entirely described by the convective term
in Eq. 2.10, whereas the electron viscosity is manifested by the electron canonical
battery term. A possible scenario is that electron inertia breaks the magnetic fields
during earlier times, but the EDR is subject to instabilities and turbulence due to
the high Reynolds number at latter times. These instabilities increase the effective
viscosity, which makes the canonical battery term break both the magnetic field and
the canonical vorticity field. Whether a particular reconnection event is affected by
mostly the convective term or by both the convective term and the canonical battery
term depends on the parameters and the time-scale of the system.
The advantage of the canonical vorticity framework is now clear. The convective
term in Eq. 2.10 signifies the convection of Qe with ue , which intuitively explains
reconnection electron physics for zero beta (no pressure), as in Chapter 2. One can
then study how a particular kinetic effect, such as electron pressure anisotropy, viscosity, or distribution function foliation [97], influences reconnection by examining
how that effect is manifested in the canonical battery term and then competes with
the convective term. This is not only much simpler than examining multiple terms
in the generalized Ohm’s law (Eq. 1.34), but also exhaustive because Eq. 2.10 is a
direct consequence of the collisionless Vlasov equation.
3.6

Summary

In summary, the electron canonical battery term completes the canonical vorticity
framework of the electron physics in magnetic reconnection. As demonstration of the
power of this approach, the growth, saturation, stability, and morphology of the EDR
have been reinterpreted, expanded, and unified within this framework. In particular,
the framework illustrates how the changes in the electron fluid closure (Eqs. 3.28
and 3.29) affect the current structure through the electron canonical battery term.
The distinction between electron-inertia-driven and pressure-tensor-driven models
of reconnection is also clearly demonstrated. The simple yet complete nature of this
framework makes it an appealing alternative to the traditional magnetic-field-based
approach to magnetic reconnection.

52
Chapter 4

STOCHASTIC ION HEATING IN MAGNETIC RECONNECTION
Magnetic reconnection is ultimately a process in which magnetic energy is converted
to particle energies. Although magnetic reconnection has been the focus of many
studies for decades, the mechanisms underlying this energy conversion from the
magnetic field to particles are still poorly understood. In particular, anomalous
ion heating much faster than conventional collisional heating has frequently been
observed in various astrophysical and laboratory reconnection events [29, 50, 65,
74, 96, 117, 118, 124, 141, 168], and resolving the fundamental cause of this heating
remains a crucial objective of reconnection research. The resolution of this problem
is an important key to tackling critical conundrums such as the coronal heating
problem [59], i.e., why the solar atmosphere is over a million degrees while the
solar 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⊥ , Tik ) [47, 70, 95, 103], (iii) non-conservation of
the ion magnetic moment µ [46, 47, 82], (iv) preferential heating of ions with
higher mass-to-charge ratios mi /qi [51, 106, 128], and (v) reduction of the heating
amount under a finite guide field (background out-of-plane magnetic field) [46].
Since anomalous ion heating occurs over a broad range of plasma parameters, the
heating mechanism is expected to involve physical quantities that are intrinsic to the
reconnection process. So far, numerous mechanisms have been proposed to explain
ion heating; examples include pick-up [46, 47], turbulent interaction [107, 138],
interactions with fluctuating electric fields [51], remagnetization [168], and shocks
or viscous damping [118]. However, the proposed mechanisms are still under
serious debate; there is no general consensus among the community on what the
fundamental ion heating mechanism is.
In this chapter, it is shown that stochastic ion heating is intrinsic to transient collisionless magnetic reconnection because stochastic ion heating will always occur at length
scales smaller than the ion skin depth di = c/ω pi when there is a perturbation of a
Harris-type magnetic equilibrium [16, 66]. It is shown via analytic methods supplemented by numerical calculations that the in-plane electric fields near the magnetic

53
separatrices (lines that separate regions of reconnected and non-reconnected fields)
satisfy the condition for stochastic ion heating. Thus, the correlation of anomalous
heating with the in-plane Hall electric fields will be established; the other four features of ion heating in reconnection will be explicitly demonstrated in Chapter 5.
The stochastic nature of ions will also be demonstrated via single-particle simulations. Comparisons to laboratory experiments support the analytical calculations
presented here. Hall electric fields have been attributed to ion energization seen in
previous numerical [7, 46, 125], observational [162] and experimental [141, 168]
reconnection studies, but the specific mechanism linking the fields to the heating
has not been identified.
4.1

Stochastic Heating

Stochastic heating occurs when a particle in a magnetic field is subject to a perpendicular potential gradient so strong that it causes a breakdown of the guiding-center
approximation [10, 11, 108, 142]. The particle then experiences a large potential
difference within an orbital period and its motion becomes chaotic. When this
happens to a large number of particles, the velocity distribution function broadens
and this broadening is observationally equivalent to heating. Stochastic heating
has been proposed to be an ion heating source in the solar wind via Alfvén wave
interactions [22, 30, 31, 142, 156–159] and has been observed in laboratory experiments [29, 108, 133] . The threshold in Coulomb gauge for ion stochastic heating
is [10, 108, 142],
α=

mi
qi |B|

∇⊥
φ =

mi
qi |B| 2

|∇⊥ · E⊥ | > 1,

(4.1)

where φ is the electrostatic potential and ∇⊥ is the gradient perpendicular to B.
Breakdown of the Guiding-Center Approximation
In order to understand this mechanism, consider particle guiding-center motion
under the electromagnetic fields

The E × B drift is
vE =

E = E ŷ sin (k y − ωt) ,

(4.2)

B = B ẑ.

(4.3)

E×B
= x̂ sin (k y − ωt) ,

(4.4)

54
and the polarization drift is
m dE⊥
qB2 dt
mE d
= ŷ 2 sin (k y − ωt) ,
qB dt
dy
mE
= ŷ 2 k
− ω cos (k y − ωt) .
dt
qB

vp =

(4.5)
(4.6)
(4.7)

Here, dy/dt = v y is the velocity in the y-direction. Since vE is in the x-direction
and v p is in the y-direction, ŷdy/dt = v p . Solving Eq. 4.7 in terms of v p gives
v p = − ŷ

ωmE cos (k y − ωt)
qB2 1 − α cos (k y − ωt)

where
α=

mkE
qB2

(4.8)

(4.9)

is the stochastic heating condition in Eq. 4.1 assuming that the particle is an ion
and |∇⊥ · E⊥ | ≃ kE. It can clearly be seen from Eq. 4.8 that when α > 1,
the denominator vanishes when cos (k y − ωt) = α−1 . However, when deriving the
polarization drift in Section 1.1, it was explicitly assumed that |dvE /dt|
dv p /dt .
Because vE is finite as in Eq. 4.4, this assumption breaks down if v p → ∞ along
with the guiding-center approximation. The drift equations in Section 1.1 therefore
cannot be used, and the magnetic moment µ is not conserved.
Chaotic Motion
It 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 entails
the spatial divergence of two particles that have an infinitesimal initial separation
in 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-direction
and the magnetic field is in the z-direction. The Lorentz force equation yields
qB
δ yÛ,
q ∂E
qB
δ yÜ =
δy −
δ x,
m ∂y

δ xÜ =

δ zÜ = 0.

(4.10)
(4.11)
(4.12)

55
Differentiating the second equation in time and substituting the first equation in,
d 3 δy
q ∂E dδy q2 B2 dδy
m ∂ y dt
d 3t
m2 dt
q ∂E qB2 dδy
d 3 δy
m ∂y
dt
d 3t

(4.13)
(4.14)

which yields an exponentially growing solution for dδy/dt when
m ∂E
= α > 1.
qB2 ∂ y

(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 mechanism
that causes particle trajectories to become chaotic in coherent fields. Thus, stochastic
heating should not be confused with mechanisms where the fields themselves are
stochastic. It should be noted, however, that there has been extensive work on ion
acceleration via stochastic interaction with magnetic islands [48, 61, 72, 90, 91,
174, 175]. The present work shows that stochastic ion heating occurs at a single
reconnection region, but it will also work in the context of stochastic fields with
multiple reconnection regions if the condition in Eq. 4.1 remains satisfied.
4.2

Relevant Equations

We begin the analysis by noting that the electric field acting on the ions can be
determined from the two-fluid electron equation of motion. Using this equation to
prescribe the electric field is equivalent to invoking the generalized Ohm’s law with
the Hall, electron inertia, and electron pressure terms all included. The normalization of quantities is the same as in Chapter 3 and thus Table 3.1; the magnitude of
the magnetic field far from the reconnection region is denoted B0 and this defines a
reference electron cyclotron frequency ωce = qe B0 /me . On normalizing length by
the electron collisionless skin depth de = c/ω pe , time by |ωce | −1 , magnetic field by
B0 , density by the background density n0 , electric field by de |ωce | B0 , and pressure
by B02 /µ0 , the two-fluid electron equation of motion can be expressed as
Ē + ūe × B̄ = −

Dūe ∇¯ p̄e
Dt¯
n̄e

(4.16)

where barred quantities are dimensionless. The pressure has been assumed to
be scalar. To reduce clutter, the bars will be dropped from now on. Under this
normalization, Eq. 4.1 becomes
∇2 φ > 1,
εB2 ⊥

(4.17)

56
where ε = me /mi is the electron-to-ion mass ratio.
Recall from Chapter 2 that assuming negligible pressure term by positing either cold
electrons (low βe ) or barotropic pressure, and invoking Faraday’s law yield the ideal
electron canonical induction equation:
∂Qe
= ∇ × (ue × Qe ) ,
∂t

(4.18)

where Qe ≡ ∇×ue −B is the electron canonical vorticity. An important consequence
of Eq. 4.18 was that Qe field lines remain connected whereas B field lines may
reconnect.
It should be noted that as seen in Chapter 3, the canonical battery term can be
important and will add an additional term to Eq. 4.18, allowing for the reconnection
of Qe . However, this effect will be disregarded in the present analysis because
omitting the canonical battery term keeps the analysis simple and enables focusing
on 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 in
the kinetic regime that includes the canonical battery term and under the presence
of 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.,
2
∇pe
∂ue
+ ue × Qe − ∇ e −
(4.19)
E=−
∂t
ne
The connected nature of Qe will be now used to simplify Eq. 4.19 in a way that
enables checking whether the condition given by Eq. 4.1 for stochastic ion heating
is satisfied in transient reconnection situations. The cold electron assumption will
be 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 electron diffusion region (EDR) can be expressed in terms of gradients of components
of the electron flow. It is seen from the definition of Qe that ∇ · Qe = 0 and it is also
seen that ∇ · ue ∼ ∇ · J ∼ ∇ · ∇ × B = 0. Using these properties and Q ex (x = 0) = 0,
expansion of the y-component of Eq. 4.18 gives an exponentially growing solution
for Q ey with a growth rate
∂uey
∂uex
≃−
∂t
∂y
∂x

(4.20)

Figure 4.1: (a) Reconnected in-plane B field lines (white) and connected in-plane
Qe 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 are
respectively in the ±x and ±y directions. The x-axis and the y-axis have different
scales to show the field lines more evidently. (b) Comparison of By2 /2 with − E x dx
and − uez By dx along the black dotted line in (a). (c) Comparison of u2ey + u2e /2
with − E y dy along the magenta line in (a).
The growth rate is always positive since ∂uey /∂ y ≃ ∂ 2 Bz /∂ x∂ y and the quadrupole
out-of-plane Hall magnetic field has the dependence Bz ∼ x y in the EDR. This
growth rate will be used to rewrite the ∂ue /∂t term in Eq. 4.19.

58
4.3

Numerical Simulation and Results

To validate the predictions of the model presented here, the EMHD simulation that
was developed in Chapter 2.2 was used. Again, a perturbation to the background
Harris-type magnetic field
By = tanh
(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 the
reconnection of B and the non-reconnection of Qe is clearly evident. Specifically,
Bx (x = 0) is finite but Q ex (x = 0) = 0, so B reconnects but Qe remains connected.
The electric field is calculated numerically using the values of ue and Qe on the
right-hand side of Eq. 4.19, and the associated electrostatic potential (color) is
plotted for reference. The potential is saddle-shaped with contours approximately
following the B field lines in agreement with previous simulations [46, 79, 125] and
laboratory experiments [168].
4.4

Stochastic Heating Condition Analysis

Inflow
We 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 relevant
to stochastic heating is E x . Although Ez is also perpendicular to By , we need not
consider Ez for the purposes of analyzing stochastic heating because ∂/∂z = 0 in
2D, so Ez is irrelevant to Eq. 4.1. Using the cold electron assumption (to be justified
later), the x-component of Eq. 4.19 becomes

∂ u2e
∂uex
+ uey Q ez − uez Q ey −
(4.22)
Ex = −
∂t
∂x 2
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) =
Q ez (y = 0) = 0 due to the antisymmetry of the outflow and the quadrupole nature
of the out-of-plane Qe fields as seen in Chapter 2. On expanding the third term
using Q ey = −∂uez /∂ x − By and the fourth term using u2e = u2ex + u2ez at y = 0, Eq.
4.22 simplifies to
E x = uez By

(4.23)

≃−

∂ By
∂x 2

(4.24)

59
In 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 and
also ∂/∂ x >> ∂/∂ y. Therefore, By2 /2 acts as an effective electrostatic potential
in the x-direction (this is equivalent to Eq. 3 in Li and Horiuchi [93]). This
conclusion regarding the effective potential is shown graphically in Fig. 4.1b, where
− E x dx and − uez By dx are compared with By2 /2, calculated from the simulation.
Outside the EDR, ∂By /∂ x −→ 0, so the approximation fails, but for the purposes
of analyzing the slope of the potential valley into which the ions fall (ion inflow
directions represented by the green arrows) and thus for the purpose of investigating
the existence of stochastic heating, By2 /2 remains a good approximation for the
electrostatic potential. The integration constant was set so that the three functions
coincide at x = 0 in Fig. 4.1b.
Equation 4.17, the condition for ion stochasticity in the inflow direction, can thus be
written as
1 ∂ 2 By
> 1.
(4.25)
By2 ∂ x 2 2
Using By (x) ≃ B0 x/λ for x < λ and de2 /ε = di2 , Eq. 4.25 in dimensioned quantities
becomes x 2 < di2 . A sufficient condition to satisfy x 2 < di2 while x < λ is
λ2 < di2 .

(4.26)

Therefore, at sub-ion-skin-depth length scales, at which collisionless reconnection
occurs, 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] studies
show that the magnetic force on the electron fluid is balanced by the electric field
corresponding to Eq. 4.23 and thus Eq. 4.24.
Outflow
We now consider the outflow (flow along the x = 0 line, magenta dashed line in
Fig. 4.1a). Because the magnitude of B is very small on this line, the cold electron
assumption in principle cannot be used. However, it will be shown later that
including 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 field
relevant to stochastic heating is E y . The magnetic field component Bx at x = 0 has

60
small amplitude and its being finite is what constitutes the reconnection. Ez again
plays no role in Eq. 4.1 because ∂/∂z = 0. The y-component of Eq. 4.19 with the
cold electron assumption invoked is

∂uey
∂ u2e
Ey = −
+ uez Q ex − uex Q ez −
(4.27)
∂t
∂y 2
Using Eq. 4.20, the first term
on the right-hand side of Eq. 4.27 is −∂uey /∂t ≃
−uey ∂uey /∂ y = −∂ uey /2 /∂ y. The second term disappears because Q ex (x = 0) =
0 since Qe does not reconnect. The third term disappears because of the antisymmetry of uex and the quadrupole nature of Q ez . Equation 4.27 now simplifies to
∂ uey + ue
(4.28)
Ey ≃ −
∂y
Thus, u2ey + u2e /2 acts as an effective potential in the y-direction. A comparison
between − E y dx and uey + ue /2 along x = 0 shows that the latter is a good
approximation, as shown in Figure 4.1c. The potential forms a hill off of which
the ions fall downwards (ion outflow directions represented by green arrows). The
integration 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 is
2 + u2
1 ∂
ey
> 1.
(4.29)
εBx ∂ y
Here the connected nature of Qe gives Q ex (x = 0) = ∂uez /∂ y − Bx ≃ uez /L y − Bx =
0, so using ∂/∂ y ∼ 1/L y = Bx /uez on Eq. 4.29 gives
1 uey + ue
> 1.
(4.30)
2u2ez
However, u2ey + u2e /u2ez ≥ 1 since u2e = u2ey + u2ez , so Eq. 4.30 reduces to
mi
> 1,
me

(4.31)

which is of course satisfied. Consequently, the stochasticity condition in the outflow
direction is always satisfied by a wide margin.
The cold electron assumption in the outflow direction will now be justified. Recalling
that p̄e = ne k BTe /(B02 /µ0 ) gives p̄e = n̄e v̄T2 e /2 where v̄T2 e = (2k BTe /me ) /(de |ωce |)2 .
2
vT e
vT2 e ∇ne
Dropping bars, the pressure term in Eq. 4.19 becomes ∇p
ne
2 ne ≃

2
vT e

. The T2e ∇n
ne term was dropped by invoking the quasi-neutrality condition;

because the normalized Poisson’s equation gives ∇ · E = − vc2 (ne − n0 ) where
Ae

v 2Ae = B2 /µ0 me ne is the electron Alfvén velocity, taking the gradient of both sides
and requiring ∇ · E to be small by quasi-neutrality yields ∇ne << v 2Ae /c2 . Using
this new pressure term in Eq. 4.19 and applying the same reasoning used to derive
Eq. 4.28, E y approximates to
∂ uey + ue + vT e
Ey ≃ −
(4.32)
∂y
Equation 4.32 implies that if there is electron heating in the exhaust region, the
potential becomes flatter and thus less conducive to stochastic heating. This is not a
problem in guide-field reconnection in which electron heating is localized near the
EDR [146]. For zero guide-field reconnection, however, if the exhaust region (finite
y) is hotter than the EDR (y ≈ 0) [166], the potential is a well centered around y = 0
which is not conducive to stochastic heating. Nevertheless, temperature effects do
not negate the stochastic condition being satisfied for the following reasons.
Consider the worst case scenario where the outflow becomes completely thermalized, i.e., u2ey + vT2 e = const. In this case, only u2e is left in Eq. 4.32 so Eq. 4.30
2
becomes ε1 2ue2 > 1, which is always satisfied because u2e ≥ u2ez .
ez

Furthermore, previous studies are consistent with the cold electron assumption.
Several numerical studies [79, 125] have shown that |eΦ/Te | — which corresponds
to ∼ u2e /vT2 e — increases to be 10 ∼ 20 inside the saddle potential, which means
that the electron flow is cold and laminar. A previous experimental study [168]
measured the plasma potential including electron temperature effects and showed
that the potential is a hill in the y-direction with the peak at y = 0, confirming the
shape given in Fig. 4.1(c).
The fact that Eqs. 4.26 and 4.31 do not depend on the magnetic field signifies
that stochastic heating is intrinsic to collisionless reconnection. φ in Eq. 4.17
has a dependence on B in such a way that this dependence cancels out B in the
denominator. Thus, Eq. 4.17 is always met given that the assumptions in the present
analysis are met as well.
4.5

Test Particle Simulation

In order to verify ion stochastic heating, 6000 test ion particle trajectories were
simulated using an implicit particle integrator [15]. The details of the algorithm

Figure 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 ions
at t = 580 |ωce | −1 , and (d) their stochastic counterparts. (e), (f) Zoom-ins of (c)
and (d), respectively. (g) Distribution of |∆ yÛ | 2 for the coherent case (blue) and
the stochastic case (red). (h) Time-dependent 3D separation distances between two
selected particles with the minimum initial separation distance, 0.015de , under the
stationary electric field (blue line; particles are red dots in (b)) and the growing
electric field (red line; particles are red dots in (f)).

63
are described in Appendix D. Trajectories were calculated for both stationary and
growing electric fields. These two cases should respectively give ordered and
stochastic 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 small
spatial separation was chosen because stochasticity is defined by particles’ having
disparate trajectories with slightly different initial conditions. The results presented
in 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 and
stochastic situations. Simulations with higher mass ratios (20, 50, 100) have also
been conducted and exhibit similar qualitative behavior; mass and time scalings are
subject to future investigations.
The growing electric field was the time-dependent numerically calculated field, and
the stationary electric field was chosen to have the value of the time-dependent
field at t = 5 |ωce | −1 . A small value for this reference time was chosen because
the particles were started at rest from different positions. This means that they all
have different initial total energies, so a small potential was chosen to keep the ions
initially cold in order to explicitly show the thermal energy gain due to stochastic
heating. 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 margin
so the stochastic evolution in this direction is more evident.
Particle Trajectories and Phase Mixing
Figure 4.2a shows ten selected ion trajectories (red) in the stationary electric field
case, 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 have
Speiser-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 coherent
motion. In the growing electric field case (i.e., Fig. 4.2b), the trajectories clearly
diverge and undergo phase mixing, the signature of stochastic motion. Stochasticity
is further demonstrated by the central trajectory (white) which starts at x = 0 in
both 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 unstable
indicating stochasticity, where the seed for the instability is the ever-present small
numerical error.

64
Ion Phase Space
Examination of the ion phase space indicates chaotic motion. Figure 4.2c shows the
positions in y phase space of all ions at t = 580 |ωce | −1 for the stationary electric
field 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 the
right 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 the
minimal separation of 0.015de for the stationary electric field case, and those in
Fig. 4.2d represent their stochastic counterpart (growing electric field). The timedependent 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 particles
in the stochastic case separate rapidly from one another, a behavior which is the
characteristic of chaotic motion and in accordance with the concept of Lyapunov
exponents [100].
Effective Heating
The 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 outflow
velocity |∆ yÛ | 2 , a proxy for thermal energy. Significant heating is clearly seen for the
stochastic case. The distribution is non-Maxwellian, however, because collisions
are not included.
Initial Temperature
In reality, ions have a finite initial temperature. Another simulation was conducted
where the ions start with a thermal velocity of 0.01 de |ωce |, and the results are
shown in Fig. 4.3. The phase space as shown in Fig. 4.3b is much broader than that
in 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 in
contrast to Fig. 4.3c. Also, in comparison with Figs. 4.2c and 4.2d, it can be seen
that 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 show
the contrast between coherent and stochastic ion motion, and the results presented
here reflect the initial stage of stochastic evolution. A temporal limitation exists due
to the time scale separation between the ions and electrons, so the simulation must

Figure 4.3: Positions in y phase space of ions that started with a thermal velocity
of 0.01 de |ωce | in (a) the coherent case and (b) the stochastic case. Distribution of
|∆ yÛ | 2 in (c) the coherent case and (d) the stochastic case.

be halted before the ions are exhausted from the reconnection region. This also
means that the ions have yet to gain all the available energy from this mechanism.
4.6

Discussion

Heating Rate
An order-of-magnitude estimate for the heating rate and strength in a reconnection
situation will be given as follows. A proxy for the exponential heating rate from this
mechanism in a reconnection situation is its Lyapunov exponent
qi E
(4.33)
γE =
mi a

66
where a is the length scale of the electric field E [10]. Here,
qi B0 B⊥
qi E
mi E
mi a
mi B0 qi B⊥2 a

(4.34)

and noting that the quantity inside the square root is equivalent α of the
q stochastic
B⊥ di
B⊥
heating criterion (Eq. 4.1), γ ≈ ωci B0 Lx for the inflow and ≈ ωci B0 mmei for the
outflow. Assuming B⊥ . B0 and L x . di , the theoretical heating rate γ is in the
range ωci < γ < ωci ωce . This is much faster than the MHD time scale governing
the macrophysics of the solar environment.
As noted in Sanders et al. [133], the maximally achieved ion thermal velocity
due to stochastic heating is vT = ωci a(α + 1.9), where α is defined in Eq. 4.1
and thus satisfies α > 1. Using a ≃ di , the upper bound on the local beta is
βlocal = vT /v A = 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 upstream
field, the global beta may still stay sub-unity. A more detailed analysis on the heating
amount is subject to future investigations.
Heating Profile
Our results are consistent with a recent particle-in-cell calculation of ion temperature
profile in collisionless reconnection. Haggerty et al. [63, Figure 2] shows a sharp
ion temperature increase across the magnetic separatrix where the potential drop
occurs, and a further increase in the ion outflow jet. This is congruous with the fact
that 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 more
dramatic along the outflow.
Comparisons to Experiments
The claim that Eq. 4.1 is satisfied in collisionless reconnection is also supported
by experimental observations. In the Magnetic Reconnection eXperiment [166],
the measured in-plane electric field was 700 V/m with a 1 cm length scale. Using
the upstream magnetic field B0 = 0.1 kG, Eq. 4.1 becomes 14.6 >> 1 and so the
condition for ion stochastic heating is satisfied by a wide margin. The condition
was also met in the spontaneous collisionless reconnection event in the Caltech
MHD-driven jet experiment [29].
The calculated heating rate also agrees with experiments. Defining γobs to be the inverse e-folding time scale of the ion temperature based on experimental observations,

67
γobs ≈ 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 the
former and γ ≈ 1–100 × 106 s−1 for the latter. Therefore, stochastic ion heating is
fast enough to account for the observations in both experiments.
4.7

Summary

It has been shown that ion stochastic heating, which had already been established as
an important heating mechanism in the solar wind, atmosphere, and flares, is also
inherent to collisionless magnetic reconnection. The connected nature of electron
canonical vorticity has been exploited to show that the electric fields near the magnetic separatrix satsify the stochastic heating criterion. The approximations for the
electric field were verified by an electron fluid simulation, and ion stochasticity was
demonstrated by test particle simulations. Comparisons to laboratory reconnection
experiments also support the existence of this mechanism.

68
Chapter 5

KINETIC VERIFICATION OF STOCHASTIC ION HEATING IN
MAGNETIC RECONNECTION
In Chapter 4, it was demonstrated via a two-fluid, zero-beta analysis of the generalized Ohm’s law that stochastic heating is intrinsic to anti-parallel (zero guide field)
collisionless reconnection. Although the demonstration provided useful insights
into the fundamental ion heating mechanism in reconnection, a fully self-consistent
kinetic verification of this mechanism was not demonstrated. Also, four of the five
features of ion heating in reconnection, namely temperature anisotropy (Ti⊥ , Tik ),
µ violation, preferential heating of higher m/q particles, and reduction of the heating
amount in the presence of a guide field, were not explicitly illustrated in terms of
stochastic heating.
In this chapter, it is unambiguously shown via exact kinetic analyses and particle-incell simulations that stochastic heating [10, 108, 133, 142] is the main ion heating
mechanism in a generic collisionless reconnection process up to moderate guide
fields and exhibits all five features of anomalous ion heating.
Recall from Chapter 4 that the criterion for stochastic ion heating is given by a single
dimensionless parameter α (x, t) where
α (x, t) =

mi
qi |B|

∇⊥
φ > 1;

(5.1)

here ∇⊥ is the gradient perpendicular to the local magnetic field B = B (x, t), and
φ = φ (x, t) is the local electrostatic potential. Equation 5.1 provides a spatiotemporal prediction for stochastic heating. Stochastic heating involves the breakdown of the guiding-center approximation and causes associated non-conservation
of the magnetic moment µ. Since µ involves v⊥ , this mechanism preferentially
increases T⊥ . In addition, according to Eq. 5.1, ions with higher mi /qi more easily
satisfy 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 mechanism
in magnetic reconnection.
This chapter fills the gap between the prediction of stochastic ion heating in magnetic
reconnection in Chapter 4 and a self-consistent verification by confirming the exis-

69
tence and inherent features of stochastic ion heating through fully kinetic analyses
and simulations.
5.1

Stochastic Heating in the Harris Equilibrium Plasma Sheath

We begin with an analysis of the existence of stochastic heating in the Harris
equilibrium sheath [66], which is often used as the initial condition for reconnection
analyses and simulations. Although the Harris-type hyperbolic tangent magnetic
field profile has been used in previous chapters, the exact Harris equilibrium is more
complicated and will now be briefly derived.
Harris Equilibrium Plasma Sheath
The Harris equilibrium plasma sheath is a kinetically exact specific solution for
a steady-state velocity distribution function that simultaneously satisfies Poisson’s
equation, Ampère’s law, and the Vlasov equation (Eq. 1.13). A sheath refers to
the region of plasma sandwiched between regions of oppositely-directed magnetic
fields. Using the specific distribution function, φ and A can be calculated, and from
these the equilibrium electric and magnetic field can be derived. For a more detailed
derivation, see Appendix E.
The equilibrium is initially solved in a frame where ions and electrons are moving
at equal and opposite velocities in the z direction, i.e., Viz = −Vez = V. Assuming
uniform T = Ti = Te ,
φ=0
(5.2)
in this frame, so
E = 0.

(5.3)

The equilibrium magnetic field is determined to have the profile [66]
By (x) = 2 µ0 n0 k BT tanh ,

(5.4)

where
2c
λD
λ=
2c
−2
λ−2
Di + λ De
c 0 k B T
n0 e2

(5.5)
(5.6)

(5.7)

70
is the sheath half-thickness and n0 is the peak density of the sheath. The density
profile is
ni (x) = ne (x) = n0 cosh−2 .
(5.8)
Lorentz tranforming to a frame where Viz = 0, which is approximately the centerof-mass frame (i.e., the lab frame due to mi
me ), a lab-frame electric field with
the following profile develops [66]:
x
E x (x) = −V By (x) = −2V µ0 n0 k BT tanh
(5.9)
where we have assumed V
c, so the Lorentz factor γ = 1. The magnetic field
stays 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 field
E x,Hall (x) = −Jz (x) By /n (x) e = −2V By (x) in the lab frame. In the vicinity
of the reconnection region, the in-plane electric field is mainly balanced by the
Hall component [93, 171], so whether Eq. 5.9 satisfies Eq. 5.1 is important for
determining the existence of stochastic heating.
Stochastic Ion Heating Condition
To make the analysis more general, a guide field Bz = bg By (x → ∞) is now included
as well where bg is the ratio of the guide field to the asymptotic value of By . Since
only E x (x), By (x), and Bz are involved, Eq. 5.1 becomes
∂E x (x)
mi
2
> 1,
∂x
qi By (x) + Bz

(5.10)

which, after inserting the Harris profiles (Eqs. 5.4 and 5.9), becomes
h

λ2
2 x
2 x
2 2 cosh
tanh
+ bg < 1,
di

(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 where
2 1 − b2 + 1 +
λ̄
1 − 2λ̄2 b2g 2λ̄2 + 1
λ̄
x̄sh = ln
(5.12)
λ̄2 1 + b2g
A more detailed derivation of this solution is in Appendix E. Figure 5.1a shows
contours of x̄sh λ̄, bg , and the inset plots x̄sh as a function of bg for λ̄ = 0.5, 1, 2.

71
(a) xsh

1.50
1.25

0.200

0.5

0.75

1.0

1.5

2.0

2.5

0.5

0.0

1.0

0.5

0.100

0.600

0.00

1.000

0.50

0.25

1.00

1.0

0.5

0.50

0.00
0.0

0.0

0.50

1.500

40
0.

0.75

0.0

0.75

0.25

1.00

λ = 0.5
λ=1
λ=2

1.00

xsh/λ

0.4
0.2

0.30

1.25

1.75

λ = 0.5
λ=1
λ=2

2.000

0.200

1.50

(b) xsh/λ

2.00

0.6

xsh

1.75

0.100

2.00

0.25
3.0

0.00
0.0

0.5

1.0

0.2

1.5

2.0

2.5

3.0

Figure 5.1: (a) Contours of x̄sh λ̄, Bg . Inset plots x̄sh as a function of Bg for three
λ̄ 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 that
is 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 intrinsic
to a Harris sheath. Figure 5.1b shows that the fractional heating extent increases
as the sheath becomes thinner. However, Figs. 5.1a and 5.1b show that for each λ̄
there exists a maximum guide field value above which stochastic ion heating does
not occur (red dashed lines). This corresponds to the argument of the square root in
√ −1
Eq. 5.12 becoming negative when bg > 2λ̄ , 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 to
an evolving reconnection sheet since the Harris sheath does not by itself involve
reconnection. However, because the Harris electric field (Eq. 5.9) is comparable to
the Hall electric field, and the reconnecting magnetic field profile is closely related
to the Harris profile [164], one can extrapolate a general trend that the fractional
extent of stochastic ion heating decreases as the guide field or the sheath width
increases.

72
Quantity

Normalization
−1
ωci
di
di ωci = v Ai
B0
di ωci B0 = v Ai B0
n0
B0 /µ0
mi v 2Ai
mi v 2Ai /B0

Table 5.1: Dimensional quantities and their normalization parameters in Chapter 5.

5.2

Particle-in-cell verification

Setup
In order to test the prediction that stochastic ion heating is intrinsic to collisionless
reconnection, fully kinetic particle-in-cell simulations were conducted using the
SMILEI code [41]. The simulations were conducted in 2D, i.e., ∂/∂z → 0. Lengths
were normalized to di , time to ωci , velocity to v Ai = di ωci , B to the upstream B0 ,
density to the peak density n0 , T to mi v 2Ai , and the magnetic moment µ to mi v 2Ai /B0 .
A summary of the normalizations used in the simulation in this Chapter is detailed
in Table 5.1.
Double periodic Harris sheaths were employed as initial conditions, i.e.,

x̄ − x̄max /2
x̄
− tanh
−1
By = B0 tanh
λ̄
λ̄
and

(5.13)

x̄
−2 x̄ − x̄max /2
+ cosh
+ nb ,
(5.14)
n = n0 cosh
λ̄
λ̄
where x̄max is the simulation box size in the x direction, nb = 0.2 the relative
upstream density, λ̄ = 1 the half-thickness, and T = Ti = Te the temperature. Only
one of the two sheaths is examined; the double sheath facilitates the application of
periodic boundary conditions. We work in the lab-frame, so an initial Harris-type
electric field in the form of Eq. 5.9 is employed. The ion-to-electron mass ratio was
mi /me = 100 and the grid size was 1024 × 2048 = 40.96di × 81.92di . The number
of particles per cell ranged from 100 to 600 depending on the local density, and
c/v A = 20. Although x̄sh depends on both λ̄ and bg , changing λ̄ requires significant
alterations in simulation parameters such as the domain, and the current sheet tends

−2

73
to be more unstable for smaller λ̄. Thus, in order to facilitate the analysis, bg was
chosen to be the variable parameter.
From now on, the bars will be dropped and only normalized quantities will be used
unless specified otherwise.
Calculation of Ti⊥ and Tik
Pressure and temperature are not variables that are intrinsic to the particle-in-cell
algorithm (Section 3.3) and therefore must be post-processed. The mean ion flow
velocity is first calculated as
ui =
vi fi d 3 vi,
(5.15)
where fi is calculated from Eq. 3.35.
Now, the ion pressure tensor is defined as
pi =
v0i v0i fi d 3 vi,

(5.16)

where v0i = vi − ui is the random part of the ion velocity. Consider the following
integral:
vi vi fi d vi = (ui + v0i )(ui + v0i ) fi d 3 vi,
(5.17)
= (ui ui + ui v0i + v0i ui + v0i v0i ) fi d 3 vi,
(5.18)
= ui ui
fi d vi + v0i v0i fi d 3 vi
(5.19)

= ui ui + pi,
where

(5.20)

v0i fi d 3 vi = 0 by the definition of v0i which is the random part of vi . Therefore,
pi =
vi vi fi d 3 vi − ui ui,
(5.21)

where the right-hand side can be calculated from the variables explicitly used in the
PIC algorithm.
Now, in terms of pi⊥ and pik ,

pi = pi⊥ I + (pik − pi⊥ ) B̂ B̂,

(5.22)

74
where B̂ is the local unit vector in the direction of the local magnetic field B. It is
easy to see that

pik = B̂ · pi · B̂,

(5.23)

B · pi · B
B2

(5.24)

Since pik is now known, pi⊥ can be calculated by simply using the fact that the trace
of a tensor does not change under coordinate transformations, i.e.,

Tr pi = 2pi⊥ + pik,
or

(5.25)

Tr pi − pik
pi⊥ =
Tik and Ti⊥ can then be found simply by dividing pik and pi⊥ by ni =
respectively.

(5.26)

fi d 3 vi ,

Hall Electric Fields
Figure 5.2 shows B (black lines) and E x (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 the
periodicity of the domain has not yet affected the local system. It is well-known that
the Hall term in the generalized Ohm’s law is important in collisionless reconnection
[16]. Thus, strong in-plane Hall electric fields E x ∼ (J × B) x (color) develop; these
have also been seen in previous studies [7, 47, 93, 125, 162, 171]. Although a finite
guide field alters the structure, E x is generally pointing toward x = 0, meaning that
ions are falling down a potential valley along the inflow. A Hall E y (not plotted)
also develops but is smaller in spatial extent and magnitude.
Comparison of the Stochastic Heating Criterion to the Ion Temperature Profile
α as defined by Eq. 5.1 can be calculated from the electric and magnetic field
information. Figure 5.3 shows ln α; regions where ln α is above zero (red and
yellow) represent the spatial extent of predicted stochastic ion heating. It is seen
that 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⊥ + Tik /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, as

75
(a) bg = 0

x/di

−1

−5

(b) bg = 0.1

x/di

−1

−5

x/di

−1

−5

(d) bg = 0.5

x/di

−1

−5

−15

−10

−5

y/di

Figure 5.2: B (black lines) and E x (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 shows
that 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 guide
field is an observed feature of ion heating in reconnection.
Temperature Anisoptropy
Figures 5.5 and 5.6 show Ti⊥ and Tik , respectively. It can clearly be seen that the two
temperature profiles are vastly different, confirming the temperature anisotropy that
is an observed feature of ion heating in reconnection. Also, Ti⊥ is more consistent
with the prediction in Fig. 5.3 than Tik . Although Ti⊥ is generally lower in magnitude
than Tik , the spatial profile of Ti resembles that of Ti⊥ more because Ti⊥ contributes
to Ti twice as much as Tik does (recall that Ti = (2Ti⊥ + Tik )/3).

76
(a) bg = 0

x/di

−1

−5

−2

(b) bg = 0.1

x/di

−1

−5

−2

x/di

−1

−5

−2

(d) bg = 0.5

x/di

−1

−5

−15

−10

−5

y/di

−2

Figure 5.3: ln α for bg = 0, 0.1, 0.3, and 0.5, respectively.

5.3

Confirmation of Stochastic Motion

The correlation between perpendicular heating and the stochastic criterion involves
the non-conservation of the ion magnetic moment µ = mi vc2 /2B, where vc is the
cyclotron velocity that must be calculated in the frame moving with the guidingcenter velocity [11, pages 96-98]. However, the guiding-center under zero guide
field 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-conservation
measures the departure from guiding center motion and thus the breakdown of
the drift hierarchy. Stochastic heating involves this very breakdown [10], so the
correlation between µE×B and α indicates the existence of stochastic heating.
Another indication of stochastic heating is the Lyapunov exponent [100], which
describes the rate of separation of two initially infinitesimally close particles. As
seen in Chapter 4, a positive Lyapunov exponent is characteristic of chaotic behavior.

77
(e) bg = 0

1.00

x/di

0.75

0.50
0.25

−5

0.00

(f) bg = 0.1

1.00

x/di

0.75

0.50
0.25

−5

0.00

(g) bg = 0.3

1.00

x/di

0.75

0.50
0.25

−5

0.00

(h) bg = 0.5

1.00

x/di

0.75

0.50
0.25

−5

−15

−10

−5

y/di

0.00

Figure 5.4: Ti for bg = 0, 0.1, 0.3, and 0.5, respectively.
µE×B violation
To verify stochastic heating at a single-particle level, test-particle simulations were
conducted with the algorithm in Appendix D. Figure 5.7a shows two typical ion
outflow trajectories (black solid and dashed line) with an initial spatial separation
of 0.04di for bg = 0. Both ions undergo oscillatory motion in the x-direction
under the Hall electric field (color) [7, 47, 162, 168, 171]. Figure 5.7b shows the
time-dependent spatial ion separation, which exhibits divergence and thus a positive
Lyapunov exponent. Figure 5.7c shows µE×B (blue) and ln α (red) along the particle
trajectory represented by the black solid line in Fig. 5.7a. The red dashed line
represents 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 even more obvious in Figs.
5.7d, 5.7e, and 5.7f, which are respectively the same as Figures 5.7a, 5.7b, and 5.7c

78
(a) bg = 0

1.5

x/di

1.0
0.5
−5

0.0

(b) bg = 0.1

1.5

x/di

1.0
0.5
−5

0.0

1.5

x/di

1.0
0.5
−5

0.0

(d) bg = 0.5

1.5

x/di

1.0
0.5
−5

−15

−10

−5

y/di

0.0

Figure 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 in
this case). In Fig. 5.7e, it is apparent that the spatial separation of the ions does
not diverge, illustrating non-stochastic behavior. In Fig. 5.7f, ln α rarely goes above
zero, and µE×B is relatively conserved.
5.4

Heavy Ions

Finally, Equation 5.1 predicts that ions with larger mi /qi more easily satisfy the
stochastic heating criterion — an observed feature of ion heating in reconnection. A
simulation containing a mix of heavy ions (mi /me = 500) of density 0.1n and light
ions (mi /me = 100) of density 0.9n for bg = 0.3 was run. Figure 5.4a shows ln α for
the heavy ions, which, because of their larger mi /qi , satisfy the stochastic heating
criterion across a broader range compared to the light ions in Fig. 5.3c. Figure
5.4b 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
(a) bg = 0

1.5

x/di

1.0
0.5
−5

0.0

(b) bg = 0.1

1.5

x/di

1.0
0.5
−5

0.0

1.5

x/di

1.0
0.5
−5

0.0

(d) bg = 0.5

1.5

x/di

1.0
0.5
−5

−15

−10

−5

y/di

0.0

Figure 5.6: Tik 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 stochastic
mechanism and which will be subject to future investigations. Figure 5.4c shows
that Ti⊥ agrees well with the prediction from Fig. 5.4a, hinting that the additional
heating comes from Tik .
Single-particle trajectories of the heavy ions confirm that stochastic heating is occurring. Figure 5.7g is the same as Figure 5.7d, but for the heavy ions. The separation
divergence (Fig. 5.7h) and the kicks in µE×B in correlation with α (Fig. 5.7i) verify
chaotic behavior. At tωci > 20, ln α is positive only sporadically, so the kicks are
smaller, whereas at tωci < 20, ln α stays positive for longer times, so the kicks are
bigger.
5.5

Summary

The study presented in this chapter provides strong confirmation that stochastic
heating is the fast ion heating mechanism in collisionless magnetic reconnection

80
up to moderate guide fields. The heating is consistent with calculations based on
the properties of the in-plane Hall electric fields that intrinsically develop during
reconnection. Ion temperature profiles were found to agree with the predictions.
The stochastic nature of ions was confirmed through the examination of Lyapunov
exponents and the correlation between µE×B violation and the stochastic heating
criterion. Heavier ions were found to be heated more strongly because they more
easily satisfy the criterion.

Figure 5.7: (a) Two test ion trajectories (black solid and dashed lines) and E x (color)
under bg = 0. (b) The time-dependent spatial separation between the ions in (a). (c)
µE×B (blue) and ln α (red) along the particle trajectory represented by the black solid
line in (a). The red dashed line represents the stochastic heating criterion, above
which 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.

(a) ln(α) @ mi/me = 500

x/di

−1

−5

x/di

1.0

0.5

−5

0.0

x/di

−2

(b) Ti @ mi/me = 500

1.5
1.0

0.5
−5

−15

−10

−5

y/di

0.0

Figure 5.8: (a) ln α, (b) Ti , (c) Ti⊥ for ions with mi /me = 500 under bg = 0.3.

83
Chapter 6

PITCH-ANGLE SCATTERING OF ENERGETIC PARTICLES BY
COHERENT WHISTLER WAVES
In Chapter 2, it was seen that whistler waves are generated as a result of collisionless
magnetic reconnection. Because reconnection is ubiquitous in magnetized plasma
environments, whistler waves are also ubiquitous in the Earth’s magnetosphere
[26, 62, 132, 151], Jupiter’s magnetosphere [92, 135, 152], and Saturn’s magnetosphere [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 and
magnetospheric whistler waves is important since the interaction can change the
pitch-angle of the particles, potentially scattering them into the loss cone of a magnetic mirror configuration such as the Earth’s dipole magnetic field. Because the
escaped energetic particles can cause pulsating auroras at the Earth’s poles and
energetic particles in general can damage spacecraft, this interaction has been the
focus 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 of
particle energization and pitch-angle scattering. Resonant interaction arises when
ω − kvz =

(6.1)

Here, ω is the wave frequency, k is the wavenumber parallel to the background
magnetic field B0 which is oriented in the z direction, vz is the parallel particle
velocity, and Ω = qB0 /m is the cyclotron frequency of the particle with charge q
−1/2
and mass m. Also, γ = 1 − v 2 /c2
is the particle Lorentz factor where v is the
particle speed and c is the speed of light. Kennel and Petschek [80] first quantified
the scattering mechanism by which incoherent whistler waves lead to velocity space
diffusion, and numerous studies have further developed this mechanism [2, 81,
101, 153]. However, recent spacecraft measurements indicate that the observed
chorus bursts are, in fact, extremely coherent and that these waves, especially largeamplitude ones (δB/B0 ∼ 0.01 where δB is the wave magnetic field), are directly
linked to electron energization, loss, and microbursts [4, 24, 28, 55, 153, 154]. This
linkage suggests that a non-diffusive process could be governing what is observed.

84
There has thus been a continuing and substantial theoretical effort to investigate the
dynamics of energetic particles under coherent whistler waves. Bortnik et al. [19]
numerically investigated ad hoc the coherent interaction between large-amplitude
whistler waves and relativistic particles. Lakhina et al. [87] showed via calculations
of pitch-angle diffusion coefficients that coherent chorus subelements can cause
rapid pitch angle scattering, although Lakhina et al. [87] used diffusion coefficients
calculated from incoherent whistler waves [80] and used non-relativistic equations of
motion whereas the actual wave-particle interaction involves relativistic particles (10
keV to MeV [24, 154]). Bellan [9] presented an exact analytical calculation involving
a relativistic particle in a right-handed circularly polarized electromagnetic wave.
This calculation showed that a certain class of particles undergo quick, drastic pitchangle scattering depending on whether the individual particle’s initial conditions
meet a certain criterion, which will be discussed in the next section. Also note
that other studies have investigated this single-particle problem via various methods
[21, 56, 126, 130]. However, an analysis of the importance of this mechanism for
a distribution of particles has not yet been done. To demonstrate importance, one
must show that a significant fraction of the particles in the distribution experiences
this drastic scattering. If this can be demonstrated, then the particle interaction with
coherent whistler waves will be a dominant pitch-angle scattering mechanism.
In this chapter, the analysis presented in Bellan [9] is extended to the relativistic
generalization of a thermal distribution of particles; the generalization is prescribed
by the Maxwell-Jüttner distribution [78]. It is found that for parameters relevant to
magnetospheric chorus, coherent right-handed circularly polarized waves propagating parallel to the background magnetic field trigger large, non-diffusive pitch-angle
scatterings for a significant fraction (1% − 5%) of the energetic particles. The scaling of this fraction with the wave amplitude may also explain the association of
relativistic microbursts to large-amplitude chorus [24]. A new condition for large
pitch-angle scattering is also presented; this condition is a certain range related
to Eq. 6.1, but may or may not include exact resonance depending on the particle
initial conditions. Test-particle simulations corroborate the predictions made by this
analysis. 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 in
this chapter and that this simplified approximation effectively misses critical details
of the wave-particle interaction. The present study illustrates that coherent whistler
waves are an important cause of non-diffusive pitch-angle scattering and provides
an accurate condition for this scattering.

85
6.1

Two-Valley Motion Review

Let us begin with a brief review of the large pitch-angle scattering mechanism presented in Bellan [9]. A thorough comprehension of this single-particle mechanism
is essential for understanding the ensuing analysis presented here. It is assumed that
the wave is right-handed circularly polarized and travels parallel to a uniform background magnetic field, so the total magnetic field can be expressed as B = B0 ẑ + B̃
where
B̃ = κB0 [ x̂ sin (k z − ωt) + ŷ cos (k z − ωt)] .
(6.2)
Here κ is the wave amplitude relative to the background B0 . Faraday’s law determines
the wave electric field to be
Ẽ = − ẑ × B̃ = B̃ [ x̂ cos (k z − ωt) − ŷ sin (k z − ωt)] .

(6.3)

The relativistic Lorentz force equation determines the motion of a charged particle:
q Ẽ
+β×B ,
(6.4)
(γβ) =
dt
m c
where β = v/c and γ = 1 − β2

−1/2

In Bellan [9], a left-handed circularly polarized wave was used although the study
was intended for right-handed waves. However, the result therein is unaffected by
this apparent error because the sign of the particle charge was unspecified. Although
it was not explicitly stated, the analysis was carried out assuming that the charge
is 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 a
right-handed polarization. Therefore, the theory in Bellan [9] describes waveparticle interactions between positively charged particles and left-handed waves,
and equivalently between negatively charged particles and right-handed waves — or
electrons and right-handed whistler waves. This equivalence can also be seen using
charge-parity-time symmetry, which is a fundamental law of any Lorentz-invariant
system [58]; making the changes z → −z and t → −t in Eqs. 6.2 and 6.3 changes
the sense of rotation of the wave, and the relevant physics must be equivalent when
the change q → −q is made.
In this chapter, the analysis in Bellan [9] with the left-handed wave and positively
charged particles will be used for two reasons. First, the analysis can then be kept
general for any particle with any sign of charge. Second, the derivation of a separate
theory for negatively charged particles will merely be a matter of some sign changes

86
and is not worth the additional complexity in understanding the core points of this
chapter.
In Bellan [9], a "frequency mismatch" parameter
ξ = 1 + αγ (nβz − 1)

(6.5)

was defined, where α = ω/Ω is the normalized frequency, βz = vz /c is the normalized parallel velocity, and n = ck/ω is the refractive index. Equation 6.1 is
satisfied when ξ = 0, so ξ is a measure of the departure from resonance. An exact
rearrangement of Eq. 6.4 leads to an equation of motion for a particle moving in
ξ-space [9]:
1 d2ξ
∂ψ
(6.6)
=− ,
02
Ω dt
∂ξ
where

ξ02
1 4
ξ2
02
ψ (ξ) = ξ + κ −
− sκ sin φ0
− κ02 ξ

(6.7)

is the pseudo-potential for ξ-space motion. Here the primed quantities are calculated
in the wave frame, i.e., a frame moving with a velocity ẑω/k. The subscript 0 refers
to the value at the initial time t = t 0 = 0 and there are two parameters, namely s
−1/2
and φ0 . The parameter s is defined as s = αnβ⊥0 γ0 /γT where γT = 1 − n−2
is the Lorentz factor of the wave. The parameter φ0 is defined as the initial angular
orientation of the perpendicular velocity in the x − y plane, i.e., the angle between
β⊥0 and Ẽ(t = 0, z = 0). The shape of the pseudo-potential is entirely determined
by 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 fixed
parameters in the present analysis.
Multiplying Eq. 6.6 by dξ/dt 0 and integrating with respect to t 0 yields the particle
pseudo-energy,
2
dξ
+ ψ (ξ) ,
(6.8)
W=
2Ω02 dt 0
which is a constant of the motion. For certain initial conditions, ψ (ξ) consists of
two valleys separated by a hill in between. If the initial particle pseudo-energy
is sufficiently large to go over the hill between the two valleys, then the particle
undergoes two-valley motion in ξ-space. This motion involves large changes in ξ
and thus in βz , β⊥ and the pitch-angle θ pitch = tan−1 β⊥ /βz .

87
6.2

Two-Valley Motion Condition

Let us now derive the conditions for two-valley motion for a given particle. The
conditions consist of two parts: ψ(ξ) must first be two-valleyed, and the particle
must have sufficient pseudo-energy to overcome the hill between the two-valleys.
The initial particle kinetic pseudo-energy can be expressed as [9]
2
dξ
= s2 κ02 cos2 φ0,
(6.9)
02
2Ω dt t 0=t=0 2
so the total pseudo-energy is
ξ04 ξ02 0 0
1 2 02
+ κ (κ − s sin φ0 ) − κ02 ξ0 .
W = s κ cos φ0 −

(6.10)

We write Eq. 6.7 as ψ (ξ) = ξ 4 /8 + bξ 2 /2 − κ02 ξ where b = κ02 − ξ02 /2 − sκ0 sin φ0 .
Then dψ/dξ = ξ 3 /2 + bξ − κ02 , so one extremum is at small ξ ≃ κ02 /b and two
extrema are at large ξ ≃ ± −2b. Since d 2 ψ/dξ 2 = 3ξ 2 /2 + b, for b < 0 the large
extrema 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 at
ξ ≃ κ02 /b. Figure 6.1a shows an example of a two-valley ψ (ξ) for which b < 0, and
Fig. 6.1b shows a one-valley ψ (ξ) for which b ≥ 0.
We now make the assumption
κ0
s,

(6.11)

which will be shown in Section 6.4 to be appropriate for relevant magnetospheric
situations. Then, b ≃ −ξ02 /2 − sκ0 sin φ0 is negative for
ξ02 ≥ −2sκ0 sin φ0 .

(6.12)

All particles having sin φ0 > 0 satisfy this equation because ξ02 is non-negative.
Particles having sin φ0 ≤ 0 satisfy Eq. 6.12 only if they are in a certain distance
away from exact resonance (ξ = 0).
Now, inserting ξ = κ02 /b in Eq. 6.7, we have the height of the hill to be ψmax ≃
−κ04 /(2b). Therefore, the particle has enough pseudo-energy to cross over the hill
if
ξ4 ξ2
κ04
1 2 02
s κ cos2 φ0 − 0 + 0 κ0 (κ0 − s sin φ0 ) ≥ κ02 ξ0 −
(6.13)
2b
We now assume and justify later that the terms on the right-hand side of Eq. 6.13 are
much smaller than those on the left-hand side. Using Eq. 6.11, Eq. 6.13 becomes
ξ04 ξ02 0
1 2 02
s κ cos φ0 −
− κ s sin φ0 ≥ 0,

(6.14)

1.0

1e−4(a) Two-valley ψ(ξ)

(b) One-valley ψ(ξ)

3 1e−4

0.0

0.5

−0.5

b=-0.008
−0.2 −0.1 0.0

0.1

0.2

b=0.031
−0.2 −0.1 0.0

θpitch

(∘) Two-valley θpitch(t)
120

110

100

1e−4

200

t/Ω−1

400

600

(e) χ(ξ)

1e−4

200

t/Ω−1

400

600

(f) χ(ξ)

1.0

0.5
0.0

0.2

(d) One-valley θpitch(t)

120

0.1

−0.2 −0.1 0.0

0.1

0.2

−0.2 −0.1 0.0

0.1

0.2

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-dependent
pitch-angle of the particle undergoing two-valley motion, and (d) that of the particle
undergoing one-valley motion. The wave parameters were κ = 0.01, α = 0.25, and
n(α) = 18 from Eq. 6.27. (e), (f) The approximated pseudo-potentials χ obtained
by keeping only the term involving sκ0 in Eq. 6.7 for the respective particles.

89
whose solution is
ξ02 ≤ 2sκ0 (1 − sin φ0 ) .

(6.15)

Now we derive the conditions for which the assumptions regarding Eq. 6.13 are
valid. This is done by using the solution (i.e., Eq. 6.15) obtained under the
assumptions and deriving the conditions for which the terms on the right-hand side
of 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 is
O s2 κ02 except for the κ02 ξ02 /2 term which is ignored by Eq. 6.11. On the right√
hand side, κ02 ξ0 = O sκ05 so it can be ignored if κ0
s3 . Examining the second
term, κ04 /2b = O κ03 /s because b = O (sκ0), so it can be ignored if κ0
s. Since
κ0
1 for linear waves, κ0
s3 and κ0
s are both true for s ≥ 1, and κ0
s3 is
a stronger statement than κ0
s if s < 1. Therefore, for κ0
s3 —which will later
be demonstrated to be valid for relevant magnetospheric parameters—the following
gives the condition for which a particle undergoes two-valley motion and thus a
large pitch-angle scattering:
− 2sκ0 sin φ0 ≤ ξ02 ≤ 2sκ0 (1 − sin φ0 ) .

(6.16)

Equation 6.16 is one of the main results of this chapter. For φ0 ≥ 0, Eq. 6.16
becomes Eq. 6.15 and specifies a certain range around ξ0 = 0. However, for φ0 < 0
that statistically represents half of the particle population, Eq. 6.16 does not include
ξ0 = 0, which means that particles further away from exact resonance undergo twovalley motion and thus large pitch-angle scattering. Therefore, Eq. 6.16 specifies
the 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 has
enough 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 the
one-valley pseudo-potential. The particle in Fig. 6.1c experiences a much larger
change in pitch-angle than that in Fig. 6.1d. The rate of change of the pitch-angle
in Fig. 6.1c is also very large; the wave period is Twave Ω = 2π/α ≃ 25, so the
pitch-angle changes by ∼ 15◦ in tΩ ≃ 40 or in about one to two wave periods.

6.3

Distribution of ξ

The initial particle distribution in ξ-space will now be derived. The subscript zero
will henceforth be dropped because only the initial conditions are being examined.
The thermal distribution is assumed to be the Maxwell-Jüttner distribution [78]

90
with an isotropic temperature. This distribution is a considerable simplification,
and repercussions of this simplification and possible remedies will be discussed
in Section 6.6. This is the relativistic generalization of the Maxwell-Boltzmann
distribution and can be expressed in terms of the Lorentz factor γ as
γ
γ 2 1 − 1/γ 2
fγ =
exp − ,
(6.17)
θK2 (1/θ)
where θ = k BT/mc2 is the normalized temperature and Kn is the modified Bessel
function of the second kind of order n. Using γ = 1 + p2 /m2 c2 = 1 + ρ2 where
ρ = p/mc is the normalized particle momentum, Eq. 6.17 can be expressed as
1 + ρ2
exp −
(6.18)
fρ =
4πθK2 (1/θ)
Integrating Eq. 6.18 in ρz and over all angles gives fρ⊥ :
ρ⊥ 1 + ρ2⊥ © 1 + ρ2⊥ ª
®.
f ρ⊥ =
K1
(6.19)
θK2 (1/θ)
Note that fρ is defined in 3D ρ-space so that fρ d ρ = 1, whereas fρ⊥ is defined
in 1D ρ⊥ -space so that fρ⊥ dρ⊥ = 1. Integrating Eq. 6.18 in ρ x and ρ y gives fρz :
1 + ρ2z ª
zª
1 +
® exp −
®,
(6.20)
f ρz =
2K2 (1/θ)
where fρz dρz = 1. The details of the derivations of fρ⊥ and fρz are given in
Appendices F.1 and F.2, respectively.
Now, noting that γβ = γv/c = p/mc = ρ, the mismatch parameter (Eq. 6.5) can
be expressed as
ξ = 1 + α (nρz − γ) .
(6.21)
The probability distribution of having a specific ξ is obtained by multiplying the
probability distribution of having a certain γ by that of having the corresponding ρz
which yields the specified ξ, and then integrating over all γ (full derivation given in
Appendix F.3). The solution is
∫ ∞ 2p
2 (γ, ξ)
1 + ρ2z (γ, ξ) ª
γ 1 − 1/γ ©
® exp −
® dγ,
fξ =
1+
2αnK22 (1/θ)
(6.22)

91
(b) fξ for varying α

(a) fξ for varying θ
= 0.01
θ = 0.04
θ = 0.16
θ = 0.64

1.5

fξ

1.0

2.0
1.5

0.5
0.0

= 0.05
α = 0.1
α = 0.2
α = 0.4

−2

0.75

1.0

0.50

0.5

0.25

0.0
−4

=4
=8
n = 16
n = 32

1.00

0.00
−4

−2

−4

−2

Figure 6.2: fξ for different (a) θ, (b) α and (c) n values. The default values are
θ = 0.1, α = 0.25, and n = 10. The black dashed line is the resonant condition
ξ = 0.
where ρz (γ, ξ) = [(ξ − 1) /α + γ] /n is a rearrangement of Eq. 6.21 and
Given θ, α and n, Eq. 6.22 is an integral solution for fξ .

fξ dξ = 1.

Figure 6.2 shows fξ for different (a) θ, (b) α and (c) n values. The default values
are θ = 0.1, α = 0.25, and n = 10, where α = 0.25 and θ = 0.1 are relevant
values for the dayside outer magnetosphere [153], and n = 18 ∼ 10 from the
whistler dispersion relation (Eq. 6.27). The black dashed vertical line represents
the resonant condition ξ = 0 (or equivalently Eq. 6.1). As θ, α, and n increase
from zero, fξ broadens and more particles are resonant. After a certain threshold,
however, too much broadening leads to the decrease of the local magnitude of
fξ (ξ = 0) and reduces the number of resonant particles. Increasing α significantly
changes the mean value of ξ as well, raising this threshold higher.
6.4

Fraction of Particles Undergoing Two-Valley Motion

Before calculating the fraction of particles undergoing two-valley motion, the probability 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 relevant distribution is that of ρ⊥ and sin φ. Equation 6.19 prescribes fρ⊥ , and assuming
that φ is isotropic, the probability distribution of Φ = sin φ is the Arcsine(-1,1)
distribution,
fΦ = √
(6.23)
π 1 − Φ2
for Φ ∈ (−1, 1).
We now have all the ingredients to calculate the fraction of particles that undergo
two-valley motion in ξ-space and thus experience large pitch-angle scattering. This
fraction can be found by calculating the probability that both Eqs. 6.12 and 6.15

92
(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 = 2ακ0 n2 − 1 so that 2sκ0 sin φ = aρ⊥ Φ and
2sκ0 (1 − sin φ) = aρ⊥ (1 − Φ), the probability of two-valley motion is
∫ √
p+ =

ρ⊥ =0

Φ=0

aρ⊥ (1−Φ)

fρ⊥ fΦ

ξ=− aρ⊥ (1−Φ)

fξ dξdΦdρ⊥ .

(6.24)

In the opposite case where Φ ≤ 0, the probability of two-valley motion is,
∫ √
∫ √

p− =

ρ⊥ =0

− −aρ⊥ Φ

fρ⊥ fΦ
Φ=−1

fξ dξ +

− aρ⊥ (1−Φ)

aρ⊥ (1−Φ)

−aρ⊥ Φ

fξ dξ 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 whistler
wave dispersion relation, which, for parallel propagation and Ω p /Ω
1 where Ω p
is the electron plasma frequency, is
Ω2p /ω2
c2 k 2
|Ω| /ω − 1
ω2

(6.26)

In terms of the dimensionless variables used in this chapter, this becomes
n= p

Ω p /Ω
α (1 − α)

(6.27)

which can be used to express n (α) if Ω p /Ω is specified. Using parameters in
Tsurutani et al. [153] (ne ≃ 10 cm−3 ,B0 ≃ 125 nT), we obtain Ω p /Ω ≃ 8; this value
will 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 from
5.11 keV to 5.11 MeV). Since the parameter range is determined, the conditions for
which the assumption κ0
s3 that was used to derive Eq. 6.16 is true can now be
determined. Because n
1, κ0 = κ/γT = κ 1 − 1/n2 ≃ κ and s = α n2 − 1ρ⊥ ≃
p
αnρ⊥ . From Eq. 6.27 it follows that αn = Ω p /Ω α/(1 − α). We now compare
the largest value of κ to the lowest value of s3 , which involves the smallest values
of α and θ. For θ
1, the most likely ρ⊥ is θ (see Appendix F.4). Thus, the
condition κ0
s3 can be expressed as
!3
Ωp
αθ

(6.28)
Ω 1−α

93
101

(a) ptv for κ = 0.0001

100
10−1
10−2

0.25

0.50

0.75

0.0064 101
0.0056
0.0048
0.0040 10
0.0032
0.0024 10−1
0.0016
0.0008
0.0000 10−2

(b) ptv for κ = 0.0010

0.25

0.50

0.75

0.0200 101
0.0175
0.0150
0.0125 10
0.0100
0.0075 10−1
0.0050
0.0025
0.0000 10−2

0.25

0.50

0.75

0.0640
0.0560
0.0480
0.0400
0.0320
0.0240
0.0160
0.0080
0.0000

Figure 6.3: ptv as a function of α and θ for different κ values.

κ 2/3
2
Ω p /Ω (α/(1 − α))

(6.29)

Inserting α = 0.1 and κ = 0.01 shows that κ0
s3 is valid if θ
0.0066. Thus,
0.01 ≤ θ ≤ 10 is consistent with κ0
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 significant
fraction (1%−5%) of particles undergo two-valley motion and thus large pitch-angle
scattering. However, ptv decreases at high θ (θ & 1), and this phenomenon is related
to the decrease of the local magnitude of fξ (ξ = 0) if there is too much broadening
of fξ , as shown in Fig. 6.2. Figure 6.3 also shows that ptv (α, θ) has more or less the
same shape across a wide range of κ but its magnitude is proportional to κ. This
is because the limits of the ξ integrals in Eqs. 6.24 and 6.25 scale as a ∼ κ, so if
the integration range is sufficiently small so that the integrand may be approximated
by a linear function, it follows that ptv ∝ κ.
6.5

Numerical Verification

The analytical predictions presented so far will now be verified via numerical simulations. A computer code was written which solves Eq. 6.4 and dx/dt = cβ using
the fully implicit Runge-Kutta method of the Radau IIA family of order 5 [64] in the
scipy.integrate.solve_ivp package in Python 3.7. This particular method was used
because it yielded the smallest numerical error out of the available methods in the
Python package, measured by the drift of the average value of the pitch-angle over
time which should be zero in principle (see, e.g., Fig. 6.1c where the particle’s
pitch-angle oscillates around a stationary average value). The error was quantified
by using the statistics of the 10,000 particles in Fig. 6.5c. The Radau method with
a time step ∆t = 0.2 yielded a median value for the pitch-angle drift of 0.07◦ with a

94
(a) Two-Valley Predictions (ϕ = π/4)

1.0
0.8

ρ⟂

1.5

(b) App oximate P edictions (ϕ = π/4)
1.0
0.8

1.5

1.0

0.4

0.5

0.2
0.0

0.0
−0.4

1.0

0.2

0.5

0.0

0.4

0.5

0.0

−0.2

1.5

0.6

−0.4

ρz

−0.2

ρz

(d) Two-Valley Predictions (ϕ = −π/4) (e) App oximate P edictions (ϕ = (π/4)

ρ⟂

1.5

1.0

0.8

1.5

0.6

1.0

0.4

0.5
0.0

0.2

(0.4

(0.2

ρz

) (ρ Δ ρ ) (ϕ = π/4)
(c) Δθpitch
⟂ z

0.0

0.5
0.0
−0.4

−0.2

ρz

(0.4

(0.2

ρz

) (ρ Δ ρ ) (ϕ = (π/4)
(f) Δθpitch
⟂ z

1.5

1.0

0.2

0.5

0.0

10
(0.4

(0.2

ρz

Figure 6.4: (a) Regions of initial momentum space (dark green) that satisfy the
unapproximated 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 a single particle trajectory for a range of initial
particle 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 parameters
were 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 vast
majority of the particles. The electromagnetic fields were prescribed by Eqs. 6.2
and 6.3, which is a simplified model of a whistler wave. The code was parallelized
with the multiprocessing package.
It will first be verified that particles which satisfy Eq. 6.16 and thus undergo twovalley motion experience large pitch-angle scattering. 2,500 particle trajectories
were numerically integrated, and the initial particle momenta were scanned in the
range ρ⊥ ∈ [0, 2], ρz ∈ [−0.5, 0], and φ = π/4, −π/4. The wave amplitude was
κ = 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 the
unapproximated two-valley criteria (Eqs. 6.13 and b < 0) for φ = π/4. Figure 6.4b
shows regions of this space that satisfy the approximated criterion (Eq. 6.16). The
regions 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 predictions
of large pitch-angle scattering. The colors in Fig. 6.4c show the pitch-angle range
that a particle undergoes for each point in (ρ⊥, ρz ) space; this pitch-angle range is

95
(a∘ κ = 0.0001

2000 4000 6000 8000 10000

Particle Number

Red = 1.52Δ

10.0
7.5
5.0
2.5
0.0

2000 4000 6000 8000 10000

Particle Number

Red = 4.29Δ

Δθpitch ( ∘

(c∘ κ = 0.0100

(b∘ κ = 0.0010

12.5

Red = 0.43Δ

30
20
10

2000 4000 6000 8000 10000

Particle Number

Figure 6.5: Pitch-angle range of 10,000 particles whose initial momenta were
randomly sampled from Eq. 6.18 for different κ values. Red points represent
particles that meet the two-valley criterion (Eq. 6.16), and the text inside represents
the percentage of red particles. The red horizontal lines represents the median
∆θ ◦pitch of the red particles

defined by the absolute difference between the maximum and minimum pitch-angles
along the particle trajectory. For example, the particle in Fig. 6.1c has a pitch-angle
range of ∼ 15◦ , and that in Fig. 6.1d has a pitch-angle range of ∼ 3◦ . Figures 6.4d-f
are the same as Figs. 6.4a-c except for φ = −π/4. It can be clearly seen that if
a particle’s initial momentum satisfies the two-valley criteria, it undergoes a large
pitch-angle scattering.
The blue curves in Fig. 6.4 represent the resonance conditionq(Eq. 6.1; ξ = 0). The

curve is found by solving ξ = 1 + α(nρz − γ) = 1 + α(nρz − 1 + ρ2⊥ + ρ2z ) = 0 for
ρ⊥ (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 and
large scattering. This fact is consistent with Eq. 6.16 which qualitatively states that
for φ < 0, the condition for two-valley motion and large scattering does not include
ξ = 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 sampled
from Eq. 6.18 were respectively integrated for κ = 0.0001, 0.001, and 0.01. Other
parameters were α = 0.25, n = 18, and θ = 0.1.
Figure 6.5 shows the pitch-angle range (in degrees) of the randomly sampled particles for different κ values. Red points represent particles that meet the two-valley
criterion (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 ranges
are 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 points

0.15
0.10
0.05
0.00

2000 4000 6000 8000 10000

Particle Number

(c∘ κ = 0.0100

(b∘ κ = 0.0010

Red = 1.52Δ

1.5
1.0
0.5
0.0

2000 4000 6000 8000 10000

Particle Number

Δθpitch ( ∘ per Twave

Red = 0.43Δ

Δθpitch ( ∘ per Twave

(a∘ κ = 0.0001

Red = 4.29Δ

15
10

2000 4000 6000 8000 10000

Particle Number

Figure 6.6: Pitch-angle change per wave period of the respective simulations in
Fig. 6.5. The red horizontal lines respectively represent the median value of the
pitch-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). However, 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 have
pseudo-energies that are just short of overcoming the two-valley hill, so they "almost" undergo two-valley motion and experience substantial pitch-angle scattering.
Therefore, we conclude that ptv is a lower-bound for the fraction of particles with
large pitch-angle scattering.
Even if two-valley motion were to cause large pitch-angle scattering, the mechanism
would 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 twovalley motion to occur. Figure 6.6 shows the pitch-angle change within a single wave
period for the respective simulations in Fig. 6.5. Tsurutani et al. [153] observed
in the outer magnetosphere coherent chorus elements with amplitudes κ ≃ 0.0016
that are 0.1 ∼ 0.5 s long with a frequency of ∼ 700Hz. These elements consisted
of subelements or packets lasting 5 ∼ 10 ms, corresponding to about 3.5 to 7 wave
periods. κ ≃ 0.0016 approximately corresponds to Fig. 6.6b, which shows that
red particles can reach their median pitch-angle range (∼ 5◦ from Fig. 6.5b) in five
wave periods on average. For κ = 0.01 (Fig. 6.6b), this rate is even faster as the red
particles can reach their median pitch-angle range of ∼ 15◦ (Fig. 6.5c) in about two
wave periods.
6.6

Discussion

The results presented here may help explain the the association of large-amplitude
whistler waves to relativistic microbursts (∼ 1 MeV) [24] and may explain the lack of

97
(a) Two-Valley Predictions (ϕ = π/4)

1.0
0.8

ρ⟂

1.5

(b) App oximate P edictions (ϕ = π/4)
1.0
0.8

1.5

0.6

0.6
1.0

0.4

0.5

0.2
0.0

0.0
−0.4

1.0

0.4

0.5

0.0

−0.2

−0.4

ρz

−0.2

0.5

0.0

1.0

ρ⟂

0.8

1.5

0.6

0.6
1.0

0.4

0.5
0.0

0.2

(0.4

(0.2

ρz

0.0

0.5
0.0
−0.4

−0.2

ρz

1.0

ρz
1.0

1.0

1.5

0.2

(d) Two-Valley Predictions (ϕ = −π/4) (e) App oximate P edictions (ϕ = (π/4)
1.5

) (ρ Δ ρ ) (ϕ = π/4)
(c) Δθpitch
⟂ z

20
(0.4

(0.2

ρz

) (ρ Δ ρ ) (ϕ = (π/4)
(f) Δθpitch
⟂ z

1.5

1.0

0.2

0.5

0.0

20
10
(0.4

(0.2

ρz

Figure 6.7: Same as Fig. 6.4, but for κ = 0.02.
such energetic microbursts in small-amplitude chorus [154]. Particle energization is
not a subject of this Chapter and thus will not be discussed; it will be assumed that the
particles are first energized by some mechanism that yields a relativistic distribution,
and then the ensuing pitch-angle dynamics are studied in order to concentrate on
one topic. It should be noted, however, that energization and pitch-angle scattering
may occur simultaneously.
In Fig. 6.3, for small amplitudes (0.0001 ≤ κ ≤ 0.001), only up to 0.5% of particles
in a distribution with a temperature of ∼ 1 MeV (corresponding to θ ≃ 2) interact
with 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 the
wave amplitude κ; i.e., as the wave amplitude increases, more particles, including
energetic particles, satisfy the two-valley condition.
The interaction of large-amplitude waves with relativistic particles is further explained 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. 6.7
are much broader in phase space than those in Fig. 6.4. This is important because
in Fig. 6.4, relativistic particles with ρ & 1 must have large initial pitch-angles to
interact with the wave since the two-valley condition is a narrow range related to the
exact resonance condition, and thus these particles must undergo extremely large
pitch-angle scatterings in order to jump into the loss cone. However, in Fig. 6.7, the
range for two-valley motion is much increased, allowing for relativistic particles with

98
smaller initial pitch-angles to interact with the wave. The deviation of the two-valley
condition from the exact resonance condition is because the range in Eq. 6.16 scales
with κ. Furthermore, the pitch-angle range itself is significantly increased in Fig.
6.7. Therefore, a larger wave amplitude allows for relativistic particles with lower
initial pitch-angles to interact with the wave, while simultaneously increasing the
amount of pitch-angle scattering; these two effects lead to more relativistic particles
being pitch-angle scattered into the loss-cone.
There are a few limitations to the present analysis that may be subject to future
work. First, the Maxwell-Jüttner distribution is a simplification and should not be
considered as a distribution representing the entire electron population. The actual
distribution is a sum of these Maxwellians or other functions such as the kappa
distribution [123]. If the actual distribution can be expressed as a weighted sum
of Maxwelll-Jüttner distributions, then the total fraction of particles that undergo
two-valley motion is the sum of the partial fractions for each distribution. On the
other hand, if the actual distribution is another sufficiently simple function, then an
analysis 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, whereas
observations indicate that electron temperature in the magnetosphere in general is
anisotropic and electron distribution functions can be more complex than simple
anisotropic distributions [94]. The transition to an anisotropic Maxwell-Jüttner
distribution is outlined in Livadiotis [98] and Treumann and Baumjohann [150].
Third, the wave is assumed to have exact parallel propagation, whereas many instances of magnetospheric chorus involve oblique propagation [6, 134]. Also,
chorus typically exhibits frequency and amplitude changes over a short time period [153], but the model presented here is based on a plane wave with a fixed
frequency and wavenumber (Eqs. 6.2 and 6.3). However, including obliquity and
variable frequency makes the analysis considerably more complicated and so would
be inappropriate for an inaugural analysis.
6.7

Comparison to Second-order Trapping Theory

A popular theory describing wave-particle interactions is the second-order trapping
effect 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 the
theory. However, it will now be shown that this previous theory is an approximation

99
of the theory presented here; this approximation effectively misses the critical twovalley nature of the pseudo-potential.
Omura et al. [113] use the following coupled equations for non-relativistic speeds:
dζ
= k(vz − VR ),
dt
ω2
(vz − VR ) = t (sin ζ + S),
dt

(6.30)
(6.31)

where

ω−Ω
(6.32)
ζ is the angle between v⊥ and B̃, ωt = kv⊥ Ωκ is the trapping frequency, and S
is a parameter that is equal to zero when the background magnetic field is spatially
uniform and ω is a constant. Therefore, setting S = 0, differentiating Eq. 6.31 in
time, and using Eq. 6.30,
VR =

ωt2
dζ
d2
cos
(v
dt
dt 2
= ωt2 (vz − VR ) cos ζ .

(6.33)
(6.34)

Letting γ = 1 in Eq. 6.5 and rearranging shows that
ξ=

(vz − VR ) ,

(6.35)

so Eq. 6.34 becomes
d2ξ
= ξωt2 cos ζ,
dt
2
ξ 2
=−
− ωt cos ζ ,
∂ξ
∂ χ(ξ)
1 d2ξ
=−
∂ξ
Ω dt

(6.36)
(6.37)
(6.38)

where χ(ξ) = −ξ 2 ωt2 cos ζ/2Ω2 is the pseudo-potential of this system.
Now, let us examine the term involving sκ0 in Eq. 6.7 assuming γ0 = γT = 1;
−sκ sin φ0

ξ2
ξ2
= −αnκ β⊥0 sin φ0 ,
ω ck v⊥0
ξ2
=−
sin φ0 ,
Ωω c
ωt0
= − 2 sin φ0 ,
= χ0 (ξ),

(6.39)
(6.40)
(6.41)
(6.42)

100
because ζ and φ are related by ζ = φ − π/2, so cos ζ = sin φ. χ0 (ξ) is χ(ξ) except
that v⊥0 and φ0 are used instead of v⊥ and φ, and the relationship is similar for ωt0
and ωt . Therefore, χ(ξ) results from keeping only the sκ0 term in ψ(ξ). This is
important because χ(ξ) only describes either a trapping or a non-trapping potential
but not a two-valley potential.
Figure 6.1e and 6.1f plot the approximated pseudo-potentials χ(ξ) for the particles
in Fig. 6.1a and 6.1b, respectively. For both particles, χ(ξ) is clearly a one-valley
potential, 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 in
Fig. 6.1b . Therefore, if the theory in Omura et al. [113] were to be used, it would be
impossible to distinguish between the two particles which clearly have an extremely
large difference in the amount of pitch-angle scattering.
Another important problem with the second-order trapping theory is that the timedependence 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 that
combining 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 no
approximation regarding time-dependence needs to be made. This is an extremely
important point because this time-dependence of v⊥ gives the two-valley potential
whereas treating it as a constant does not. This fact can be more explicitly illustrated
by 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):
B̃0⊥
1 d 2 β0z
0 B̃⊥ B̃⊥
ξβ
Ω0 dt 02
B0
B0 B0

(6.43)

The second-order trapping theory effectively drops the last term in Eq. 6.43 and
ignores the time-dependence of the first term on the right-hand side. This leads to
1 d2ξ
γ0 0 B̃0⊥
αn
ξβ ·
Ω0 dt 02
γT ⊥ B0

(6.44)

which is equivalent to Eq. 6.38 if γ0 = γT = 1 is assumed. However, Eq. 35 of
Bellan [9] states that
B̃0⊥0
B̃0⊥
γT 2
β⊥ ·
= β⊥0 ·
0 ,
B0
B0
2αnγ0

(6.45)

101
which means that treating v⊥ as a constant effectively misses the ξ-dependence in
Eq. 6.45, which is the reason for the two-valley shape of the pseudo-potential.
For example, neglecting the ξ02 term in Eq. 6.45 leads to erroneous conclusions
regarding the shape of the potential near ξ = 0. In Fig. 6.1e, χ(ξ) is a valley
because −sκ0 sin φ0 is positive in this case. However, the correct pseudo-potential
ψ(ξ) in Fig. 6.1a is a hill near ξ = 0 because −ξ02 /2 − sκ0 sin φ0 in Eq. 6.7 is negative
in this case. Also, the ξ 2 term in Eq. 6.45, which leads to the positive ξ 4 term in Eq.
6.7, prevents the pseudo-potential from diverging to −∞ as ξ → ±∞. This prevents
the particle ξ from veering off to infinity; this phenomenon is unphysically allowed
if the approximated χ(ξ) is used and sin φ0 > 0. The term linear in ξ in Eq. 6.7
which affects the asymmetry of the two-valleys is also neglected in χ(ξ). The fact
that v⊥ is not constant can be explicitly seen in Fig. 5g of Bellan [9], where v⊥ of a
particle 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 timedependent wave frequencies, S is finite in Eq. 6.31 and this may have an important
role in the system additional to the effects described in the present chapter.
6.8

Summary

The interaction of a relativistically-consistent thermal distribution of particles with
a coherent right-handed circularly polarized wave has been investigated. Departure
from wave-particle resonance for each particle is expressed by a frequency mismatch
parameter ξ, where ξ = 0 represents perfect resonance. An exact rearrangement of
the relativistic particle equation of motion shows that ξ follows pseudo-Hamiltonian
dynamics with an associated pseudo-potential ψ (ξ). If ψ (ξ) has two-valleys separated by a hill, and the particle has enough pseudo-energy to overcome the hill, then
the particle undergoes two-valley ξ-space motion that produces a large, non-diffusive
pitch-angle scattering.
An accurate condition for two-valley motion and thus for large pitch-angle scattering
has been derived; this condition is related to but may or may not include the exact
resonance condition (Eq. 6.1), and the range of this condition scales with the
wave amplitude. Assuming that the particle distribution is Maxwell-Jüttner, which
is a relativistic generalization of the Maxwell-Boltzmann distribution, for typical
magnetospheric parameters a significant fraction (1 − 5%) of the particles undergoes
two-valley motion. The pertinent analysis can potentially be used for the actual local
electron distribution, which may not be exactly Maxwellian. Numerical simulations

102
confirm the analytical results. The scaling of the fraction of interacting particles
with the wave amplitude may also explain the association of relativistic microbursts
to large-amplitude chorus. The present theory is more accurate and exact than the
widely-used second-order trapping theory as second-order trapping theory fails to
take into account two-valley motion.

103
Chapter 7

MAGNETIC FIELD DIAGNOSTIC USING TWO-PHOTON
DOPPLER-FREE LASER-INDUCED FLUORESCENCE
As can be seen in Chapters 2 through 5, fast magnetic reconnection involves ion skin
depth length scales ∼ di as well as electron skin depth length scales ∼ de . Therefore,
any experimental diagnostic that probes magnetic fields in fast reconnection must
be able to resolve these scales. However, di and de in many situations are minute
quantities compared to the global scale L of the plasma, as can be seen in Table
7.1. This scale constraint as well as other limitations to existing magnetic field
diagnostics impose the need for a new, alternative diagnostic.
A widely-used magnetic field diagnostic is the Faraday pickup coil, which is simply
a loop of an arbitrary number of turns made of a conducting wire. Because Faraday’s
law states that the voltage induced in the loop is proportional to the rate of change of
magnetic flux, a Faraday pickup coil can change fluctuating magnetic fields into an
electrical signal. Because of the simplicity of their design and application, these coils
are frequently used to probe magnetic fields in a plasma [29, 67, 129]. However,
when probing smaller length scales, the area through the loop must inevitably
become smaller. This smaller area leads to smaller magnetic flux and thus a smaller
signal-to-noise ratio, which limits the spatial dimensions of the coil. Another
important limitation is that the coil, being a physical wire, essentially perturbs the
Plasma Type
Flaring Solar Loops
Solar Active Regions
Fusion Plasmas
Earth’s Magnetotail
Caltech MHD-driven Jet Experiment (Argon)
Magnetic Reconnection Experiment (MRX)

L (m)
107 − 108
107 − 108
1 − 10
108 − 109
10−1

n (m−3 )
1017
1015 − 1016
1021
109
1022

di (m)
1 − 10
10−2
104
10−2

de (m)
10−2
10−2 − 10−1
10−4
102
10−4

0.5

1020

10−1

10−3

Table 7.1: Plasma parameters of regions where magnetic reconnection occurs or 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 specified otherwise, di is the ion skin depth assuming
that the ion species is hydrogen. Referenced from Refs. [29, 74–76, 109, 127]

104
plasma; the effect of this perturbation obviously becomes larger as the region of
interest becomes smaller. These constraints hinder the usage of pickup coils for
investigating magnetic fields at small length scales.
Another magnetic field diagnostic uses Faraday rotation, which is an effect where
the plane of polarization of a linearly polarized light wave rotates when passing
through a magnetic field in a medium. The amount of this rotation is proportional
to the strength of the magnetic field parallel to the light propagation direction, so
this method has also been widely used to probe magnetic fields [18, 140]. This
method induces little to no physical perturbation to the plasma and can probe small
length scales because the size of the light source is arbitrary. However, because
the 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 is
not 3D-localized.
In this chapter, a new magnetic field diagnostic based on the laser-induced fluorescence (LIF) technique is presented. This diagnostic is essentially non-perturbing,
3D-localized, and can probe arbitrary scale lengths limited only by the diameter
of the laser source. The particular scheme suppresses Doppler broadening of the
spectral lines, enabling the resolution of Zeeman splitting, which can then be used to
infer the magnetic field information. The theoretical background is first presented,
followed by a description of a repetitively pulsed radio-frequency plasma source
that was constructed in order to test the diagnostic. Preliminary results are then
presented.
7.1

Sources of Spectral Broadening

Several textbook sources of spectral broadening will now be described; these effects
can be analyzed to yield various properties of the diagnosed plasma [38].
Doppler Broadening
Doppler broadening of a spectral line arises due to the thermal motion of atoms. It
is first assumed that the distribution function f is a 1D Maxwellian in the direction
of the laser beam, i.e,

f (v)dv =

mv 2
exp −
dv,
2πk BT
2k BT

(7.1)

where m is the atom mass, 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 where

105
c is the speed of light, the laser wavelength is Doppler shifted by
v
λ = λ0 1 + ,
where λ0 is laser wavelength in the lab frame, or
λ − λ0
v=c
λ0
dv = dλ.
λ0
Inserting these into Eq. 7.1, we have
mc2
mc2
f (λ)dλ =
exp −
[λ − λ0 ] ,
2πk BT λ02
2k BT λ02

(7.2)

(7.3)
(7.4)

(7.5)

which has the standard deviation

k BT
λ0 .
(7.6)
mc2
This effect is called the Doppler broadening effect. Note that if the atoms are moving
at a finite average velocity, a Doppler shift in the center wavelength occurs as well.
σλ =

Zeeman Splitting
When an atom is subject to an external magnetic field, the interaction between the
two effectively splits any energy level to multiple levels. One transition line between
two energy levels then splits into multiple transition lines. The energy perturbation
is
∆E = µB g j m j |B| ,
(7.7)
where µB = e~/2me is the Bohr magneton, m j is the total angular momentum in
the z-direction, B is the external magnetic field, g j is the Landé g-factor which is a
function of the total angular momentum j, the orbital angular momentum l, and the
spin angular momentum s. One energy level therefore splits into multiple levels,
each corresponding to certain possible values of the quantum numbers. Since the
amount of the splitting depends on the magnitude of B, this amount can be used to
infer the strength of the magnetic field.
Natural Broadening
Natural broadening inevitably arises due to the Heisenberg uncertainty principle,
which effectively relates the lifetime of a particular energy level to the uncertainty
of its energy value. The broadening is thus
∆E ≃ ,

(7.8)

106
where τ is the lifetime of the energy level. The broadening profile is Lorentzian.
Stark Broadening
Stark broadening is analogous to the Zeeman effect in that the electric field instead
of the magnetic field perturbs the energy levels. Stark broadening is significant for
high plasma densities. For hydrogen lines, Stark broadening scales as ∼ n2/3
e where
ne is the electron density, whereas for non-hydrogen lines it scales as ∼ ne [173].
The broadening profile is Lorentzian.
Instrumental Broadening
Instrumental 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 Broadening
Saturation broadening, or power broadening, occurs when the laser intensity is so
high that the population of the base level becomes equal to the excited level. At
this saturated point, no further absorption of the laser by the plasma can occur. A
spectral absorption probability typically peaks at the center wavelength and is lower
at the wings of the profile. Therefore, if the laser intensity is above a threshold, the
plasma can no longer absorb the photons at the center wavelength but can absorb
those further away from the center wavelength. This effectively leads to a broadened
line profile with the width [38]
B12 ρ(ω12 )
(7.9)
γs = w 1 +
A12
where w is the original profile width, A12 and B12 are respectively Einstein coefficients 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 . The
broadening profile is Lorentzian.
Comparison of Broadening Effects
The measured spectral profile is a convolution of all the broadening effects. Depending on the plasma parameters, different effects contribute in different amounts.
For example, Table 7.2 shows the contribution of different broadening effects on
the 394.539 nm (vacuum; 4 D5/2
→4 D7/2 ) Ar II line for two different plasmas.
The Caltech MHD-driven plasma jet [109] has a relatively high density and tem-

107
Excitation Level
Laser
Fluorescence

Base Level
De-excitation
Level

Figure 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 a
lower density and temperature, and so Doppler broadening is comparable to Zeeman
splitting while Stark broadening is negligible. Natural broadening (not shown) is
negligible for both cases.
7.2

Two-Photon Doppler-Free Laser-Induced Fluorescence

Laser-Induced Fluorescence
Laser-induced fluorescence, or LIF, is an active spectroscopy method in which a
laser source excites the electron population at a certain energy level to a higher
level [39]. These electrons then de-excite to other lower energy levels, emitting
fluorescent signals that can be analyzed to obtain various information such as the
temperature, density, and magnetic field of the diagnosed plasma. Figure 7.1
graphically illustrates the scheme.

Caltech Jet
Experiment
InductivelyCoupled
Plasma

ne (m−3 ) Ti (eV)
1022
1019

0.05

Stark (pm)
2.2

Doppler (pm)

Zeeman (pm)
0.7

2.2 × 10−3

0.5

0.7

Table 7.2: Comparison of different broadening effects in two types of plasmas:
the Caltech MHD-driven Jet Experiment [29, 105, 109], and a pulsed, inductivelycoupled, radio-frequency plasma [32]. Stark broadening is calculated by using the
experimentally measured values in Konjević et al. [83] and using the scaling ∝ ne
for Argon [173]. Zeeman splitting is calculated by assuming m j and g j are of order
1, and B = 0.1 T.

108

Lab Frame

Atom Frame
(1+v/c)
(1-v/c)

Figure 7.2: Two-photon interaction of an atom in the lab frame and in the atom
frame.
Because the absorption of the laser is dependent on the laser wavelength, the population of the excited level and thus the fluorescence intensity is wavelength-dependent.
Therefore, the full spectrum can be obtained by scanning the laser wavelength around
the spectral line.
The fluorescence intensity depends on the population of the excited level, which in
turn depends on the population of the base level and the laser intensity. Therefore, a
highly populated base level needs to be chosen in order to maximize the fluorescence
signal. An obvious choice is the ground state of the atom, but the transition energy
between the ground state and the first excited state is typically too high for any
simple, conventional laser. Therefore, a metastable level is typically chosen. By
definition, there are no allowed first-order transitions to a lower energy level from
metastable levels, so these levels have relatively long lifetimes and so are highly
populated.
Two-Photon LIF
As in can be seen in Table 7.2, Doppler broadening can dominate over Zeeman
splitting so that the latter is unresolvable. Therefore, in order to resolve Zeeman
splitting and thus obtain the magnetic field information, Doppler broadening must
be eliminated. Two-photon laser-induced fluorescence achieves this feat by using
two 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 counterpropagating photons of wavelength λ. Then, in the frame of the atom, one of the
photons is blue-shifted while the other is red-shifted. If both photons are absorbed

109

5s P
3/2

410.499 nm
393.430 nm
393.984 nm

4p D

5/2

4p D5/2

442.724 nm
394.539 nm

4s P

3/2

3d D
7/2

Figure 7.3: Two-photon laser-induced fluorescence scheme for Argon II. Displayed
wavelengths are vacuum values.
by the atom, then the absorbed energy is, assuming v
c,
λ(1 + v/c) λ(1 − v/c)
v
c
1− +1+ ,
2c
= ,

(7.10)
(7.11)
(7.12)

which does not depend on velocity. Therefore, Doppler broadening is cancelled out
in this scheme.
Figure 7.3 shows the proposed two-photon LIF scheme in Argon II. Two counterpropagating laser beams at 393.984 nm pumps the metastable 3d 4 D7/2 level (level
1) to the 5s2 P3/2 level (level 3) via a virtual level (level 20). The intermediate level
(level 2) is not necessary for two-photon transition but allows it to be achieved with
relatively low-power lasers; if the intermediate level does not exist, then high-power
lasers are necessary. This fact can be seen by examining the two-photon transition
probability A13 [39]:
A13 ∝

γ13 I1 I2
[ω13 − ω1 − ω2 − v · (k1 + k2 )]2 + (γ13 /2)

D12 · ê1 · D23 · ê2 D12 · ê2 · D23 · ê1
ω12 − ω1 − v · k1 ω12 − ω2 − v · k2

(7.13)

where ωi , ki , êi and Ii are respectively the angular frequency, the wavevector, the
polarization vector, and the intensity of the i-th laser beam, and ωi j , γi j , and Di j

110

Figure 7.4: Example of two-photon LIF in Rubidium. (a) The original Dopplerbroadened spectrum. (b) Result after applying two photons. (c),(d),(e),(f) Zoom-ins
of respective peaks, showing hyperfine splittings. Reproduced from Jacques et al.
[77]. © European Physical Society. Reproduced by permission of IOP Publishing.

are the transition angular frequency, the natural linewidth, and the dipole matrix
element [38] between energy levels i and j. It can be seen that the second factor
increases as ω1 and ω2 get closer to ω12 .
Two-Photon LIF in Rubidium
There 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 Zeeman
splitting of the hyperfine structure of Rubidium.
Figure 7.4 shows the result of applying two-photon LIF in Rubidium. The Dopplerbroadened 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.

111

Figure 7.5: Two-photon LIF observation of Zeeman splitting of a Rubidium hyperfine 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-Frequency
Plasma Source

A new plasma source was designed and built as the subject of diagnosis of the
proposed two-photon LIF diagnostic. This source is based on the battery-powered
pulsed inductively-coupled radio-frequency (RF) plasma source previously developed by Chaplin [32]. The prior source uses a battery rack to charge a capacitor to
a high voltage, which is then discharged through a homemade RF generator circuit
to deliver an electrical power of 2.7 kW to the load. This power creates a plasma
of 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 corresponds to a 0.02% duty cycle. Also, it cannot last for more than about 20 hours
before the batteries need to be replaced. Therefore, it is not suitable for experiments
that 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 on
indefinitely. The circuit was designed so that the capacitor can be repetitively
pulsed at varying frequencies depending on the pulse length. The final design was
such that the source was pulsed for 250 µs at 60 Hz, corresponding to a duty cycle
of 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.

112
RF Driver Circuit

DC Power Supply
+5V

27.12MHz
Oscillator

+15V

DRF1301
I Limiter

47uF

I V

13.56MHz
RF OUT
IV
Probe
Transformer

Optical
Gate
Fuse

VARIAC
Transformer
0-130V

Voltagex2
Transformer

Bridge
Rectifier

Optical
Converter

Antenna
+Plasma

Timing
Circuit

Figure 7.6: A schematic diagram of the RF plasma source.

RF Generator
A commercial power supply converts wall power to +5 V and +15 V DC outputs. The
+5 V output is used to power the crystal oscillator which provides the fundamental
frequency of 27.12 MHz. The +15 V output is used to power a Microsemi DRF1301
power MOSFET hybrid chip containing two power MOSFETs. A current limiter was
put in place to prevent the chip from drawing excessive current from the power supply.
Without this current limiter, the power supply would heat up to extremely high
temperatures (> 100 C◦ ). The two power MOSFETs inside DRF1301 effectively act
as 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-down
converter to power a cooling fan at +12 V.
Pulsed Power Driver
A VARIAC (variable AC) transformer rated for 2 kW and 20A converts wall power
to AC voltages between 0-130 V. This voltage is then doubled through another
transformer. A diode bridge rectifier then flips the negative values of the AC voltage
to positive values (sin ωt → |sin ωt|). The output of the rectifier is used to charge a
47 µF capacitor rated for 300 V, and because the rectifier does not allow backward
current, it can only charge, not discharge, the capacitor. Therefore, the capacitor
is charged to a voltage of 2 times the VARIAC transformer voltage — recall that
AC voltages are rms values — in about 4 ms, which is a quarter of the wall power
period.
The gate for the circuit is made optical in order to isolate it from the high-power
components. When the gate is open, the main 47 µF capacitor is discharged through

113
one of the two MOSFETs in an alternating manner. This oscillating discharge is
picked 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 Probe
An IV probe measures the current and voltage outputs of the RF generator. The
current was measured by an Ion Physics CM-10-M current monitor. The voltage
was measured by a 10x or a 1000x voltage probe, depending on the voltage level of
the 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 by
replacing the impedance matcher and the antenna by a resistor and then probing the
current and voltage profiles across it. Because the impedance of a resistor is purely
real, there should theoretically be no phase delay between I and V. Therefore, any
phase delay across the resistor measured by the IV probe can be considered to be
instrumental and thus can be subtracted out from future measurements.
Impedance Matcher
In order for maximal power to be transferred to the load, the load impedance must
be equal to the the complex conjugate of the source impedance [32]. Because the
source impedance is unknown in the homemade RF generator, the load impedance
must be adjusted. Instead of expensive commercial impedance matchers, two binary
arrays of ceramic capacitors serve as variable parallel and serial capacitances — Cp
and Cs — constituting the impedance matcher. The binary arrays could be adjusted
by using copper jumpers to "activate" each of the ceramic capacitors with values of
1 pF, 2 pF, 4 pF, and so on, providing nearly any capacitance value up to Cp = 2604
pF and Cs = 886 pF with 1 pF resolution.
Antenna and Plasma
A half-turn helical (HTH) [139] antenna design is used. Other types of antennas such
as Nagoya type III antennas [34] were tested as well, but the HTH antenna proved
to be the most efficient in terms of plasma generation. The antenna wraps around a
glass tube fed into a vacuum chamber and determines the spatial dimensions of the
plasma. Its radius is about 1.27 cm, and its length is about 7.5 cm.

114

Figure 7.7: The resultant plasma discharge.

Timing Circuit
The timing circuit consists of a trigger generator that determines the frequency of
the pulse, which is fed into a pulse generator that determines the gate length of the
pulse. 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 Setup

Figure 7.8 shows the basic experimental setup for two-photon laser-induced fluorescence.
Laser Excitation System
A tunable diode laser (Sacher Lasertechnik LYNX Littrow Cavity) is controlled by
a laser controller, which is in turn controlled by a personal computer (PC). The laser
wavelength is controlled by the laser current, temperature, and the piezo voltage,
but the piezo voltage was mainly used to scan the wavelength during the main
measurements. The other parameters were varied to stabilize the laser during the
scans. The diode laser produces a laser beam with a maximum power of 30 mW
and 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 is
focused into a multimode optical fiber and is collimated at the output. Therefore, by
providing a TTL signal that turns the RF signal on and off at the lock-in frequency
flock-in via the lock-in amplifier, the output laser beam is modulated (turned on and
off) at flock-in .
A 30:70 beamsplitter directs 30% of the output beam into a wavemeter for wavelength

115
Optical
Fiber
Slit
Diode Laser
w/ OI

M2
L3

L2
AOM

L1
PMT

Laser
Controller

RF
Generator
TTL

Lock-in Amplifier

Filter
BS

Collimator

Lock-in
Signal
PC
w/ Labview
Program

Wavemeter

Wavemeter Switch
(For Two Lasers)

Figure 7.8: Experimental setup for two-photon laser-induced fluorescence. L stands
for lens, M for mirror, BS for beam-splitter, AOM for acousto-optic modulator, OI
for optical isolator, and PMT for photo-multiplier tube. The purple color is the
end-on view of the plasma.

reference. The rest of the beam is directed through the plasma, bounces off of a
mirror (M2), and is redirected back into the plasma, providing counter-propagating
photons.
Fluorescence Collection System
The laser-induced fluorescence from a single point in the plasma passes through
a lens (L2) and is made into a collimated beam, which is in turn passed through
a band-pass optical filter of the desired wavelength. The beam is then focused
by another lens (L3) into a battery-operated photo-multiplier tube (PMT; Hamatsu
H10721-110). In order to maximize the detection of the signal, it is imperative to
use a lens to collimate the fluorescence first and then pass it through the filter. This
is because the wavelength band of the filter is usually rated at a 90◦ incident angle
and 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 at
flock-in , provided that the fluorescence rise and decay times are not long compared

116
to Tlock-in = 1/ flock-in . Thus, the PMT signal is fed into the input of the lock-in
amplifier for averaging. The output of the lock-in amplifier is transferred to the PC
via a data acquisition system (National Instruments USB-6001).
Wavelength Feedback and Scanning System
A LabVIEW program was written that scans the laser wavelength around a center
value at set intervals and collects lock-in data while locking the laser wavelength
at each set value. The wavelength locking is performed by feeding the real-time
wavemeter measurement back to the laser piezo voltage via the PC. The user inputs
the center wavelength, the scan range, the number of data points, and the averaging
time, and these are translated into an array of wavelength values for which the data
is to be taken. Then, the laser wavelength is sequentially locked at each wavelength
value for the duration of the averaging time, and the data from the lock-in is recorded
at the end of each averaging time.
For example, let us assume that the center wavelength is 394.538 nm, the scan range
is ±4 pm, the number of data points is 200, and the averaging time is 15 seconds. The
scan interval is thus 8/200 = 0.04 pm. So, the program first locks laser wavelength
at 394.534 nm for 15 s, records the value of the lock-in output, and then moves
the laser wavelength to 394.53404 nm. The process then continues until the laser
wavelength reaches 394.542 nm.
Two-Photon LIF with Different Colors
The setup can be altered so that two separate counter-propagating laser beams are
respectively tuned to the transition wavelengths of the first (level 1 → 2) and the
second (level 2 → 3) transitions. In this case, a second laser + AOM + fiber system
is 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 optical
fibers that respectively measure the wavelength of the first and the second laser are
then fed into a wavemeter switch — a device that allows a single wavemeter to
measure multiple wavelengths simultaneously.
7.5

One-Photon LIF Results

In 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) was
measured for different background Argon pressures. This allowed for the testing

117
of the alignment and the overall setup because the expected signal is theoretically
higher for one-photon transition.
Figure 7.9 shows the LIF profiles for background Argon pressures of 1, 5, and 20
mTorr, respectively. Red lines are fitted Voigt profiles, which are a convolution of
Gaussian and Lorentzian profiles; i.e.,
∫ ∞
V(λ; σ, γ) =
G(λ0; σ)L(λ − λ0, γ)dλ0,
(7.14)
−∞

where

(λ − λ0 )2
G(λ; σ) = √
exp −
2σ 2
2πσ 2

(7.15)

and

(7.16)
π[(λ − λ0 )2 + γ 2 ]
Here, γ is the Lorentzian width, σ is the Gaussian width, and λ0 is the center
wavelength.
L(λ; γ) =

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 ion
temperatures are respectively 1230±186 K and 756±189 K. Because the input power
is fixed, the more ions there are (i.e., higher density), the less power is distributed to
individual ions. Therefore, the temperature decreases as the background pressure
increases. However, it can be seen that the temperature at 1 mTorr is lower than
that at 5 mTorr. This is because the ion-electron collision frequency becomes lower
as the density decreases. Since the RF power is primarily transferred to the much
lighter electrons, if the number of ion-electron collisions decreases, then electrons
transfer less energy to ions.
The fact that the turnaround in the ion temperature is observed can actually be used to
estimate the electron (and thus the ion) density. Ion-electron collisions will become
negligible when the mean-free-path l mfp is comparable to the plasma length l plasma ,
which is about 5 cm. From the measured amount of Ar II, the electron temperature
is estimated to be around Te = 3 eV. The highest electron density at which collisions
are negligible can therefore be estimated to be
ne =

σl plasma
2
2π
20Te
l plasma
ln Λ

(7.17)

= 3 × 1018 m−3,

(7.19)

(7.18)

118

Fluorescence (Arb.)

(a) 1 mTorr (Scan No. 75)

−1
−2

γ = 0.769±0.134 pm
σ = 0.391±0.119 pm
Ti = 428±259 K
λ0 = 394.53890±0.00002 nm

−3
−4
394.536 394.538 394.540 394.542

Fl orescence (Arb.)

(b) 5 mTorr (Scan No. 77)

−1
−2

γ = 0.000±0.102 pm
σ = 0.663±0.050 pm
Ti = 1230±186 K
λ0 = 394.53891±0.00001 nm

−3
−4

394.536 394.538 394.540 394.542

Fl orescence (Arb.)

−1
−2

γ = 0.099±0.128 pm
σ = 0.520±0.065 pm
Ti = 756±189 K
λ0 = 394.53885±0.00001 nm

−3
−4

394.536 394.538 394.540 394.542
λ (nm)

Figure 7.9: One-photon LIF results for 1, 5, and 20 mTorr. The laser wavelength
is scanned around the 394.5 nm line in Fig. 7.3 and the plots show the subsequent
442.7 nm fluorescence as a function of the laser wavelength. 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.

119
where σ is the collision cross-section, ln Λ is a constant of order 10, and Te is in
units of eV. Above this density, collisions start becoming significant, so electrons
start imparting their energies to ions. Previously measured electron densities in a
similar setup yielded ∼ 1019 m−3 at ∼ 10 mTorr, with which the present estimate
agrees well.
The line broadening at 1 mTorr is dominated by a Lorentzian profile. Because the
Stark width and the natural width for the given parameters are both ∼ 2 fm and
so are negligible, it can be concluded that the broadening comes from saturation
broadening. This makes sense because if the density is lower, there is a smaller
number of atoms that can be excited, so the laser intensity threshold for population
saturation becomes lower.
The measured saturation broadening can be used to calculate the laser bandwidth
— the spectral uncertainty of the laser wavelength. The Einstein coefficients are
related by
λ3 A12
(7.20)
B12 = 0
8πh
where λ0 is the transition wavelength and h is Planck’s constant. The spectral
density is related to the laser intensity by
I=
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 have
λ03 I
B12 ρ
(7.22)
A12
8πhc∆ν
Using Eq. 7.9, we have
λ3 I
∆ν = 0
(7.23)
8πhc γs /w 2 − 1
In Fig. 7.9a, the total broadened of profile is γs = γ 2 + σ 2 = 0.863 ± 0.173,
and w ≃ σ because other sources of Lorentzian broadening, namely Stark and
natural broadening, are negligible. Using the values λ0 = 394.539 nm, I = 10
mW/(π × 1 mm2 ) because the laser power at the output of the collimator is ∼ 10
mW and the beam radius is ∼ 1 mm, we have ∆ν = 10.1 ± 9.7 MHz. Although the
propagated error is very large, the mean value indicates a laser wavelength bandwidth
of ∆λ = 5.25 fm, which is very small compared to other dominant effects such as
Doppler broadening.

120

393.98358

393.98360

393.98362 393.98364
λ (nm)

393.98366

393.98368

Figure 7.10: The expected locations of the Zeeman splitted transition lines involving
the 1 → 3 transition in Fig. 7.3 for |B| = 50 G. The red line is the original un-splitted
line.
7.6

Two-Photon LIF Preliminary Results

In order to observe the two-photon transition, the laser wavelength was shifted to
the average of the transition wavelengths λ12 = 394.539 nm and λ23 = 393.430 nm
in Fig. 7.3, which is 393.984 nm. The 442.7 nm optical filter that was used in onephoton LIF was replaced by a 410.5 nm filter in order to measure the fluorescence
from level 3 to level 5 in Fig. 7.3. A permanent magnet was also placed near the
plasma in order to provide a source of magnetic field. The presence of the magnetic
field increased the ionization fraction of the plasma source. This could be seen both
visibly as the color of the plasma turned bluer (a large fraction of Ar II spectral
lines are around 400 nm) and measurably as the natural emission intensity of the
410.499 nm line increased. The explanation is that the magnetic field traps the
ionizing electrons for a longer period of time compared to a situation where there is
no magnetic field, so they impart more energy to the ions.
The strength of the magnetic field at the location of the collection point was measured
by a Hall effect sensor to be around 50 G. Figure 7.10 shows the expected locations
of the Zeeman splitted transition lines from level 1 to level 3 for |B| = 50 G. The
distance between each line scales linearly with the strength of the magnetic field.
Because the selection rule for one-photon Zeeman transition is m J = 0, ±1, that for
two-photon Zeeman transition is m J = 0, ±1, ±2.
Figure 7.11 shows the two-photon LIF results of two separate scan instances at
1 mTorr. The red line is the theoretical center transition wavelength, and the

121

Scan No. 82 (blue) and 83 (green)
Fluorescence (Arb.)

4.5

5.0

4.5

5.5
6.0

4.0

6.5

3.5

7.0
393.982

393.984

393.986

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 the
expected Zeeman splitting for |B| = 50 G.
grey area is the expected Zeeman splitting for |B| = 50 G. Although saturation
broadening (∼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 because
the 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. This
shift was actually observed in the one-photon LIF results (Fig. 7.9), where the
measured 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 that
the predicted fractional wavelength shift due to finite plasma density is 7 × 10−10 ,
which is negligible compared to other shift effects. Significant broadening allows
the detection of a signal even when the laser wavelength is reasonably far from the
center value.
It can be seen from Fig. 7.11 the measured peak is shifted to the left, and that there
is broadening comparable to that in Fig. 7.9a. Line fitting the measurements to
Voigt profiles yielded too big of an error to warrant any analysis. Nevertheless, the
peak near 393.9836 nm is evident from the two measurements.

122
7.7

Suggestions for Further Work

Due to the COVID-19 pandemic in the year 2020 and subsequent restricted access
to the laboratory, no further experiments could be done to improve upon the results
that have been presented so far. Therefore, some suggestions for further work will
now 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 the
measurements in Fig. 7.11. This allows for the measurement of the complete line
profile.
Second, once it is confirmed that the peaks shown in Fig. 7.11 are indeed twophoton LIF signals with saturation broadening, the background gas pressure can
then 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 of
the 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 of
Zeeman splitting can be altered by changing the proximity of the magnet, or simply
by using a different source of magnetic field such as solenoids. Two-color LIF can
also be conducted to increase the signal. The polarization of the laser beam may
also be controlled to alter the selection rules.
7.8

Summary

In this chapter, a new laser magnetic field diagnostic was conceived and preliminary
results were presented. The method uses the two-photon laser-induced fluorescence
(LIF) technique, which allows for the elimination of Doppler broadening arising
from finite temperature. The riddance of the dominant broadening effect enables
the resolution of Zeeman splitting, which can be used in turn to infer the magnetic
field information.
A repetitively-pulsed, inductively-coupled high-power RF plasma source was newly
developed as the subject of diagnosis. This source improved upon a previously built
battery-powered source and is wall-powered.
One-photon LIF results revealed various properties of the plasma as well as the
laser. The temperature of the plasma was measured via Doppler broadening to be
around 103 K, and this value was dependent on the background gas pressure. The
laser bandwidth was measured via saturation broadening to be around 10 MHz.
Finally, Preliminary two-photon LIF results were presented. Initial results indicate

123
that there is two-photon excitation around the expected wavelength. Although
further developments were thwarted by the COVID-19 pandemic, suggestions for
improvements on this diagnostic were presented.

124
Chapter 8

SUMMARY
The progression, properties, and progenies of magnetic reconnection were studied,
and a method of probing the magnetic field in a plasma was partially developed in
this thesis.
8.1

Progression

Canonical Vorticity Framework
An intuitive framework for explaining the progression of magnetic reconnection was
first presented. In order to describe various facets of the process, the framework
utilizes a quantity called the electron canonical vorticity Qe , which is perfectly frozen
into the electron flow while the magnetic field B is not frozen-in and thus reconnects.
A careful analysis of the dynamics of a flux tube defined by Qe intuitively explains
why magnetic reconnection is an instability, how electrons are accelerated, and how
whistler 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 section
of a Qe flux tube is subject to a stronger out-of-plane electron flow at the center
than 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 that
generates more electron flow toward the center. The process then feeds back to
itself, generating an instability.
As the instability progresses and the flux tube is pulled in the out-of-plane direction, it
is also lengthened and narrowed in order to conserve its volume, just like a stretching
rubber band. This decrease in the circumference of the flux tube, combined with
magnetic field cancellation due to magnetic reconnection, increases the electron
flow velocity, i.e., accelerates electrons.
Finally, the spatial gradient in the electron flow twists up the Qe , and this twist is
propagated as whistler waves. This is analogous to the generation mechanism of
torsional Alfvén waves.

125
Completion of the Framework
By including the "canonical battery" effect, the canonical vorticity framework of
magnetic reconnection was completed down to first principles. Assuming that collisions are ignored, this battery effect is the only non-ideal term that relaxes the
topological constraint on Qe . Different kinetic effects such as electron viscosity and
pressure anisotropy are manifested by the canonical battery term and significantly
affect the reconnection process. The distinction between a Sweet-Parker-type diffusion process and a purely growing instability could be made by the significance
of the canonical battery term. Two-fluid and particle-in-cell simulations aided the
analysis.
8.2

Properties

Stochastic Ion Heating in Magnetic Reconnection
Out of the various observed properties of magnetic reconnection, its accompaniment by anomalous ion heating much faster than conventional heating has been a
longstanding dilemma.
Stochastic heating — a mechanism in which particles subject to sufficiently strong
electric fields undergo chaotic trajectories — is fast enough to account for the
observed heating rate. Using the established canonical vorticity framework, the
Hall electric fields that develop during reconnection could be calculated. It was
then shown that these fields satisfy the stochastic ion heating criterion, and that the
ions involved indeed undergo chaotic trajectories. This results in stochastic motion
in phase space, which is equivalent to heating. Single-particle simulations were
conducted to verify stochastic motion.
Kinetic Verification of Stochastic Ion Heating
The stochastic ion heating mechanism during magnetic reconnection was then kinetically verified through analytical calculations and particle-in-cell simulations. First,
it was shown that stochastic ion heating is intrinsic to the Harris equilibrium current
sheet, which is profoundly related to magnetic reconnection. Then, it was verified via particle-in-cell simulations that the Hall electric fields that develop during
magnetic reconnection indeed satisfy the stochastic ion heating criterion even in the
kinetic regime, and that the ion heating profile agreed well with the prediction from
the 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 consistent
with stochastic ion heating.

126
It was thus confirmed that stochastic ion heating is an intrinsic ion heating mechanism in magnetic reconnection.
8.3

Progenies

A very important progeny of magnetic reconnection is whistler waves, the generation mechanism of which had already been discussed using the canonical vorticity
framework. The interaction of whistler waves with relativistic particles, specifically
how these waves scatter the pitch-angle of the particles, was examined.
A previous study showed that exact rearrangements of the particle equation of motion
under a circularly polarized wave leads to a pseudo-Hamiltonian equation of motion
of a frequency mismatch parameter ξ. If the pseudo-potential of this motion is twovalleyed, and the particle has enough pseudo-energy to undergo periodic motion
in these two valleys, then the particle was shown to undergo drastic pitch-angle
scattering.
Here, the conditions were derived for which the particle undergoes two-valley motion and thus undergoes drastic scattering. The conditions are a certain range related
to the exact resonance condition but may or may not include this condition depending on the initial conditions of the particle. It was then shown that if the particle
distribution is assumed to be the Maxwell-Jüttner distribution, which the relativistic generalization of the Maxwell-Boltzmann distribution, a significant fraction of
the particles undergo large scattering for typical magnetospheric parameters. The
scaling of this fraction with the wave amplitude suggests that relativistic microburst
events may be explained by the two-valley mechanism. The widely-used secondorder trapping theory is shown to be an approximation of the theory presented here
and actually misses critical details of the wave-particle interaction.
8.4

Probe

A new magnetic field diagnostic using two-photon Doppler-free laser-induced fluorescence was described and developed to some extent. By using two counterpropagating laser beams, Doppler broadening of a spectral line can be cancelled
out, enabling the resolution of Zeeman splitting. This splitting can then be used to
extract the magnetic field information.
A repetitively-pulsed, inductively-coupled high-power RF plasma source was newly
developed as the subject of diagnosis. One-photon LIF scans of this plasma showed
that the temperature of the plasma was around 103 K. The laser bandwidth was also

127
measured via saturation broadening to be around 10 MHz.
Preliminary two-photon LIF results were presented. Initial results indicate that
there is two-photon excitation around the expected wavelength. Although further
developments were thwarted by the COVID-19 pandemic, improvements on this
diagnostic were suggested.

128

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Appendix A

WHISTLER WAVE ANALYTICAL SOLUTION
The analytical solution for a whistler wave driven by a ring current source will
now be derived. Relevant normalizations are described in Table 2.1. It will be
assumed that all linear quantities have the dependence ∼ eik·x−iωt . The linearized
dimensionless equations are, under uniform B0 = ẑ,
∂B
∂t
∇ × B = −ue + Iδ (r − a) δ (z) e−iΩt φ̂,
∂ue
= −E − ue × ẑ,
∂t

∇×E=−

(A.1)
(A.2)
(A.3)

where Iδ (r − a) δ (z) e−iΩt φ̂ is the ring current source with frequency Ω, normalized
source strength I, and radius a. Also, Due /Dt = ∂ue /∂t + ue · ∇ue ≃ ∂ue /∂t
because ue is a linear quantity. Because the background magnetic field and the ring
current source has no dependence on φ, there is cylindrical symmetry; therefore,
the wavevector k = kr r̂ + k z ẑ, i.e., k φ = 0, and k 2 = kr2 + k z2 . Fourier analyzing in
space, the source term S (r) = δ (r − a) δ (z) turns into
∫ ∞ ∫ 2π ∫ ∞
S (k) =
δ (r − a) δ (z) e−ikr r cos(kφ −φ)−ikz z r dr dφdz,
(A.4)
r=0

φ=0

∫ ∞ ∫ 2π

z=−∞

δ (r − a) e−ikr r cos φ r dr dφ,

r=0 φ=0
∫ 2π
−ikr a cos φ
φ=0

dφ,

= 2πaJ0 (kr a) ,

(A.5)
(A.6)
(A.7)

where Jn is the Bessel function of the first kind of order n. Fourier analyzing the
time 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, gives
k × 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)

150
We will now define I (2π)2 aJ0 (kr a) δ (Ω − ω) = α to reduce clutter. Taking k× of
the Eq. A.10:
ik × (k × B) = −k × ue + αk × φ̂,
−ik 2 B = −k × ue + αkr ẑ − αk z r̂,

(A.12)
(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 − k z ue + (k · ue ) ẑ,

(A.15)

−iωk × ue = −ωB − k z ue .

(A.16)

Using Eq. A.13 and Eq. A.10,
−iω ik 2 B + αkr ẑ − αk z r̂ = −ωB − k z −ik × B + α φ̂ ,
ω k 2 + 1 B = ik z k × B − iαωk z r̂ − αk z φ̂ + iαωkr ẑ,
ω k 2 + 1 B = ik z k × B + C,

(A.17)
(A.18)
(A.19)

where
C = −iαωk z r̂ − αk z φ̂ + iαωkr ẑ.
Taking k× of Eq. A.19 and using Eq. A.19,
ω k + 1 k × B = ik z k × (k × B) + k × C,
ω k 2 + 1 k × B = −ik z k 2 B + k × C,
ω k 2 + 1 ik z k × B = k z2 k 2 B + ik z k × C,
h
ω k + 1 ω k + 1 B − C = k z2 k 2 B + ik z k × C,

2
2 2
ω k + 1 − k z k B = ω k 2 + 1 C + ik z k × C,
ω k 2 + 1 C + ik z k × C
B=
2
ω2 k 2 + 1 − k z2 k 2

(A.20)

(A.21)
(A.22)
(A.23)
(A.24)
(A.25)
(A.26)

Let us examine the φ direction. Since
(k × C)φ = k z Cr − kr Cz,
= −iαω k z2 + kr2 ,

(A.27)

= −iαωk 2,

(A.29)

(A.28)

151
it follows that
ω k 2 + 1 C + ik z k × C = −αω k 2 + 1 k z + k z αωk 2,

(A.30)

= αωk z,

(A.31)

= I (2π)2 aJ0 (kr a) δ (Ω − ω) ωk z .

(A.32)

The solution for Bφ is thus
−I (2π)2 aδ (Ω − ω) ωJ0 (kr a) k z
Bφ (ω, k) =
2
ω2 k 2 + 1 − k z2 k 2

(A.33)

The inverse transform in space gives
3
(A.34)
Bφ (ω, r) = −I (2π) aδ (Ω − ω) ω
2π
∫ ∞ ∫ 2π ∫ ∞
J0 (kr a) k z
eikr r cos( kφ −φ) eikz z kr dkr dk φ dk z,
2
kr =0 k φ =0 k z =−∞ ω k + 1 − k z k
(A.35)
∫ ∞ ∫ ∞
J0 (kr a) J0 (kr r) kr ikz z
e dkr dk z .
= Iaδ (Ω − ω) ωi
∂z kr =0 kz =−∞ ω2 k 2 + 1 2 − k z2 k 2
(A.36)

Let us now do a contour integral for the k z integral. Solving for the poles:

− k z2 k 2 = 0,

(A.37)

ω2 k 4 + 2k 2 + 1 − k z2 k 2 = 0,
2
ω2 k z2 + kr2 + 2ω2 k z2 + kr2 + ω2 − k z2 k z2 + kr2 = 0,

(A.38)

ω2 k z4 + 2ω2 k z2 kr2 + ω2 kr4 + 2ω2 k z2 + 2ω2 kr2 + ω2 − k z4 − k z2 kr2 = 0,
ω2 − 1 k z4 + 2ω2 kr2 + 2ω2 − kr2 + ω2 kr4 + 2ω2 kr2 + ω2 = 0,
2
ω2 − 1 k z4 + 2ω2 1 + kr2 − kr2 + ω2 1 + kr2 = 0,

(A.40)

k +1

(A.39)

(A.41)
(A.42)

whose solution for k z is
k z2 =

−2ω2

1 + kr2

+ kr2 ±

2
2
2ω2 1 + kr2 − kr2 − 4 ω2 − 1 ω2 1 + kr2
2 ω2 − 1
(A.43)

152
Here, the determinant is
2

2
D = 2ω 1 + kr − kr − 4ω ω − 1 1 + kr ,
(A.44)
2
2
2
= 4ω 1 + kr − 4ω 1 + kr kr + kr − 4ω 1 + kr + 4ω 1 + kr ,
(A.45)

= −4ω2 1 + kr2 kr2 + kr4 + 4ω2 1 + kr2

= −4ω2 1 + kr2 kr2 − 1 − kr2 + kr4,
= 4ω2 1 + kr2 + kr4 .

Therefore, there exist 4 poles at
t 2
2ω 1 + kr − kr ± 4ω2 1 + kr2 + kr4
kz = ±
2 1 − ω2
To check the complexity of the poles, we need to check the following:
2ω2 1 + kr2 − kr2 < 4ω2 1 + kr2 + kr4,

(A.46)
(A.47)
(A.48)

(A.49)

(A.50)

in which case there are at least two imaginary poles. The right-hand side is always
positive, so if the following is true,
(A.51)
2ω 1 + kr > kr2,
ω2 >

kr2
,
2 1 + kr2

(A.52)

then 2ω2 1 + kr2 − kr2 is also positive. In this case, by squaring both sides Eq.
A.50,
2
2 2
4ω 1 + kr − 4ω kr 1 + kr + kr < 4ω 1 + kr + kr4,
(A.53)
4ω4 1 + kr2 − 4ω2 kr2 < 4ω2,
(A.54)
4ω4 1 + kr2 < 4ω2 1 + kr2 ,
(A.55)
ω2 < 1,

(A.56)

< ω2 < 1, two poles are real and two poles are imaginary.
so for 2 1+k
( r2 )

In the opposite case where
ω2 <

kr2
,
2 1 + kr2

(A.57)

153
two poles are real if and only if
2ω2 1 + kr2 − kr2 < 4ω2 1 + kr2 + kr4,
q
kr2 − 2ω2 1 + kr2 < 4ω2 1 + kr2 + kr4,

(A.58)
(A.59)

which actually has the solution ω2 < 1.
Therefore, there are two real and two imaginary poles, respectively, for ω2 < 1,
which is true by assumption for whistler waves (recall from Table 2.1 that ω2 < 1
2 in dimensional quantities, which is true for whistler waves).
means ω2 < ωce
Therefore, there exist 4 simple poles which will be called A, −A, iB, and −iB, where
A, B > 0. The integral in Eq. A.36 now becomes
Bφ (ω, r) = Iaδ (Ω − ω) ωi
∫ ∞
dkr J0 (kr a) kr eikr r cos( kφ −φ)
kr =0

∂z

∫ ∞
k z =−∞

dk z

(A.60)
(A.61)

eikz z
. (A.62)
1 − ω (k z − A) (k z + A) (k z − iB) (k z + iB)

The 1 − ω2 term in the denominator exists because the coefficient of the quadratic
term of equation for k z is 1 − ω2 . Doing the complex contour integral from
k z = −∞ to +∞, we pick up half the residues at A and −A, and the full residue at
iB. With respect to the k z integral, the residues at A, −A, and iB are, respectively,
ei Az
Res (A) =
1 − ω2 (A + A) (A − iB) (A + iB)
ei Az
,
2A 1 − ω2 A2 + B2
e−i Az
,
Res (−A) =
−2A 1 − ω2 A2 + B2
e−Bz
Res (iB) =
1 − ω2 (iB + A) (iB − A) 2iB
e−Bz
.
−2iB 1 − ω2 A2 + B2

(A.63)
(A.64)
(A.65)
(A.66)
(A.67)

154
It can be inferred from Eq. A.49 that
4ω2 1 + kr2 + kr4
A2 =
2 1 − ω2
−2ω 1 + kr + kr + 4ω2 1 + kr2 + kr4
B2 =
2 1 − ω2
4ω2 1 + kr2 + kr4
A2 + B2 =
1 − ω2
2ω2

1 + kr2

− kr2 +

(A.68)

(A.69)

(A.70)

Thus, the k z integral evaluates to
i Az
sin Az e−Bz
πi
e − e−i Az e−Bz
=q
2A
iB
4ω2 1 + kr2 + kr4
4ω2 1 + kr2 + kr4
(A.71)
Doing the partial z derivative,
4ω2 1 + kr2 + kr4

− cos Az + e−Bz .

(A.72)

The solution for Bφ is now
∫ ∞

J0 (kr a) J0 (kr r) kr
−Bz
cos
dkr .
(Az)
kr =0
4ω 1 + kr + kr
(A.73)
Because A, B > 0, this is a sum of an oscillating solution and an evanescent solution.
Bφ (ω, r) = −Iπaδ (Ω − ω) ωi

Figure A.1 shows the analytical solution for Bφ in Eq. A.73 for a source with
ω = 0.35, a = 1, and an arbitrary amplitude. The red vertical line represents the
location and extent of the ring current source. The numerical integration was carried
out using the scipy.integrate.quad package in Python 3.7. The result of the integral
at r = 0 (black horizontal dashed line) does not converge fast enough and is thus
dubious. The cone-like structures emanating from the source are called "resonance
cones" [52]. These arise from the frequency dependence of the plasma dielectric
tensor; their angles have the following relation in dimensional quantities [52]:
Ω ω pe + ωce − Ω
sin2 θ =
(A.74)
2 ω2
ωce
pe
where θ is the cone angle, ω pe is the electron plasma frequency, ωce is the electron
cyclotron frequency, and Ω is the frequency of the source. In the limit ω pe
ωce ,

155

r/de

0.8

Bϕ (Arb.)

0.6

0.4

0.0

0.2
−0.2

−5
−10

−0.4
−0.6

20
z/de

−0.8

Figure 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 numerical
integration 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 becomes
sin2 θ =

Ω2
ωce

(A.75)

which shows that θ = 0.35 = 21◦ for Ω/ωce = 0.35. The cone-angle in Fig. A.1 is
approximately 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 k z :
k z2 k z2 + kr2
Ω =
2 , (A.76)
1 + kr2 + k z2
2
(A.77)
Ω2 1 + kr2 + k z2 = k z4 + k z2 kr2,
2
(A.78)
Ω2 1 + kr2 + 2Ω2 1 + kr2 k z2 + Ω2 k z4 = k z4 + k z2 kr2,
2
i
(A.79)
1 − Ω2 k z4 + kr2 − 2Ω2 1 + kr2 k z2 − Ω2 1 + kr2 = 0.
The solution for k z is
k z2 =

2Ω2 1 + kr2 − kr2 + kr4 + 4Ω2 1 + kr2
2(1 − Ω2 )

(A.80)

156

kz vs. kr

500
400

kzde

300

200
−1
100
−2
−4

−3

−2

−1

krde

Figure A.2: The fast Fourier transform of the result in Fig. A.1. The red lines
represent the dispersion relation in Eq. A.80 for Ω = 0.35.
which, for a particular value of Ω, yields a relation between k z and kr .
Figure A.2 shows the fast Fourier transform of the result in Fig. A.1. The red lines
represent the dispersion relation in Eq. A.80 for Ω = 0.35. It can be seen that the
areas 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 current
source has thus been presented.

157
Appendix B

ALGORITHM TEST USING WHISTLER WAVES
The algorithm presented in Section 2.3 should yield whistler waves similar to that
in Appendix A. A 3D code that mimics the current source in Appendix A.73 was
constructed.
Because it is difficult to realize a delta function in a numerical simulation, the current
source was approximated by a Gaussian of the form
(r − a)2 + z2
sin Ωt,
(B.1)
j φ = φ̂ exp −
2σ 2
where is the current amplitude, a is the ring radius, and σ is the ring thickness. A
Cartesian grid of dimensions (x, y, z) = (80, 80, 320) = (20, 20, 80)de was used, and
the time step was ∆t = 0.05. The source parameters were a = 1, σ = 0.5, = 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 general
spatial structure is similar to that in Fig. A.1, except for at r = 0. The cone angle is
slightly wider (θ = 0.4), which may be due to the fact that here the current source
is not a perfect delta function.

Bϕ

0.03

0.02

r/de

0.01

0.00

−0.01

−5

−0.02
−10

z/de

Figure B.1: Bφ at t = 80|ωce | −1

−0.03

158

kz vs. kr

kzde

15
10

−1

−2
−4

−3

−2

−1

krde

Figure B.2: The fast Fourier transform of the results in Fig. B.1. The red lines
represent 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 lines
represent the dispersion relation in Eq. A.80 for Ω = 0.35. The regions of high
magnitude 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 Section
2.3 produce whistler waves as expected.

159
Appendix C

CANONICAL HELICITY DENSITY AND THE LAGRANGIAN
DENSITY IN ELECTRON-MAGNETOHYDRODYNAMICS
Relevant normalizations are carried out according to Section 2.3. The normalized
electron canonical momentum and canonical vorticity are defined as, respectively,
Pe = ue − A,

(C.1)

Qe = ∇ × Pe,

(C.2)

= ∇ × ue − B.
The normalized electron equation of motion is
∂ue
+ ue · ∇ue = −E − ue × B − ∇pe,
∂t
or, in terms of Qe ,
2
∂ue
+ ∇ e = −E + ue × Qe − ∇pe,
∂t

(C.3)

(C.4)

(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 electrostatic
potential φ = 0, so E = −∂A/∂t. Examining the time evolution of the normalized
electron canonical helicity density κQ = Pe · Qe ,
∂κQ ∂Pe
∂Qe
· Qe + Pe ·
(C.6)
∂t
∂t
∂t
∂ue ∂A
· Qe + Pe · ∇ × (ue × Qe ) ,
(C.7)
∂t
∂t
: ue × Qe · Qe = 0

∂ue
+ E · Qe + ∇ · ([ue × Qe ] × Pe ) + (ue ×
(C.8)
) · (∇ × Pe ),
e
∂t

2
ue
0
+
ue× Qe − ∇pe · Qe + ∇ · (Qe [ue · Pe ] − ue [Pe · Qe ]) ,
= −∇
(C.9)
2
(C.10)
= Qe · ∇ − e − pe + ue · Pe − ue · ∇κQ,
where ∇ · Qe = ∇ · ue = 0 has been used. Expressing this in terms of the convective
derivative and using ue · Pe = u2e − ue · A,
2
DκQ
ue
= Qe · ∇
− pe − ue · A .
(C.11)
Dt

160
u2

Now we prove that L = 2e − pe − ue · A is the Lagrangian density of the system.
The Euler-Lagrange equation
∇x L −

∇u L = 0,
Dt

(C.12)

where L (x, ue, t) = L (x, ue, t) dV is the Lagrangian, must give the equation of
motion of the system (Eq. C.4). ∇ x is the spatial gradient and ∇u is the gradient in
velocity space. Here,
∇x L =
(C.13)
[−∇ x pe − ∇ x (ue · A)] dV,
(C.14)
[−∇ x pe − ue · ∇ x A − ue × (∇ x × A)] dV,
(C.15)
[−∇ x pe − ue · ∇ x A − ue × B] dV,
since only pe and A are explicitly spatially dependent. Also,
∇u L =
∇u LdV,
(ue − A) dV .

(C.16)
(C.17)

So,
(ue − A) dV + (ue − A) (ue · ds) ,
∂t
∫
∂ue ∂A
dV + ∇ x · [ue (ue − A)] dV,
∂t
∂t
∫
Due
+ E − ue · ∇ x A dV .
Dt

∇u L =
Dt

(C.18)
(C.19)
(C.20)

Therefore,
−∇ x pe − ue · ∇ x A − ue × B
dV,
− Du
Dt
∫
Due
−∇ x pe − ue × B −
− E dV,
Dt

∇x L −
∇u L =
Dt

∫ "

= 0.

(C.21)
(C.22)
(C.23)

Setting the integrand to zero gives us the original normalized equation of motion
(Eq. C.4):
Due
= −E − ue × B − ∇pe .
(C.24)
Dt

161
Therefore, L is the Lagrangian density of the system and
DκQ
= Qe · ∇L.
Dt

(C.25)

Also, integrating this equation over a volume and writing the total canonical helicity
as KQ = V κQ dV,
DKQ
Dt

Qe · ∇LdV,

(C.26)

∇ · (Qe L) dV,

(C.27)

LQe · ds.

(C.28)

∫V
∮V

Therefore, if Qe does not penetrate the volume at the boundary, KQ is a conserved
quantity.

162
Appendix D

IMPLICIT PARTICLE INTEGRATOR
An algorithm [15] will now be illustrated that implicitly solves the particle equation
of motion
dv
= qE + qv × B.
(D.1)
dt
Defining Ω = qB/m and Σ = qE/m,
dv
= Σ + v × Ω.
dt

(D.2)

Now, velocity is defined at integer time steps, and position is defined at half-integer
time steps. The time derivative of the velocity is evaluated at the half-integer time
step, i.e.,
v + v
vt+1 − vt
t+1
= f
(D.3)
∆t
vt+1 + vt
= Σt+1/2 +
× Ωt+1/2 .
(D.4)
This equation may be expressed as
vt+1 + A × vt+1 = C,
where
A=

Ωt+1/2 ∆t

(D.5)

(D.6)

and

vt
C = vt + ∆t Σt+1/2 + × Ωt+1/2 .
Now, dotting Eq. D.5 with A,

(D.7)

vnew · A = C · A.

(D.8)

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)

Crossing Eq. D.5 with A,

vt+1 − C + vt+1 A2 − A (C · A) = C × A,
C + A (C · A) + C × A
vt+1 =
1 + A2

(D.11)
(D.12)

163
which can be used to update a given vt .
The new position is given by
xt+3/2 = xt+1/2 + vt+1 ∆t.

(D.13)

164
Appendix E

HARRIS EQUILIBRIUM PLASMA SHEATH DERIVATION
The 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 is
only in the x-direction, i.e., only ∂/∂ x is finite, and that A = Az ẑ. It then follows
that B = By ŷ and E = E x x̂. Equations to solve are
∂ fσ
∂ fσ qσ
(E + vσ × B) ·
∂x
∂vσ
∂E x
1 Õ
fσ d 3 vσ,
qσ
∂x
0 σ
Õ ∫
∂By
fσ vz d 3 vσ .
= µ0
qσ
∂x
0 = vx

(E.1)
(E.2)
(E.3)

In terms of the electrostatic and vector potentials φ and Az ,
∂ fσ qσ
∂ fσ
(E + vσ × B) ·
∂x
mσ
∂vσ
∂2 φ
1 Õ
fσ d 3 vσ,
=−
qσ
0 σ
∂ x2
Õ ∫
∂ 2 Az
qσ
fσ vz d 3 vσ .
= −µ0
∂x
0 = vx

(E.4)
(E.5)
(E.6)

Now, because the system is independent of y and z coordinates, the canonical
momenta p y and pz are constants of the motion, as well as the total energy. In other
words,
1
(E.7)
W = m v x2 + v y2 + vz2 + qφ,
p y = mv y,
(E.8)
pz = mvz + qAz

(E.9)

are constants of motion for each species. Now consider the distribution function

mσ
fσ =
2πk BTσ

3/2

mσ
2q
n0 exp −
v + v y + (vz − Vσ ) +
(φ − Az Vσ ) ,
2k BTσ x
mσ
(E.10)

165
where Tσ is the temperature of each species, n0 is the reference density, and Vσ is
the average velocity of each species in the z-direction. Because
v x2 + v y2 + (vz − Vσ )2 +

2qσ
2W 2pz
+ Vσ2
(φ − Az Vσ ) =
mσ
mσ mσ

(E.11)

is a function of constants of motion only, Eq. E.10 is a solution of Eq. E.4 by
Liouville’s theorem. Thus we have
qσ
(E.12)
fσ d v = n0 exp −
(φ − Az Vσ ) ,
k BTσ
qσ
vz fσ d v = n0Vσ exp −
(E.13)
(φ − Az Vσ ) .
k BTσ
Inserting these into Eqs. E.5 and E.6,

∂2 φ
en0
(E.14)
exp −
=−
(φ − Az Vi ) − exp
(φ − Az Ve ) ,
0
k BTi
k BTe
∂ x2

∂ 2 Az
= −µ0 en0 Vi exp −
(φ − Az Vi ) − Ve exp
(φ − Az Ve ) . (E.15)
k BTi
k BTe
∂ x2
It is now assumed that Vi = −Ve = V, which can be done by moving to a frame where
the 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 is
described in Ref. [164]. Then,

en0
eφ
∂2 φ
eφ
=−
exp
Az V exp −
− exp
(E.16)
0
k BT
k BT
k BT
∂ x2

eφ
eφ
∂ 2 Az
Az V exp −
= −µ0 en0V exp
+ exp
(E.17)
k BT
k BT
k BT
∂ x2
Since the first equation is trivially solved by φ = 0, the only equation to solve is
∂ 2 Az
= −2µ0 en0V exp
Az V .
(E.18)
k BT
∂ x2
The solution to this equation is
2k BT
Vx
Az = −
ln cosh √
eV
2cλD
where
λD = 2 + 2
λDi λDe
is the Debye length.

! − 21

0 k BT
2n0 e2

(E.19)

(E.20)

166
Using Eq. E.19, the solution for the magnetic field is
∂ Az 2k BT
n0 e2
Vx
By = −
tanh √
∂x
ec
0 k BT
2cλD
Vx
= 2 µ0 n0 k BT tanh √
2cλD
Vx
= B0 tanh √
2cλD

(E.21)
(E.22)
(E.23)

The magnitude of the equilibrium magnetic field illustrates that the magnetic pressure is balance by the thermal pressure, i.e.,
B02
2µ0

= 2n0 k BT = 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 density
Vx
−2
n = n0 cosh
(E.25)
2cλD
which shows that the density is peaked at x = 0 and goes to zero as x → ±∞. This
is why the Harris equilibrium is also called the Harris "sheath."

167
Appendix F

ONE-DIMENSIONAL AND TWO-DIMENSIONAL
MAXWELL-JÜTTNER DISTRIBUTIONS AND THE
DISTRIBUTION OF ξ
F.1

Derivation of fρ⊥

In cylindrical coordinates, Eq. 6.18 is equivalent to
1 + ρ2
exp −
ρ⊥ dρ⊥ dφdρz .
fρ d ρ =
4πθK2 (1/θ)

(F.1)

Integrating in ρz gives

1 + ρ2⊥ + ρ2z ª
® ρ⊥ dρ⊥ dφdρz
fρ ρ⊥ dρ⊥ dφdρz =
exp −
4πθK
(1/θ)
ρz =−∞
ρz =−∞
(F.2)
∫ ∞
1 + ρ2⊥ + ρ2z ª
® dρz,
= ρ⊥ dρ⊥ dφ
exp −
2πθK
(1/θ)
ρz =0
(F.3)

∫ ∞

where ρ2 = ρ2⊥ + ρ2z . Defining
a2 =

1 + ρ2⊥
θ2

(F.4)

ρz

(F.5)

and
t=
the ρz -integral in Eq. F.3 becomes
2πK2 (1/θ)

∫ ∞

√
exp − a2 + t 2 dt.

(F.6)

Now we define
t = a sinh z,
(F.7)
so a2 + t 2 = a 1 + sinh2 z = a cosh z and dt = a cosh zdz. Equation F.6 is now
∫ ∞
cosh z exp (−a cosh z) dz.
(F.8)
2πK2 (1/θ) 0

168
The z-integral in Eq. F.8 evaluates to K1 (a) where Kn is the modified Bessel function
of the second kind of order n [176, Section 8.432, 1.].
Therefore, Eq. F.3 is now
1 + ρ2⊥

(F.9)

(F.10)

which is Eq. 6.19.
F.2

Derivation of fρz

In cylindrical coordinates, Eq. 6.18 is equivalent to
fρ d 3 ρ =
exp −
ρ⊥ dρ⊥ dφdρz .
4πθK2 (1/θ)
Integrating in all φ and ρ⊥ , Eq. F.11 becomes
∫ ∞
1 + ρ2⊥ + ρ2z ª
® ρ⊥ dρ⊥ dρz .
fρz dρz =
exp −
ρ⊥ =0 2θK2 (1/θ)

(F.11)

(F.12)

Letting η2 = 1 + ρ2⊥ + ρ2z while keeping ρz constant so that
ηdη = ρ⊥ dρ⊥,
we have
fρz dρz =

(F.13)

η
exp
− ηdηdρz .
η= 1+ρ2z 2θK2 (1/θ)

∫ ∞

(F.14)

Using the integral formula [176, Section 3.351, 2.]
∫ ∞

x n e−µx dx = e−uµ

k! µn−k+1
k=0

(F.15)

169
where x = η, u = 1 + ρ2z , µ = 1/θ, and n = 1 in this case, we have
1 + ρ2z ª
®,
fρz dρz =
θ + θ 1 + ρz exp −
2θK2 (1/θ)

1 +
2K2 (1/θ)

1 + ρ2z ª
1 + ρ2z ª
® exp −
®,

(F.16)

(F.17)

which is Eq. 6.20.
F.3

Derivation of fξ

ξ is defined as
ξ = 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 have
fρz (ρz )dρz = fρz (R/n)

1+
2nK2 (1/θ)
= fR (R)dR.

(F.19)

1 + R2 /n2
1 + R2 /n2
exp −
dR

(F.20)
(F.21)

Now, in order for ζ = R − γ to be true, the value of R has to equal ζ + γ for a given
value of γ. The probability distribution of this occurrence integrated over all values
of γ gives fζ :
∫ ∞
fζ (ζ) =
fγ (γ) fR (ζ + γ) dγ
(F.22)
∫ ∞ 2p
1 + (ζ + γ) /n ª
© γ + 1 + (ζ + γ)2 /n2 ª
γ 1 − 1/γ 2 ©
® exp −
® dγ.
1+
2nK22 (1/θ)
(F.23)
Finally, rearranging Eq. F.18 yields ζ (ξ) = (ξ − 1) /α so that dζ = dξ/α. It follows

170
that
fζ (ζ)dζ = fζ ([ξ − 1]/α)

dξ

(F.24)
∫ ∞ 2p
1 + (ζ (ξ) + γ)2 /n2 ª
© γ + 1 + (ζ (ξ) + γ)2 /n2 ª
γ 1 − 1/γ 2 ©
® dγdξ
1+
® exp −
2 (1/θ)
2αnK
(F.25)
= fξ (ξ)dξ.

(F.26)

Writing ρz (γ, ξ) = (ζ (ξ) + γ) /n yields a more compact expression:
∫ ∞ 2p
2 (γ, ξ)
1 + ρ2z (γ, ξ) ª
γ 1 − 1/γ ©
® exp −
® dγ,
fξ (ξ) =
1+
2αnK22 (1/θ)
(F.27)
which is Eq. 6.22.
F.4

Derivation of non-relativistic fρ⊥

From Eq. F.10,
ρ⊥ 1 + ρ2⊥

(F.28)

ρ2⊥
1 + ρ⊥ ≃ 1 +

(F.29)

For ρ⊥
1,

For θ
1, it is seen that [160, Section 7.23]
πθ −1/θ
K2 (1/θ) ≃
and for small θ
1 and ρ⊥
1,
© 1 + ρ2⊥ ª
1 ρ2⊥
K1
® ≃ K1 θ + 2θ
¬ s
πθ
1 ρ2⊥
exp − −
θ 2θ
2 + ρ2⊥

(F.30)

(F.31)

(F.32)

171
so to lowest order,

2
ρ2⊥
1+
exp − ⊥
2θ
2
ρ⊥
exp − ⊥ .
2θ

ρ⊥
f ρ⊥ ≃

(F.33)
(F.34)

The most likely ρ⊥ value given this probability distribution function is
ρ⊥,ML = θ.

(F.35)