UNIVERSITY OF CALIFORNIA
Los Angeles
Charged Particle Energization and Transport
in the Magnetotail during Substorms
A dissertation submitted in partial satisfaction of the
requirements for the degree Doctor of Philosophy
in Physics
by
Qingjiang Pan
2015
ABSTRACT OF THE DISSERTATION
Charged Particle Energization and Transport
in the Magnetotail during Substorms
by
Qingjiang Pan
Doctor of Philosophy in Physics
University of California, Los Angeles, 2015
Professor Maha Ashour-Abdalla, Chair
This dissertation addresses the problem of energization of particles (both electrons and ions)
to tens and hundreds of keV and the associated transport process in the magnetotail during
substorms. Particles energized in the magnetotail are further accelerated to even higher energies
(hundreds of keV to MeV) in the radiation belts, causing space weather hazards to human activities
in space and on ground. We develop an analytical model to quantitatively estimate flux changes
caused by betatron and Fermi acceleration when particles are transported along narrow high-speed
flow channels from the magnetotail to the inner magnetosphere. The model shows that energetic
particle flux can be significantly enhanced by a modest compression of the magnetic field and/or
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shrinking of the distance between the magnetic mirror points. We use coordinated spacecraft
measurements, global magnetohydrodynamic (MHD) simulations driven by measured upstream
solar wind conditions, and large-scale kinetic (LSK) simulations to quantify electron local
acceleration in the near-Earth reconnection region and nonlocal acceleration during plasma earthward transport. Compared to the analytical model, application of the LSK simulations is much less restrictive because trajectories of millions of test particles are calculated in the realistically determined global MHD fields and the results are statistical. The simulation results validated by the observations show that electrons following a power law distribution at high energies are generated earthward of the reconnection site, and that the majority of the energetic electrons observed in the inner magnetosphere are caused by adiabatic acceleration in association with magnetic dipolarizations and fast flows during earthward transport. We extend the global
MHD+LSK simulations to examine ion energization and compare it with electron energization.
The simulations demonstrate that ions in the magnetotail are first nonadiabatically accelerated in the weak field region close to the reconnection site, and then adiabatically accelerated in the high- speed flow channels as they catch up with and ride on the earthward propagating dipolarization structures. The nonlocal adiabatic acceleration mechanism for ions is very similar to that for electrons. However, the motion of energetic electrons is adiabatic except in very limited regions near the reconnection site while the motion of energetic ions is marginally adiabatic in the dipolarization regions. The simulations also show that the earthward transport of both species is controlled by the high-speed flows via the dominant ExB drift in the magnetotail. To understand how the power law electrons are initially produced in the magnetotail, we use an implicit particle- in-cell (PIC) code to model the processes in the near-Earth reconnection region. We find that the power law electrons are produced not in the reconnection diffusion region, but in the immediate
iii downstream of the reconnection outflow in the course of dipolarization formation and intensification. Our study illustrates that during substorms, particles are accelerated via a multi- step process, including local acceleration in the reconnection region and nonlocal acceleration during the earthward transport, and the multi-step acceleration occurs on multiple spatial scales ranging from a few kilometers (the scale of electron diffusion region) to more than ten Earth radii
(the transport scale).
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The dissertation of Qingjiang Pan is approved.
George Morales
Christopher T. Russell
Raymond J. Walker
Maha Ashour-Abdalla, Committee Chair
University of California, Los Angeles
2015
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To my mother
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TABLE OF CONTENTS
ABSTRACT ii
DEDICATION vi
LIST OF FIGURES xi
LIST OF SYMBOLS xiii
ACKNOWLEDGEMENTS xiv
VITA xvii
1. Background and Purpose of this Study 1
1.1. Introduction ...... 1
1.2. Acceleration by Magnetic Reconnection ...... 7
1.3. Acceleration during Plasma Earthward Transport ...... 13
1.4. Purpose of this Study ...... 24
1.5. Structure of the Dissertation ...... 26
2. Theory of Adiabatic Acceleration of Charged Particles 28
2.1. Introduction ...... 28
2.2. Characteristics of Adiabatic Particle Orbits ...... 29
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2.3. An Analytical Model of Adiabatic Acceleration ...... 33
2.3.1. Motivation ...... 33
2.3.2. The Adiabatic Acceleration Model ...... 34
2.3.3. Comparisons with Observations ...... 38
2.3.4. Discussions ...... 49
3. Modeling Electron Energization and Transport in the Magnetotail during
a Substorm 52
3.1. Introduction ...... 52
3.2. Observations of the March 11, 2008 Substorm Event ...... 52
3.2.1. Geotail Observations of the Solar Wind ...... 52
3.2.2. THEMIS Observations in the Magnetotail ...... 54
3.3. Simulation Methodology ...... 60
3.4. MHD and Electron LSK Simulations of the March 11, 2008 Substorm Event ...... 62
3.4.1. MHD Simulation Results ...... 62
3.4.2. LSK Simulation Results and Comparisons with Observations ...... 67
3.5. Discussions ...... 75
3.6. Conclusions ...... 79
4. Modeling Ion Energization and Transport Associated with Magnetic
Dipolarizations during a Substorm 81
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4.1. Introduction ...... 81
4.2. Observations of the February 07, 2009 Substorm Event ...... 81
4.3. MHD Simulation of the February 07, 2009 Substorm Event ...... 88
4.4. Ion LSK Simulation of the February 07, 2009 Substorm Event ...... 92
4.4.1. LSK Simulation Set-up ...... 92
4.4.2. LSK Simulation Results and Comparisons with Observations ...... 93
4.5. Conclusions and Discussions ...... 100
5. A Comparison Study of Ion and Electron Energization and Transport
Mechanisms during a Substorm 103
5.1. Introduction ...... 103
5.2. Comparisons of the Electron and Ion LSK Simulation Set-up for the February
07, 2009 Substorm Event ...... 104
5.3. Simulation Results ...... 107
5.3.1. Comparisons of Simulation Results with Observations ...... 107
5.3.2. Comparisons of Ion and Electron Acceleration Mechanisms ...... 111
5.4. Conclusions and Discussions ...... 121
6. Particle-in-cell (PIC) Simulation of Electron Acceleration by Magnetic
Reconnection 125
6.1. Introduction ...... 125
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6.2. Simulation Methodology ...... 126
6.3. Simulation Results and Comparisons with Observations ...... 130
6.3.1. Reconnection Structure ...... 130
6.3.2. Electron Acceleration ...... 134
6.4. Conclusions ...... 147
7. Conclusions and Problems for the Future 149
7.1. Conclusions ...... 149
7.2. Unsolved Problems and Future Work ...... 159
APPENDIX 1: Particle Sources for LSK Simulations 165
BIBLIOGRAPHY 169
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LIST OF FIGURES
1.1 A schematic of the magnetospheric convection driven by magnetic reconnection ...... 2
2.1 Energy flux and power law index at THEMIS P2 in the March 11, 2008 event ...... 41
2.2 Energy flux and power law index at THEMIS P4 in the March 11, 2008 event ...... 42
2.3 Betatron and Fermi acceleration in the March 11, 2008 event...... 44
2.4 Energy flux and power law index at THEMIS P1 in the February 27, 2009 event ...... 46
2.5 Energy flux and power law index at THEMIS P4 in the February 27, 2009 event ...... 47
2.6 Betatron and Fermi acceleration in the February 27, 2009 event ...... 48
3.1 Geotail observations of the solar wind on March 11, 2008 ...... 53
3.2 THEMIS locations and the observed magnetic fields in the March 11, 2008 event ...... 55
3.3 THEMIS P2 observations in the March 11, 2008 event ...... 57
3.4 THEMIS P4 observations in the March 11, 2008 event ...... 59
3.5 Snapshots of the MHD and LSK simulations for the March 11, 2008 event ...... 64
3.6 Two types of electron source distributions for the LSK simulation ...... 68
3.7 Comparison of the energy flux for THEMIS P2 in the March 11, 2008 event ...... 71
3.8 Comparison of the energy flux for THEMIS P4 in the March 11, 2008 event ...... 72
3.9 Comparison of the differential flux as a function of energy for THEMIS P2 ...... 73
3.10 Comparison of the differential flux as a function of energy for THEMIS P4 ...... 74
3.11 Electron acceleration mechanism during transport in the March 11, 2008 event ...... 76
3.12 Electron transport in the March 11, 2008 event ...... 78
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4.1 Solar wind measured by WIND and satellite positions in the February 07, 2009 event ...83
4.2 THEMIS P2 observations in the February 07, 2009 event ...... 85
4.3 THEMIS P3 observations in the February 07, 2009 event ...... 87
4.4 Snapshots of the MHD and ion LSK simulations for the February 07, 2009 event ...... 90
4.5 Comparison of observations and simulations for P3 for the February 07, 2009 event ...... 94
4.6 Characteristics of a representative test ion ...... 98
5.1 Comparisons of observations with simulation results for THEMIS P3 ...... 110
5.2 A Snapshot of the MHD and LSK simulations for the February 07, 2009 event ...... 113
5.3 Characteristics of a representative ion ...... 115
5.4 Characteristics of a representative electron ...... 117
5.5 Kappa in the February 07, 2009 event ...... 121
6.1 Reconnection rate normalized by the upstream Alfvén velocity and magnetic field ...... 132
6.2 Snapshots of the reconnection at t=3.05 sec and t=5.25 sec ...... 133
6.3 Electron heating and production of energetic electrons at t=3.05 sec and t=5.25 sec .....136
6.4 Dipolarization and reconnection outflow at the center of the current sheet (Z=0) ...... 138
6.5 Normalized electron distribution functions at the DF and in the EDR ...... 141
6.6 Electron velocity distributions at the DF and in the EDR at t=5.25 sec...... 144
6.7 Densities of electrons and ions from the current sheet and background ...... 146
7.1 Distribution functions produced in the model of particle multistep energization on
multiple scales in the magnetotail during substorms ...... 157
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LIST OF SYMBOLS t time u guiding-center parallel velocity r position uE guiding-center EB drift velocity v particle velocity uB guiding-center gradient drift velocity V fluid velocity uc guiding-center curvature drift f distribution function velocity q electrical charge uac guiding-center acceleration drift m mass velocity
E electric field vA Alfvén speed E electric field strength or resistivity kinetic energy magnetic field compressional factor
B magnetic field contraction factor of bounce distance B magnetic field strength between mirror points c speed of light in vacuum solid angle particle gyro radius pitch angle w particle gyration velocity gyro phase
W kinetic energy kappa parameter
c particle gyro frequency D Debye length s unit length along field line pe plasma frequency M magnetic moment di ion inertial length R guiding-center position
de electron inertial length uGC guiding-center velocity
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ACKNOWLEDGEMENTS
I wish to first thank my thesis advisor, Professor Maha Ashour-Abdalla. Her guidance, advice and enthusiasm have been invaluable to me every step of my graduate study and research.
She has supported my graduate study and research by providing the best research tools, by creating good opportunities for academic activities, and most of all, by pointing to me a correct direction of research. Every time I came up with unrealistic and “ambitious” ideas, she reminded me of current research status in space physics. This is a very important reason that I could conduct good scientific projects and graduate in time. I thank her also because she has offered me a fair amount of freedom in details of my work and has encouraged me to explore new area. I have been grateful to her and hopefully will “become more mature in the future” (in her words).
I am grateful to Professor Raymond J. Walker, who has also advised me every step of my research. His patience and kindness have been influential to me. He has helped me revised every manuscript, poster, presentation slide, and of course this dissertation. He has taught me how to write in an elaborating fashion. He and Maha constantly suggested me writing what progresses I have made instead of criticizing others’ work. I would like to thank Professor George Morales, who taught me plasma physics. His clear and thorough teaching is apparently invaluable to my research. He has also been kind enough to give me advice on multiple occasions and to serve on my thesis committee. I would like to thank Professor Christopher Russell and Professor Christoph
Niemann for serving on my thesis committee and offering advice on my research. Professor
Giovanni Lapenta was very generous to let me use his state-of-the-art implicit PIC code and has helped me tremendously in finishing Chapter 6 of the thesis. I appreciate Professor Richard Sydora for his advice and hospitality during my visit at University of Alberta.
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I learned from scientists in the space plasma simulation group. Dr. Dave Schriver developed the electron LSK code, which has been an indispensable tool in my research. He has also helped me understand particle motion in the magnetotail. Dr. Mostafa El-Alaoui has helped me on the MHD simulations and provided excellent MHD fields that are used in this dissertation.
Dr. Robert Richard was kind enough to read some of my manuscripts and provided good comments.
I am grateful to Haoming Liang, who has been a good friend and my companion of research. He has helped me deal with numerous details.
I would like to thank Greg Kallemeyn in helping me edit part of the thesis and some of my manuscripts.
Most of all, I wish to thank my parents. Their unconditional love and support make me feel fortunate. My love, gratitude and debt to them are beyond measure.
The research in this dissertation was support by a Magnetospheric Multiscale Mission
Interdisciplinary Scientist grant (NASA grant NNX08AO48G at UCLA), Geospace grant (NASA grant NNX12AD13G) and NASA grant NNX10AQ47G. We acknowledge V. Angelopoulos and the THEMIS Science Support team for the use of data and software (TDAS) from the THEMIS
Mission, and specifically, C.W. Carlson and J.P. McFadden for the use of ESA data, D. Larson and R.P. Lin for the use of SST data, D.L. Turner for calibration of SST data, K.H. Glassmeier, U.
Auster, and W. Baumjohann for the use of FGM data, J.W. Bonnell and F.S. Mozer for the use of
EFI data, and A. Roux and O. LeContel for the use of SCM data. Part of the computations were carried out with the support of the NASA Advanced Supercomputing Facility at the Ames
Research Center, and the Gordon supercomputer at San Diego. The Gordon supercomputer is part of the Extreme Science and Engineering Discovery Environment (XSEDE) project, which is supported by National Science Foundation grant OCI-1053575. We would like to acknowledge
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high-performance computing support from Yellowstone (ark:/85065/d7wd3xhc) provided by
NCAR's Computational and Information Systems Laboratory, sponsored by the National Science
Foundation. This work used computational and storage services associated with the Hoffman2
Shared Cluster provided by UCLA Institute for Digital Research and Education's Research
Technology Group.
Chapter 2, Chapter 3, Chapter 4, Chapter 5, and Chapter 6 are respectively reproduced with
significant modifications from manuscripts Pan et al. [2012], Pan et al. [2014a], Pan et al. [2014b],
Pan et al. [2015a], and Pan et al. [2015b]. I would like to thank my co-authors and the publishers
for this matter. Future use of the content of these copyrighted materials shall not infringe the
copyright of their respective owners.
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VITA
Education 07/2010 B.S. (with honors), in Space Physics, Peking University 03/2012 M.S. in Physics, UCLA
Selected Awards 2007-2010 “Zeng Xianzi” Scholarship 2007-2010 Xin Changcheng Scholarship 2007-2009 Chancellor Fund for Undergraduate Student Research 2007-2008 Scholarship for Excellent Academic Performance
Research Experience 09/2007-06/2010 Undergraduate Student Research Assistant Peking/Beijing University, Beijing, China Advisors: Prof. Zuyin Pu and Prof. Lun Xie Analyzed ultra-low frequency waves data obtained by conjugate NASA THEMIS multi-spacecraft and IMAGE ground-based magnetometers.
10/2012-12/2012 Visiting Scholar University of Alberta, Edmonton, Canada Collaborator: Prof. Richard D. Sydora Simulated and analyzed magnetic reconnection and plasma waves with an implicit particle-in-cell (PIC) code.
09/2010-present Graduate Student Researcher University of California, Los Angeles, USA PhD thesis topic: charged particle energization and transport in the magnetotail during substorms. Advisors: Prof. Maha Ashour-Abdalla and Prof. Raymond J. Walker Responsibilities include: Modeling particle energization and transport in the terrestrial space with a parallelized magnetohydrodynamics (MHD) code and large- scale kinetic (LSK) codes. Developing a code for particle sources of the LSK simulations. Modeling particle energization by processes near reconnection sites with a parallelized implicit PIC code. Developing theoretical plasma physics models to quantitatively predict effects of particle energization. Calibrating, analyzing and visualizing multi-spacecraft time series data from NASA and ESA missions, including THEMIS, CLUSTER, Geotail; and using the satellite data to test and validate (or invalidate) simulation and analytical models.
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Publications Pan Qingjiang, Maha Ashour-Abdalla, Mostafa El-Alaoui, Raymond J. Walker, and Melvin L. Goldstein (2012), Adiabatic Acceleration of Suprathermal Electrons Associated with Dipolarization Fronts, JGR-Space Physics, 117, A12224, doi:10.1029/2012JA018156.
Pan Qingjiang, Maha Ashour-Abdalla, Mostafa El-Alaoui, Raymond J. Walker (2014), Electron Energization and Transport in the Magnetotail during Substorms, JGR-Space Physics, 119, doi:10.1002/2013JA019508.
Pan Qingjiang, Maha Ashour-Abdalla, Raymond J. Walker, Mostafa El-Alaoui (2014), Ion Energization and Transport Associated with Magnetic Dipolarizations, GRL-Space Physics, 41, doi:10.1002/2014GL061209.
Melvyn L. Goldstein, Maha Ashour-Abdalla, Adolfo F. Viñas, John Dorelli, Deirdre Wendel, Alex Klimas, Kyoung-Joo Hwang, Mostafa El-Alaoui, Raymond J. Walker, Qingjiang Pan, Haoming Liang(2015), Mission Oriented Support and Theory (MOST) for MMS—the Goddard Space Flight Center/UCLA Interdisciplinary Science Program, Space Science Reviews, DOI 10.1007/s11214- 014-0127-6 (book chapter).
Pan Qingjiang, Maha Ashour-Abdalla, Raymond J. Walker, Mostafa El-Alaoui (2015a), A Comparison of Electron and Ion Energization and Transport Mechanisms in the Magnetotail during Substorms, submitted to JGR-Space Physics.
Pan Qingjiang, Maha Ashour-Abdalla, Raymond J. Walker, Giovanni Lapenta (2015b), Production of Power Law Electrons: Magnetic Field Diffusion or Dipolarization, in preparation for publication.
Talks Pan Qingjiang, Energization of Electrons Associated with Dipolarization Fronts, space physics seminar, University of Alberta, Edmonton, Canada, November 2012.
Pan Qingjiang, Ion Energization and Transport in the Magnetotail during Substorms, the NASA Magnetospheric Multiscale Mission (MMS) meeting, University of Iowa, Iowa city, USA, March 2014.
Pan Qingjiang, Particle Energization and Transport in the Magnetotail during Substorms, space physics seminar, UCLA, Los Angeles, USA, April 2014.
Posters Pan Qingjiang, Maha Ashour-Abdalla, Raymond J. Walker, What Breaks Magnetic Field Lines- A Revisit of Reconnection Theory, the 10th International School/Symposium for Space Simulations (ISSS-10), Banff, Alberta, Canada, July 2011.
Ashour-Abdalla, Maha., Mostafa El-Alaoui, David Schriver, Pan Qingjiang, Robert Richard, Meng Zhou, and Raymond J. Walker, Electron Acceleration Associated with Earthward
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Propagating Dipolarization Fronts, AGU Fall meeting, San Francisco, California, USA, December 2011.
Pan Qingjiang, Maha Ashour-Abdalla, Mostafa El-Alaoui, Raymond J. Walker, David Schriver, and Robert L. Richard, Adiabatic Acceleration of Suprathermal Electrons Associated with Dipolarization Fronts, Geospace Environment Modeling (GEM) meeting, Snowmass, Colorado, USA, June 2012.
Pan Qingjiang, Maha Ashour-Abdalla, Mostafa El-Alaoui, Raymond J. Walker, David Schriver, and Robert L. Richard, On the Importance of Magnetic Reconnection and Adiabatic Acceleration for Suprathermal Electrons Associated with Dipolarization Fronts, AGU Fall meeting, San Francisco, California, USA, December 2012.
Pan Qingjiang, Maha Ashour-Abdalla, Mostafa El-Alaoui, Raymond J. Walker, Electron Energization and Transport in the Magnetotail during a Substorm, Geospace Environment Modeling (GEM) meeting, Snowmass, Colorado, USA, June 2013.
Pan Qingjiang, Maha Ashour-Abdalla, Mostafa El-Alaoui, Raymond J. Walker, Electron Energization and Transport in the Magnetotail during a Substorm, AGU Fall meeting, San Francisco, California, USA, December 2013.
Pan, Qingjiang, Maha Ashour-Abdalla, Raymond J. Walker, Mostafa El-Alaoui, Ion Energization and Transport in the Magnetotail during Substorms, Geospace Environment Modeling (GEM) meeting, Portsmouth, Virginia, USA, June 2014.
Pan Qingjiang, Maha Ashour-Abdalla, Raymond J. Walker, Mostafa El-Alaoui, A Comparison Study of Electron and Ion Energization and Transport Mechanisms in the Magnetotail during Substorms, AGU Fall meeting, San Francisco, California, USA, December 2014.
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CHAPTER 1
Background and Purpose of this Study
1.1. Introduction
The problem of how particles in the magnetosphere are energized to tens of keV to MeV
has been under intensive investigations since the beginning of the space age [Kivelson and Russell,
1995, and references therein]. These magnetospheric electrons and ions originate from the solar wind and the ionosphere with typical initial energies of a few eV. The energization occurs through a multistep process occurring in multiple regions: (1) the bow shock formed in front of the magnetopause, through which the super-Alfvénic solar wind is slowed down to adjust to the magnetospheric obstacle; (2) the dynamic magnetotail and its connection to aurora; and (3) the radiation belts located in the inner magnetosphere where particles are trapped. This dissertation addresses the problem of energization of particles (both electrons and ions) to tens and hundreds of keV [e.g. Baker et al., 1981] and the associated transport process in the magnetotail during substorms.
In a two-page paper, Dungey [1961] proposed for the first time an open magnetospheric convection model driven by magnetic reconnection at the magnetopause and in the magnetotail.
This model has been a major framework to study the dynamics of the Earth’s magnetosphere.
Figure 1 (top panel) shows a schematic view of the open magnetospheric convection model
[Hughes, 1995]. For simplicity, we assume that the interplanetary field is predominantly southward.
Then the magnetic field driven by the solar wind flowing against the front of the magnetosphere will be approximately antiparallel to the geomagnetic field on the other side of the magnetopause.
1
1’ bow shock 2 3 4 magnetosheath magnetopause
5
6 1 7 9 8 jet solar reconnection 7’ wind 6’
5’
3’ 2’ 4’ z 1’ x
y
Vi By>0 Ve By<0
δe δi
By<0 By>0 yellow: IDR green: EDR Le Li
Figure 1.1. A schematic of the magnetospheric convection driven by magnetic reconnection. (Top)
Flow of plasma within the magnetosphere. The numbered field lines show the succession of
2 configuration of the geomagnetic field line (1) after reconnection with an interplanetary magnetic field line (1’) at the magnetopause. Field lines 6 and 6’ reconnect at an X-line in the tail, after which the field line returns to the dayside at lower latitudes, adapted from Hughes [1995]. (Bottom)
A schematic of the night side reconnection region. In the upstream inflow region, electrons and ions are magnetized and convected with the magnetic field toward the diffusion region. The ion inflow is diverted in the ion diffusion region (IDR), where ions are demagnetized and electrons are magnetized. After the ion inflow diversion, the magnetized electrons continue to flow toward the center, until they become demagnetized in the electron diffusion region (EDR). The separation of electron and ion flows in the IDR generates the quadrupole Hall magnetic fields in the Y- direction, adapted from Drake and Shay [2006].
Reconnection occurs between field line 1 and 1’. The newly reconnected field line is dragged by the solar wind successively via the numbered locations 2, 3, and 4, entering the magnetopause on the night side (location 5). The field line is reconnected at location 6, becoming a closed one. It then returns to the Earth through locations 7, 8 and eventually to the dayside through location 9, completing a global magnetospheric convection cycle. The reconnection on the magnetopause is asymmetric, while the reconnection on the night side tends to be symmetric about the current.
During geomagnetic quiet times, the average location of the night side reconnection is at
XRGSM~110 E [McPherron, 1991]. The geocentric solar magnetospheric (GSM) coordinate system is used in this dissertation. In this system, the X-direction points to the Sun, the Z-direction points to northern magnetic pole, and Y-direction completes the orthogonal system according to the right-hand rule, pointing to the west or dusk side.
3
Unlike the steady large-scale magnetospheric convection, magnetospheric substorms are
dynamic processes, in which magnetic energy is explosively released and transferred to particle
kinetic energy [e.g. Akasofu, 1964; McPherron, 1972; McPherron et al., 1973; Russell and
McPherron, 1973; Baker et al., 1981; McPherron, 1991; Lui, 1996; Baker et al., 1996;
Angelopoulos et al., 2008, and references therein]. Consider the magnetosphere as a system. The solar wind drives the system through dayside reconnection and the magnetosphere responds internally in a passive fashion through convection throttled by the distant tail reconnection. During the substorm growth phase, the external solar wind driving exceeds the capacity of internal
convection, resulting in accumulation of magnetic flux in the magnetotail. The accumulated
nightside magnetic flux increases magnetic pressure and energy, compressing and stretching the
magnetotail. In the subsequent expansion phase, the stretched configuration becomes unstable to
tearing. A new neutral line, called the near-Earth X-line, typically forms at
30RXE GSM 20 R E , sometimes as close as XRGSM~10 E [McPherron, 1991], where the
stretched field lines reconnect. The near-Earth reconnection changes field line topology and
generates Alfvénic outflow jets. As the jets propagate to (and away from) the Earth, the magnetic
pressure and energy stored in the stretched tail is released via an abrupt reconfiguration of the
magnetic field to a dipole-like shape. The near-Earth reconnection, the earthward flow jets, the
global dipolarization of the magnetic field, and the associated complex processes closer to the
Earth (e.g. disruption of the cross-tail electric current) give rise to the auroral activity associated with substorms [Akasofu, 1964]. The abrupt reconfiguration of the tail magnetic field following an intensive external driving and instability triggering is analogous to the sawtooth oscillation in a tokamak plasma [Biskamp, 2005]. Eventually, substorm expansion exhausts excess flux and energy in the tail, the X-line retreats to distant tail and the magnetosphere is stabilized during the
4
substorm recovery phase. This qualitative framework of substorms is called the near-Earth neutral
line (NENL) model (see the review by Baker et al. [1996]). The NENL model seems to be
generally accepted by a majority of the magnetospheric community as the basic model of the
processes in the tail during substorms. The data and simulations in this dissertation are consistent
with this model. However, the details of the model have been under intensive debate mainly
because satellite observations are only conducted at a few points. There are also substorm models
that do not emphasize the role of the near-Earth magnetic reconnection. Interested readers are
encouraged to read the comprehensive review on substorms by McPherron [1991] and a more
recent review by Sergeev et al. [2012]. Note the overview presented in Figure 1.1 for
magnetospheric convection is also applicable to the substorm expansion phase, with three caveats.
First, the night side reconnection shown should be understood as the near-Earth reconnection X-
line. Second, in the beginning of the expansion phase, due to reconnection in the distant tail
( XRGSM~110 E ) and the newly formed near-Earth X-line, plasmoids with a transverse scale of
tens of RE are generated and ejected tailward, which are not shown in the figure. Third, the figure does not show variations in the Y-direction; during substorm expansion phase, the azimuthal variations are expected to be substantially different from that during steady magnetospheric convection (details are in section 1.3).
In the global substorm model, the critical near-Earth reconnection occurs on a microscopic scale. Figure 1.1 (bottom panel) shows schematically the reconnection region. In the inflow region, ions and electrons are magnetized and convected with magnetic field toward the diffusion region.
The ion inflow is diverted and ejected in the ion diffusion region (IDR), where ions are demagnetized whereas electrons are magnetized. After the ion inflow diversion, the electrons continue to flow with the magnetic field toward the center, and they are demagnetized in the
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electron diffusion region (EDR). The separated electron and ion flows in the IDR constitute the
electric Hall currents, which generate quadrupole Hall magnetic fields By near the separatrices
[Sonnerup, 1979]. The IDR width i is several times the ion inertial length di ; its length Li is
tens of times di . The EDR width e is a few times the electron inertial length de and the EDR
length Le is a few to tens of times the di , shorter than but comparable to Li [Daughton et al.,
2006]. Note that close to each of the separatrices, there are counter-streaming fast electron beams consisting of an incoming beam toward the X-line in the inflow region and an outgoing beam away from the X-line in the outflow region, which also contribute to the Hall electric currents [Hoshino et al., 2001; Pritchett, 2001]. The beams are not depicted in the diagram.
The near-Earth magnetic reconnection plays an instrumental role in the magnetic-to-kinetic energy transfer during substorms. On one hand, close to the X-line, magnetic energy is directly
converted to kinetic energy via localized magnetic diffusion, resulting in plasma heating and
production of energetic electrons. On the other hand, the topological change of the tail field lines by reconnection enables subsequent particle energization when the reconnection outflow jets
propagate to the Earth. Accordingly, particle acceleration mechanisms in the magnetotail proposed
by previous studies can be categorized in two classes: those operating near the reconnection
diffusion region and those occurring during plasma earthward transport (see sections 1.2 and 1.3 for detailed references). The former class is termed local acceleration, and the latter class is termed
nonlocal acceleration. We stress that there is no clear distinction between local acceleration and
nonlocal acceleration in the Earth’s magnetotail. In addition, some of the studies emphasizing local
acceleration considered energization in the immediate reconnection outflow region, while some of
the studies on nonlocal acceleration also discussed particle energization resulting from
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reconnection acceleration. Nevertheless, we feel it is an approximate framework to organize the
large and growing volume of literature on particle energization in the magnetotail. Therefore, we
review local and nonlocal acceleration separately in the following two sections.
1.2. Acceleration by Magnetic Reconnection
The suggestion that magnetic reconnection could be a significant cause of particle
acceleration in cosmic plasmas was first made by Giovanelli [1947]. In the Earth’s magnetotail,
magnetic reconnection is believed to be particularly important in accelerating particles, because
the magnetotail-like field reversal across the current sheet provides favorable conditions for
reconnection, through which magnetic energy can be rapidly transferred to particle kinetic energy.
Using measurements by the Explorer 47 (IMP7) satellite, Sarris et al., [1976] reported particle
bursts with proton energy E p 0.29MeV and electron energy EMeVe 0.22 in the plasma sheet
at X ~ 35RE , and suggested that one of the possible acceleration mechanisms was the annihilation of magnetic field lines at the region of the neutral line. Using Explorer 34
measurements at X ~ (20 30)RE , Terasawa and Nishida [1976] reported energetic electron
bursts of EMeVe 0.3 ~ 1.0 in association with the southward turning of Z-component of the
local magnetic field. They suggested that these electrons are accelerated at the neutral line and
trapped in the magnetic flux rope tailwards of the neutral line. Similar electron bursts of
EkeVe 200 in association with southward turning of Bz and tailward plasma jets have also been
observed at X ~ 30RE with the IMP8 spacecraft [Baker and Stone, 1976, 1977]. Möebius et al.
[1983] reported measurements of energetic protons of E p 20 500keV and energetic electrons
7
of EkeVe 75 from ISSE-1 and ISSE-2 satellites and suggested that the energetic particles were accelerated near an earthward moving X-line in the event analyzed. Energetic particle bursts in these early measurements have been related to magnetic reconnection. However, no direct evidence of magnetic reconnection were reported in association with the energetic particles.
Recent observations have made significant progresses in directly connecting energetic
particles with reconnection characteristics. Øieroset et al. [2002] reported an event in which the
WIND spacecraft at X ~ 60RE detected energetic electrons (up to 300 keV) in the IDR with Hall
reconnection characteristics. They found no enhancements of energetic ion fluxes in this event.
Hoshino et al. [2001] identified an event of earthward crossing of a reconnection X-line by the
Geotail spacecraft at X ~ 24RE , and found that a power law distribution of suprathermal
electrons (EkeVe 20 ) that extended from the Maxwellian distribution of hot thermal electrons
(2-3 keV) was generated by the magnetic reconnection. Using data obtained from Cluster
spacecraft at X ~ 16RE and YR ~ 8 E , Imada et al. [2007] identified a reconnection event and
found that electrons were accelerated to about 127 keV via a two-step acceleration process, namely
initial acceleration at the X-line and subsequent acceleration in the immediate downstream reconnection outflow region. Note that the 127 keV energy was the highest energy channel
presented in the study (same for the studies with Cluster data discussed below). The energy channel
was determined by the RAPID instrument onboard Cluster [Wilken et al., 2001]. The two-step
acceleration scenario derived from an event study by Imada et al. [2007] is consistent with their
earlier statistical study of the average profiles of energetic and thermal electrons in the magnetotail
reconnection regions by using Geotail data, which showed that due to the Earth’s dipole field
energetic electron flux is much stronger in the earthward side of X-line than in the tail side [Imada
8 et al., 2005]. Their later statistical study with Geotail data found that electrons are efficiently accelerated to E 38 keV in a thin current sheet during fast reconnection events [Imada et al.,
2011]. Chen et al. [2008] analyzed Cluster data during a reconnection event and found that electrons are accelerated in multiple magnetic islands (flux ropes). Using data from Cluster
spacecraft at X ~ 17RE , Retinò et al. [2008] reported the spacecraft crossed a flux rope in the
IDR. They found that the flux of electrons is largest within the flux rope where they are mainly directed perpendicular to the magnetic field. At the magnetic separatrices, the fluxes are smaller, but the energy spectra are harder and electrons are mainly field aligned. Similar events were reported by Wang et al. [2010] and Huang et al. [2012a]. During these events, energetic electron flux is larger in a downstream magnetic island than that in the reconnection current sheet. Even though all the aforementioned Cluster observations were limited to below 127 keV, they did show that the flux increases at 127 keV were weak, suggesting electron acceleration upper limit was close to that energy. To summarize, the overwhelming evidence in recent observations show electrons are accelerated to tens of keV to about a hundred keV by processes close to the X-line, while there are very few corresponding observations of energetic ions.
A number of analytical models of particle energization by reconnection have been developed. The simplest analytical models are stationary [Bulanov and Sasorov, 1976; Burkhart et al., 1990; Vekstein and Priest, 1995]. These models assume simple magnetic field
BeBzL00()zz BxL () xx e with characteristic lengths Lx and Lz in the X- and Z-directions,
and a constant and uniform electric field along the Y-direction E E0e y . The basic idea is to solve particle trajectories in the prescribed stationary fields (differential equations), and to estimate particle asymptotic energy gain from the electric field when they are thrown out of the
reconnection region by the finite Lorentz force vByz . A more complicated class of models
9
consider the effects related to the formation of the X-line in the course of substorm expansion,
which can be described as the growth of an ion tearing mode [Galeev et al., 1978, 1979]. The
growth of the ion tearing mode generates an inductive electric field that can accelerate particles in
the vicinity of the X-line. With appropriate assumptions of the length of the X-line and the
evolution of the dynamic fields, it is estimated that the inductive electric field can accelerate both
electrons and protons to MeV energies in the Earth’s magnetotail [Zelenyi et al., 1984, 1990a].
Surprisingly, models like these are the only analytical ones for considering particle acceleration
by magnetic reconnection. Various mechanisms of particle acceleration near the diffusion region
are derived from numerical simulations and/or suggested by observational data. These models
include:
a. acceleration by a reconnection electric field in the vicinity of the X-line. Essentially,
all the aforementioned analytical models deal with this mechanism, which was later
confirmed by kinetic simulations [Ricci et al., 2003; Pritchett, 2006a, 2006b].
b. acceleration by an electric field parallel to the magnetic field; the parallel electric field
is supported by electron density cavities and/or pressure anisotropy [Drake et al., 2003,
2005; Egedal et al., 2005, 2012, 2013]. This mechanism operates favorably near the
reconnection separatrices, where the fast electron beams are subjected to streaming
instabilities, leading to nonlinear turbulence and generating electron density cavities
[Goldman et al., 2008].
c. acceleration due to particle Speiser motion in the diffusion region. Speiser motion was
originally proposed for particle acceleration in the tail current sheet with Bx reversal
across current sheet, a weak Bz and a constant Ey [Speiser 1965, 1967]. However,
because in the classical two-dimensional reconnection picture, the geometry in the
10
diffusion region satisfies the conditions for Speiser motion (recall the fields in the
aforementioned stationary models). It was suggested that electrons and ions can be
accelerated to their respective Alfvén velocities in the IDR [Shay et al., 2001; Hoshino
et al., 2001].
d. acceleration due to gradient and curvature drift along the electric field ( Ey ) direction
in the immediate downstream pileup region [Hoshino et al., 2001; Imada et al., 2007].
This mechanism is similar to the adiabatic acceleration during plasma earthward
transport farther away the reconnection site (see section 1.3). The difference is the
distance between the acceleration site and the X-line, and therefore the nature of
particle motion is different. Particle motion tends to be nonadiabatic close to the X-line,
and becomes more adiabatic farther away.
e. wave-particle interaction [Cattell et al., 1994; Okada et al., 1994; Shinohara et al.,
1998; Cairns and McMillan, 2005]. This mechanism is mainly speculated from
observations of waves with frequency close to the lower-hybrid frequency in the
reconnection region. However, it was suggested that hybrid waves scatter electrons to
larger pitch angles so that they can gain more energy in the equatorial plane via
mechanism (d) [Hoshino et al., 2001].
f. acceleration by a reconnection electric field when particles are trapped in the O-point
type field geometry (magnetic islands) [Vasyliunas, 1980; Mattaeus et al., 1984;
Goldstein et al., 1986].
g. acceleration due to contraction of magnetic islands [Drake et al., 2006], coalescence of
magnetic islands [Pritchett, 2008; Oka et al., 2010], and particle stochastic motion
11
across multiple magnetic islands [Drake et al., 2006; Hoshino, 2012]. These
mechanisms are derived from simulations.
The observational evidence for mechanisms (f) and (g) are that there are high-
energy particles, in particular high-energy electrons, in multiple magnetic islands (flux
ropes in 3D) near the reconnection region [Chen et al., 2008; Retinò et al., 2008; Wang
et al., 2010; Huang et al., 2012a], although it less definitive that these electrons are
accelerated by the magnetic islands.
h. surfing/surfatron acceleration [Sagdeev and Shapiro, 1973]. This mechanism describes
particle acceleration by the reconnection electric field ( Ey ) when they are trapped in
the current sheet. The trapping is due to the polarization electric field ( Ez ) developed
in the boundary layer between the current sheet and the lobe for thin current sheet
reconnection [Hoshino, 2005]. The polarization electric field ( Ez ) is induced in
association with the Hall reconnection [Birn et al., 2001; Nagai et al., 2001].
Each of these models favors a particular kind of scenario, and obviously has been and will
continue evolving as satellite measurements improve and computer capacity increases. To a large
extent, all the aforementioned models (except for (e)) are derived from two-dimensional (2D)
reconnection studies. Three-dimensional (3D) reconnection has been shown to be substantially
different from 2D reconnection [Daughton et al., 2011]. For example, with a finite guide field, the
3D evolution is dominated by the formation and interaction of helical flux ropes (corresponding
to magnetic islands in 2D), resulting from secondary instabilities within the electron layers. New
flux ropes spontaneously appear within these layers, leading to a turbulent evolution. How particles
are energized in realistic 3D dynamics remains poorly understood. In addition to the reduction of
dimensionality, simulations from which these mechanisms are derived are often compromised by
12
small simulation domains and unrealistic proton-to-electron mass ratio. With small simulation
domains, it is difficult to develop a large picture of particle energization. An artificial mass ratio,
along with other unrealistic physical parameters such as the magnitude of the Alfvén speed and
particle thermal speed in terms of the speed of light, pose difficult questions on interpreting the
final particle energies and quantifying particle energization. Complicated by the use of unrealistic
physical parameters, the simulation results referenced above were usually not vigorously
compared with observations in the same units. Relativistic electrons with energies close to the rest
mass energy are easily produced in these simulations [e.g. Hoshino et al., 2001; Drake et al., 2005;
Pritchett, 2006a, 2006b], while the aforementioned observations in or very close to the diffusion region showed that electrons are accelerated only to tens of keV to about a hundred keV energy range in the reconnection diffusion region [Øieroset et al., 2002; Imada et al., 2005, 2007; Chen
et al., 2008; Retinò et al., 2008; Wang et al., 2010; Imada et al., 2011; Huang et al., 2012a].
1.3. Acceleration during Plasma Earthward Transport
While magnetic reconnection in the magnetotail is actively examined as a major candidate
for particle energization during substorms, studies of particle energization beyond the reconnection
region focus on dipolarizations and substorm injections.
Dipolarization refers to a reconfiguration of the magnetosphere from a stretched tail-like
to a dipole-like configuration. The leading edges of dipolarizations, known as dipolarization fronts
(DFs), are characterized by rapid increases in the north-south component of the magnetic field
()Bz and are frequently observed in the magnetotail during substorms [Russell and McPherron,
1973; Sergeev et al., 1996; Ohtani et al., 2004; Runov et al., 2009; Schmid et al., 2011; Fu et al.,
13
2012a]. In addition to the characteristic increase of Bz , common features of a DF passing a
spacecraft include a dip of Bz ahead of the increase of Bz , a drop in the plasma density, a decrease
in plasma beta, an increase in plasma temperature, a decrease in plasma pressure and an increase
in magnetic pressure, and a decrease in plasma entropy [Hwang et al., 2011]. The hot and tenuous plasmas in the flux tubes following DFs are plasma bubbles characterized by low plasma entropy
[Wolf et al., 2009 and references therein]. From an MHD perspective, DFs are tangential discontinuities [e.g. Sergeev et al., 1996; Fu et al., 2012b], which are related to the nonlinear slow
mode [Kivelson and Russell, 1995]. However, DFs have their kinetic structure, resulting from
diamagnetic drift of electrons and ions in the presence of a steep magnetic field gradient. Naturally,
the thickness of DFs is comparable with the ion inertial length or the ion gyro radius, whichever
is larger [e.g. Runov et al., 2009; Sergeev et al., 2009]. From a multi-fluid perspective, physics related to the Hall term in the generalized Ohm’s law governs the structure of DFs [Biskamp, 2005].
The Hall electric current near DFs has been confirmed by observations [Zhou et al., 2009; Zhang et al., 2011; Fu et al., 2012b].
Dipolarizations are often accompanied by high-speed earthward (and tailward) flows
[Baumjohann et al., 1990; Angelopoulos et al., 1992]. These flows (~ 1 min) known as bursty bulk
flows (BBFs) are often embedded in intervals of enhanced plasma flows (~ 10 min) and are
believed to be the primary mechanism for the earthward transport of mass, energy and magnetic
flux in the magnetotail [Angelopoulos et al., 1994]. Previous studies have suggested that the width
of these flow channels is 23 RE [Angelopoulos et al., 1996; Sergeev et al., 1996; Nakamura et al., 2004]. Dipolarizations and flows are associated with large transverse electric fields [e.g.
Sergeev et al., 2009; Fu et al., 2012b] and strong wave activity whose frequency ranges from the
lower-hybrid frequency to the electron gyro frequency [Zhou et al., 2009].
14
DFs and associated flows observed in the near-Earth plasma sheet at XRGSM~10 E , are often interpreted as magnetic flux pileup associated with flow braking [Hesse and Birn, 1991;
Shiokawa et al., 1997; Baumjohann et al., 1999; Runov et al., 2012] and current disruption [Lui et al., 1988]. Transient DFs in the mid-tail are interpreted as signatures of nightside flux transfer events (NFTE) resulting from sporadic and spatially localized reconnection [Sergeev et al. 1992;
Fu et al., 2013]. DFs have been found in both quasi-local and global MHD simulations [e.g.
Wiltberger et al., 2000; Birn et al., 2004a, 2011; El-Alaoui et al., 2012, 2013], hybrid simulations
[Krauss-Varban and Karimabadi, 2003] and PIC kinetic simulations [Sitnov et al., 2009; Sitnov and Swisdak, 2011]. Dipolarizations in global MHD simulations are intensified as the reconnection jets brake in the inner magnetosphere, whereas the dipolarizations in particle-in-cell (PIC) simulations are more pulse-like and transient, and are intensified near the reconnection region.
Other mechanisms such as the ballooning interchange instability [Hurricane et al., 1996; Pritchett and Coroniti, 2010] may also be important for dipolarization formation and for determining their azimuthal extent.
Large increases in high-energy fluxes of electrons associated with dipolarizations and high- speed flows have been observed by the Cluster spacecraft and Time History of Events and
Macroscale Interactions during Substorms (THEMIS) spacecraft [e.g. Runov et al., 2009; Asano et al., 2010; Vaivads et al., 2011; Runov et al., 2011]. For most of these dipolarization events, as fronts pass by an observing spacecraft, the electron energy fluxes shift to higher energy. This is observed as a simultaneous increase of the high-energy electron fluxes (tens of keV to hundreds of keV) and decrease of the low-energy electron fluxes (a few keV or less) [e.g. Deng et al., 2010;
Hwang et al., 2011]. For some DF events, the energy fluxes of high-energy electrons increase as the fronts arrive, while in other events the high-energy electron energy fluxes decrease just before
15
the fronts arrive and then increase. For some events the high-energy electron pitch angle
distributions peak near 90o, while others peak at smaller pitch angles (<45o). Both types of
distributions can appear in the inner magnetosphere and the mid-tail plasma sheet, depending on
the particular event [e.g. Deng et al., 2010; Fu et al., 2011]. However, flux increases associated
with DFs are not limited to electrons. Ion fluxes that demonstrate similar increases associated with
DFs were observed by the THEMIS spacecraft for the February 27, 2009 event [Runov et al., 2009;
Deng et al., 2010]. (Note that the THEMIS ion observations do not allow us to differentiate
between ion species.) Observations of high-energy flux increases associated with DFs for both
electrons and ions were also presented in a multi-event study, in which the high-energy ion flux
increases on a time scale of tens of seconds, whereas the electron flux increases within a few
seconds [Runov et al., 2011]. While it remains unclear whether these similarities and differences
between electron fluxes and ion fluxes are universal across events, that fluxes of both species
increase upon the arrival of DFs and high-speed flows indicates that certain energization and
transport mechanisms operate for both species in these events.
As a consequence of particle energization and transport in the magnetotail, injections of
energetic particles are frequently observed in the inner magnetosphere. Early observations of
injections at geosynchronous orbit during substorms led to the idea of an “injection boundary”, an
azimuthal boundary moving earthward fromX ~ (9 12)RE to the geosynchronous orbit under
an enhanced cross-tail electric field [McIlwain, 1974; Mauk and McIlwain, 1974]. The injection
boundary covers the region where fluxes become simultaneously enhanced at different energies,
called dispersionless injection. The dispersionless injection boundary/region was later found to be
azimuthally narrow [Belian et al., 1978; Reeves et al., 1991; Gabrielse et al., 2014]. Energetic
particles observed azimuthally distant from the injection boundary get there by energy-dependent
16
drifts and hence appear dispersed in energy, with electrons drifting eastward [Arnoldy and Chan,
1969; Pfitzer and Winckler, 1969] and ions drifting westward [Bogott and Mozer, 1973]. When
the injected particles become trapped and complete full drift orbits, drift echoes are observed as
injections of multiple times with progressively increased dispersion [Lanzerotti et al., 1967]. Note
that to avoid the effect of energy-dependent drifts, dispersionless injections are caused either by simultaneous acceleration of particles across energies in the injection region or by energy-
independent transport of accelerated particles from other regions. The injection boundary idea was
subsequently further developed. Birn et al. [1997a] demonstrated statistically that a spatially-
dependent pattern exists for dispersionless injections, which can be explained as two injection
boundaries—one for electrons, one for ions—that are offset from each other. Typically, ion
dispersionless injections are inclined towards dusk (~3 hours before midnight) and electron
dispersionless injections towards dawn (~2 hours after midnight). Simultaneous injections of both
species are observed near midnight. This pattern was confirmed by Thomsen et al. [2001], who
used two geosynchronous satellites to show that the pattern occurs not only statistically, but also
for individual injection events. Energetic particle injections and magnetic field dipolarizations are
found to occur concurrently [Sauvaud and Winckler, 1980]. Sharp inner fronts of substorm
injections (injection fronts) have been observed to coincide with DFs in the inner magnetosphere
[Sergeev et al., 1998].
Notably, the observations of substorm injections are mainly in the inner magnetosphere,
within the geosynchronous orbit in particular. In contrast, observations of high-energy fluxes of
electrons associated with dipolarizations and fast flows are in the tail as well as in the inner
magnetosphere. Baker et al. [1979] made an important connection between observations of
energetic particle bursts in the tail and observations of substorm injections. They conjectured that
17
in events with no drift echoes under extremely strong solar wind driving, Ey in the plasma sheet
is strong; the impulsive bursts of energetic protons ( EMeV 0.3 ) observed in the plasma sheet at
XR~18 E are caused by magnetic reconnection; these energetic protons are convected to the
inner magnetosphere rapidly without significant gradient drift, so they are observed as
dispersionless injections at the geosynchronous orbit. In contrast, in drift echo events under
relatively weaker solar wind driving, the transverse electric field Ey is relatively weaker, the
earthward transport is slower, and protons may often gradient drift substantially during their transport, resulting in dispersed substorm injections for lower energy protons ( E 200 keV ).
Various adiabatic and nonadiabatic acceleration mechanisms have been proposed to
account for energetic particles in the magnetotail. The adiabatic acceleration mechanisms include
betatron acceleration as particles are transported from a weak magnetic field region in the tail to a
stronger field region in the inner magnetosphere [e.g. Tverskoy, 1969; Kivelson et al., 1973;
Ashour-Abdalla et al., 2011; Pan et al., 2012], and Fermi acceleration resulting from contraction
of field line length between mirror points during convection [e.g. Tverskoy, 1969; Sharber and
Heikkila, 1972; Pan et al., 2012; Ashour-Abdalla et al., 2013]. Particles, especially ions can gain
or lose energy via nonadiabatic motion in the tail current sheet. The tail current sheet is
characterized by a magnetic field reversal across the current sheet, a weak magnetic field normal
to the current sheet and a dawn-dusk electric field. Particles in the current sheet undertake Speiser
motion consisting of bounce motion within the current sheet and gyration about the normal
magnetic field, until they are eventually ejected out of the current sheet along the magnetic field
line. During the trapped motion in the current sheet, ions (electrons) drift in the Y-direction
(negative Y-direction), so they gain energy from the dawn-dusk electric field before they are
ejected out [Speiser, 1965, 1967; Lyons and Speiser, 1982; Lyons, 1984]. In the magnetotail, ions
18
can also gain or lose energy when they encounter the current sheet multiple times, through resonant
orbits [Burkhart and Chen, 1991; Büchner, 1991; Ashour-Abdalla et al., 1993, 1995; Zelenyi et
al., 2007] and quasi-adiabatic orbits [Zelenyi et al., 1990b]. These particle energization
mechanisms were reviewed by Birn et al. [2012]. Regarding energization related to dipolarizations, wave-particle scattering has been suggested to be important for electron acceleration and heating
because lower-hybrid waves, electron cyclotron harmonics and whistler waves have been observed
in association with dipolarizations, [e.g. Deng et al., 2010; Khotyaintsev et al., 2011; Huang et al.,
2012b], although it is not clearly demonstrated that wave-particle interaction can result in electron
energy gain. Ions can be trapped or quasi-trapped by the electric potentials caused by moving
dipolarizations [Artemyev et al., 2012; Ukhorskiy et al., 2013]. The quasi-trapped ions can gain
energy by encountering and reflecting from DFs multiple times. This process is similar to ion
acceleration by perpendicular shocks.
Particle tracing calculations have been used to investigate the particle energization in the magnetotail and the resultant substorm injections in the inner magnetosphere. Using 3D fields
resembling dipolarization events, including a transient electric field surge, and a magnetic field
whose time scale of variation is comparable with ion gyro period, Delcourt and Sauvaud [1994]
calculated nonadiabatic ion trajectories and showed that ions are accelerated to high energies by
the rapidly changing magnetic field and the transient electric field. Li et al. [1998] and Zaharia et
al. [2000] calculated particle guiding-center orbits in the equatorial plane, using assumed localized
earthward propagating electromagnetic pulses in a fixed background magnetic field. They found
that particles transported from the tail to the inner magnetosphere undertake betatron acceleration,
resulting in dispersionless injections at the geosynchronous orbit. Zaharia et al. [2004] extended
their earlier model by including the background field evolving from a nondipolar (stretched) to a
19
dipole shape as the pulse propagates to the Earth. The resultant energization was increased and
more realistic due to the “dipolarization” of the background magnetic field. Birn et al. [1997b]
examined ion (proton) acceleration by calculating ion orbits in the dynamic electric and magnetic
fields of a 3D MHD simulation. The MHD fields included the generic characteristics of neutral
line formation and dipolarization. The energetic proton flux changes obtained from the test proton
orbits agreed well with observations that demonstrate rapid ion flux increases at energies above
about 20 keV during substorm injections. The protons are accelerated by the transverse electric
field associated with the dipolarization. The acceleration mechanism is equivalent to the betatron
effect. In a complementary study, Birn et al. [1998] examined electron acceleration using the
electron guiding-center approximation in the equatorial plane and the same 3D MHD fields. Their
simulations were able to reproduce observed energy-dependent characteristics of substorm
electron injections, and showed that the dipolarization region earthward of the reconnection site is
more significant than reconnection in accelerating electrons. They later extended the electron
orbits to include nonequatorial drifts [Birn et al., 2004b]. A more recent study by Birn et al. [2013]
confirmed that the major acceleration mechanisms responsible for the energetic electrons injected
into the inner magnetosphere are betatron acceleration and Fermi acceleration associated with the
dipolarization. Ions are accelerated in a similar fashion, despite that ion orbits are nonadiabatic.
Since 1990s, the UCLA space plasma simulation group has been developing the particle
tracing technique for ions [Ashour-Abdalla et al., 1990] and electrons [Schriver et al., 1998] to
study three types of problems. The first type concerns the formation of magnetospheric structures
such as the central plasma sheet, the plasma bulk flows, the plasma sheet boundary layer (PSBL),
and the “beamlets” within the PSBL [Ashour-Abdalla et al., 1992a, 1993, 1995, 1996, 1999a,
1999b, 2000, 2005; Peroomian and Ashour-Abdalla, 1995, 1996]. The second type of problem
20
addressed is the origin of the ion-to-electron temperature ratio [Schriver et al., 1998]. Below we
review the literature on the third type problem, which is energization of particles to tens and
hundreds of keV during substorms.
Regarding ion energization, Ashour-Abdalla et al. [1992b, 1992c, 1994] calculated ion
trajectories in a 2D Tsyganenko [1989] empirical magnetic field model and a uniform dawn-dusk electric field. They showed that the different regions of the plasma sheet could be characterized by
varying values of the adiabaticity parameter, , which is defined as the square root of the ratio of
the curvature radius of the magnetic field line ( Rcurv ) to the ion gyro radius ( ), namely
Rcurv [Büchner and Zelenyi, 1989]. Both Rcurv and are determined locally at the ion
position. In the region near the X-line, 1 and the ions can be described as quasiadiabatic or
nonadiabatic. In the other extreme, close to the Earth or away from the current sheet, 1 and ion motion is adiabatic. Between these two regions, in the wall region, ~1 and the ions are chaotic [Büchner and Zelenyi, 1989]. In the wall region with a dawn-dusk electric field, ions gain
energy continuously as they are demagnetized and move rapidly along the dawn-dusk direction.
Ion acceleration in the wall region was further demonstrated in a substorm event study in which ion trajectories were calculated in a more sophisticated electromagnetic field derived from a global
MHD simulation driven by realistic upstream solar wind conditions [Ashour-Abdalla et al., 2009].
In that study, Ashour-Abdalla et al. [2009] also showed that the resultant high-energy ions were observed by the THEMIS spacecraft just after substorm onset. The technique of tracing a large number of particle trajectories in a given electromagnetic field and then calculating collective quantities such as the distribution function and energy flux is called large-scale kinetic (LSK) simulations [Ashour-Abdalla et al., 1993, 2005]. Zhou et al. [2011] used the global MHD+LSK method to examine ion injections during a substorm event. They found that the injected energetic
21
ions observed at X ~ 7RE were accelerated in two regions. One region was around the near-
Earth X-line (X ~ 20RE ), where particles were mostly accelerated nonadiabatically by strong
electric fields. The other consisted of several localized regions with ~1 3 betweenX ~ 7RE
and X ~ 18RE , where particles were accelerated in nonadiabatic motion by the potential electric
field. This latter mechanism is the aforementioned nonadiabatic acceleration in the wall region.
Pan et al. [2014b] applied the MHD+LSK method to examine ion energization associated with
magnetic dipolarizations and found that ions that originated near the reconnection site initially
gained energy nonadiabatically, and then gained energy adiabatically as the ions caught up with
and then rode on the earthward propagating dipolarizations. Pan et al. [2014b] also demonstrated
that high-speed flows in narrow channels controlled the earthward transport of ions in the outer
magnetosphere due to the dominance of the EB drift compared to the gradient and curvature
drifts.
The MHD+LSK simulation scheme has also been applied to study electron energization in
the magnetotail during substorms. Ashour-Abdalla et al. [2011] simulated a substorm event that
occurred on February 15, 2008. They found that only low-energy electrons (e.g. 6–12 keV) were
present near the reconnection site and most high-energy electrons (e.g. 41–95 keV) were generated
in association with dipolarizations and fast flows. The electron fluxes from the LSK simulation,
which used Maxwellian distributions as electron sources, were consistent with the measured fluxes
from THEMIS P4 at X ~ 9.8RE . They suggested that reconnection produced low-energy
electrons and the electromagnetic fields associated with the DF accelerated electrons to high
energy via nonlocal betatron process. However, in the February 15, 2008 event, there were no
observations in the tail that could be used to quantify high-energy electrons near the reconnection
22
site. Furthermore, using Maxwellian sources in the LSK simulation may have underestimated the
flux of high-energy electrons resulting from reconnection. Therefore it was not clear how much of
the acceleration occurred near the reconnection site and how much of it occurred as the DF
propagated toward the Earth in that particular substorm. To continue this effort, Pan et al. [2014a]
studied a substorm event that occurred on March 11, 2008 and demonstrated that adding a high-
energy power law tail with E~10 keV to the LSK source distribution was necessary to reproduce
energetic electrons observed by THEMIS P2 at X ~ 14.6RE , suggesting that acceleration near
the reconnection region was important in generating these suprathermal electrons. Meanwhile, by
comparing THEMIS P2 measurements in the tail and P3/P4 measurements at X ~ 10RE , Pan et
al. [2014a] also showed that nonlocal acceleration during plasma earthward transport was
responsible for the majority of the high-energy electrons observed in the inner magnetosphere. In
light of the convincing evidence for nonlocal electron acceleration in the magnetotail during substorms, Liang et al. [2014] examined the nonlocal acceleration mechanism in two very different substorm events that occurred on February 15, 2008 and August 15, 2001. They found that in the
February 15, 2008 event, the high-speed flows in narrow channels produced by azimuthally localized reconnection swept the electrons and adiabatically accelerated them, leading to pancake-
like electron distributions ( f ()vfv () ) in the inner magnetosphere. In contrast, in the August
15, 2001 event, an X-line extending across the tail was formed and the flows were slow. The electrons were nonadiabatically accelerated in the weak field region close to the X-line, resulting
in cigar-like electron distributions ( f ()vfv ( ) ).
In the event studies of particle energization from the reconnection region to the inner
magnetosphere during substorms, electron and ion acceleration were addressed separately. Pan et
al. [2015a] used THEMIS measurements and MHD+LSK simulations to investigate energization
23
and transport mechanisms for both species during a substorm event. The LSK simulation results
showed that thermal ions and electrons (a few keV) observed at the dipolarizations originated from
a relatively wide region of the tail near the reconnection site and were convected to the inner
magnetosphere. Higher-energy particles (tens of keV up to ~100 keV) were produced far away
from the reconnection site by the perpendicular electric fields associated with the dipolarizations
and accompanying high-speed flows. Electrons that originated from the reconnection site were
adiabatically accelerated during earthward transport, and surprisingly ions undertook adiabatic acceleration in a manner similar to that of electrons. However, electron motion was adiabatic in
the magnetotail except in very limited regions close to the X-line, while ion motion was marginally
adiabatic in the dipolarization regions. It was demonstrated that high-speed flows in narrow
channels controlled the earthward particle transport of both electrons and ions in the magnetotail
due to the dominant EB drift in this event.
We point out that the aforementioned particle tracing calculations differ in the electromagnetic fields in which particle trajectories are calculated. The fields are either from
relatively simple analytical description [Delcourt and Sauvaud, 1994; Li et al., 1998; Zaharia et al., 2000, 2004], from empirical models [Ashour-Abdalla et al., 1992b, 1992c, 1994], from generic
MHD fields [Birn et al., 1997b, 1998, 2004, 2013], or from event-dependent global MHD fields
[Ashour-Abdalla et al., 2009; Zhou et al., 2011; Ashour-Abdalla et al., 2011; Pan et al., 2014a,
2014b; Liang et al., 2014; Pan et al., 2015a].
1.4. Purpose of this Study
24
The question addressed in this dissertation is where and how charged particles are
accelerated to tens and hundreds of keV in the magnetotail during substorms. The purpose of our
present study is to integrate simultaneous THEMIS spacecraft measurements in the tail and in the
inner magnetosphere, theoretical models, global MHD+LSK simulations, and implicit PIC
simulations, to quantify both electron and ion energization by various processes and incorporate
the energization processes into a global and complete picture. The computer simulations and
spacecraft measurements are complementary to each other for understanding particle energization
and transport on multiple scales. Specifically, the coordinated observations by identical THEMIS
spacecraft in the magnetotail and in the inner magnetosphere allow us to quantify energization by
processes near the X-line and energization during earthward plasma transport. They provide
information for the LSK particle sources, and constraint MHD+LSK and PIC simulations. The
MHD+LSK simulations provide a global picture of the energization and transport processes that
are responsible for the high-energy particle fluxes observed at a few points by THEMIS spacecraft.
The implicit PIC simulations resolve kinetic effects and allow us to model processes near the X-
line in a relatively large simulation domain with mass ratio and physical parameters that are close
to the realistic values [Brackbill and Forslund, 1982; Vu and Brackbill, 1992; Lapenta et al., 2006;
Markidis et al., 2010 and references therein; Lapenta, 2012]. The large simulation domain allows
us to quantify nonlocal acceleration in the reconnection outflows and clarify the relationship
between local acceleration in the diffusion region and nonlocal acceleration during transport. The
PIC simulations provide explanations to the high-energy fluxes that are observed by THEMIS
spacecraft in the tail. The high-energy fluxes in the magnetotail are generated by processes near
the reconnection, which cannot be properly handled by the MHD+LSK simulations. Throughout
our study, we provide comparisons of simulation results with observations if possible. “In the end,
25
the proof of the value of computer modeling will come from the comparison of its results with
experiments or observations.” (John Dawson, 1985)
Particle energization and transport in the magnetotail is an interesting and challenging
physics question because it involves processes on various scales from a few kilometers (the EDR
scale) to tens of RE (the transport scale). Moreover, developing a quantitative and global picture
of electron and ion energization is also practically desirable because tens to hundreds of keV
electrons injected from the tail are thought to be the source of electrons further accelerated to even
higher energies in the radiation belts [e.g. Green and Kivelson, 2004 and references therein; Horne
et al., 2005]. To model very energetic electrons (hundreds of keV to MeV) trapped in the radiation
belts correctly, we must understand and be able to model the particle sources injected from the
magnetotail. In addition, the tens to hundreds of keV injected ions are one major component of the sources of the ring current [e.g. Thomsen et al., 1998; Nose et al., 2001]. The ring current is a major element that determines the inner magnetospheric dynamics during magnetic storms. The sources of the very energetic particles and development of the ring current are critical issues of space weather because the very energetic particles and the ring current can influence the performance and reliability of space-borne and ground-based technological systems, e.g. causing disruption to satellite operations, telecommunications, navigation, and electric power distribution grids, leading to a variety of socioeconomic losses [Bothmer and Daglis, 2007 and reference therein].
1.5. Structure of the Dissertation
26
Following this introduction, in Chapter 2 we first briefly review the characteristics of
adiabatic particle orbits. The “classical” derivation of equation of guiding-center motion was presented by Northrop [1963] and Banõs [1967]. We comment on the implications of the theory with regard to numerical modeling of particle energization in the magnetotail. Following this review, we present a simple model of adiabatic acceleration to estimate flux changes when particles are transported from the magnetotail to the inner magnetosphere. The model and comparisons of its predictions with observations were published by Pan et al. [2012].
The following three chapters present studies of substorm events, focusing on electron
energization (Chapter 3), ion energization (Chapter 4), and a comparison between them (Chapter
5). The central goal is to use the coordinated THEMIS spacecraft measurements and the global
numerical MHD+LSK simulations to quantify charged particle energization on the meso to global
scale and identify the major acceleration mechanisms. These three chapters are reproduced with
modifications from Pan et al. [2014a], Pan et al. [2014b] and Pan et al. [2015a], respectively.
Chapter 6 deals with electron acceleration on the micro to meso scale in the critical near-
Earth reconnection region. As will be pointed out in Chapter 3, high-energy electrons that follow
a power law distribution near the reconnection X-line are indispensable for achieving consistency
between the global MHD+LSK simulation results and the THEMIS measurements. The problem
of how these power law distributed electrons are generated is examined by using the state-of-the-
art implicit PIC code. The simulation results can also be found in the manuscript by Pan et al.
[2015b].
In Chapter 7, we summarize our findings, discuss caveats of the present study and unsolved
problems, and point out possible major progresses in the future.
27
CHAPTER 2
Theory of Adiabatic Acceleration of Charged Particles
2.1. Introduction
In this chapter, we first briefly review orbit characteristics for particles undertaking
adiabatic guiding-center drifts. The “classical” derivation of guiding-center equation of motion
was presented by Northrop [1963] and Banõs [1967] and will not be repeated here. Instead, the
results will be quoted. The guiding-center theory, as part of the gyrokinetic theory, has received
considerable efforts and progresses in the plasma physics research related to fusion. The classical
derivation is based on physical intuition rather than mathematical rigorousness. The readers are
encouraged to read literature in fusion research, e.g. Brizard and Hahm [2007] for a comprehensive
review of gyrokinetic theory, in which the authors described how to overcome mathematical
vagueness in the procedure of averaging the gyro phase and how to preserve symmetries (e.g.
conservation of energy and phase space) of the guiding-center Hamiltonian system in modern
derivation. Note that the classical results are corrected to the lowest order [Northrop, 1963]. We
comment on implications of the theory with regard to numerical modeling of particle energization
in the magnetotail.
Following this review, we present a simple model to estimate flux changes when particles
are adiabatically transported from the magnetotail to the inner magnetosphere. The model and
comparisons of the model predictions with spacecraft measurements were published by Pan et al.
[2012].
28
2.2. Characteristics of Adiabatic Particle Orbits
“The understanding of the individual orbits is not the same as complete understanding of
plasma physics. The science, or the art, of treating cooperative phenomena must be added to the
thorough discussion of the motion of the individual particles. But it is a basic step in the development of plasma physics to lay the foundation by discussing individual particle motions.
The result is, as always when meaningful progress is made in physics, that a seemingly involved
subject is beginning to show signs of regularity and therefore simplicity. ” (Edward Teller, 1963)
The origins of the adiabatic theory of charged particle motion can be traced back to Alfvén
in the 1940s [Alfvén, 1950]. The fundamental assumption of adiabatic theory is that a separation
of scales of particle motion exists, e.g. between gyration and guiding-center drift with respect to
ambient fields. The derivation of the guiding-center equation of motion relies on a small parameter
for the Taylor expansion of the Lorentz-force equation about the guiding-center position. The
small parameter used by Northrop [1963] and Banõs [1967] is L , where mv0 c qB0 is
1 the particle gyro radius with typical velocity v0 in the magnetic field B0 , and L ln B is the
characteristic spatial scale of the magnetic field variation. Because mv00 c qB L m q , mq is a proxy for the small parameter in the following discussion where physical quantities are not normalized by their characteristic ones [Northrop, 1963]. Expanding the Lorentz-force equation of motion about the guiding-center position and averaging out the gyro phase, we have the equation
for the guiding-center drift velocity uGC for non-relativistic particles
29
dR u GC dt Mc cBE (2.1) q e1 2 e1u () B mc ee112 uE uu uueEE1 u u uu EE qt s t s where Rrρ is the guiding-center position, eB1 B is a unit vector in the direction of the
dR magnetic field, u e is the guiding-center parallel velocity, M is the magnetic moment (the dt 1
cEe first adiabatic invariant), u 1 is the EB drift velocity, and e is the projection E B s 1 of the gradient operator in the magnetic field direction. The magnetic moment is defined as
2 q mw2 M c , where and w are particle gyro frequency and gyro velocity, respectively. 22cB c
The right-hand-side quantities of equation (2.1) are evaluated at the guiding-center position. The drift velocity in the perpendicular direction includes the EB drift term on the order of (1) or
() , depending on the order of perpendicular electric field, the gradient drift term on the order of () , and the “acceleration” drift on the order of () (terms in the square bracket). The
mu2 c ee contribution by u 11 (the first term in the square bracket) is termed curvature drift c qB s
[e.g. Chen, 2006].
The perpendicular drift velocity depends on electromagnetic fields and guiding-center parallel velocity. To follow particle guiding-center motion in given fields, we need update the parallel velocity, which is given by
mdu M B m 2 Eu uueEE 1 () (2.2) qdt q s q t s
30
mu2 mu2 The equation for the change of particle kinetic energy WMBE is given by 22
1(,)dW dRR M B t ER(,)t ( 2 ) (2.3) qdt dt q t The change of the first adiabatic invariant to the lowest order is
1 dM () 2 (2.4) qdt
Note that the right hand side of equation (2.4) is to the order of () 2 rather than () . It is because M q on the left hand side contains a small parameter mq~() . The same situation occurs in equation (2.3).
In the derivation of these equations, it is required that E ~() and that
tu~~ suE acting on e1 and uE are at most (1) . It should be understood that the orderings of these quantities are not necessarily expressed in term of the same small parameter, instead there should be one small parameter for each of them. However, if we adopt the parameter for the Taylor expansion to be the maximum of these small parameters, i.e. take the strongest constraint, the theory is still valid. Let’s examine some of the constraints relevant to particle motion in the magnetotail.
1 B First, it is obvious that B,T 1 is required, since an appreciable change in the c Bt magnetic field on the time scale of gyration shall invalidate the guiding-center picture. This constraint is important for ions in intensifying dipolarizations because ion gyro period (~seconds) is comparable to dipolarization intensification time scale.
31
Second, the time scale of particle parallel traveling a radius of curvature of the magnetic field line should be much larger than the gyration time scale, namely
1 uuuue1 1 e1 221 , where Rc is the radius of curvature of the ccccsR w s
magnetic field line, and Rc is an adiabaticity parameter [Büchner and Zelenyi, 1989]. For a 90 pitch angle particle, u 0 , therefore a small kappa puts no constraint. For a small pitch angle particle on the equatorial plane, its parallel velocity is large, its kappa value is small, hence its parallel motion imposes a strong constraint on the adiabaticity of particle motion.
Third, the time scale of perpendicular EB drift across a characteristic length should be
uE ln B cEb larger than gyration time scale, namely ~12 B . In the presence of ccB high-speed flows, the perpendicular convection electric field is large, so the perpendicular drift imposes a strong constraint.
The adiabatic theory has important implications on test particle simulations (the method is used in the following three chapters). First, the theory sets forth a series of criteria for particle
1 B u cEb motion to be adiabatic. They are B,T 1, ~12 , ~12 B and c Bt w B c
1/2 1/4 5 (empirical). Since B,T m , m , m , m for given parallel and perpendicular energies, ion motion is more likely to become nonadiabatic than electron’s. This requires that we solve Lorentz-force equation for ions in the magnetotail, but we may solve the guiding-center equation for electrons. Second, if the adiabaticity criteria are satisfied, the guiding- center drift velocity may be estimated as
32
cMcmucEe e2 e e uuuu~~11 B 1 1. The EB drift does not depend on GC E B c B Bq qBs energy or charge, whereas the gradient and curvature drifts depend on both of them. For electrons and ions with the same parallel and perpendicular energies, their guiding-center velocities are the same, but they gradient and curvature drift in opposite directions. In the magnetotail, localized high-speed flows carry strong dawn-dusk convection electric fields. If the EB drift is dominant, electrons and ions are transported along the flow channels in the tail. On the other hand, if gradient and curvature drifts dominate, the major feature of transport is that electrons drift eastward and ions drift westward in the magnetosphere. The relative magnitude of the EB drift compared to the gradient and curvature drifts varies with particle kinetic energy. We will show the relative magnitude of the drifts is of great importance in determining how particle gain energy in the
1 dW M B magnetotail. Third, consider the adiabatic energy gain equation uE . The first qdtGC q t term on the right-hand-side (RHS) is charge-dependent, so in order for electrons and ions to gain energy via the first term, they need to drift in opposite directions, namely ions need to drift parallel to the dawn-dusk electric field in the magnetotail, whereas electrons need to do the opposite. This is made possible by the charge-dependent gradient and curvature drifts. The second term is charge- independent (the charges on both sides are cancelled), so an intensification of magnetic field (e.g. dipolarization) will accelerate both electrons and ions. These applications to the Earth’s magnetosphere are discussed in the following three chapters.
2.3. An Analytical Model of Adiabatic Acceleration
2.3.1. Motivation
33
Large increases in the energy fluxes of high-energy electrons (tens of keV to hundreds of keV) are frequently reported in association with DFs. It is unclear from observations how much of the acceleration occurs as the DFs propagates toward the Earth and how much occurs near the reconnection site. Recent studies using data from the four Cluster satellites indicate that electrons can be accelerated by multi-step processes near the site of magnetic reconnection as well as far away from the diffusion region in the same events [Asano et al., 2010; Vaivads et al., 2011].
In this section, we explore the collective effect on electron fluxes when electrons are adiabatically transported from the magnetotail to the inner magnetosphere. Based upon the analysis of adiabatic acceleration, we attempt to qualitatively clarify the roles of magnetic reconnection and adiabatic acceleration in producing suprathermal electrons associated with DFs in the inner
magnetosphere (XGSM~-10 RE ). In section 2.2.2 we discuss the adiabatic acceleration model. In section 2.2.3 we present comparisons of the model predictions with data from the THEMIS spacecraft during two geo-active intervals (March 11, 2008 and February 27, 2009). In section
2.2.4 we discuss these observations in terms of what we would expect for acceleration as the DFs propagate earthward and use those inferences to consider what may be happening nearer the reconnection site.
2.3.2. The Adiabatic Acceleration Model
In the Earth’s magnetosphere, the time scales of changes in the magnetic and electric fields are much larger than time periods of the gyration of a particle about a magnetic field line, the bounce motion between mirror points, or the azimuthal drift of a particle about the Earth. There are three adiabatic invariants corresponding to these three types of motion, respectively [Northrop,
34
1963; Roederer, 1970; Schulz and Lanzerotti, 1974; Green and Kivelson, 2004]. The first invariant
(magnetic moment) is given by
P 2 M (2.5) 2mB where P is the relativistic momentum perpendicular to the magnetic field. The increase in P due to a slowing varying B is termed betatron acceleration. The second adiabatic invariant involves the parallel momentum integral along the bounce motion between mirror points
JPds (2.6) where P and ds are the relativistic momentum parallel to the magnetic field and the distance
along a field line, respectively. An increase in P due to a decrease of the distance between mirror points is usually referred to as Fermi acceleration.
The azimuthally integrated differential flux jE(,,,) r t and distribution function
f (,,,)Et r are functions of kinetic energy E , pitch angle , position r and time t . The relation of differential flux and distribution function is [Schulz and Lanzerotti, 1974; Lyons and Williams,
1986]
j(,,,)Et r fE(,,,) r t (2.7) P2 where P is the relativistic momentum.
From Liouville’s theorem, which states that the particle distribution function in phase space is constant along the particle characteristic trajectory without collisions or diffusion caused by wave-particle interaction, namely
jE11111(,,,) rr t jE 2 (, 2 222 ,,) t 22constant (2.8) PP12
35 where the subscripts 1 and 2 indicate that the quantities are evaluated at different positions and times.
We assume that particles undergo slow magnetic field compression given by
B 2 (2.9) B1 where is the magnetic field compression factor. Assume the first adiabatic invariant is conserved
2 P,2 2 (2.10) P,1 and PP sin , we have
22 P22sin 22 (2.11) P11sin Assume the second adiabatic invariant is conserved, the effect of Fermi acceleration on particles due to contraction of mirror distance is
P ,2 (2.12) P,1 where PP cos . is the ratio of the distances between mirror points for a particular particle on different mirror bounces, which we call the contraction factor. should vary for particles with different pitch angles, energies and positions; however, in this model it is assumed to be uniform in order to obtain analytical estimates (see the discussion section below). Hence,
P cos 22 (2.13) P11cos With betatron and Fermi acceleration operating simultaneously, a particle trajectory in phase space is given by equations (2.11) and (2.13),
2222 2 PP21(cos 1 sin) 1 2 (2.14) 2 sin 1 sin 2 22 2 cos11 sin
36
Combining equations (2.8) and (2.14), we have
22 2 jE2(,,,)( 2 222rr t cos 1 sin)(,,,) 11 jE 1 111 t (2.15) where the relationship of the pitch angles and energies indicated by the subscripts follow from
22 22 equation (2.14) after noting that PEmcEc(20 )/, where m0 is the particle rest mass.
To quantify the spectra of particles, we assume that the differential flux of energetic particles varies as a power law in energy [Øieroset et al., 2002; Imada et al., 2007]; that is
j(,,,)EtCWtErr ( ,,,)n (2.16) where W is kinetic energy range with EW . The exponent n is the power law index, which quantifies the slope of differential flux as a function of energy. If we insert the power law into
22 22 equation (2.15) and use non-relativistic approximation PEmcEcmE(200 )/2 , we obtain:
jE20222(,,,) r t
22 2 E0 ( cos111 sin )jt (22 2 , 111 ,r , ) cos11 sin n 22 2 E0 ( cos110111 sin )CW ( , ,r , t )22 2 (2.17) cos11 sin 22 2nn 1 ( cos1101110 sin )CW ( , ,r , t ) E 22 2n 1 (cos1110111 sin)jE (,,, r t) Thus,
jE20222(,,,) r t 22 2n 1 (cos11 sin) (2.18) jE10111(,,,) r t
The differential flux at energy E0 with EW00 is larger than its source flux at E0 by a factor of
2 2 2 n1 ( cos 1 sin 1 ) . An important implication of this is that adiabatic enhancement of the differential flux depends strongly on the power law index of the source flux. For example, if the power law index of suprathermal electrons associated with DFs is in the range -4 to -6 and the
37 compression factor is 1.5 when transported from the outer magnetosphere to the inner magnetosphere, the differential flux can increase by a factor of about 7 to 17. However, if the power law index is, say, between 0 and -1 in low energy range, the differential flux will increase only by a factor of about 1 to 2 under the same compression. Moreover, adiabatic acceleration does not change the power law index because
jE20222(,,,) r t 22 2nn 1 (cos1101110 sin)CW (,,,) r t E (2.19) n CW(,,,)02220 r t E where
22 2n 1 CW(,,,)(cos0222rr t 1 sin) 1 CW (,,,) 0111 t (2.20)
2.3.3. Comparisons with Observations
We have selected two DF events observed by the THEMIS satellites. The first event was on March 11, 2008. The second event was on February 27, 2009 and has been studied extensively
[Runov et al., 2009; Deng et al., 2010; Ge et al., 2011]. Our approach is to use differential flux data at a spacecraft in the tail (usually, THEMIS P1 or P2) to calculate the resultant adiabatic accelerated flux at a spacecraft closer to Earth (usually P4), and then compare the result with the data at P4. The method assumes that the electrons at the distant spacecraft are the same set of electrons that are observed closer to Earth. With that assumption, we can determine if the electron flux observed nearer to the Earth is consistent with adiabatic acceleration of the flux observed at the more distant satellite. Note that electrons also drift mainly in the Y-direction due to magnetic field curvature and gradient. However, the large convective electric field drives electrons earthward from P1 (or P2) to P4 within 1~2 minutes. The width of fast flow channels in
magnetosphere is typically a few RE [e.g. Sergeev et al., 1996; Nakamura et al., 2004], we estimate
38 electron drift in the Y-direction to be about one RE , which is smaller than the flow channel width, hence electrons tend to stay in the same fast flow channel as they are transported earthward.
To more easily estimate the effects of betatron and Fermi acceleration we selected quasi- perpendicular electrons to evaluate betatron acceleration and quasi-parallel electrons for Fermi
acceleration. For betatron acceleration, we selected electrons such that 1 ~ 90 2 ~ 90 so that
2 2 P2 P1 and EE21 . Then the equation relating the differential fluxes of quasi-perpendicular electrons at different locations becomes
jE21(,90,,)rr 2211 t jE (,90,,) 11 t (2.21)
2 2 2 2 For Fermi acceleration, the criteria were 1 ~ 0 2 ~ 0 so that P2 P1 E21 E , and
22 jE21221111( ,0,rr , t ) jEt ( ,0, , ) (2.22)
A. Event #1 March 11, 2008
THEMIS P2 at (XYZ , , )GSM = (-14.7, 5.4, -1.8) RE and P4 at (XYZ , , )GSM = (-10.4, 5.3,
-1.6) RE probed similar structures and they were on the same flow channel in a MHD simulation
[Pan et al., 2014a]. The first three panels in Figure 2.1 are data from P2 showing in the top panel
128 Hz resolution magnetic field observations from the Flux-Gate Magnetometer (FGM) [Auster et al., 2008] in GSM coordinates. The differential energy fluxes observed by P2 are presented in the second panel. The lowest three energy channels of the fluxes plotted were measured by the
Electrostatic Analyzer (ESA) instrument [McFadden et al., 2008], and other channels in the energy range from 26 keV to 113 keV were obtained by the Solid State Telescope (SST) instrument
[Angelopoulos, 2008]. Median values giving the approximate energy of each of the energy channels are summarized in the legend on the right. The power law index of electron differential flux from SST data (26keV to 113keV) was calculated by using a least squares fit to the differential
39 flux as a function of energy on a log-log scale. Red and green lines plotted along with the power law index are plus or minus one standard deviation. They provide upper and lower limits to the power law index, indicating the error of the fit. Note that in the theoretical section we discussed the changes in power law distributed differential fluxes. The differential fluxes are different from the differential energy fluxes in the second panel. The DF of interest observed by P2 was
characterized by a sudden intensification of Bz at 06:22:58 UT after modest bumps and dips. The front was followed by large transient fluctuations. Starting from 06:20:00 UT, concomitant with a bump/dip in the magnetic field, the energy fluxes gradually decreased to their minima just before the front arrived. At the end of this decrease, the energy fluxes were about half of those at 06:20:00
UT. This decrease was followed by a significant but gradual increase (by a factor of 5~6 in about
2 minutes) associated with the fronts. The power law index decreased and reached a minimum of
-3.5 as the front passed by. This change in power law index means that flux changes at different energies were different. After the transient fluctuations in the magnetic field, the power law index returned to -2.5 after 06:26 although the energy fluxes stayed at higher levels.
Figure 2.2 shows data from P4; the format is the same as Figure 2.1. The DF arrived at P4 at 06:23:53. The energy fluxes increased by almost an order of magnitude and did so more abruptly than at P2 as the front passed by. After the DF passed (~06:28) the energy fluxes at P4 returned to the level before the front’s arrival. The power law index was roughly anti-correlated with the energy fluxes. It decreased to -4.5 at the front and increased to -2.5 as the energy fluxes decreased to the pre-front level. The large magnitude of the power law index at the front indicates the differential flux is steeper than that of the preexisting plasma sheet electrons. The flux of suprathermal electrons at P4 is larger than that at P2 by a factor of 6 to 7 in the fronts. The power law index during the fluctuation period at P4 was smaller than that at P2 by about 0.8.
40
P2 20 Bx 10
0 By [nT] B (gsm) -10 Bz -20 15keV(ESA) 107 20keV 26keV 106 31keV(SST) 41keV 105 52keV 65keV [eV/s/cm^2/sr/eV] 104 93keV Differential energy flux -1.5 index -2.0
-2.5 lower limit -3.0
Powerlaw index -3.5 upper limit -4.0 X(Re) -14.8 -14.7 -14.7 Y(Re) 5.4 5.3 5.3 Z(Re) -1.7 -1.8 -1.8 hhmm 0620 0625 0630 2008 Mar 11
Figure 2.1. Energy flux and power law index at THEMIS P2 in the March 11, 2008 event. From top to bottom showed are magnetic field, electron differential energy flux and suprathermal electron power law index. The first three channels of energy flux are measured by the Electrostatic
Analyzer (ESA), and the other channels by the Solid State Telescope (SST).
41
P4 30 Bx 20
10 By [nT] 0 B (gsm) -10 Bz -20 15keV(ESA) 107 20keV 26keV 106 31keV(SST) 41keV 105 52keV 65keV [eV/s/cm^2/sr/eV] 104 93keV Differential energy flux -1.5 -2.0 index -2.5 -3.0 lower limit -3.5
Powerlaw index -4.0 upper limit -4.5 X(Re) -10.4 -10.4 -10.4 Y(Re) 5.4 5.3 5.3 Z(Re) -1.6 -1.6 -1.6 hhmm 0620 0625 0630 2008 Mar 11
Figure 2.2. Energy flux and power law index at THEMIS P4 in the March 11, 2008 event. Same format as in Figure 2.1.
Figure 2.3 (top panel) shows the differential fluxes of electrons. The blue, green lines show the fluxes observed by P2 and P4, respectively. The red line is the predicted flux at P4 assuming betatron acceleration using the flux at P2 as the source. The data points of electron fluxes shown were picked from the positions when the magnetic field reached its peak during the dipolarization.
Those times are 06:22:59 UT at P2 and 06:23:55 UT at P4. Quasi-perpendicular electrons were selected for the betatron acceleration calculation. The compression ratio of the total magnetic field was 1.3 due to either a global change of magnetic field or a local compression inside a growing
42 pileup region [Fu et al., 2011]. Pitch angle ranges of P2, 1 and P4, 2 were selected to be
1270 110 . The calculated flux of electrons in the energy of 3 keV-200 keV at P4 agrees with the data at P4. The flux at P4 significantly increases (by a factor of 6 to 7) in the high energy range although the compression of magnetic field from P2 to P4 is just by a factor of 1.3. Because the magnitude of the power law index of the electron flux is relatively large, a slight shift in the x- axis on the log-log plot of flux as function of energy generates a large flux increase.
The observed flux of electrons below ~3 keV at P4 is half an order smaller than the calculated one. Several possible factors could lead to this discrepancy. First, whistler-mode waves with electric wave field ~5 mV/m were detected by P4 from 06:23:55.6 UT to 06:23:56.4 UT (not shown); those waves might scatter electrons at large pitch angles to small pitch angles within seconds due to pitch-angle scattering near the resonant energy, which was a few keV in this case
[Khotyaintsev et al., 2011]; this is consistent with the data that, in this time period, electrons below
~3 keV are predominantly in the parallel direction. Second, Bx and By were about -10 nT for P2 and P4 at the DF, which indicates that the satellites were not in the center of plasma sheet, leading to the variation of electron density.
Figure 2.3 (bottom panel) shows the differential fluxes of electrons assuming Fermi acceleration. Again the blue, green lines give observations from P2 and P4, The red line is the predicted flux at P4 assuming Fermi acceleration using the flux at P2 as the source. Quasi-parallel
electrons were used (12020). 1.5 from P2 to P4 gave the best fit. For suprathermal electrons, the calculated flux fits the observations very well and the discrepancy at low energy is modest.
43
106
P2 observation P4 observation betatron operating 104
102
100 Differential flux (#/s/cm^2/ev/sr) 10-2
10-4 102 103 104 105 Energy (eV)
106
P2 observation P4 observation Fermi operating 104
102
100 Differential flux (#/s/cm^2/ev/sr) 10-2
10-4 102 103 104 105 Energy (eV)
Figure 2.3. Betatron and Fermi acceleration in the March 11, 2008 event. The top (bottom) panel shows differential flux of quasi-perpendicular (quasi-parallel) electrons. The blue and green lines are observations from P2 and P4, respectively, and the red line is the predicted flux at P4 using the flux at P2 as the source assuming betatron acceleration (top) and Fermi acceleration (bottom).
44
B. Event #2 February 27, 2009
We applied our theory to the well-studied February 27, 2009 event. The figures of this event are in the same format as those for the March 11, 2008 event. THEMIS P1, P2, P3 and P4 observed similar DF signatures and were in the same flow channel in a MHD simulation (see
Figure 12-d in Ge et al. [2011]). As seen in Figures 2.4-2.6, significant flux increments of high energy electrons are associated with DFs and are consistent with predicted fluxes from adiabatic acceleration. The electron fluxes shown in Figure 2.6 were picked from 07:51:29 UT at P1 and
07:54:15 UT at P4. The compression ratio of the total magnetic field from P1 to P4 was 1.6 and contraction factor was 2.0 . There are several differences in this event. First, the energy fluxes of suprathermal electrons increased immediately when the fronts arrived at both spacecraft
(Figures 2.4-2.5). Unlike the event on March 11, 2008, there was no decrease in the energy fluxes before the increase. Second, the calculated differential fluxes from betatron and Fermi acceleration are consistent with the observed ones (Figures 2.6); the discrepancy between the observations and theory for quasi-perpendicular electrons at low energy is much smaller than on March 11, 2008.
This might be because P1 and P4 were both in the central plasma sheet. More importantly, the spectrum of suprathermal electrons associated with the DF captured by P1 (Figure 3.4) was very
similar to those observed closer to the Earth by P2 at -15 RE (not shown) and P4 at 10RE (Figure
3.5). P1 was at 20RE , which was close to the location of the reconnection site in the MHD simulation [Ge et al., 2011], suggesting these suprathermal electrons were produced by magnetic reconnection and convected with the reconnection jets to the location of P1. As the reconnection jets propagated earthward from P1 to P4, the high-energy electron fluxes were further enhanced.
45
In Figure 2.6 the predicted betatron and Fermi acceleration fluxes at high energy respectively increased by factors of 5 and 10 from P1 to P4.
P1 30 Bx 20
10 By [nT] B (gsm) 0 Bz -10 15keV(ESA) 6 10 20keV 105 26keV 4 31keV(SST) 10 41keV 103 52keV 102 65keV [eV/s/cm^2/sr/eV] 1 93keV Differential energy flux 10 -3 index -4
-5 lower limit
-6
Powerlaw index upper limit -7 X(Re) -20.1 -20.1 -20.1 -20.0 Y(Re) -0.6 -0.6 -0.6 -0.6 Z(Re) -1.5 -1.5 -1.5 -1.5 hhmm 0750 0752 0754 0756 2009 Feb 27
Figure 2.4. Energy flux and power law index at THEMIS P1 in the February 27, 2009 event.
Same format as in Figure 2.1.
46
P4 40 Bx 20
0 By [nT] B (gsm) -20 Bz -40 15keV(ESA) 6 10 20keV 105 26keV 4 31keV(SST) 10 41keV 103 52keV 102 65keV [eV/s/cm^2/sr/eV] 1 93keV Differential energy flux 10 -3.5 -4.0 index -4.5 -5.0 lower limit -5.5
Powerlaw index -6.0 upper limit -6.5 X(Re) -11.1 -11.1 -11.1 -11.1 Y(Re) -1.7 -1.8 -1.8 -1.8 Z(Re) -2.4 -2.4 -2.4 -2.4 hhmm 0750 0752 0754 0756 2009 Feb 27
Figure 2.5. Energy flux and power law index at THEMIS P4 in the February 27, 2009 event.
Same format as in Figure 2.1.
47
106
P1 observation P4 observation betatron operating 104
102
100 Differential flux (#/s/cm^2/ev/sr) 10-2
10-4 102 103 104 105 Energy (eV)
106
P1 observation P4 observation Fermi operating 104
102
100 Differential flux (#/s/cm^2/ev/sr) 10-2
10-4 102 103 104 105 Energy (eV)
Figure 2.6. Betatron and Fermi acceleration in the February 27, 2009 event. Same format as in
Figure 2.3.
48
2.3.4. Discussions
Particle transport and nonlocal acceleration associated with DFs is complicated in part because of structures embedded within the fronts. There is coupling between processes on multiple spatial and temporal scales [e.g. Sergeev et al., 2009; Deng et al., 2010]. Analysis is further complicated because we have observations at only a limited number of points. Despite this complexity we believe that THEMIS observed suprathermal electrons that originated near the outer satellites (P1 or P2) and subsequently were detected in the inner tail region near P4. This interpretation seems reasonable because of similarities in the observed DFs and because the virtual satellites were in the same flow channel in the MHD simulations. Our analysis of the data, in combination with theory, reveals that adiabatic enhancement of flux strongly depends on the slope of the source differential flux. In particular, for suprathermal electrons large increases in flux can be induced by modest magnetic field compressions and modest contractions of distance between mirror points.
What cause magnetic compressions during transport? Consider the magnetic field evolution
dcB BVBV()() ( B ) (2.23) dt 4 This equation describes the change in the magnetic field in the frame moving with the MHD flow.
The RHS includes terms of plasma compression, flow gradients along a field line, and diffusion.
Intuitively, compression (V 0 ) increases the magnetic field whereas diffusion decreases it.
By neglecting the diffusion term because it is insignificant in the region of flux pileup [e.g. Pan et al., 2014b], the frozen-in condition is satisfied. We then have VEB and
ddtBBVBV()() . As the plasma propagates toward the inner magnetosphere, it is slowed and compressed, i.e. V 0 , therefore B 0 and 1.
49
Adiabatic acceleration theory predicts invariance of the power law index during adiabatic transport. In the observations, the power law indices were roughly anti-correlated with the energy fluxes. The power law indices at the DFs changed within 0.8 from P2 (or P1) to P4. Thus, the suprathermal electrons associated with dipolarizations characterized by these power law indices can be traced back to the source regions of DFs, which are most likely near the reconnection region.
We suggest that a combination of local processes near the magnetic reconnection site and nonlocal adiabatic acceleration during earthward transport is responsible for the high flux of suprathermal electrons associated with DFs in the inner magnetosphere.
We simplified the effect of Fermi acceleration by setting the contraction factor to be uniform for electrons with different energies and pitch angles, which is determined by contraction of mirror point distance during earthward propagation. However, a more detailed analysis would need to take into account of the non-uniformity of the parameters that describe Fermi acceleration because electron trajectories in non-dipole and dynamic magnetic fields are expected to be complex, which makes it difficult to assign a simple contraction factor to account for the effect of
Fermi acceleration in the variation of the electron flux. Moreover, immediately downstream of reconnection outflow, electrons are expected to behave nonadiabatically because of the small radius of curvature of magnetic field lines and large magnetic gradient [e.g. Lyons, 1984; Büchner and Zelenyi, 1989; Schriver et at., 1998; Hoshino et al., 2001; Imada et al., 2007]. The process in this region cannot be addressed by the analytical model because it is not clear exactly how far P1
(or P2) is from the diffusion region and how to quantify the nonadiabatic effects. In the next chapter, we will refine our calculation by using a large-scale kinetic (LSK) simulation, in which we follow a large number of electron trajectories in the time-dependent electric and magnetic fields derived from a global MHD simulation [Ashour-Abdalla et al., 2005]. That study will enable us to include
50 the effects due to nonadiabatic motion in the outflow region immediately downstream from the reconnection site and demonstrate nonlocal acceleration for an ensemble of particles. A statistical study using THEMIS data should provide clues that reveal the relationship between the structures in DFs and reconnection, and thereby help us understand production of high-energy particles in these processes. Our present analysis suggests, however, that such detailed simulation and statistical studies will not fundamentally change the physics described here that is both local processes near reconnection site and nonlocal adiabatic enhancement in the downstream region contribute to the observed characteristics of suprathermal electrons in the inner magnetosphere.
51
CHAPTER 3
Modeling Electron Energization and Transport in the Magnetotail
during a Substorm
3.1. Introduction
Having gained insights about nonlocal acceleration from the simple theoretical model
presented in Chapter 2, we now apply the more powerful global MHD+LSK calculations to the
March 11, 2008 event. The goals are to determine the major electron energization processes in the
magnetotail and incorporate their contributions into a global scenario under realistic
magnetospheric conditions, and especially to quantify the local energization near the reconnection
sites and nonlocal energization during earthward transport. In section 3.2, we present the solar
wind observations from the Geotail spacecraft, which was upstream from the bow shock and the
THEMIS observations in the magnetotail. The simulation methodology is discussed in section 3.3.
The MHD+LSK results are presented in section 3.4. Electron energization mechanisms and
transport features are discussed in section 3.5. We summarize in section 3.6.
3.2. Observations of the March 11, 2008 Substorm Event
3.2.1. Geotail Observations of the Solar Wind
Figure 3.1 shows the solar wind conditions on March 11, 2008 observed by the Geotail
spacecraft at (X ,YZ , )GSE (15, 2, -1) R E (upstream of the sub-solar point of the bow shock) in the geocentric solar ecliptic (GSE) coordinate system. The GSE system has its X-axis pointing from
52
the Earth towards the Sun and its Y-axis is chosen to be in the ecliptic plane pointing towards dusk.
Its Z-axis is parallel to the ecliptic pole. The pertinent interval is indicated by the two black vertical
dashed lines (05:40-06:51 UT). The earthward component of the solar wind velocity Vx was about
680km/s (fast solar wind) for the interval starting at least from 02:00 UT to 08:00 UT. The IMF
Z-component Bz was northward and fluctuated around 2nT from 02:00 UT to 05:40 UT and
turned southward at about 05:40 UT (first black vertical dashed line) and then stayed around -2nT
until 06:50 UT (second black dashed line), when it turned northward again. A sizable substorm
onset occurred around 05:51 UT where the AE index substantially increased (blue vertical dashed
line in the last panel), and it continued to increase to a peak of about 700nT at 06:40UT.
Geotail Observation of Solar Wind 6 4 2 0 Bx (nT) -2 -4 -6 6 4 2 0 By (nT) -2 -4 -6 6 4 2 0 Bz (nT) -2 -4 -6 200 Vz 0 -200 Vy Vi -400 (km/s) -600 Vx -800 5 4 3 P
(nP) 2 1 0
AL 500
0 AU (nT) Index Geomagnetic -500 AE
hhmm 0200 0400 0600 0800 2008 Mar 11
53
Figure 3.1. Geotail observations of the solar wind on March 11, 2008. From top to bottom, the
first three panels show X-, Y-, and Z-components of the magnetic field in GSE coordinates, the
fourth panel shows solar wind velocity, and the fifth panel shows solar wind dynamic pressure.
The last panel gives the geomagnetic indices from the World Data Center (WDC) for
Geomagnetism.
3.2.2. THEMIS Observations in the Magnetotail
During the substorm interval, THEMIS P2, P3 and P4 were in the nightside magnetosphere.
THEMIS P1 was in the magnetosheath, and P5 was in the radiation belt region. Because we are
interested in the transient dipolarization structures in the magnetotail, we focus on the data
obtained by P2, P3, and P4. There were at least three dipolarization events, as indicated by the
vertical dashed lines in Figure 3.2. They occurred during the time intervals 05:50-06:10 UT, 06:20-
06:40 UT and 06:40-07:20 UT. The first dipolarization occurred during the beginning of the
expansion phase, while the second and third dipolarizations occurred during the expansion phase.
It is interesting to notice that there were strong magnetic field oscillations before the onset of the
expansion phase. In this chapter, we study the second dipolarization event, because the Bx values are small from 06:20 UT to 06:30 UT, indicating that the spacecraft were close to the plasma sheet.
54
-5 10
0 P5 5
P3 5 P2 0 P4 P5 P3
Z_gsm (Re) P4 Y_gsm (Re) P2 10 -5
15 -10 5 0 -5 -10 -15 -20 5 0 -5 -10 -15 -20 X_gsm (Re) X_gsm (Re)
THEMIS Observation of Magnetic Field 40 Bz 20
0 By P2
B_GSM (nT) -20 Bx -40 40 Bz 20
0 By P3
B_GSM (nT) -20 Bx -40 40 Bz 20
0 By P4
B_GSM (nT) -20 Bx -40 hhmm 0500 0600 0700 0800 2008 Mar 11
Figure 3.2. THEMIS locations (top) and the observed magnetic fields (bottom) in the March 11,
2008 event.
55
During the interval from 06:20 UT to 06:30 UT, THEMIS spacecraft P2 was at
(X ,YZ , )GSM (-14.7, 5.4, -1.8) RE , P3 was at (X ,YZ , )GSM (-10.7, 4.5, -1.6) RE , and P4 was
at (X ,YZ , )GSM (-10.7, 4.5, -1.6) RE . Figure 3.3 shows THEMIS P2 observations from 06:20
UT to 06:30 UT. From top to bottom showed are 128 Hz resolution magnetic field observations
from the Fluxgate Magnetometer (FGM) [Auster et al., 2008] in GSM coordinates, ion bulk
velocity calculated from measurements by the Electrostatic Analyzer (ESA) instrument
[McFadden et al., 2008] and Solid State Telescope (SST) instrument [Angelopoulos, 2008], the
differential energy flux of high-energy electrons (>26keV) from SST, the differential energy flux
of low-energy electrons (<26keV) from ESA, the electric wave fields from the Electric Field
Instrument (EFI) [Le Contel et al., 2008], and the magnetic wave fields from the Search Coil
Magnetometer (SCM) [Bonnell et al., 2008]. The black and red dashed lines in the last two panels
are the electron gyro frequency fce and the lower hybrid frequency fcef ci , respectively. The DF
of interest observed by P2 was characterized by the sudden intensification of Bz at 06:22:58 UT
(vertical dashed line), following a series of modest bumps and dips. The front was followed by a region of large transient fluctuations. The earthward flow was large and increasing. The ion bulk flow velocity was 300 km/s and gradually increased to 600 km/s. Starting from 06:22 UT, concomitant with a bump and dip in the magnetic field, the high-energy electron flux gradually decreased to its minimum just before the front arrived. This decrease was followed by a gradual
increase as the front passed by P2. The flux reached its maximum value at 06:26 UT when the
dipolarized magnetic field became relatively steady. The thermal electron energy flux behaved similarly. Electromagnetic waves between the lower hybrid frequency and the electron gyro frequency became active as the front passed by.
56
THEMIS P2 Observation 20 bz 10 0 by FGM B (nT) -10 bx -20 600 vz 400 200 vy
Vi (km/s) 0
ESA+SST vx
-200 ) -1 8 10 eV 107 -1 5 6 sr 10 10 -2 5 (eV) SST
10 cm DEflux 104 -1 103
10000 108 (eV s 107 6 1000 10
(ev) 5 ESA 10 DEflux 104 100 103 1.00 1000
100 0.10 EFI E (Hz) 10 (mV/m) 1 0.01 0.100 1000
100 0.010 (nT) SCM B (Hz) 10
1 Sun Apr 13 16:00:51 2014 0.001 X_gsm(Re) -14.8 -14.7 -14.7 Y_gsm(Re) 5.4 5.3 5.3 Z_gsm(Re) -1.7 -1.8 -1.8 hhmm 0620 0625 0630 2008 Mar 11
Figure 3.3. THEMIS P2 observations in the March 11, 2008 event. Shown from top to bottom are the magnetic field observations in GSM coordinates, the ion bulk velocity, the differential energy flux of high-energy electrons (>26keV), the differential energy flux of low-energy electrons
(<26keV), the electric wave fields, and the magnetic wave fields. The black and red dashed lines in the last two panels are respectively the lower hybrid frequency and the electron gyro frequency.
57
THEMIS P4 observations are shown in Figure 3.4 in the same format as Figure 3.3.
Because P3 and P4 were close to each other and their data are similar, we show only P4 data. As
the front arrived at 06:23:53 UT (vertical dashed line), P4 detected the transient DF structure and
subsequent large fluctuations, which are similar to the P2 observations. Compared with the P2 observations, the earthward flow slowed down to 200 km/s. Unlike the flux observed by P2, the
energy flux measured by P4 remained steady before the front arrived, and then suddenly shifted to
lower-energy for about 20 seconds and then shifted back to higher energy. At this time, the high-
energy electron flux was well above the level before the front arrival. The wave activity began as the front arrived and persisted during the subsequent period of magnetic field fluctuations.
Due to the ambiguity between spatial and temporal changes in the measurements, there are two possibilities for the increase in the high-energy electron fluxes: (1) hot and tenuous plasmas
are transported into the regions surrounding the satellites, while the local cold and dense plasmas
are excluded by electromagnetic fields; (2) the cold and dense plasmas are locally energized by
electromagnetic fields into hot and tenuous ones. The validity of either interpretation depends on
particle inertia. Ions move with the magnetic field if their gyro radius i is much smaller than the
spatial scale of the magnetic field gradient B / B ; otherwise they are loosely connected with
magnetic field lines. The scale of the DF structure is on the order of the ion inertial length and
comparable with the ion gyro radius [e.g. Runov et al., 2009; Sergeev et al., 2009]. Hence, it is
likely that ions near the front are not tightly bound to the magnetic field lines. In contrast, electrons
have a smaller inertia than ions, and the scale of the magnetic field gradient is much larger than
their gyro radius e , so they tend to follow the magnetic field lines they initially gyrate about. An
equivalent argument can be made in terms of plasma fluids: the validity of either interpretation
depends on whether the fluids (electron or ion) are frozen-in with the magnetic field. Near the DF,
58
ions with larger gyro radii are demagnetized, while electrons are convected with the magnetic field.
This discrepancy between ion and electron fluid motions results in an appreciable Hall current
given by the generalized Ohm’s law [Biskamp, 2005]. The Hall current near DFs has been
confirmed by observations [Zhou et al., 2009; Zhang et al., 2011; Fu et al., 2012b]. Thus we expect that the hot and tenuous electrons are transported to the satellite locations while interaction of preexisting ions with DFs is important.
THEMIS P4 Observation 30 bz 20 10 by
FGM 0 B (nT) -10 bx -20 600 vz 400 200 vy 0 Vi (km/s)
ESA+SST -200 vx -400 ) -1 8 10 eV 107 -1 5 6 sr 10 10 -2 5 (eV) SST
10 cm DEflux 104 -1 103
10000 108 (eV s 107 6 1000 10
(ev) 5 ESA 10 DEflux 104 100 103 1.00 1000
100 0.10 EFI E (Hz) 10 (mV/m) 1 0.01 0.100 1000
100 0.010 (nT) SCM B (Hz) 10
1 Sun Apr 13 16:19:10 2014 0.001 X_gsm(Re) -10.4 -10.4 -10.4 Y_gsm(Re) 5.4 5.3 5.3 Z_gsm(Re) -1.6 -1.6 -1.6 hhmm 0620 0625 0630 2008 Mar 11
59
Figure 3.4. THEMIS P4 observations in the March 11, 2008 event. The same format as in Figure
3.3 is used.
3.3. Simulation Methodology
In order to develop a quantitative and global scenario of particle energization and transport
in the magnetotail during substorms, we employ simulations to complement the satellite data. The
vastness of the space, the longtime duration of events, and the multi-scale nature of the
magnetospheric plasmas, prevent making a global kinetic simulation. In order to obtain realistic
global electromagnetic fields while capturing kinetic features related to individual particles, we
use a scheme that combines a global MHD model and a LSK simulation. The MHD model [Raeder
et al., 1998, 2001; El-Alaoui et al., 2001, 2009] driven by the measured solar wind conditions
provides a global realistic description of the magnetospheric electromagnetic fields. In LSK
simulation, large numbers of particle trajectories are followed in the MHD electromagnetic fields
to obtain a global picture of the particle behavior [Ashour-Abdalla et al., 1993]. The particle orbits
are limited by the MHD field temporal resolution. High frequency waves are omitted from the
MHD approximations, so their effects on the particle trajectories are not included.
It might be argued that LSK as a test particle method is not self-consistent because: (1)
calculations of particle trajectories do not provide feedback to the calculations of moments and
fields, and (2) interactions between test particles are not included. However, a feedback process in
the scheme of MHD+LSK simulations would be redundant because the electromagnetic fields
assume a MHD approximation, which provides a self-consistent description with certain
limitations. Incorporating interactions between test particles is also unnecessary because these
interactions have been included via the force of the mean electromagnetic fields. To better
60
understand this, let us consider the testing of a large number of particles in the electromagnetic
fields derived from a fully kinetic simulation. First, it is unnecessary to provide feedback from the
test particles to the calculation of the fields because the fields are self-consistent. Second, the test
particle orbits are exact because the interactions between particles have been incorporated through
the field calculations.
The remaining issue concerning the MHD+LSK simulations is the problem of
normalization. For the test particle method, this involves converting the individual particle
information recorded by virtual detectors such as recording time, position, particle energy, and
pitch angles into collective physical quantities such as the differential flux and distribution function,
which can be directly compared with observations. This has been discussed previously [Ashour-
Abdalla et al., 1993; Kress et al., 2007; Richard et al., 2009]. The directional differential flux
JE(,,, rD , tD ) can be calculated as
ND ()nddtvA 1 JE(,,,r , t ) C MHD LL (3.1) DD i1 NLDDD dAn v, dt dEd
where E is particle energy, is pitch angle, is azimuthal angle, rD is detector location, tD is
detection time, N L ( ND ) is the number of launched (detected) test particles, ()nv MHD is the
product of MHD number density and velocity, dA L ( dA D ) is particle launch (detector) area, nv,D
is particle velocity direction at detection, dtL ( dtD ) is launch (detection) time interval, and
dddsin is the solid angle differential. C is a constant on the order of unity. Similarly,
the directional differential energy flux JEeDD ( , , ,r , t ) is
ND ()nddtvA E JE(,,,r , t ) C MHD LL (3.2) eDD i1 NLDDD dAn v, dt dEd
61
Because in the magnetotail, the weak magnetic field may limit the accuracy of the pitch angle
information, we average the flux over the solid angle. The averaged differential flux JE ( ,rD , tD )
and the averaged differential energy flux JEeDD ( ,r , t ) are
C ND ()nddtvA1 JE(,r , t ) MHD LL (3.3) DD 4 i1 NLDDD dAn v, dt dE
CEND ()nddtvA JE(,r , t ) MHD LL (3.4) eDD 4 i1 NLDDD dAn v, dt dE
3.4. MHD and Electron LSK Simulations of the March 11, 2008 Substorm Event
3.4.1. MHD Simulation Results
We modeled the electromagnetic fields using a MHD simulation of the magnetosphere
driven by solar wind conditions from the Geotail spacecraft shown in Figure 3.1. The MHD
simulation results provide the global three-dimensional time-varying electric and magnetic fields
for the Earth's magnetosphere. The GSE coordinate system was used in the MHD simulation. The
MHD simulation data (and LSK simulation data shown later) were processed and will be presented
in a GSE-maximum pressure surface system. In this system, the Z-axis value of maximum pressure
surface is used to replace the virtual spacecraft Z-axis location in the simulation. For instance,
THEMIS P2 location at 06:20 UT is (X ,YZ , )GSE (-14.8, 4.0, -3.0) R E , the virtual detector of P2
in simulation is set as (,,)X YZGSE (-14.8, 4.0, Z mp ) R E , where Zmp is the corresponding
maximum pressure surface location. This system is used because (1) electromagnetic fields and
plasma moments vary significantly within a few RE in the magnetotail along Z-direction of GSE
coordinate and MHD simulation might not accurately reproduce the variation along Z-axis [Raeder
et al., 2001], and (2) the maximum pressure surface is a good approximation to the center of the
62
plasma sheet on the night side [Ashour-Abdalla et al., 2002]. The system is applied when
spacecraft measurements are performed close to the center plasma sheet.
Figure 3.5 (left column) shows nine snapshots from the MHD simulation results at
06:21:40 UT, 06:22:00 UT, 06:22:20 UT, 06:22:40 UT, 06:23:00 UT, 06:32:00 UT, 06:23:40 UT,
06:24:00 UT, and 06:24:20 UT. The color-coded variable is the north-south component of the
magnetic field Bz on the maximum pressure surface. DFs moving earthward can be seen as
increases in the Bz component (red region). The DF of interest was formed at 06:21:40 UT at
X GSE~-14R E and on the dawn side of P2. Bz was ~5 nT, which is relatively weak. The DF
propagated along a flow channel. The earthward flow speed was 400-600 km/s. The flow channel
was limited to a small region in the Y-direction, about 2-3 RE wide. The earthward flow originated
from a region of (,)X YRGSE ~(-21,2) E , where a flow reversal occurred. This flow reversal region is where the near-Earth reconnection occurred in the global MHD simulation. Considering the locations of the DF formation and the flow reversal, it appears that the DF strength did not grow to an appreciable level near the reconnection region, although the flow is generated by magnetic
reconnection. The front was intensified during its earthward propagation. Bz was ~10 nT at
06:23:00 UT, and it was ~14 nT at 06:23:40 when the front encountered P3 and P4 at
X GSE~ -10.4R E . The front merged with the preexisting strong magnetic field at P3 and P4
locations. The MHD results indicate that THEMIS P2 was on the dusk edge of the DF when the
front passed it. We point out that in general dipolarizations in our global MHD simulations are
formed and intensified in the region far away from the reconnection X-line, as a consequence of
flow braking and plasma compression.
63
Bz and flow 062140UT DEflux 41-95keV 062120-062140UT 1.0 20 107
-4 -4 )
6 -1 0.8 10 -2 15 -2 eV -1 5 0 0 10 sr 0.6 10 -2
2 2 4 cm P2 P2 10 -1 0.4 P3 P3 Y(RE) Y(RE) 4 5 Bz(nT) 4 103 60.2 P4 6 P4 0 2 80.0 8 10 DEflux(eV s 10 1.0 1.2 1.4-5 10 1.6 1.8 102.01 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 300 km/s X(RE) X(RE)
Bz and flow 062200UT DEflux 41-95keV 062140-062200UT 1.0 20 107
-4 -4 )
6 -1 0.8 10 -2 15 -2 eV -1 5 0 0 10 sr 0.6 10 -2
2 2 4 cm P2 P2 10 -1 0.4 P3 P3 Y(RE) Y(RE) 4 5 Bz(nT) 4 103 60.2 P4 6 P4 0 2 80.0 8 10 DEflux(eV s 10 1.0 1.2 1.4-5 10 1.6 1.8 102.01 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 300 km/s X(RE) X(RE)
Bz and flow 062220UT DEflux 41-95keV 062200-062220UT 1.0 20 107
-4 -4 )
6 -1 0.8 10 -2 15 -2 eV -1 5 0 0 10 sr 0.6 10 -2
2 2 4 cm P2 P2 10 -1 0.4 P3 P3 Y(RE) Y(RE) 4 5 Bz(nT) 4 103 60.2 P4 6 P4 0 2 80.0 8 10 DEflux(eV s 10 1.0 1.2 1.4-5 10 1.6 1.8 102.01 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 300 km/s X(RE) X(RE)
64
Bz and flow 062240UT DEflux 41-95keV 062220-062240UT 1.0 20 107
-4 -4 )
6 -1 0.8 10 -2 15 -2 eV -1 5 0 0 10 sr 0.6 10 -2
2 2 4 cm P2 P2 10 -1 0.4 P3 P3 Y(RE) Y(RE) 4 5 Bz(nT) 4 103 60.2 P4 6 P4 0 2 80.0 8 10 DEflux(eV s 10 1.0 1.2 1.4-5 10 1.6 1.8 102.01 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 300 km/s X(RE) X(RE)
Bz and flow 062300UT DEflux 41-95keV 062240-062300UT 1.0 20 107
-4 -4 )
6 -1 0.8 10 -2 15 -2 eV -1 5 0 0 10 sr 0.6 10 -2
2 2 4 cm P2 P2 10 -1 0.4 P3 P3 Y(RE) Y(RE) 4 5 Bz(nT) 4 103 60.2 P4 6 P4 0 2 80.0 8 10 DEflux(eV s 10 1.0 1.2 1.4-5 10 1.6 1.8 102.01 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 300 km/s X(RE) X(RE)
Bz and flow 062320UT DEflux 41-95keV 062300-062320UT 1.0 20 107
-4 -4 )
6 -1 0.8 10 -2 15 -2 eV -1 5 0 0 10 sr 0.6 10 -2
2 2 4 cm P2 P2 10 -1 0.4 P3 P3 Y(RE) Y(RE) 4 5 Bz(nT) 4 103 60.2 P4 6 P4 0 2 80.0 8 10 DEflux(eV s 10 1.0 1.2 1.4-5 10 1.6 1.8 102.01 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 300 km/s X(RE) X(RE)
65
Bz and flow 062340UT DEflux 41-95keV 062320-062340UT 1.0 20 107
-4 -4 )
6 -1 0.8 10 -2 15 -2 eV -1 5 0 0 10 sr 0.6 10 -2
2 2 4 cm P2 P2 10 -1 0.4 P3 P3 Y(RE) Y(RE) 4 5 Bz(nT) 4 103 60.2 P4 6 P4 0 2 80.0 8 10 DEflux(eV s 10 1.0 1.2 1.4-5 10 1.6 1.8 102.01 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 300 km/s X(RE) X(RE)
Bz and flow 062400UT DEflux 41-95keV 062340-062400UT 1.0 20 107
-4 -4 )
6 -1 0.8 10 -2 15 -2 eV -1 5 0 0 10 sr 0.6 10 -2
2 2 4 cm P2 P2 10 -1 0.4 P3 P3 Y(RE) Y(RE) 4 5 Bz(nT) 4 103 60.2 P4 6 P4 0 2 80.0 8 10 DEflux(eV s 10 1.0 1.2 1.4-5 10 1.6 1.8 102.01 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 300 km/s X(RE) X(RE)
Bz and flow 062420UT DEflux 41-95keV 062400-062420UT 1.0 20 107
-4 -4 )
6 -1 0.8 10 -2 15 -2 eV -1 5 0 0 10 sr 0.6 10 -2
2 2 4 cm P2 P2 10 -1 0.4 P3 P3 Y(RE) Y(RE) 4 5 Bz(nT) 4 103 60.2 P4 6 P4 0 2 80.0 8 10 DEflux(eV s 10 1.0 1.2 1.4-5 10 1.6 1.8 102.01 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 300 km/s X(RE) X(RE)
Figure 3.5. Snapshots of the MHD and LSK simulations for the March 11, 2008 event. Left column: magnetic field Z-component and flow vectors (black arrows) on the maximum pressure surface from the MHD simulation. The locations of THEMIS P2, P3, and P4 are also shown. Right
66
column: differential energy flux of electrons in the range of 41keV to 95keV from the LSK
simulation and flow vectors from the MHD simulation on the maximum pressure surface for the
same time intervals as the left column.
3.4.2. LSK Simulation Results and Comparisons with Observations
We present the LSK simulation results in two subsections. In subsection 3.4.2.1, we
compare the differential energy fluxes observed by THEMIS and simulated by LSK and discuss
the implications concerning energization near the reconnection site. In subsection 3.4.2.2, we
present the DF from the MHD simulation and electron energization from the LSK simulation side-
by-side and discuss the electron energization during earthward transport.
3.4.2.1. Electron Energization due to Reconnection
In principle, to quantify the acceleration due to processes near the reconnection site, we
can trace electron trajectories backwards in time starting from THEMIS P2 location, which is
closer to the reconnection site in the simulation, and thereby examine the electron flux near the
reconnection site. Conversely, if the electron flux observed by THEMIS P2 is reproduced by LSK
via forward pushing of the prescribed electron source launched near the reconnection site, then the
prescribed source probably represents a reasonable flux due to reconnection. We have adopted the
forward pushing method and performed a LSK simulation with two types of source distributions.
The first type of source distribution includes only thermal electrons obeying a Maxwellian
distribution whose one dimensional thermal energy is 1 keV (hereafter, it is referred as 1 keV
Maxwellian distribution). The second type of source distribution is a combination of the 1 keV
Maxwellian distribution and a power law distribution at high energies. The power law distribution
67
has a lower-energy boundary Emin 9 keV and a power law index n 4.5 (Figure 3.6 and also see
Appendix 1 for details). The combined distribution is close to the electron distribution measured
by THEMIS P2. Previous observations near the reconnection region suggested high-energy
electrons follow a power law distribution [Øieroset et al., 2002; Imada et al., 2007]. Theoretical
study suggested that power law distributions can be generated by localized electric fields in
collisionless plasmas [Morales and Lee, 1974]. The combination of a Maxwellian distribution of
thermal electrons and a power law distribution of high-energy tail is a special approximation of
the generalized Lorentzian (Kappa) distributions [Vasyliunas, 1968; Summers and Thorne, 1991].
This combination was adopted because it helps handle high-energy electrons.
102
Maxwell distribution
100 Powerlaw distribution
10-2 f(E) 10-4
10-6
10-8 0.01 0.10 1.00 10.00 100.00 1000.00 E(keV)
Figure 3.6. Two types of electron source distributions for the LSK simulation. The first type
includes only thermal electrons of 1 keV Maxwellian distribution (black line); the second type
includes both thermal electrons and high-energy electrons that obey a power law distribution (blue
line).
68
In the LSK simulation, 37500 electrons following Maxwellian distribution and 4000 electrons obeying power law distribution were launched for every 20 seconds from 06:13 UT to
06:29 UT. The electrons were initially launched uniformly from a planar region given by
-18REGSEEXR -16 , 15REGSEEYR, ZGSE 2.5 . This region is on the MHD maximum pressure surface. By inspection of MHD simulation, this region is also the origin of the earthward flow near the reconnection site. The electron trajectories were followed by using a combination of full particle and guiding-center calculations [Schriver et al., 2011]. We used the adiabaticity parameter that relates to electron trajectory (not the in Kappa distribution) to determine whether full particle dynamics was necessary. The parameter is defined as the square root of
the local magnetic field radius of curvature divided by the local gyro radius [Büchner and Zelenyi,
1989]. If was decreased to below 10, we switched from the guiding-center approximation to full
particle dynamics; if was increased to above 15, we performed the opposite switch. The data
about the electrons were collected with virtual detectors that were placed throughout the simulation
domain. We calculated fluxes and energy fluxes by using these detector records and equations (3.3)
and (3.4) above. Notice that flux contribution by different sources can be conveniently added up.
Figure 3.7 shows the differential energy flux for THEMIS P2. The second panel contains
P2 observed energy flux. The energy ranges of 5 channels are covered by both the ESA and SST
instruments. The third panel gives the energy flux from the LSK simulation with only thermal electrons as the source, and the fourth panel shows the results using both the thermal electron source and the high-energy electron source. It is clear that the electron flux calculated by using only a Maxwellian source disagrees with the THEMIS P2 observations at high energies. In contrast, the energy flux calculated by using the source including both thermal Maxwellian and high-energy power law electrons is consistent with the observations. This consistency is manifested in three
69
aspects: the trend in time, the flux levels, and that fluxes of different energies move together. It is
not surprising that including high-energy electrons significantly improves comparison between the
simulation and observations if we consider: (1) P2 is not far from the source region where the electrons were launched; (2) the electron flux observed by P2 has a high-energy tail; and (3) no magnetic structures such as DFs formed between the electron source region and the P2 location, therefore substantial acceleration is unlikely to occur between the source location and P2. These comparisons show that the simulation including high-energy electrons with a power law distribution is the correct one. More importantly, it suggests that electrons were accelerated to up to 95 keV near the reconnection site. The power law distribution reasonably quantifies the production of high-energy electrons by processes near the reconnection site.
Figure 3.8 shows the energy flux for THEMIS P4 in the same format as in Figure 3.7. Two characteristics are noticeable. First, unlike the observed and simulated fluxes at THEMIS P2, the energy flux at P4 is different for different energy channels. Below 12 keV, the flux fluctuates and generally does not increase, but for channels above 12 keV, the flux significantly increases at
06:24:00 UT in observations and at 06:23:00 UT in the simulation. Second, adding a power law distribution of high-energy electrons to the source in the LSK simulation improves the comparison between the observed and simulated fluxes, especially for the 25-41 keV channel. However, this improvement is less significant than that seen in the P2 data. This is expected because the flux of source high-energy electrons is much smaller than that observed at P4 when the front passed by.
For example, the 41-95 keV flux is on the level of 1056 ~10 eV/(cm 2 s sr eV) at P2 while on the
level of 1067 ~10 eV/(cm 2 s sr eV) at P4. The majority of the high-energy electrons at P4 are not
from convection of the source. Instead, they are from acceleration of lower-energy electrons by
the dipolarization during earthward transport. This will be discussed in the next subsection.
70
Magnetic field and DEflux (P2) 20 bz 10 0 by (nT)
B(GSM) -10 bx -20 ) 9 -1 10 2-6keV 8 eV 10
-1 6-12keV 7 sr 10 -2 12-25keV 6
cm 10 25-41keV -1
Observed 5 10 41-95keV 104 (eV s 9 10 108 107 106 (MXL ele) Simulated 105 104 9 10 108 107 106
Simulated 105 4 (MXL-PWL ele) 10 X_gsm(Re) -14.8 -14.7 -14.7 Y_gsm(Re) 5.4 5.3 5.3 Z_gsm(Re) -1.7 -1.8 -1.8 hhmm 0620 0625 0630 2008 Mar 11
Figure 3.7. Comparison of the energy flux for THEMIS P2 in the March 11, 2008 event. From
top to bottom, the first panel shows the measured magnetic field. The second panel shows THEMIS
P2 observations. Energy ranges of 5 channels are covered by both the ESA and SST instruments.
The third panel gives the simulated differential energy flux using only Maxwellian thermal
electrons as the source, and the fourth panel gives the result including both Maxwellian thermal
electrons and high-energy electrons. Because in the MHD simulation, the flow associated with DF
arrives at P2 about 30 seconds earlier than observation, the simulated flux is also about 30 seconds
ahead of the corresponding observed flux, as indicated by the vertical lines.
71
Magnetic field and DEflux (P4) 30 bz 20 10 by
(nT) 0 B(GSM) -10 bx -20 ) 9 -1 10 2-6keV 8 eV 10
-1 6-12keV 7 sr 10 -2 12-25keV 6
cm 10 25-41keV -1
Observed 5 10 41-95keV 104 (eV s 9 10 108 107 106 (MXL ele) Simulated 105 104 9 10 108 107 106
Simulated 105 4 (MXL-PWL ele) 10 X_gsm(Re) -10.4 -10.4 -10.4 Y_gsm(Re) 5.4 5.3 5.3 Z_gsm(Re) -1.6 -1.6 -1.6 hhmm 0620 0625 0630 2008 Mar 11
Figure 3.8. Comparison of the energy flux for THEMIS P4 in the March 11, 2008 event. The same format is used as in Figure 3.7. The simulated flux is about one minute ahead of the corresponding observed flux, as indicated by the vertical lines.
Before moving on to the next subsection, we show the comparison of the differential fluxes as functions of energy between the observations and the simulation. The differential fluxes are calculated by using equation (3.3). Figure 3.9 shows a comparison of the differential fluxes from
P2 observations and the LSK simulation at 06:24:00 UT. Because the traces of the energy flux from the THEMIS P2 data and the LSK simulation are similar in Figure 3.7, we selected the same times in observation and simulation. The differential flux as a function of energy from the LSK
72
simulation matches well with the THEMIS data. Figure 3.10 shows a comparison of the differential
flux for P4 at 06:25:00 UT. As shown in Figure 3.8, the flux change in LSK simulation occurs
about one minute before the change in the THEMIS P4 observations, so the simulated flux from
about one minute ahead was selected for comparison. Figure 3.10 shows that the simulated flux is consistent with the corresponding THEMIS P4 data. These remarkable agreements validate the
MHD+LSK simulations and demonstrate the ability to use LSK simulations to reproduce the differential fluxes in a wide range of energies.
105
P2 Obs(062400UT) P2 Sim(062340-062400UT)
) 4
-1 10 P2 Sim(062400-062420UT) eV -1 sr -1
s 3
-2 10
102
101 Differential flux (cm
100 102 103 104 105 Energy (eV)
Figure 3.9. Comparison of the differential flux as a function of energy for THEMIS P2. The black line shows the measured flux by P2 at 06:24:00 UT, the blue line shows the simulated flux at
06:23:40-06:24:00 UT, and the red line shows the simulated flux at 06:24:00-06:24:20 UT.
73
105
P4 Obs(062500UT) P4 Sim(062320-062340UT)
) 4
-1 10 P4 Sim(062340-062400UT) eV -1 sr -1
s 3
-2 10
102
101 Differential flux (cm
100 102 103 104 105 Energy (eV)
Figure 3.10. Comparison of the differential flux as a function of energy for THEMIS P4. The
black line shows the measured flux by P4 at 06:25:00 UT, the blue line shows the simulated flux
at 06:23:20-06:23:40 UT, and the red line shows the simulated flux at 0623:40-06:24:00 UT.
3.4.2.2. Electron Energization during Transport
In Figure 3.5 (right column), the differential energy flux data for the energy range 41 keV
to 95 keV are plotted on the maximum pressure surface at 20 second intervals from 06:21:40 UT
to 06:24:20 UT. Each plot corresponds to the snapshot of the MHD simulation shown next to it in
the left column. Two key points can be drawn from Figure 3.5.
First, it is evident that the high-energy electron flux at P3 and P4 increased by almost an
order of magnitude compared with the flux near P2 as the front arrived. This increase occurred far
away from the reconnection sites. The high-energy electrons at P3 and P4 were from acceleration
during earthward transport, rather than convection of the high-energy source.
74
Second, the high-energy electrons at P3 and P4 were generated where and when the
dipolarization formed and intensified, as indicated by Bz from the MHD simulation shown in the
left column for Figure 3.5. Before the dipolarization, the flux close to the source region was
510eV/(cmssreV)52 for the 41-95 keV energy range. As the dipolarization formed at about
06:21:40 UT at X GSE~-14R E and continued to intensify till 06:22:40 UT at X GSE~-12R E , electrons were energized in about one minute. The flux of 41-95 keV electrons increased to
310eV/(cmssreV)62 during that interval. After that, the flux of high-energy electrons
propagated to P3 and P4, and were accumulated in the region near X GSE~-11R E as the MHD flow
was braked and diverted. At X GSE~-11R E , the electron flux flowed toward the dawn side, resulting
from magnetic field gradient and curvature drifts.
3.5. Discussions
To identify the energization mechanism during earthward transport, we followed the flow
and inferred the trajectory of the DF on the maximum pressure surface. Then we calculated the
differential flux along this trajectory. The results are summarized in Figure 3.11. The blue line is
the simulated differential flux at 06:21:40-06:22:00 UT at the position of
(X ,YX ) (P4 5 RYR E , P4 2.5 E ) . (,)X PP44Y is virtual P4 location in the simulations. This
represents the electron source before the formation of the front. The green line is the simulated
flux at a relative position of (X ,YX ) (P4 2 RYR E , P4 0.5 E ) at 06:22:20-06:22:40 UT, which is
where and when the DF in the MHD simulation formed and intensified. The red line is the
simulated flux at the position of P4 at 06:23:20-06:23:40 UT, when the front propagated to P4.
The black line is the flux measured by P4 at 06:25:00 UT. It is clear that the acceleration of the
75
test particles (from blue line to green line) occurred where the DF formed and intensified.
Furthermore, two key points are noticeable. First, the energization is nearly uniform for all
electrons with energy above 1 keV. Second, the magnetic field increased from ~5nT to ~12nT, but the flux of high-energy electrons increased by almost an order of magnitude. The acceleration is
rather effective in increasing the high-energy flux. These two points support that the acceleration
is adiabatic, as showed in the previous chapter.
105 P4 observation source distribution energized distribution
) 4
-1 10 P4 simulation eV -1 sr -1
s 3
-2 10
102
101 Differential flux (cm
100 102 103 104 105 Energy (eV)
Figure 3.11. Electron acceleration mechanism during transport in the March 11, 2008 event. The
blue line is the simulated flux at position of (X ,YX ) (P4 5 RYR E , P4 2.5 E ) at 06:21:40-06:22:00
UT, the green line is simulated flux at a relative position of (X ,YX ) (P4 2 RYR E , P4 0.5 E ) at
06:22:20-06:22:40 UT, the red line is the simulated flux at P4 at 06:23:40-06:24:00 UT, and the
black line is the flux observed by P4 at 06:25:00 UT.
76
Now let’s discuss electron transport. Figure 3.12 shows the Z-component of the magnetic
field, low-energy (2-6 keV) electron flux and high-energy (41-95 keV) electron flux in a broader region. Electrons closely followed the flow channels, both in earthward and tailward directions.
Low- and high-energy electrons followed a similar transport pattern determined by the flows. This similarity across energies demonstrates that the energy-independent E×B drift played a central
role in electron transport in the magnetotail. Close to the Earth, low-energy electrons were
accumulated and penetrated the region of strong magnetic field while high-energy electrons circled
around it. The high-energy electrons dawnward drifted more significant than low-energy electrons.
This is because close the Earth, energy-dependent gradient and curvature drifts are the main
component of electron guiding-center motion. We point out that the transport features are
statistical. Specifically, many test electrons were launched uniformly in a region covering the flow
channel, hence the transport was averaged across the flow channel. In addition, the flux on
maximum pressure surface is bounce averaged. That the flux closely follow the flows in the
magnetotail does not to require that E×B drift is much larger than curvature/gradient drift all the
time for test electrons.
77
1.0 Bz and flow 062340UT -10 20
-5 15
0 10 P3 P2 Y(RE) 5 5 Bz(nT) 0.8 P4
10 0
15 -5 -5 -10 -15 -20 -25 -30
DEflux 2-6keV 062320-062340UT -10 0.6 108 )
-5 -1 eV -1
106 sr 0 -2 cm
P3 P2 -1 Y(RE) 5 P4 104
10 0.4 DEflux(eV s 102 15 -5 -10 -15 -20 -25 -30
DEflux 41-95keV 062320-062340UT -10 107
6
10 )
-5 -1 eV
0.2 -1 105 sr
0 -2
4 cm
P3 P2 10 -1 Y(RE) 5 P4 103
10 102 DEflux(eV s
150.0 101 -51.0 -101.2 -15 1.4 -20 1.6-25 1.8-30 2.0 300 km/s X(RE)
Figure 3.12. Electron transport in the March 11, 2008 event. From top to bottom, the first panel is
the Z-component of the magnetic field and flow vectors (black arrows) on the maximum pressure
surface from the MHD simulation at 06:23:40UT. The locations of THEMIS P2, P3, and P4 are
78
also shown. The second panel is differential energy flux of electrons in the range of 2 keV to 6
keV accumulated during 06:23:20-06:23:40 UT from the LSK simulation and flow vectors from
the MHD simulation on the maximum pressure surface. The third panel is for 41 keV to 95 keV electrons, same format is used as in the second panel.
Finally, let's look at the simulation results in the context of the THEMIS wave observations.
First, electromagnetic whistler mode waves with electric wave fields ~5 mV/m were detected by
P4 from 06:23:55.6 UT to 06:23:56.4 UT (not shown here), and their frequencies are about
0.7 ~ 0.9 fce . Previous studies showed whistler waves associated with DFs can scatter electrons
near the resonant energy [e.g. Khotyaintsev et al., 2011]. However, the resonant energy is a few
keV for this event, which is well below the energy range we are considering (tens of keV). Second,
electrostatic waves with electric fields ~2 mv/m and frequencies ~ 1.2 fce were also detected from
06:23:23 UT to 06:23:24 UT and from 06:23:25 UT to 06:23:26 UT by P3. Similar waves detected at DFs by THEMIS have been reported [Zhou et al., 2009]. Considering the short presence and small amplitude, they are unlikely to energize electrons significantly. Third, as can be seen in the last two panels in Figure 3.3 and Figure 3.4, there are broad band waves with frequencies close to
the lower hybrid frequency fcef ci associated with the DFs. According to the resonance
condition, they are expected to affect ion orbits. These waves are not incorporated in the MHD
simulation, hence not in the LSK simulation. The good agreements between the MHD+LSK
simulations and observations suggest that including the effect of these waves is unlikely to change
the electron energization and transport picture in this event.
3.6. Conclusions
79
Using analyses of coordinated multi-point observations, a realistic global MHD simulation,
and an electron LSK simulation, we have developed a global and quantitative picture of electron
energization and transport in the magnetotail during a substorm event.
The global scenario of electron energization is as follows: the electrons are initially
accelerated near the reconnection site, and subsequently, they are further accelerated adiabatically
by convection electric fields far away the reconnection site. For the March 11, 2008 event, the
electron energy flux for the energy range 41-95keV is 510eV/(cmssreV)52 due to
acceleration near the reconnection site; it is further increased to 310eV/(cmssreV)62 due to
adiabatic acceleration of lower-energy electrons. The adiabatic enhancement occurs when and
where the DF forms and intensifies. This two-step acceleration process has been previously
suggested by observational and analytical studies [e.g. Asano et al., 2010; Vaivads et al., 2011;
Pan et al., 2012]. Our simulations provide quantitative evidence supporting this picture.
The global transport of electrons occurs as follows: in the magnetotail, the electron fluxes follow along the flow channel both in earthward and tailward directions, with small diffusion in configuration space. This is because the energy-independent E×B drift driven by strong
convection electric fields is statistically dominant in the flow channels. The gradient and curvature
drifts become dominant in the inner magnetosphere (within X ~ 11RE in the March 11, 2008
event). As a consequence, the high-energy electrons drift towards the dawn side.
80
CHAPTER 4
Modeling Ion Energization and Transport Associated with Magnetic
Dipolarizations during a Substorm
4.1. Introduction
In this chapter, we extend the global MHD+LSK simulation scheme to examine ion
energization in the magnetotail during a substorm event that occurred on February 07, 2009. The
main task is to develop a global scenario of ion energization and transport associated with magnetic
dipolarizations under realistic magnetospheric conditions. Since in the next chapter the February
07, 2009 event is also used for a comparison of energization and transport mechanisms between
electrons and ions, the other task in this chapter is to present observations and MHD simulation
results. This chapter is organized as follows: the observations are presented in Section 4.2; the
MHD simulation results are presented in Section 4.3; the ion LSK simulation results are discussed
in Section 4.4; the simulation results are summarized in Section 4.5.
4.2. Observations of the February 07, 2009 Substorm Event
During the time of interest (03:00-05:00 UT) the WIND spacecraft was located at
(X ,YZ , )GSM (202, 73, 38) R E . Figure 4.1 shows its measurements of the solar wind. The IMF
was southward and weak staring from 00:10 UT (the black vertical line). The solar wind was slow;
Vx was about 320 km/s. Geomagnetic indices show a weak substorm onset at ~03:40 UT. The peak
AE index was ~ 120 nT. The solar wind observed by the WIND spacecraft needs about 55 minutes
to propagate toX 25RE , the upstream boundary of the global MHD simulation. During the
81
substorm of interest (03:00-05:00 UT), the five THEMIS spacecraft were all on the dawn side of
the magnetotail. At time 04:00 UT, THEMIS P4, P5, and P3 were located close to each other at
X 8.4RE , X 8.5RE and X 9.4RE down tail. P2 was located at X 18.6RE , and the
P1 was located atX 30.6RE . The geosynchronous spacecraft GOES-10, GOES-11, and GOES-
12 were respectively near 0, 19, and 23 h magnetic local time. They were on the dusk side of the
THEMIS fleet. Lyons et al. [2012] investigated the relation between aurora streamers observed by
THEMIS all sky imagers and dipolarization and flow activities in the tail in this substorm event.
Aurora streamers are longitudinal finger-like structures. They determined the substorm onset to be at 03:47:12 UT (the blue vertical line in Figure 4.1) near the longitude of GOES-10. Subsequently,
the aurora activity expanded westward to GOES-12 location and eastward to P4, P5, and P3 locations. They found that GOES-10 did not observe dipolarizations when the substorm onset there,
instead significant dipolarization was observed by P3, P4, and P5 at ~ 04:06 UT in association
with aurora streamers. Oka et al. [2011] showed that after the dipolarization seen by P3, P4, P5 at
~ 04:06 UT, the pressure in the inner magnetosphere increased. This pressure increase propagated
tailward and was detected by P2 at X ~ 18.6RE at ~ 04:14 UT. Meanwhile, the tail seen by P3,
P4 and P5 became more stretched. At ~ 04:18 UT, P1 at X 30.6RE detected a reversal of flow
from tailward to earthward, suggesting the X-line retreated to P1 location at that time. They
concluded that the pressure increase after dipolarizations eventually caused the X-line to move
tailward.
82
WIND Observation of Solar Wind 6 4 2 0 Bx (nT) -2 -4 -6 6 4 2 0 By (nT) -2 -4 -6 6 4 2 0 Bz (nT) -2 -4 -6 100 Vz 0 -100 Vy Vi -200 (km/s) -300 Vx -400 4 3
P 2
(nPa) 1 0 150 AL 100
50 AU (nT) Index 0
Geomagnetic -50 AE -100 hhmm 0000 0200 0400 0600 2009 Feb 07
-10
-5 P2 P3 P1 P5 0 G10 G12 P4
Y_gsm (Re) 5 G11
10 0 -10 -20 -30 X_gsm (Re)
Figure 4.1. Solar wind measured by WIND and satellite positions in the February 07, 2009 event.
(Top) Solar wind data. From top to bottom showed are the IMF, the solar wind velocity, the proton
83
number density, the dynamic pressure, and geomagnetic indices. The black vertical line is 00:10
UT and the blue vertical line is 03:47:12 UT, adapted from El-Alaoui et al. [2013]. (Bottom)
Projection of spacecraft trajectories onto the equatorial plane between 01:00 UT and 07:00 UT
(marked points are at 04:06 UT).
Since we are interested in particle energization in the earthward direction of the
reconnection site, we present the detail observations from P2 at X ~ 18.6RE in Figure 4.2. The vertical line indicates an earthward propagating reconnection jet. The measured electric field spikes were about 20 mV/m. The measured electric field was larger than the convection electric
field ( VBi ), indicating P2 was close to the IDR. The measured electron density is smaller than
the ion density, probably due to overkill of photoelectron effect. The photoelectrons were removed
before calculating the electron density. The energy below which the electrons were regarded as
photoelectrons is represented by the black line in electron energy flux (the last panel). Although
the substorm event was weak, the earthward flow reached 500-600km/s. The local Alfvén speed
3 with BnT~10 and ncmi ~0.1 was vkmsA ~ 689 / . Hence the outflow velocity was close to the
local Alfvén speed. The electron temperature was TkeVe ~ 2 and the ion temperature was
TkeVi ~3 4 . The temperature ratio was TTie / ~ 1.5 2 . The temperature of both species in the
reconnection outflow region were larger than that in the lobe region, suggesting that the plasmas
were heated in the reconnection outflow. The total pressure was almost constant, even though the
magnetic pressure and plasma pressure varied significantly. The high-energy electron distribution
5 in the outflow region is fitted to a power law distribution fe ()EE with E 11 keV . The high-
6 energy ion distribution is fitted to fi ()EE with E 25 keV .
84
THEMIS P2 observation 20 Bz 10 0 By (nT) -10 B(GSM) Bx -20 20 Ez 10
0 Ey
(mV/m) -10 E(GSM) Ex -20 10 5 0 -5 (mV/m) -10 Evixb(GSM) 1.00 Ne 0.10 Ni Density (cm^-3) 0.01 600 400 200
(km/s) 0 Vi(GSM) -200 8 Ti/Te 6 Te T 4 (keV) 2 Ti 0 1.000 Pb 0.100 Pe Pi
(nPa) 0.010
Pressure Pt 0.001 106 107 105 106 104 105
3 4
ion 10 10
Eflux 2 3
10 ) 1 10 10 102 -1 6 8 105 107 eV 10 10 -1 4 6 10 105 sr 3 -2 10 104
Eflux 2 10 10 3 cm electron 10 -1 101 102 X_gsm(Re) -18.6 -18.6 -18.6 -18.6 Y_gsm(Re) -2.5 -2.6 -2.7 -2.8 (eV s Z_gsm(Re) -4.2 -4.2 -4.2 -4.2 hhmm 0350 0400 0410 0420 2009 Feb 07
Figure 4.2. THEMIS P2 observations in the February 07, 2009 event. From top to bottom showed are the magnetic field in GSM coordinates, the measured electric field, the convection electric
field ( VBi ), the density, the ion bulk velocity, the temperature, the pressure, the ion energy
flux, and the electron energy flux. The vertical line indicates an earthward propagating
reconnection jet.
85
Figure 4.3 shows the observations of the dipolarization of interest by THEMIS P3 in the
inner magnetosphere. THEMIS P4 and P5 measurements are similar to those from P3. The electric
field associated with the dipolarization was mainly in the X- and Y- direction, due to large Vx and
Vy . The measured electron density was higher than the ion density, probably resulting from
secondary electron effect in the ESA instrument. The ion temperature was TkeVi ~ 6 8 and the
electron temperature was Te ~ 3 4 keV , larger than those prior the arrival of the front. The tenuous
plasmas carried by the dipolarization to the inner magnetosphere were also hotter than the plasmas in the reconnection outflow at P2, suggesting that substantial heating occurred during earthward transport. The magnetic pressure change was balanced by the plasma pressure change, resulting in
an almost constant total pressure across the DF. High-energy ( 25keV ) ion and electron energy
fluxes measured by the SST instrument increased as the front arrived. However, the electron flux
gradually shifted to higher-energy beginning from 03:59 UT, ~6 min prior the arrival of the DF.
Recall the aforementioned analyses by Lyons et al. [2012]. This gradual increase was likely caused by the substorm activity that occurred on the dusk side of P3 location prior to the observation of the dipolarization. We found that the electron flux spectra before the dipolarization were dispersed, i.e. higher-energy flux increased earlier than lower-energy flux, suggesting that the flux increase
resulted from energy-dependent dawnward gradient and curvature drifts. In contrast, the change
in ion fluxes prior to the dipolarization was much weaker, suggesting after injection from the tail,
ions drifted (duskward) farther away from P3. We will examine the high-energy particles in detail
when we discuss LSK simulation results.
86
THEMIS P3 observation 30 Bz 20 10 By
(nT) 0
B(GSM) -10 Bx -20 15 Ez 10 5
Ey 0 (mV/m) E(GSM) -5 Ex -10 10 5 0 -5 (mV/m) -10 Evixb(GSM) 1.0 Ne
Ni Density (cm^-3) 0.1
100 0
(km/s) -100 Vi(GSM) -200 8 Ti/Te 6 Te T 4 (keV) 2 Ti 0 1.000 Pb 0.100 Pe Pi
(nPa) 0.010
Pressure Pt 0.001 106 107 105 106 104 105
3 4
ion 10 10
Eflux 2 3
10 ) 1 10 10 102 -1 6 8 105 107 eV 10 10 -1 4 6 10 105 sr 3 -2 10 104
Eflux 2 10 10 3 cm electron 10 -1 101 102 X_gsm(Re) -9.2 -9.3 -9.3 -9.4 Y_gsm(Re) -1.7 -1.9 -2.1 -2.2 (eV s Z_gsm(Re) -3.4 -3.4 -3.4 -3.4 hhmm 0350 0400 0410 0420 2009 Feb 07
Figure 4.3. THEMIS P3 observations in the February 07, 2009 event. The same format is used as in Figure 4.2. The vertical line indicates the arrival of the DF.
87
4.3. MHD Simulation of the February 07, 2009 Substorm Event
The global MHD simulation used the WIND observation of the solar wind as input at the
upstream boundary at X 25RE . The simulation results were validated and described by El-
Alaoui et al. [2013]. They found that the MHD simulation reproduced the dipolarizations and high-
speed flows observed by P3, P4 and P5 in the magnetosphere reasonably well. Detailed analysis
of the simulation revealed a global picture of the magnetotail dynamics for this substorm event.
Specifically, the substorm related reconnection occurred at X ~20 RE and extended only
partway across the magnetotail; the localized reconnection generated high-speed outflow
propagating earthward (and tailward) in narrow channels; multiple magnetic dipolarizations
formed and intensified at the earthward ends of the flow channels; together with flows, the
dipolarizations propagated toward a region with a strong magnetic field in the inner magnetosphere
at X ~7 RE that acted like a “wall”, where the dipolarized magnetic fluxes merged with the
strong magnetic field and piled up, while the flows were slowed and subsequently diverted,
forming large vortexes.
Figure 4.4 shows the snapshots of the Z-component of the magnetic field ( Bz ) and the Y-
component of the electric field ( Ey ) every 40 sec from 04:04:00 UT to 04:09:20 UT. At 04:04:00
UT, a small dipolarization structure intensified at (,)~(11,2)X YR E (white arrow). It
propagated earthward and merged with a strong dipole magnetic field at X ~ ( 7,2) RE at
04:06:40 UT. The dipolarization was driven by a high-speed flow in a narrow channel. The flow
originated in a region at X ~20 RE and YR~0 7 E , where the reconnection occurred. The flow
curved first toward the dawn side and then toward the dusk side, and finally slowed and diverted
as it encountered the strong-field wall. Subsequently, a larger magnetic dipolarization intensified
88
near 04:06:40 UT at (X ,YR ) ~ ( 12,0) E (black arrow), and gradually increased to 30 nT at
04:10:00 UT as it propagated along a path dawnward of the earlier dipolarization. This larger
dipolarization was also driven by a similar curved flow. The characteristic flow speed was 300
km/s. The characteristic width of the flow channels was about 23 RE . A large Ey with a peak
value of ~5-6 mV/m coincided with these two dipolarizations. The electric field in an MHD
simulation is given by EVBj , where V is the MHD flow velocity, is the resistivity,
and j is the current [Raeder et al., 1998]. We checked that the large electric field associated with
the dipolarization structures was dominated by the VB term, which represents the convection
electric field carried by flows. This point can also be confirmed by a simple estimate: a 20 nT
dipolarized magnetic field in a 300 km/s flow channel induces a 6 mV/m electric field. Similar as
those in the global MHD simulation of the March 11, 2008 event presented in Chapter 3, the
simulated dipolarizations in the February 07, 2009 event also formed and intensified far away from
the reconnection region, resulting from flow braking and plasma compression.
89
Bz and flow Eflux(>25keV) and flow Ey and flow -10 1.0 -10 -10 6 30 107 -5 -5 -5 4 /sr) 20 106 2 0 0 0 2 10 5 Y(Re) Y(Re) Y(Re) 10 Bz(nT) Ey(mV/m) 040400 UT 5 0 5 104 5 0 Eflux(keV/s/cm
10 -10 10 103 10 -2 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 0.8 X(Re) X(Re) X(Re) -10 -10 -10 6 30 107 -5 -5 -5 4 /sr) 20 106 2 0 0 0 2 10 5 Y(Re) Y(Re) 10 Y(Re) Bz(nT) Ey(mV/m) 040440 UT 5 0 5 104 5 0 Eflux(keV/s/cm
10 -10 10 103 10 -2 -5 0.6-10 -15 -20 -25 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 X(Re) X(Re) X(Re) -10 -10 -10 6 30 107 -5 -5 -5 4 /sr) 20 106 2 0 0 0 2 10 5 Y(Re) Y(Re) Y(Re) 10 Bz(nT) Ey(mV/m) 040520 UT 5 0 5 104 5 0 Eflux(keV/s/cm 10 0.4 -10 10 103 10 -2 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 X(Re) X(Re) X(Re) -10 -10 -10 6 30 107 -5 -5 -5 4 /sr) 20 106 2 0 0 0 2 10 5 Y(Re) Y(Re) Y(Re) 10 Bz(nT) Ey(mV/m) 040600 UT 5 0 5 104 5 0 0.2 Eflux(keV/s/cm 10 -10 10 103 10 -2 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 X(Re) X(Re) X(Re) -10 -10 -10 6 30 107 -5 -5 -5 4 /sr) 20 106 2 =19.6keV
test 0 0 0 2 10 5 Y(Re) Y(Re) 10 Y(Re) Bz(nT) Ey(mV/m) 5 0.0 0 5 104 5 0 1.0 1.2 1.4 1.6 Eflux(keV/s/cm 1.8 2.0 3 040640 UT W 10 -10 10 10 10 -2 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 300km/s X(Re) X(Re) X(Re)
90
Bz and flow Eflux(>25keV) and flow Ey and flow -10 1.0 -10 -10 6 30 107 -5 -5 -5 4 /sr) 20 106 2 =19.3keV
test 0 0 0 2 10 5 Y(Re) Y(Re) Y(Re) 10 Bz(nT) Ey(mV/m) 5 0 5 104 5 0 Eflux(keV/s/cm
3 040720 UT W 10 -10 10 10 10 -2 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 0.8 X(Re) X(Re) X(Re) -10 -10 -10 6 30 107 -5 -5 -5 4 /sr) 20 106 2 =23.3keV
test 0 0 0 2 10 5 Y(Re) Y(Re) Y(Re) 10 Bz(nT) Ey(mV/m) 5 0 5 104 5 0 Eflux(keV/s/cm
3 040800 UT W 10 -10 10 10 10 -2 -5 0.6-10 -15 -20 -25 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 X(Re) X(Re) X(Re) -10 -10 -10 6 30 107 -5 -5 -5 4 /sr) 20 106 2 =32.1keV
test 0 0 0 2 10 5 Y(Re) Y(Re) 10 Y(Re) Bz(nT) Ey(mV/m) 5 0 5 104 5 0 Eflux(keV/s/cm
3 040840 UT W 10 0.4 -10 10 10 10 -2 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 X(Re) X(Re) X(Re) -10 -10 -10 6 30 107 -5 -5 -5 4 /sr) 20 106 2 =36.1keV
test 0 0 0 2 10 5 Y(Re) Y(Re) 10 Y(Re) Bz(nT) Ey(mV/m) 5 0 5 104 5 0 0.2 Eflux(keV/s/cm 3 040920 UT W 10 -10 10 10 10 -2 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 X(Re) X(Re) X(Re) -10 -10 -10 6 30 107 -5 -5 -5 4 /sr) 20 106 2 =37.9keV
test 0 0 0 2 10 5 Y(Re) Y(Re) 10 Y(Re) Bz(nT) Ey(mV/m) 5 0.0 0 5 104 5 0 1.0 1.2 1.4 1.6 Eflux(keV/s/cm 1.8 2.0 3 041000 UT W 10 -10 10 10 10 -2 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 300km/s X(Re) X(Re) X(Re)
91
Figure 4.4. Snapshots of the MHD and ion LSK simulations for the February 07, 2009 event. Left
column: The Z-component of the magnetic field from the MHD simulation. Middle column:
Integrated energy flux with energy above 25 keV from the ion LSK simulation. Right column: The
Y-component of the electric field from the MHD simulation. Flow vectors (black arrows) from the
MHD simulation are plotted over the magnetic field, energy flux and electric field. All of the
quantities are plotted on the maximum pressure surface. Each row of plots corresponds to the same
time, which is labeled next to the Y-axis on the left. The representative ion trajectory, shown in
Figure 4.6, is superimposed on these quantities (white lines). The solid circle on each line
represents the ion location at the corresponding time. The energy of the test ion is also labeled next
to the Y-axis.
4.4. Ion LSK Simulations of February 07, 2009 Substorm Event
4.4.1. LSK Simulation Set-up
Applying the electric and magnetic fields derived from the MHD simulation, we performed
an ion LSK simulation. The ion LSK code solves the Lorentz-force equation of motion by using a
fourth-order Runge-Kutta method. For the LSK simulation ion source in the magnetotail, we
adopted a distribution that combines a Maxwellian distribution representing the thermal ions and
a power law distribution representing the high-energy tail. Two parameters, the temperature
(4Ti keV ) of the Maxwellian distribution and the power law index (n=6) were derived from ion
distribution observed by THEMIS P2 in the reconnection outflow at X 18.6RE during this
event. For the simulation, 80,000 ions obeying an isotropic Maxwellian distribution and 80,000
ions obeying an isotropic power law distribution were launched separately every 20 sec from 03:50
UT to 04:20 UT. The ions were launched uniformly in space from a surface given by
92
-19 REEXR-17 , 5 REEYR10 , and Z Zmp , where Zmp is the maximum pressure surface location. The maximum pressure surface is a good approximation to the center of the plasma sheet on the night side [Ashour-Abdalla et al., 2002]. This launch location was selected because by inspecting the flows and magnetic field from the MHD simulation, we found that the high-speed flows originated there. The ion data were collected using virtual detectors placed throughout the simulation domain. Energy fluxes were calculated by using the detector data and
equation (3.4).
4.4.2. LSK Simulation Results and Comparisons with Observations
4.4.2.1. Ion Energy Fluxes
In order to validate the LSK simulation, we compare our simulation results with THEMIS
observations. Because THEMIS P3, P4 and P5 were within 1.5 RE to each other they observed
similar high-energy flux increases, only the comparison for P3 is shown in Figure 4.5. The key
observational features upon the arrival of the dipolarization are the increase of Bz , the earthward
flow, and the dramatic increase of high-energy ion fluxes in contrast to the modest change of low-
energy fluxes. The dipolarization of interest accompanied by the earthward flow is reproduced by
the global MHD simulation. Compared to observations, the simulated dipolarization is weaker,
and the magnetic field and flow speed have fewer variations on the shorter time-scale. The
simulation reproduces the observed dramatic enhancement of the high-energy ion fluxes. Similar
to the observations, the simulated low-energy flux changes little.
93
30 Bx 20 10
By B
(nT) 0 -10 Bz -20 200 Vx 100
Vy V 0
(km/s) -100 Vz )
-1 -200 107 25-37keV
eV 6
-1 10 5 37-48keV
sr 10 48-65keV -2 104 3 65-77keV cm (SST) 10 77-116keV -1 Ion eflux 102 101 116-175keV 7
(eV s 10 1.5-2.7keV 106 5 2.7-4.6keV 10 104 4.6-8.0keV 3
(ESA) 10 8.0-13.8keV Ion eflux 102 13.8-25keV 101 30 Bx 20 10 By B
(nT) 0 -10 Bz -20 200 Vx 100 Vy V 0
(km/s) -100 Vz -200 107 25-37keV 106 5 37-48keV 10 48-65keV 104 3 65-77keV (SST) 10 77-116keV Ion eflux 102 101 116-175keV 107 1.5-2.7keV 106 5 2.7-4.6keV 10 104 4.6-8.0keV 3
(ESA) 10 8.0-13.8keV Ion eflux 102 13.8-25keV 101 X_gsm(Re) -9.2 -9.3 -9.3 -9.4 Y_gsm(Re) -1.7 -1.9 -2.1 -2.2 Z_gsm(Re) -3.4 -3.4 -3.4 -3.4 hhmm 0350 0400 0410 0420 2009 Feb 07
Figure 4.5. Comparison of observations and simulations for P3 in the February 07, 2009 event.
From top to bottom, the first four panels are observed magnetic field, plasma velocity, high-energy ion flux from the Solid State Telescope (SST), low-energy ion flux from the Electrostatic Analyzer
(ESA). The next four panels are the corresponding simulated quantities at the virtual P3 position
94
in the simulation domain. The spike in the observed flux in between 04:03 UT and 04:04 UT is
because of the attenuator set-up in the SST detector. The vertical lines indicate the arrival of the
dipolarization of interest. The ~2 minute difference between the observed DF and the simulated
DF is reasonable, see Raeder et al. [1998, 2001], El-Alaoui et al. [2001, 2009].
To examine the energization and transport process, the integrated energy flux with energy
greater than 25 keV derived from the LSK simulation is plotted in Figure 4.4 (middle column).
The integrated energy flux is defined as JE(,,)r tdE, which gives the total energy flux above 25keV e
25 keV and has the unit of keV s-1 cm -2 sr -1 . Three features are remarkable for the total energy
flux. First, the flux intensity pattern was similar to that of Bz . Enhancements of the high-energy
flux coincided with the two dipolarizations described in section 4.3 (as indicated by the arrows in
each snapshot). Quantitatively, near the reconnection site, the flux was on the order of
~106-1-2-1 keV s cm sr , it increased to ~107-1-2-1 keV s cm sr as the two dipolarizations
intensified during their propagation to the inner magnetosphere. This illustrates that dipolarizations
are powerful accelerators acting far distant from the reconnection site. Second, in the magnetotail
( X ~9 RE ), the high-energy flux closely followed the flow channels, which is a combined
effect of acceleration and convection. Specifically, the reconnection outflow carried a large
convection electric field ( Eyy~(VB )as described above) which pushed the ions in the flow direction via the EB drift. When pushed along the flows to the dipolarization regions, the ions
were accelerated almost adiabatically (described below) to higher energy and contributed to the high-energy flux in the flow channels. Third, that the flux (statistical quantities) closely followed the flow channels is evidence that the energy-independent EB drift dominated along the flow
95
channels. The flows slowed at the “wall” ( X ~7 RE ), where the ion gradient and curvature drifts
became significant so the ions moved toward the dusk side.
4.4.2.2. Ion Trajectories
In this section, we examine ion trajectories to identify the acceleration mechanism. We
focus on the ions that contribute to the high-energy flux enhancements associated with the
dipolarizations showed in Figure 4.4. The trajectory of a representative ion and its characteristics
are presented in Figure 4.6. This ion was launched near the reconnection site at 04:06:20 UT at
X 18.3 RE and YR 0.27 E . It had an initial energy of 13.1 keV and a pitch angle near 90 .
This initial energy is in the high-energy tail of the plasma sheet distribution. We choose a high- energy tail ion because dipolarizations act as adiabatic accelerators; for ions to be accelerated to high energy, their initial energy needs to be sufficiently large. As the ion was pushed toward the
Earth (until X ~9 RE ), its energy increased to ~40 keV. The ion was accelerated through a two- stage process. The first-stage acceleration occurred from 04:06:20 UT to 04:07:26 UT in the region
of X ~ 13 RE (before the first blue vertical dashed line). Due to the weakness of the magnetic
field (frequently below 10 nT), the first adiabatic invariant was violated during this interval. Thus,
the ion was nonadiabatically accelerated and the energy gain was ~10 keV. The second-stage
acceleration occurred from 04:07:26 UT to 04:10:00 UT (between the two blue vertical dashed
lines), when the ion drifted into a region with a magnetic field greater than 20 nT. Its orbit became
stable and had a regular bounce motion. The first adiabatic invariant was conserved and the ion
was adiabatically accelerated. The scenario of the two-stage process is manifested by the parameter. According to Büchner and Zelenyi [1989], in magnetotail-like field reversals, the first adiabatic invariant is conserved if 1 because the frequencies of the gyration about magnetic
96
field lines and the bounce motion about the current sheet are well separated. As these two
frequencies approach each other, 1, the interference of gyration and bounce motion breaks
the first adiabatic invariant. Note that the radius of curvature of the magnetic field line is determined by the magnetic field from the MHD simulation, and the energy dependence of the
RqBRcurv curv 1/4 kappa is W , therefore, this two-stage acceleration process works for i 2mW
a broad energy range of ions having similar trajectories. Such a two-stage acceleration process also
has been reported in studies of electrons in the magnetotail [e.g. Ashour-Abdalla et al., 2013].
Let’s examine the physics of the acceleration mechanism. The electric field was perpendicular to the magnetic field throughout the motion. The ion exchanged energy with the perpendicular electric field. It gained energy during one-half gyration and lost energy during the other half, as implied by the oscillations in the energy and the vector product of the electric field and the ion velocity. Because of this energy exchange, the periodic oscillation in the first adiabatic invariant was significantly reduced by subtracting the energy related to the EB term. The energy
11 associated with the EB drift is calculated as WmumV 22~~1 keV, which is about E×B 22E MHD
one tenth of the ion total energy. The net effect of the energy exchange was that the ion energy
gain exceeded its energy loss as it moved into regions with a stronger magnetic field. This process
is known as betatron acceleration [Northrop, 1963; Baños, 1967; Birn et al., 2013]. As discussed
in Chapter 2, the gain of kinetic energy W can be expressed as:
1 dW M B uE (4.1) qdtGC q t
where uGC is the guiding center velocity, M is the first adiabatic invariant and q is the electric
charge. The energy gain results from two effects. The first is due to the guiding center motion
97
along the electric field direction. The second is caused by the temporal variation of the magnetic field.
5 Rx -10 0 -5 Ry -10 R(Re) -15 Rz -5 -20 40 30 04:07:26 20 0 04:10:00 W(keV)
10 Y(Re) 0 10 8 6 5 4 Kappa 2 0 3.0 10 2.5 subtract 2.0 ExB -5 -10 -15 -20 -25
(keV/nT) 1.5 X(Re) T 1.0
/B no subtract 0.5 10 per 0.0 W 40 Bx 20 By 0 Bz 5 B(nT) -20 Bt -40 8 6 Epar 0 4 Z(Re) 04:10:00 2 Eper E(mv/m) 0 -2 15 -5 04:07:26 10 5 0 -5 -10
E*V(keV/s) -10 -15 -5 -10 -15 -20 -25 hhmm 0408 0410 2009 Feb 07 X(Re) Figure 4.6. Characteristics of a representative test ion. Left column: from top to bottom, the variables are the position vector, the total energy, kappa, the first adiabatic invariant (the green
(black) line is the perpendicular energy over the total magnetic field before (after) subtraction of
energy associated with the EB drift), the magnetic field, the electric field, and the vector product
of the electric field and the ion velocity. The interval between the two vertical dashed blue lines is
04:07:26-04:10:00 UT. Right column: the trajectory of the test ion. The two panels show projection of the orbit onto the X-Y and X-Z planes. The blue solid circles correspond to the initial and final points of the 04:07:26-04:10:00 UT interval.
98
To examine how exactly the ion are accelerated by the electric field associated with the
dipolarization, the trajectory is superimposed on the magnetic field, the ion energy flux and the
electric field shown in Figure 4.4. At 04:06:20 UT, the aforementioned second dipolarization had
already formed near X ~12 RE . The ion launched near the reconnection site was quickly pushed
toward the curved dipolarization path. From 04:06:20 UT to 04:70:20 UT, the ion gained its energy
nonadiabatically because of the weak magnetic field. At 04:07:20 UT, the magnetic field at the
particle location is greater than 20 nT. The ion motion became adiabatic at that time. As the ion
continued to move earthward, at about 04:08:40 UT, it caught up with the dipolarization structure.
The ion was able to catch up with the dipolarization structure because the EB drift along the
flow channel was fast due to the large Ey . This fast drift was in contrast to the slow propagation
of the dipolarization structure. The dipolarization was slow because of plasma compression. The
adiabatic energy gain during this catching-up process from ~04:07:20 UT to ~04:08:40 UT was due to EB drift toward a stronger magnetic field. As noted by Birn et al. [2013],
M uE u B , namely the energy gained by a particle as it moves into a stronger magnetic B q E
field due to EB drift, can be expressed as a gradient drift in an electric field direction. After
catching up with the DF the ion moved with the dipolarization. The ion energy further increased
as the dipolarization continued to intensify. This energy gain was due to the temporal variation of
the magnetic field strength ( Bt 0 ). Finally, as the dipolarization merged with the strong
magnetic field region, the ion gradient drifted duskward. Its energy changed little.
In the previous section, we presented the integrated energy flux of high-energy ions side by side with the magnetic field and electric field while in this section we examined the characteristics of an ion trajectory. Because this ion trajectory is in the region of intense high-
99 energy flux associated with the dipolarization, it represents typical ions that are energized by the dipolarization. Conversely, the high-energy flux increase gives a statistical measure of the energy gained by ions via the dipolarization. Therefore, the acceleration scenario developed by examining the single ion trajectory complements and is consistent with that from inspecting the high-energy flux in the previous section.
4.5. Conclusions and Discussions
Using data from the realistic MHD and ion LSK simulations, we presented a global scenario of ion energization and transport. We found:
1. Most of the high-energy ion flux enhancements are due to nonlocal energization by the
dipolarizations, which are driven by high-speed flows in narrow channels.
2. Ions originating from the reconnection site undergo a two-stage energization process. Not
far from the reconnection site, where the magnetic field is weak, the ions are
nonadiabatically accelerated. Subsequently, they adiabatically gain energy as they catch up
with and ride on the earthward propagating dipolarizations.
3. In the magnetotail, the high-speed flows control ion transport via the EB drift, whereas
close to the Earth, ions gradient and curvature drift toward the dusk side.
Let’s compare these results with previous studies. First, previous analytical studies suggested that ions upstream of DFs can be captured by the fronts and accelerated via trapping
(resonant) and quasi-trapping interaction with the fronts [Artemyev et al., 2012; Ukhorskiy et al.,
2013]. Our simulations, as well as the simulations by Birn et al. [2013], show that most of the high-energy ions are generated in the dipolarized regions, i.e. behind the fronts. They are
100
accelerated during their earthward drift to stronger magnetic field. Second, Birn et al. [2013]
performed test particle simulations in electric and magnetic fields from a generic MHD simulation.
The MHD simulation reproduces the magnetotail reconnection, localized flow bursts and
dipolarizations during substorms. The test particle simulations reproduce a rapid rise of energetic
particle fluxes upon the arrival of dipolarizations. Ions are nonadiabatically accelerated by the
electric fields associated with dipolarizations. Our study complements and extends their results. In
our global MHD simulation driven by realistic upstream solar wind conditions, the flow speed and
width, the magnetic and electric fields, and the dynamics of the system are realistically determined
and event-dependent. In our LSK simulation, ions originating in the flows are first nonadiabatically
accelerated and then adiabatically accelerated as they catch up with and ride on the dipolarizations.
Furthermore, the ions closely follow the flow channels in the tail because of the dominant EB drift; therefore ion transport in the magnetotail is directly controlled by the flows. We have estimated drift velocities for the representative ion and found that in presence of high-speed flows, the EB drift velocity is much larger than the curvature and gradient drift velocities. It appears
that the ions simulated by Birn et al. [2013] drift across the flow channel much faster (see Figure
3 therein). They did not discuss ion collective transport in detail. The differences in the results and
emphases of the results perhaps stem from differences in the methodology. Specifically, the
electric and magnetic fields, which govern the adiabaticity of particle motion and magnitude of
particle drifts, are appreciably different in the two MHD simulations. In addition, our LSK
simulation traces many particles forward in time from a source region near the reconnection site
and collects them with virtual detectors. This is more efficient for tracking particle transport, but
less efficient for finding all the sources of particles than the backward tracing method used by Birn
et al. [2013].
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The difference between the ion earthward motion and dipolarization propagation has significant implications. First, without this difference, ions would need to ride exactly on dipolarizations in order to gain energy through dipolarization intensification. The difference allows
more ions to gain energy by catching up with the dipolarizations from behind and gain energy as
they drift toward stronger magnetic field regions. The difference is made possible by the large
electric fields carried by the dynamic high-speed flows and plasma compression at the
dipolarization regions. Second, ions can gain more energy by catching up with the dipolarizations
than by riding on them because the catching-up ions originate from the reconnection site in the
magnetotail, where the magnetic field is weaker than in the dipolarizations intensification regions
(usually several RE away from the reconnection site in MHD simulations [El-Alaoui et al., 2013]).
Third, this difference and the dynamic nature of flows require us to launch particles continuously
and uniformly in time so that the fluxes and distribution functions become statistically meaningful.
These measures have been taken in our MHD+LSK simulations.
102
CHAPTER 5
A Comparison Study of Ion and Electron Energization and
Transport Mechanisms during a Substorm
5.1. Introduction
Compared with studies by Delcourt and Sauvaud [1994], Birn et al. [1997b, 1998], Li et al. [1998], Zaharia et al. [2000, 2004] and Birn et al. [2004b, 2013], one major improvement of studies by Ashour-Abdalla et al. [2011], Pan et al. [2014a, 2014b], and Liang et al. [2014] is that we used fields from global MHD modeling in which upstream boundaries are set by event- dependent solar wind measurements. However, there has been no event-study that compares electron and ion acceleration in the magnetotail for the same substorm event, which is of great interest considering the observed differences and similarities in particle flux characteristics (see section 1.3). This chapter aims to apply the global MHD+LSK simulation scheme to the February
07, 2009 substorm. The emphases are the differences and similarities between ions and electrons in terms of the observations, the simulation approaches, and the simulation results. This chapter is organized as follows: the simulation methodology is presented in Section 5.2; the LSK simulation results are presented in Section 5.3; the simulation results are summarized and their implications are discussed in Section 5.4. Note that since the MHD results for the February 07, 2009 event were described by El-Alaoui et al. [2013] and in Chapter 4, we will not repeat the detailed discussions.
In Chapter 4, we have presented the ion LSK simulation results for this event; some of those ion
LSK results are included in this chapter to enable a detailed comparison between the ion and electron results.
103
5.2. Comparisons of the Electron and Ion LSK Simulation Set-up for the February 07, 2009
Substorm Event
We combine a global MHD model, an ion LSK simulation, and an electron LSK simulation
to model magnetospheric dynamics and particle energization for the February 9, 2009 substorm
event. Table 1 provides a summary of the LSK simulations for this event. The electromagnetic
fields used for test particles are from the global MHD simulation [Raeder et al., 1998, 2001; El-
Alaoui et al., 2001; El-Alaoui et al., 2009]. Solar wind obtained from the WIND spacecraft at
(X ,YZ , )GSM (202, 73, 38) R E was used to drive the MHD simulation. The MHD simulation
results were validated and described by El-Alaoui et al. [2013] and in Chapter 4. The LSK code
for ions (protons) solves the Lorentz-force equation of motion by using a fourth-order Runge-Kutta
method. The particle mover in the electron LSK code switches between guiding-center and
Lorentz-force equation. The switch is determined by the adibaticity parameter of electron motion
Rcurv , where Rcurv is the curvature radius of the magnetic field, and is the electron gyro
radius [Büchner and Zelenyi, 1989]. Both Rcurv and are determined locally at the position of the
electron. The thresholds of switch ( 7 and 5 ) are based on the study of particle orbit characteristic in the tail-like field reversal [Büchner and Zelenyi, 1989]. Note that if 57 , no switch occurs and electrons are pushed forward in time by using the same method as in their prior step. A detailed description of the electron LSK code is given by Schriver et al. [2011].
For the particle sources in the LSK simulations, we use a distribution that combines a thermal Maxwellian distribution with a high-energy power law tail [Pan et al., 2014a]. The combination of a Maxwellian distribution for thermal electrons and a power law distribution for the high-energy tail is a special approximation of a broad class of generalized Lorentzian (Kappa)
104
distributions [Vasyliunas, 1968; Summers and Thorne, 1991]. This combination was adopted
because it helps handle high-energy electrons. The parameters for the distribution are estimated by
using THEMIS P2 data atX 18.6RE , which was earthward of the magnetic reconnection site
according to the analysis of the observational data by Oka et al. [2011]. Note that the temperature
ratio TTie/ 2 was derived from reconnection outflow plasma measured by P2 near the center of
plasma sheet during this event. To determine launch times of particle sources, we estimate particle
convection time from the launch location to the region of interest. In this event, the convection
time from the reconnection region to the inner magnetosphere (~10 RE ) is ~5 minutes with a flow speed of ~200km/s. The dipolarization of interest was observed at ~04:06:00 UT, therefore we launched particles starting at 03:50:00 UT to ensure sufficient time for convection. The number of source electrons are based on the simulation described in Chapter 3. Note that 3,024 power law source electrons per 20 seconds are sufficient to determine the effect of high-energy electron source on producing the high-energy electron fluxes in the regions of interest. Virtual detectors are placed throughout the simulation domain. In this study, we use the information recorded by the maximum pressure surface detector, which approximates the location of the center of the plasma sheet in the magnetotail [Ashour-Abdalla et al., 2002]. In the LSK simulations, we need to convert information about individual particles recorded by the virtual detectors (e.g. recording time, position, energy, and pitch angle) into collective physical quantities (e.g. energy fluxes and distribution functions). This so-called normalization process has been discussed in Chapter 3.
105
Table 5.1. Set-up of the ion and electron LSK simulations for the February 07, 2009 substorm
Proton Electron
Electromagnetic field A global MHD simulation [El-Alaoui et al., 2013];
Solar wind monitor: WIND spacecraft
Particle mover Lorentz-force equation Switch to guiding-center
equation if 7 ;
Switch to Lorentz-force
equation if 5
[Schriver et al., 2011]
Particle sources An isotropic Maxwellian An isotropic Maxwellian
(Ti 4 keV ) and an isotropic (Te 2 keV ) and an isotropic
power law tail power law tail
( fE( ) E6 , E 25 keV ) ( fE( ) E5 , E 11 keV )
[Pan et al., 2014a, 2014b]
Particle launch location -19 REE XR -17 ; 5 REEYR 10 ; Z Zmp , Zmp is the
Maximum pressure surface [Ashour-Abdalla et al., 2002]
Launch times Every 20 seconds from 03:50:00 Every 20 seconds from
to 04:20:00 03:50:00 to 04:13:00
Number of source 80,000 Maxwellian protons; 19,440 Maxwellian electrons; particles per 20 seconds 80,000 power law protons 3,024 power law electrons
Diagnostics Detectors on the maximum pressure surface, and planes at fixed X,
Y, Z locations.
106
Normalization CEND ()nddtvA JE(,r , t ) MHD LL eDD 4 i1 NLDDD dAn v, dt dE
[Ashour-Abdalla et al., 1993; Richard et al., 2009]
5.3. Simulation Results
5.3.1. Comparisons of Simulation Results and Observations
In Figure 5.1 we compare our simulation results with THEMIS observations. Because
THEMIS P3, P4 and P5 were within 1.5 RE to each other, they observed similar high-energy flux increases, so only the comparisons for P3 are shown. The Geocentric Solar Magnetospheric (GSM) coordinate system is used. The first four panels summarize the observations. The key observational
features of the dipolarization of interest are the increase of Bz associated with the earthward flow, and the dramatic increase of high-energy ion fluxes (e.g. 37keV-48keV and 77keV-116keV) in contrast to the modest decrease of the low-energy fluxes (e.g. 4.6keV-8.0keV) upon the arrival of the dipolarization. Like the low-energy ion flux, the low-energy electron flux (e.g. 1.94keV-
3.36keV) change is modest. However, the high-energy electron fluxes demonstrate an “anomaly”: the 17.5keV-23.0keV electron flux gradually increases for ~5 min before the dipolarization front arrival, and then increases further after the dipolarization front passes, although the latter increase is less dramatic than the high-energy ion flux increase. The 58keV-73keV electron flux shows enhancement after the dipolarization; there is no high temporal resolution data before 04:05:10
UT. The increase prior to the dipolarization front in the electron flux is likely due to the substorm activity that occurred west of THEMIS P3 location prior to the observation of the dipolarization.
Lyons et al. [2012] showed a series of aurora activities associated with substorm onset at 03:47:12
UT and subsequent substorm expansion west of all the THEMIS spacecraft. THEMIS P3, P4 and
107
P5 did not enter the active region until ~04:02:06 UT, after which they observed the dipolarization
and earthward flows. Two other observed features (not shown) also support our interpretation of
the “anomaly”: (1) the electron flux spectra before the dipolarization are dispersed, namely higher-
energy flux increases earlier than lower-energy flux, suggesting that the flux increase resulted from
energy-dependent gradient and curvature drifts; (2) a weaker change is observed for ion fluxes
prior to the dipolarization, suggesting that after injection from the tail, ions drifted (westward)
farther away from THEMIS P3.
The next four panels of Figure 5.1 show the corresponding simulated quantities at the virtual P3 position in the simulation domain. The dipolarization accompanied by the earthward flow is reproduced by the global MHD simulation. Compared to observations, the simulated dipolarization is weaker, and the magnetic field and flow speed have fewer variations on the shorter time-scale. Two features about the ion fluxes are notable. First, the low-energy (4.6keV-8.0keV) flux changes modestly beginning from 04:00:00 UT, which is consistent with the observed flux.
The simulation does not reproduce the dip right after the dipolarization. This is probably because
compared to observations, the Bz increase is less in the MHD simulation so that the sweeping
effect of dipolarization front is less effective. However, the low-energy flux provides a benchmark for the LSK simulations, suggesting that particles are launched sufficiently early to allow convection to the inner magnetosphere. Second, the simulation reproduces the observed dramatic enhancement of the high-energy ion fluxes as the dipolarization arrives. Similar to that seen in the observational data, the increase of 36keV-46keV flux in the electron LSK simulation from
04:02:00 UT to 04:06:20 UT is due to electron acceleration associated with an earlier
dipolarization on the west of the virtual spacecraft location (see Figure 4.4 in Chapter 4). After
04:06:00 UT, the 36keV-46keV and 58-73keV fluxes increase upon the arrival of the
108
dipolarization front of interest. This increase is less dramatic than that seen in the simulated high-
energy ion fluxes, which is consistent with the observations. We point out that the remarkable difference between the low-energy fluxes and the high-energy fluxes is captured by the simulations,
suggesting that the simulations have correctly preserved the major underling physical process. This process is described in the next section.
109
30 Bx 20 10
By B
(nT) 0 -10 Bz -20 200 Vx 100
Vy V 0
(km/s) -100 Vz )
-1 -200 107 eV 6 4.6-8.0keV
-1 10
sr 5
-2 10 37-48keV 104 cm
-1 3 Ion eflux 10 77-116keV 102 8
(eV s 10 7 1.94-3.36keV 10 6 10 17.5-23.0keV 105 4 Ele eflux 10 58-73keV 103 30 Bx 20 10 By B
(nT) 0 -10 Bz -20 200 Vx 100 Vy V 0
(km/s) -100 Vz )
-1 -200 107 eV 6 4.6-8.0keV
-1 10
sr 5
-2 10 37-48keV 104 cm
-1 3 Ion eflux 10 77-116keV 102 8
(eV s 10 7 1.94-3.36keV 10 6 10 36-46keV 105 4 Ele eflux 10 58-73keV 103 X_gsm(Re) -9.3 -9.3 -9.3 Y_gsm(Re) -1.9 -2.0 -2.1 Z_gsm(Re) -3.4 -3.4 -3.4 hhmm 0400 0405 0410 2009 Feb 07
110
Figure 5.1. Comparisons of observations with simulation results for THEMIS P3. From top to
bottom, the first four panels show the observed magnetic field, plasma velocity, ion differential
energy fluxes, and electron differential energy fluxes from THEMIS P3. For ions, the 4.6keV-
8.0keV energy flux is measured by the Electrostatic Analyzer (ESA) [McFadden et al., 2008], whereas the 37keV-48keV and 77keV-116keV energy fluxes are measured by the Solid State
Telescope (SST) instrument [Angelopoulos, 2008]. The spike in the observed 37keV-48keV and
77keV-116keV ion fluxes in between 04:03 UT and 04:04 UT is due to the attenuator set-up in the
SST detector. For electrons, the 1.94keV-3.36keV and 17.5keV-23.0keV fluxes are obtained by the ESA instrument, and the 58keV-73keV flux is obtained by the SST instrument. There is no burst mode data for the 58keV-73keV electron flux before 04:05:10 UT. The next four panels show the corresponding simulated quantities at the virtual P3 position in the simulation domain. The
vertical lines indicate the arrival of the dipolarization of interest. The ~2 minute difference between
the observed dipolarization and the simulated one is reasonable, see Raeder et al. [1998, 2001],
El-Alaoui et al. [2001, 2009].
5.3.2. Comparisons of Ion and Electron Acceleration Mechanisms
In order to identify the physical processes underlying the observed fluxes, we examine
particle energy fluxes and characteristics of representative particle trajectories. The energy fluxes
give a statistical measure of the particle energy changes, whereas the single particle trajectories
enable us to determine the acceleration mechanisms.
Figure 5.2 shows a snapshot of the MHD and LSK simulations at 04:09:00 UT, in which
the magnetic and electric fields are plotted along with the particle energy fluxes. The first row
111 shows Bz and Ey from the MHD simulation. The second (third) row displays the ion (electron) high-energy and low-energy fluxes from the LSK simulations.
Regarding the energy fluxes, two critical remarks are necessary. First, for each species, the low-energy particles are spread cross the magnetotail and are convected from the reconnection site to the inner magnetosphere along the flow channels, whereas the majority of high-energy particles are present in association with the dipolarization. The high-energy flux concentration at the dipolarization location is a consequence of nonlocal acceleration, rather than convection, because the earthward convection driven by the energy-independent EB× drift would result in the same pattern for the high- and low- energy fluxes, which is not the case.
Second, the convection-dominated low-energy flux pattern is the same for both species, and both high-energy electrons and high-energy ions are associated with the dipolarization, suggesting that the underlying nonlocal energization and transport mechanisms apply to both
species. A notable difference is that in the region inside of X ~10 RE , the high-energy ion fluxes flow toward the dusk side, whereas high-energy electron fluxes flow to the dawn side. This difference is less obvious for the low-energy fluxes. This is because in the inner magnetosphere the energy- and charge-dependent gradient and curvature drifts comprise the major component of particle guiding-center drift (see below).
112
Bz1.0 and flow 040900UT Ey and flow 040900UT -10 -10 6 30 -5 -5 4 20 P3 P3
0 0 2
Y(Re) 10 Bz(nT) Ey(mV/m) 0.8 5 0 5 0
10 -10 10 -2 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 300km/s X(Re) X(Re) Ion eflux(37keV-48keV) Wtest=32.9keV Ion eflux(4.6keV-8.0keV) 6 7 -10 0.6 10 -10 10
105 106 -5 -5 /sr/keV) /sr/keV)
4 2 5 2 P3 10 P3 10
0 0
Y(Re) 103 104
5 5 102 103
0.4 Eflux(keV/s/cm Eflux(keV/s/cm
10 101 10 102 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 X(Re) X(Re) Ele eflux(36keV-46keV) Wtest=35.8keV Ele eflux(1.94keV-3.36keV) -10 107 -10 108
0.2 106 107 -5 -5 /sr/keV) /sr/keV)
5 2 6 2 P3 10 P3 10
0 0
Y(Re) 104 105
5 5 103 104 Eflux(keV/s/cm Eflux(keV/s/cm
10 0.0 102 10 103 -5 1.0-10 -15 -201.2 -25 1.4 -5 1.6-10 -15 -201.8 -25 2.0 X(Re) X(Re)
113
Figure 5.2. A snapshot of the MHD and LSK simulations for the February 07, 2009 event. First
row: Z-component of the magnetic field ( Bz ) and Y-component of the electric field ( Ey ) from the
MHD simulation. Black arrows represent flow vectors. Second row: energy fluxes of high-energy
(37-48keV) and low-energy (4.6-8.0keV) ions from the ion LSK simulation. Third row: energy fluxes of high-energy (36-46keV) and low-energy (1.94-3.36keV) electrons from the electron LSK simulation. All the quantities are plotted on the maximum pressure surface. The representative ion
(electron) trajectory, shown in Figure 5.3 (Figure 5.4), is superimposed on the fields and high- energy ion (electron) flux diagrams. The white line shows the ion trajectory and the black line shows the electron trajectory. The solid circles represent particle locations at 04:09:00 UT. The
energies of the test particles (Wtest ) are labeled at the top of the high-energy flux plots. The location
of the virtual spacecraft P3 is represented by the black square.
To clearly identify acceleration mechanisms, we examine individual particle trajectories.
Figure 5.3 is a reproduction of Figure 4.3 in Chapter 4, summarizing the characteristics of a
representative ion of a large pitch-angle from the ion LSK simulation. The trajectory of this ion is
also superimposed on the fields and the 37keV-48keV ion flux in Figure 5.2 above (white line).
This ion was launched near the reconnection site at 04:06:20 UT at (X ,YR ) ( 18.3,0.27) E . The
ion first undergoes nonadiabatic acceleration from 04:06:20 UT to 04:07:26 UT (before the first
vertical line) in the weak field region with X ~ 13RE , where the gyro radius is comparable to
the characteristic scale of the magnetic field variation. It then gains energy adiabatically from
04:07:06 UT to 04:10:00 UT (between the vertical lines) as it drifts earthward toward a stronger
magnetic field region. The adiabatic acceleration mechanism is characterized by the relatively
114 large adiabaticity parameter ( ~4) and the stable first adiabatic invariant ( M ~1keV / nT ).
During the adiabatic acceleration process, the ion exchanges energy with the perpendicular electric field. It gains energy during one-half gyration and loses energy during the other half, as implied by the oscillations in the energy and the vector product of the electric field and the ion velocity.
The net effect of the energy exchange is that the ion energy gain exceeds its energy loss as it moves into regions with a stronger magnetic field. This process is known as betatron acceleration
[Northrop, 1963; Baños, 1967; Birn et al., 2013]. A more detailed description of the trajectory and characteristics of this ion was given in the previous chapter (section 4.4.2.2).
5 Rx -10 0 -5 Ry -10 R(Re) -15 Rz -5 -20 40 30 04:07:26 20 0 04:10:00 W(keV)
10 Y(Re) 0 10 8 6 5 4 Kappa 2 0 3.0 10 2.5 subtract 2.0 ExB -5 -10 -15 -20 -25
(keV/nT) 1.5 X(Re) T 1.0
/B no subtract 0.5 10 per 0.0 W 40 Bx 20 By 0 Bz 5 B(nT) -20 Bt -40 8 6 Epar 0 4 Z(Re) 04:10:00 2 Eper E(mv/m) 0 -2 15 -5 04:07:26 10 5 0 -5 -10
E*V(keV/s) -10 -15 -5 -10 -15 -20 -25 hhmm 0408 0410 2009 Feb 07 X(Re) Figure 5.3. Characteristics of a representative ion. Left column: from top to bottom, the variables are the position vector, the total energy, kappa, the first adiabatic invariant (the green (black) line is the perpendicular energy over the total magnetic field before (after) subtraction of the energy
115 associated with the EB drift), the magnetic field, the electric field, and the vector product of the electric field and the ion velocity. The time interval between the two vertical dashed blue lines is
04:07:26-04:10:00 UT. Right column: projections of the ion trajectory onto the X-Y and X-Z planes. The blue solid circles correspond to the initial and final points of the 04:07:26-04:10:00
UT interval.
Figure 5.4 shows the characteristics of a typical electron from the electron LSK simulation.
Its trajectory is also plotted in Figure 5.2 (black line). In a very limited region at
(X ,YR ) ( 20,10) E before 04:04:20 UT (the first vertical line), the electron motion is nonadiabatic. Its energy increases from a few keV to ~30 keV from 04:04:20 UT to 04:09:00 UT
(between the vertical lines) as its motion quickly becomes adiabatic ( 5). The first adiabatic invariant is M ~1.1keV / nT . The electron gains energy from the perpendicular convection electric field. The parallel electric field is negligible. Note that in the electron LSK code, the electron with 5 is pushed according to the guiding-center drift equation and the first invariant does not change. The increase of electron energy is numerically realized by conserving the first invariant. Because the electron undertakes adiabatic motion, the guiding-center drift velocities were conveniently calculated and extracted from the simulation. From 04:04:20 UT to 04:09:00
UT, the X-component of the EB drift velocity is about 200km/s. Because the MHD flow is curved, a substantial Y-component (~100 km/s) is also present. The gradient drift is about 100-
200 km/s for a short period from 04:04:20 UT to 04:06:00 UT. The curvature drift velocity for this quasi-perpendicular electron shows some spikes with peaks of 200 km/s when the electron encounters the current sheet during its bounce motion. It is clear that the EB drift velocity is consistently much larger than the gradient and curvature drift velocities in the tail. Therefore, the
116
electron closely follows the flow channel, as demonstrated by the trajectory in Figure 5.2. The gradient drift later on dominates the electron guiding-center motion in the inner magnetosphere
for this quasi-perpendicular electron. After ~04:11:00 UT, the gradient drift velocity is about 100
km/s insideX 8 RE while the EB and curvature drift velocities are negligible. We have
checked that the curvature drift is comparable to the gradient drift in the inner magnetosphere for
electrons with relatively small pitch angles.
105 Rx -10 0 -5 Ry -10 R(Re) Rz -20-15 40 Wper -5 30 20 Wpar 10 W(keV) W 04:09:00 0 T 30 0 20 y(Re) 10 Kappa 0 4 5 04:04:20
T 3 /B 2 per
W 1 (keV/nT) 0 10 Bx 3040 20 By -5 -10 -15 -20 -25 10 Bz x(Re)
B(nT) 0 B -20-10 T 10 6 4 Epar 2 0 Eper E(mV/m) -2 5 400 Vx 200 0 Vy -200 Vz 0 Veb(km/s) -400 400 Vx z(Re) 04:09:00 200 0 Vy -200 Vz -5 04:04:20
Vgb(km/s) -400 400 Vx 200 0 Vy -200 Vz -10 Vc(km/s) -400 -5 -10 -15 -20 -25 hhmm 0405 0410 0415 2009 Feb 07 x(Re) Figure 5.4. Characteristics of a representative electron. Left column: from top to bottom, the variables are the position vector, the perpendicular, parallel and total energies, kappa, the first adiabatic invariant, the magnetic field, the electric field, the EB drift velocity, the gradient drift
velocity, and the curvature drift velocity. The interval between the two vertical dashed blue lines
117
is 04:04:20-04:09:00 UT. Right column: projections of the electron trajectory onto the X-Y and
X-Z planes. The blue solid circles correspond to the initial and final points of the 04:04:20-
04:09:00 UT interval.
In Chapter 4, we estimated the ion drift velocities and found that the EB drift is the main
component in the high-speed flow channel in the outer magnetosphere, whereas the gradient and
curvature drifts dominate in the inner magnetosphere. Because the ion LSK simulation solves the
Lorentz-force equation of motion rather than the guiding-center equation, it was inconvenient to directly calculate the drift velocities for ions. Here given electron guiding-center velocity calculation, we present a straightforward argument on the relative importance of the ion guiding-
center drifts. As discussed in Chapter 2, the guiding-center velocity (uGC ) for a particle in the
magnetotail can be approximated by:
cEe Mc e mu2 c e e uuuu~~u+ u+11 B 11 (5.1) GC E B c B qB q B s
where u is the guiding-center parallel velocity,uE , uB ,uc are respectively the EB , gradient,
mw2 and curvature drift velocities, q is the electric charge and M is the first adiabatic invariant, 2B
B e and s are respectively a unit vector and a unit length along the magnetic field [Northrop, 1 B
1963]. Given the magnetic field, the gradient and curvature drifts are determined by perpendicular
energy and parallel energies respectively. They both depend on charge. Therefore, with the similar
energy for the quasi-perpendicular test particles, the representative ion has gradient and curvature
drift velocities close to those of the representative electron, but in the opposite direction. On the
118
other hand, the EB drift is independent of the particle data. Therefore for the representative ion,
the relative importance of its EB drift compared to its gradient and curvature drifts is the same as that of the representative electron, namely the EB drift dominates in the tail while the
gradient and curvature drifts take over the guiding-center motion in the inner magnetosphere.
The change in the particle kinetic energy (W ) during adiabatic motion can be expressed
as:
1 dW M B uE (5.2) qdtGC q t
The energy gain results from two effects. The first is due to the guiding-center motion along the
electric field direction and the second is due to the temporal variation of the magnetic field
[Northrop, 1963]. The second term is independent of charge. However, the first term does depend
on charge. If both electrons and ions gain energy via a process related to the first term, one expects
them to drift in opposite directions. This is demonstrated by the trajectories superimposed on the
Ey in Figure 5.2. The electron crosses the flow channel from the dusk side to the dawn side, while
the ion does the opposite. A subtle point is that with Ey ~4 6 mV/m, the particle energy changes
by 25.6 keV-38.4 keV if it drifts 1 RE across the flow channel, so to gain tens of keV energy the
required cross-tail drift distance is much smaller in a strong localized electric field than in a weak
large-scale cross-tail electric field. In summary, except for drifting in opposite directions, ions and
electrons with similar energies and pitch angles are not only transported earthward in a similar
way, but also accelerated by the same nonlocal process during transport. This explains the similar
patterns across species seen in Figure 5.2.
119
Because the nonlocal acceleration mechanism depends critically on the adiabaticity of
particle motion, we calculate the kappa parameter on the global scale. The results are shown in
Figure 5.5. The middle (right) column shows kappa for ions/protons (electrons) obtained by fixing
the first adiabatic invariant ( M 1 keV/nT ) for the given magnetic field from the MHD simulation.
This generic first adiabatic invariant value is close to that of the representative ion and electron. It
corresponds to W ~ 30 keV particles in the dipolarized region where Btotal ~ 30 nT . It is very
interesting to note the pattern of kappa. For ions, inside of X ~ 20RE , the dipolarized regions
have larger kappa values than the non-dipolarized regions. This is because: (1) magnetic field in dipolarized regions is large so the gyro radius is small; and (2) the magnetic field is more dipole- like so the radius of field line curvature is large. Were there no dipolarizations, the ion motion
would be nonadiabatic outside of X ~8 RE . Kappa is about 3-4 near the dipolarization region,
confirming that the kappa value derived from the representative ion trajectory is typical. In contrast,
the electrons are adiabatic except in very limited regions close to the reconnection site for this
event. Note that during each cycle of particle bounce motion, kappa reaches its minimum at the
center of the current sheet, thus Figure 5.5 shows the lowest value. In this event ion acceleration
in the second stage becomes adiabatic because of the relative large kappa in the dipolarization
regions, whereas electron motion is adiabatic almost everywhere and not just in the dipolarization
structures.
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0.8
0.6 Bz and flow 040900UT Kappa [proton: 1keV/nT] Kappa [electon: 1keV/nT] -10 -10 6 -10 6 30 0.4 5 5 -5 -5 -5 20 4 4
0 0 3 0 3 0.2 10 Y(RE) Kappa Kappa Bz(nT) 2 2 5 0 5 5 1 1 0.0 10 1.0 1.2 -10 10 1.4 1.60 10 1.8 2.00 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 -5 -10 -15 -20 -25 X(RE) X(RE) X(RE) 300km/s
Figure 5.5. Kappa in the February 07, 2009 event. Left column: Z-component of the magnetic field from the MHD simulation at 04:09:00 UT for the February 7, 2009 event. Middle column:
Kappa for M 1 keV/nT protons in the magnetic field. Right column: Kappa for M 1 keV/nT electrons in the magnetic field. The black arrows represent flow vectors from the MHD simulation.
All the quantities are presented on the maximum pressure surface.
5.4. Conclusions and Discussions
By using THEMIS observations, a global MHD simulation, and ion and electron LSK simulations for the February 07, 2009 event, we attempted to develop a global scenario for particle energization in the magnetotail. Specifically, the distributions obtained by THEMIS P2 at
X ~ 18.6RE (earthward of the reconnection site) were set as the particle sources in the LSK
simulations, therefore the resultant energization by processes near the reconnection site was
quantified. We found that magnetic reconnection produces thermal particles (a few keV) and high-
energy particles that obey a power law distribution (more than 10 keV). These particles are
accelerated far away from the reconnection site (nonlocally) to tens of keV up to a hundred keV
by perpendicular electric fields associated with earthward propagating dipolarizations and fast
flows in narrow channels. The picture of nonlocal particle energization applies to both electrons
121
and ions. The nonlocal acceleration is adiabatic, which is supported by calculations of the
adiabaticity parameter and the first adiabatic invariant for typical high-energy test particles. The
adiabaticity parameter pattern reveals that electron motion is adiabatic except in very limited
regions near the reconnection site, while ion motion is marginally adiabatic in the dipolarized
regions, where the nonlocal acceleration is most effective. As the ion motion becomes adiabatic,
the nonlocal acceleration process for ions is the same as that for electrons. The charge- and energy-
independent EB drift dominates over the charge- and energy-dependent gradient and curvature
drifts in the flow channels in the magnetotail, therefore particles closely follow the flow channels.
On the other hand, the gradient and curvature drifts are the major component of the guiding-center
motion in the inner magnetosphere (within X ~ 8 RE for the February 07, 2009 event), hence
ions drift toward the dusk side and electrons drift toward the dawn side. The higher the particle energy, the more significant the azimuthal drift.
Our study shows that injections of particles during substorms are highly dependent on dipolarizations and flow channels. First, the local time of the injections is determined by the location of flow channels because acceleration by dipolarizations occurs in narrow flow channels, and because in the magnetotail both low-energy and high-energy particles are transported along
flow channels. Second, dispersionless injections occur in the flow channels because the EB drift is dominant. Outside of the injection regions, particle spectra become dispersed as a consequence of the energy-dependent gradient and curvature drifts. Third, dipolarizations are powerful accelerators due to the effectiveness of adiabatic acceleration, so the magnitude of injections depends on dipolarization strength and flow speed. In large substorms, the dipolarizations are expected to be stronger and the flow speeds are expected to be greater, so the convection electric fields are larger, and the nonlocal particle energization is therefore stronger. These findings are
122
consistent with the observed characteristics of substorm injections. Specifically, Gabrielse et al.
[2014] performed a statistical study using THEMIS data that ranged from within geosynchronous
orbit to X ~ 30RE , and found a clear correlation between injections and azimuthally localized dipolarizations, fast flows and impulsive dawn-dusk electric fields. They also found that with increased geomagnetic activity, the dipolarizations are stronger, the electric fields are larger, the injection occurrence rates are higher, the energy spectra are harder (more high-energy particles), and the aforementioned correlation is better. Injections in fast flow channels are found to be dispersionless.
It is notable that in the MHD simulation, physical processes beyond the MHD temporal and spatial scales are missing, therefore the LSK simulations cannot reproduce the shorter-scale spatial and temporal variations seen in the observed fluxes. Perhaps more importantly, using the global MHD+LSK simulation scheme, we cannot properly model the physical processes near the reconnection region. In the present study, particle sources similar to THEMIS P2 measurement
(closer to the simulated reconnection site) are used to represent the energized distributions near the reconnection site, and then the nonlocal energization from the sources to the inner magnetosphere is examined. It is not clear from the observations how far P2 was from the
reconnection site. The region from the assumed X point/line continuing a few RE away along the
reconnection outflow is important for particle energization [e.g. Speiser, 1965; Hoshino et al.,
2001; Imada et al., 2007]. Nevertheless, the combined Maxwellian and power law sources in the
LSK simulations quantify the overall particle energization due to magnetic reconnection and
processes operating within possibly a few RE to the diffusion region. The question as to which
process(es) the source particles in the LSK simulations should be attributed can be resolved by
using PIC simulations, in which the region from the diffusion region to a few RE away is covered
123 by a self-consistent particle simulation and particle energization can be thoroughly examined. Such a study is the subject of the next chapter.
124
CHAPTER 6
Particle-in-cell (PIC) Simulation of Electron Acceleration by
Magnetic Reconnection
6.1. Introduction
As demonstrated in Chapters 3-5, with the MHD+LSK simulations, particle energization
by magnetic reconnection processes can be better quantified by using distributions measured by
spacecraft closer to the assumed or simulated reconnection site as the sources than using empirical
Maxwellian distributions. In particular, in the study of the March 11, 2008 event described in
Chapter 3, we discovered that adding high-energy electrons that follow a power law distribution
near the reconnection X-line in the LSK simulation is crucial in achieving consistency between
simulated fluxes and those obtained by THEMIS P2 atX ~ 14.7RE . This suggests that the
physical processes occurring near the reconnection site are important for understanding particle
energization in the magnetotail as a whole. However, it was not clear how far the spacecraft was
from the reconnection site during that event, because the measurements showed no signatures of
X-line crossing, e.g. simultaneous reversals of Bz and Vx , or signatures related to Hall
reconnection physics, e.g. quadrupole Hall magnetic field [Øieroset et al., 2001]. As described in
Chapter 1, the region from the assumed X-line to a few RE away in the reconnection outflow is
important for particle energization. The question as to which processes the source particles in the
LSK simulations should be attributed to was not answered by the global MHD+LSK simulations,
which cannot properly model the physical processes close to the X-line because electron physics
and Hall physics rather than macroscopic MHD physics are the major players.
125
In this chapter, the problem of how these power law distributed electrons are generated
near the reconnection site is investigated by using an implicit particle-in-cell (PIC) code. By using
a relatively large simulation domain and following the dipolarization propagation, we separate and
quantify electron energization by the reconnection electric field and by dipolarization. To our
knowledge, there is no study that directly quantifies them both.
6.2. Simulation Methodology
PIC simulations solve the Maxwell-Vlasov equation system self-consistently. A PIC
simulation typically contains three components: (1) interpolating particles to the grid points and
integrating density and current on the grid points with the particle data; (2) calculating electric and
magnetic fields on the grid points by solving Maxwell’s equations given the density and current;
(3) interpolating the fields solved on the grid points to particle locations and pushing particles according to the Lorentz-force equation. Note the Lorentz-force equation for the particle equation
of motion is derived from the Vlasov equation with appropriate interpolation of particles to grid
points and interpolation of fields to particle locations [Birdsall and Langdon, 2004; Lapenta, 2012].
The equations of particle motion are coupled to Maxwell’s equations. They can be decoupled by
using an explicit scheme. That is using particle position and velocity from a prior step to calculate
the fields, and then using the newly calculated fields to update particle data. The procedure is
basically cycling the aforementioned three components in the order of (1)(2)(3)(1).
However the explicit scheme is subject to three severe numerical stability constraints [Hockney and Eastwood, 1988; Birdsall and Longdon, 2004]. First, the explicit differentiation of the
Maxwell’s equations requires that a Courant–Friedrichs–Lewy (CFL) condition must be satisfied on the speed of light, ct x. Second, the explicit discretization of the equations of motion
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introduces a constraint related to the fastest electron response time, pet 1 . Third, the
interpolation between grid and particles causes a loss of information and an aliasing instability
called finite grid instability that produces an additional stability constraint x D , where the
proportionality constant is of order one. This requires the grid spacing to be of the order of the
Debye length D or smaller. To overcome the stability constraints in the explicit numerical
scheme, a fully implicit PIC method has been developed [Langdon et al., 1983; Lapenta, 2012 and
references therein]. In the fully implicit method, the field equations and Lorentz-force equations
are coupled: the source terms (density and current) in the field equations contain particle data, and
the particle equations of motion contain field information. Because the number of particles and the
number of grid points in a PIC simulation are usually very large, it is very expensive to solve the
coupled equation system.
The aforementioned difficulties in the fully implicit PIC method are significantly reduced
by using an implicit moment method [Brackbill and Forslund, 1982; Vu and Brackbill, 1992;
Lapenta et al., 2006; Markidis et al., 2010 and references therein; Lapenta, 2012]. The cycle of
the implicit moment method is as follows: (1) predicting particle positions with present and
(unknown) future fields, and approximating the charge and current density (moments of particles)
using Taylor expansion with the predicted particle positions; (2) solving the fields with the
approximated charge and current density; (3) pushing particles using the newly computed fields.
Compared with the explicit scheme, the major change in the implicit moment method is in step (1):
the charge and current density values are extrapolated using a Taylor expansion and are expressed
in terms of the present and future fields. The implicit moment scheme is linearly unconditionally
stable and the stability constraints of the explicit scheme do not apply to it [Brackbill and Forslund,
1982]. Moreover, it has been shown that the implicit method gives an accurate estimate of the
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evolution of the system when the particle displacement per time step is smaller than the grid
spacing, vte x, where ve is electron thermal velocity [Brackbill and Forslund, 1985]. The
inequality can be satisfied with large time steps even when the grid spacing is large compared to
the Debye length. It removes the need to resolve small time scales (plasma frequencies) without
eliminating them. The unresolved scales are kept in an approximate way allowing the coupling
with slower scales that are fully resolved by the large time step [Brackbill and Forslund, 1985].
The state-of-the-art iPIC3D code implements the implicit moment method and is used in the
present study [Markidis et al., 2010 and references therein; Lapenta, 2012].
To simulate magnetic reconnection, we assume a 2D Harris equilibrium [Harris, 1962] as
2 the initial condition, with Be00(xz , ) B tanh( z / ) x and nxz(,) n0 sec( h z / ) nb , where
0.5di is the current sheet width, B0 is the lobe magnetic field, n0 is the density at the center
of the current sheet, and nnb 0.1 0 is the uniform background density representing the level of the
lobe density. The densities are for both electrons and ions. There is an uncertainty in setting the
lobe density relative to the plasma sheet density because in general spacecraft cannot accurately measure the very low energy (a few to tens of eV) electrons due to the photoelectron effect and the relatively high low-energy limits of instruments. In some other PIC simulations, the
background density is set as nnb 0.2 0 [e.g. Hoshino et al., 2001; Pritchett, 2001]. The
temperature is spatially uniform for both species, and the temperature ratio is TTie / 5 . This temperature ratio is larger than that used for the particle sources in the LSK simulations of the
February 07, 2009 event. This is because the LSK source particle temperature is for the reconnection outflow plasmas, whereas the temperature in the present study is for plasmas before
reconnection occurs. The mass ratio is mmie / 400 , which is larger than those used in typical
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explicit PIC simulations studying electron acceleration by magnetic reconnection [e.g. Hoshino et
al., 2001; Drake et al., 2005; Pritchett, 2006a, 2006b] and dipolarization formation [e.g. Sitnov et
al. 2009; Sitnov and Swisdak, 2011]. The time step size is pit 0.05 , corresponding to
pet 1 and cet 0.277 . The time step in the implicit PIC simulation is determined by the physics of interest. Our study requires us to resolve the electron gyro frequency, which is satisfied
by cet 0.277 . The plasma frequency to electron gyro frequency ratio is pe/7.2 ce . Using
3 the observed lobe magnetic field of B0 ~20nT and plasma sheet density of ncm0 ~0.2 observed
by THEMIS P2 in the February 07, 2009 event, the ratio is pe/~7.17 ce . Explicit PIC
simulations typically set this value lower, e.g. pe/~2 ce [e.g. Drake et al., 2005; Pritchett,
2006a, 2006b] because they require pet ~0.1. The grid size is x zd 0.05 i , corresponding
to x zd1 e . Because of this, the electron diffusion region (EDR) on the scale of several de
is resolved. Electron thermal velocity in terms of the speed of light is vce 0.04 , and electron
2 thermal energy is kTee 0.0016 m c . Explicit PIC simulations require x ~ D dvee c d e, so
the electron thermal speed is usually set much higher, e.g. vce ~0.2 [e.g. Drake et al., 2005;
Pritchett, 2006a, 2006b]. The simulation domain is (LLx ,zi ) (60,30) d. The system size is
significantly larger than that used in the study of dipolarization formation [e.g. Sitnov et al., 2009;
Sitnov and Swisdak, 2011]. The relatively large domain allows us to examine energization by the dipolarizations before they reach the boundaries, therefore we can differentiate electron acceleration by reconnection and acceleration by dipolarizations. The coordinate system is similar to GSM. Boundaries for particles and fields are open in the X- and Z- directions [Divin et al.,
2007], and they are periodic in the Y-direction.
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The physical parameters are set to be close to the observed ones, except for the artificial
mass ratio. With the artificial mass ratio, a nontrivial and open question is how to interpret the
simulated quantities in terms of physical quantities. In particular, we are interested in interpreting
the energetics of electrons. The electron acceleration process is strongly coupled with both ion and
electron dynamics [Hoshino et al., 2001]. We feel that with mmie / 400 , it is reasonable to
assume that the electron mass is realistic, whereas the ion mass is smaller than its realistic value.
2 With this assumption, kTee0.0016 m c 0.82 keV , close to the electron temperature
(TkeVe ~ 0.5 1.0 ) observed before reconnection during the February 07, 2009 event. The
reconnection electric field (reconnection rate) is expected to be larger than its realistic value by
about a factor of 2.14 because the reconnection rate normalized by the upstream Alfvén velocity
1/2 ( vmA ) is approximately independent of mass ratio [Shay et al., 2001]. The energy gain by electrons is therefore larger by about a factor of 2.14 at most, since direct acceleration by the reconnection electric field is only one of a few possible acceleration mechanisms. To compare with observations, we present our results in physical units by setting the lobe magnetic field
3 B0 20nT and plasma sheet density ncm0 0.2 . The lobe magnetic field B0 and the plasma
sheet density n0 are used for normalization, e.g. the Alfvén velocity is calculated as
B0 vA 0.006928c ~2078 km/s. 4 nm0 i
6.3. Simulation Results and Comparisons with Observations
6.3.1 Reconnection Structure
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The purpose of the present study is to investigate electron energization in the EDR and at the DF. Before proceeding to analyze electron energization, let’s compare the reconnection structure with previous results as a validation of our simulation. Figure 6.1 shows the magnetic reconnection rate. The reconnection rate is normalized by the upstream Alfvén velocity and the reconnection magnetic field [Shay et al., 1999]. The fast reconnection starts at t~2 sec, when the reconnection rate dramatically increases. The reconnection rate reaches its peak at t~2.9 sec. The peak value is about 0.13, which is comparable with previously reported Hall collisionless reconnection rates [Shay et al., 1999]. Subsequently, the reconnection rate gradually drops to below 0.1. Previous studies suggested two possible mechanisms that can cause the rate drop. The first one is that during the fast reconnection, an electron current layer is formed and elongated at the center of the current sheet, throttling the reconnection outflow [Daughton et al., 2006]. The second mechanism is that in the decay stage of the reconnection there are not enough inflow electrons to support the reconnection electric field, so the rate decreases accordingly [Wan and
Lapenta, 2008]. Our simulation shows elongation of the current layer when the rate drops (see
Figure 6.2(h) below). However, to determine the major underlying mechanism that is responsible for the rate drop requires much more detailed analysis, which is beyond the scope of the present study. Interested readers should read the study by Wan and Lapenta [2008], in which they compared different set-up of simulations and the resultant reconnection rates. The electron layer formed in our simulation is unstable to the tearing instability. A secondary island is generated at t~5.4 sec, and the reconnection rate increases accordingly. Formation of magnetic islands due to the unstable electron current layer and its possible effect on reconnection rate have been reported in explicit PIC simulations [Daughton et al., 2006].
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0.14
0.12
0.1
/c) 0.08 A,up v 0 0.06 (B γ
0.04
0.02
0 0 1 2 3 4 5 6 7 t(sec)
Figure 6.1. Reconnection rate normalized by the upstream Alfvén velocity and magnetic field.
The vertical line corresponds to t=5.40 sec, after which the effect of the magnetic island sets in.
Figure 6.2 shows two snapshots of the fast reconnection at t=3.05 sec and t=5.05 sec. We
select these two times because t =3.05 sec is the time when the DF and the EDR are separated (see
below), and t=5.25 sec is the time just before the magnetic island is produced. During the fast
reconnection, DFs are generated and propagate outwards. The DFs are confined to a region near
the equatorial plane (Z=0). In the magnetotail, the earthward DF is stronger than the tailward DF
due to the background magnetic field. In our simulations, because the initial magnetic field is set
to be uniform in the X-direction, the system is anti-symmetric about the X-line in the X-direction.
Hereafter we focus on the DF that propagates in the positive X-direction. The strength of the DF
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a) e)
b) f)
c) g)
d) h)
Figure 6.2. Snapshots of the reconnection at t=3.05 sec (a-d) and t=5.25 sec (e-h). (a) and (e):
magnetic field Z-component Bz ; (b) and (f): electric field Y-component E y ; (c) and (g): electron
density; (d) and (h): electron velocity X-component. The black lines on top of the color plots are
contours of magnetic field potential Ay , representing projection of magnetic field lines onto the
XZ plane. The coordinates are relative to the center of the system.
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is Bz ~ 15nT at X~0.3 Re at t=3.05 sec. It intensifies to Bz ~ 25nT at X~1.03 Re at t=5.25 sec.
The associated convection electric field at the DF is about 30 mV/m at t=3.05 sec and about 60
mV/m at t=5.25 sec. Both values are much larger than the typical observed value (~10-20 mV/m),
including those in the February 07, 2009 event (see Figure 4.2). This is mainly because the Alfvén
velocity in the simulation with lighter ions is larger by a factor of 2.14, therefore the simulated
outflow velocity and convection electric field are larger by a factor of 2.14 than the observed values.
In addition, the lobe (background) density is set to be one tenth of the current sheet density, which
is half of the value used in other studies [e.g. Hoshino et al., 2001; Pritchett, 2001]. The lower
lobe density likely increases the outflow velocity in our simulation by a factor of 2 . The dipolarization pulse sweeps the electrons, resulting in a low density plasma region behind it. The ion density (not shown) demonstrates the same feature as the electron density. As the fast reconnection proceeds, a thin electron current layer is formed and elongated near the X-line (panels
(d) and (h)). The length of the current layer at t=3.05 sec is ~0.16 Re at t=3.05 sec, and is increased to ~0.8 Re at t=5.25 sec. Close to each of the separatrices, there are counter-streaming fast electron beams consisting of an incoming beam toward the X-line in the inflow region and an outgoing beam away from the X-line in the outflow region. Fast electron beams [Hoshino et al., 2001;
Pritchett, 2001] and formation of an electron current layer [Daughton et al., 2006; Fujimoto, 2006] have been respectively demonstrated in explicit PIC simulations. Both features were also presented in the study of reconnection using an implicit PIC code called CELESTE3D (similar to iPIC3D) by Wan and Lapenta [2008].
6.3.2. Electron Acceleration
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Electron heating and acceleration to suprathermal energies are examined in this section. Figure 6.3
shows the electron temperature and energetic electrons at t=3.05 sec and t=5.25 sec, corresponding
to the snapshots shown in Figure 6.2. At t=3.05 sec, the parallel temperature is Te, ~1 2 keV along the separatrices. From t=3.05 sec to t=5.25 sec, the parallel temperature along the
separatrices is further increased to TkeVe, ~3 4 . Parallel acceleration by electric fields in the
separatrix regions is a well-known aspect of kinetic reconnection [Lapenta et al., 2014]. The
parallel electric field is supported by kinetic effects, including electron density cavities and
pressure anisotropy [Drake et al., 2003, 2005; Egedal et al., 2005, 2012]. As an indicator of the
strength of the parallel electric field, the potential drop along a separatrix for the X-point to the
Boundary system boundary is Eds 22.3 kV ( ds is a length along the separatrix). The potential drop X-point
is much larger than the corresponding parallel temperature. The strong parallel electric field produces fast electron beams [Hoshino et al., 2001; Pritchett, 2001; Lapenta et al., 2010] (also see
Figure 6.2(h) above), which are unstable to streaming instabilities, leading to nonlinear turbulence and electron heating [Goldman et al., 2008]. In contrast to the parallel temperature, the perpendicular temperature is larger in the outflow region behind the DF, indicating that perpendicular heating occurs mainly in the outflow region, although substantial heating also occurs in a thin layer near the X-line. This thin layer is within the electron current layer described above
(Figure 6.2(h)). The maximum of the temperature in the outflow region at t=5.25 sec is
Te, ~4 keV . The thermal electron distribution is isotropic in the perpendicular plane (the other
perpendicular temperature is not shown). From t=3.05 sec to t=5.25 sec, the region of electron heating expands as the reconnection jets propagate outwards. To show production of energetic
electrons, the averaged differential energy flux JEe ( ) at 25.5 keV-51 keV is calculated by using
135
a) e)
b) f)
c) g)
d) h)
Figure 6.3. Electron heating and production of energetic electrons at t=3.05 sec (a-d) and t=5.25 sec (e-h). (a) and (e): electron temperature parallel to the magnetic field; (b) and (f): electron temperature perpendicular to the magnetic field; (c) and (g): averaged differential energy flux
JEe () (including the current sheet electrons and the background electrons) in the energy range of
25.5 keV-51 keV; (d) and (h): differential energy flux of the current sheet electrons in the energy
range of 25.5 keV-51 keV.
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12E the formula JE() EfE (), where E is the kinetic energy and f ()E is the e 4 m distribution function. Similar to the pattern of perpendicular temperature, the high-energy flux concentrates in the outflow region, representing an extension of the heated distribution to suprathermal energies as a result of continuous production of energetic electrons by reconnection exhaust. However, the high-energy flux peaks at the DF, indicating the DF is a powerful accelerator for energetic electrons. Panels (d) and (h) contain the energy flux contributed only by the current sheet energetic electrons. The current sheet electrons do not enter the diffusion region; they are swept away by the DF. However, they contribute significantly to the high-energy flux at the DF (the reasons are discussed below).
To separately identify energization by reconnection and by dipolarization, we need to identify the
EDR and DF locations. The EDR is defined as the region where the electron flow speed perpendicular to the magnetic field deviates from the perpendicular E×B drift velocity. The DF
location ( X DF ) is defined as the location of the maximum Bz . Figure 6.4 shows two snapshots of
the X-components of the EB drift velocity, the electron velocity, the ion velocity, and Bz at the
center of the current sheet (Z=0). In the center of the current sheet, the X-components of velocities
are perpendicular to the magnetic field, whose X- and Y-components vanish due to anti-symmetry
about Z=0 plane. Therefore the X-components of velocities can help us identify the EDR. At the
beginning of fast reconnection, the dipolarization pulse is embedded in the EDR. As the fast
reconnection proceeds, the DF propagates outward and intensifies. At t=3.05 sec, the DF with
Bz ~15nT at X ~ 0.3RE is well separated from the EDR, whose outer edge is X ~ 0.16RE . At
t=5.25 sec, it propagates to X~1.03 RE and its peak value is Bz ~ 25nT . During the fast
reconnection the EDR is elongated. However, the elongation is not as fast as the DF propagation.
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Therefore, after t=3.05 sec, the DF is further separated from the EDR. This separation provides an
opportunity to identify acceleration of electrons by the reconnection electric field in the EDR and
acceleration by the DF during its propagation.
v and B (t = 3.05 sec) v and B (t = 5.25 sec) 4 4 x 10 x z x 10 x z 6 30 6 30 v v ExB,x ExB,x v v 4 e,x 20 4 e,x 20 v v i,x i,x B B 2 z 10 2 z 10
0 0 (nT) 0 0 (nT) z z (km/s) (km/s) x x B B v v −2 −10 −2 −10
−4 −20 −4 −20
−6 −30 −6 −30 −2.4 −1.8 −1.2 −0.6 0 0.6 1.2 1.8 2.4 −2.4 −1.8 −1.2 −0.6 0 0.6 1.2 1.8 2.4 x(Re) x(Re)
Figure 6.4. Dipolarization and reconnection outflow at the center of the current sheet (Z=0) at t=3.05 sec (left) and at t=5.25 sec (right). The green line is the X-component of the EB drift velocity, the blue line is the X-component of electron velocity, the black line is the X-component of the ion velocity, and the red line is the Z-component of the magnetic field. The vertical line represents the DF location.
In the following discussions, we will quantify electron acceleration by calculating electron distributions from the simulation. We intend to fit each electron distribution to a Maxwellian distribution or a combination of a Maxwellian distribution with a power law distribution,
depending on whether a substantial high-energy tail is present. The Maxwellian distribution fM with temperature T is defined as:
138
E 1 EE fM exp with E [0, ) (6.1) Eth22 EE th th
where ETth 2 is the one-dimensional thermal energy. The normalized power law distribution
f p can be defined as [Pan et al., 2014a, also please see Appendix 1 of the thesis]:
1/2n EE1 f p with EEE[,min max ] (6.2) EabEth th
where Emin is the lower-energy bound, Emax is the higher-energy bound, n 1.5 ,
aE min Eth and bE max Eth . We combine the power law distribution with the Maxwellian
distribution, namely
E xfM E [0, Emin ] E Eth f (6.3) Eth E yf E [ E , E ] p min max Eth
where the coefficients x and y are the weights of the Maxwellian distribution and the power law
distribution, respectively. By requiring the combined distribution and its first-order derivative to be continuous at the lower-energy bound, we have
EEmin min xfMp yf (6.4) EEth th
E n min (6.5) 2Eth
Finally, the combined distribution satisfies the normalization condition
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Emin EE xfEth d y1 (6.6) 0 M EEth th
Solving for the coefficients x and y , the combined distribution is completely determined.
We calculate the electron distributions at the DF by following the DF propagation. Figure
6.5 shows the distributions (data points) at t=0 sec, t=3.05 sec and t=6.61 sec. The electrons are
selected in the region of X DFidXX DF and 4dZee 4 d. Note ddie 20 ~ 0.08 R E. The distributions at t=0 sec and t=3.05 sec are fitted to Maxwellian distributions (the black and the blue
lines). The fitting is done by inspection of trials of different temperatures. The criteria are that the
results need to be approximately evenly spread on both sides of the fitted lines and that the results
near the maximum of the distributions are weighted more heavily because they are expected to
have least uncertainty. The electron temperature increases from initial value TkeVe ~ 0.82 to
TkeVe ~ 2.66 at t=3.05 sec. After the separation of the DF from the EDR at t=3.05 sec, electron
temperature increases a little. However, a high-energy tail is generated from t=3.05 sec to t=6.61
sec. The high-energy tail follows a power law distribution, which is a straight line in the plot with
a log-log scale. Therefore, the distribution at the DF at t=6.61 is fitted to a combined distribution
(the red line). The fitted temperature is TkeVe ~ 2.86 , the lower-energy bound is EkeVmin ~ 14.7 ,
and the power law index is n ~5.15. The power law electrons are about 2.1% of the total electron
population. The high-energy tail generated at the DF is consistent with the feature reflected in
Figure 6.3(g), in which the 25.5 keV-51 keV energy flux peaks at the DF, reaffirming that DFs are
powerful in accelerating electrons to suprathermal energies.
For a comparison, the distributions in a fixed box with 2dXii 2 d and 4dZee 4 d
are also shown in Figure 3. This box approximates the EDR at t=3.05 sec, whose length is Ldei ~ 4
140
Electron distribution (at DF) 1 10
0 10
−1 10
−2 10 )
−1 −3 10 f(keV
−4 10 t=0 sec T =0.82 keV e −5 10 t=3.05 sec T =2.66 keV e −6 10 t=6.61 sec T =2.86 keV e
−7 n=5.15; E>14.7 keV 10 −2 −1 0 1 2 3 10 10 10 10 10 10 E(keV)
Electron distribution (−0.16Re < x < 0.16Re) 1 10
0 10
−1 10
−2 10 )
−1 −3 10 f(keV
−4 10 t=0 sec −5 T =0.82 keV 10 e t=3.05 sec
−6 T =2.04 keV 10 e t=6.61 sec T =4.29 keV −7 e 10 −2 −1 0 1 2 3 10 10 10 10 10 10 E(keV)
141
Figure 6.5. Normalized electron distribution functions at the DF (top) and in the EDR (bottom).
The data points are from the simulation. The solid lines are Maxwellian distributions, except for
the red line in the top panel, which is a combination of a Maxwellian distribution and a power law
distribution. The horizontal arrow points to the lower-energy bound that delineates the power law
distribution from the Maxwellian distribution.
and whose width is ee ~ 8d . At t=3.05 sec, the temperature in the EDR is TkeVe ~ 2.04 , which
is smaller than that at the DF TkeVe ~ 2.66 , suggesting that the increase in temperature at the DF
is not due to temporal effects associated with the non-uniform reconnection rate. More importantly,
we have checked that throughout the simulation, the electron distributions in the EDR are close to
Maxwellian, without substantial high-energy tails. The electrons are heated to TkeVe ~ 4.29 at
t=6.61 sec, resulting from magnetic island formation and contraction after t~5.25 sec.
Three notes are necessary before moving on to compare the power law distribution with
observations. First, we found that the power law tail is gradually enhanced as the DF intensifies and propagates away from the EDR beginning from t=3.05 sec. The tail at t=6.61 sec is the strongest before the DF reaches the boundary of the simulation domain. Second, we do not focus on electron heating by the magnetic island near the initial X-line because extensive studies have reported that magnetic islands can be formed in the reconnection region and accelerate electrons
[e.g. Mattaeus et al., 1984; Drake et al., 2005, 2006; Hoshino, 2012]. Third, we have assumed that electrons are convected with the DF as they are magnetized outside of the EDR, so the electrons
heated in the magnetic island are not expected to enter the distribution at the DF.
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The power law distribution at the DF qualitatively agrees with observations. The
parameters derived from the electron distribution observed by THEMIS P2 at X 18.6RE in the
February 07, 2009 substorm event are TkeVe ~ 2 , EkeVmin ~11 and n ~5.5 [Pan et al., 2015a].
In this event, P2 was in the earthward reconnection outflow region, and these parameters were calculated with a DF passing the spacecraft. The parameters of electron distribution in the
reconnection outflow at X ~ 20RE inferred from a global MHD+LSK simulation during the
March 11, 2008 substorm event are TkeVe ~ 2 , Emin ~9 keV , and n ~4.5 [Pan et al., 2014a].
The simulated distribution at the DF is also comparable with that observed by Cluster spacecraft
at X ~ 16.4RE in the reconnection downstream region with a DF passing [Imada et al., 2007].
Moreover, the Cluster observations demonstrated that the electron spectrum is harder at the DF than that close to the X-line, in agreement with the simulation. The simulated distributions are not verifiable with observations from WIND spacecraft in the distant tail-reconnection region reported by Øieroset et al. [2002], because no signatures of dipolarization were observed there [Øieroset et al., 2001].
To identify the heating and acceleration mechanisms, we calculate electron velocity distributions. The results are shown in Figure 6.6. As in Figure 6.5, the distributions at the DF are
calculated from electrons in X DFidXX DF and 4dZee 4 d , and the electrons in the
EDR are from 2dXii 2 d and 4dZee 4 d. At the DF, electrons are energized predominantly in the perpendicular direction (panel (a)). The distribution is isotropic in the perpendicular plane (panel (b)). Note we have also checked the distribution at the DF as a function
of perpendicular energy (f (E ) ) and as a function of parallel energy ( f ()E ), and found that the acceleration is in the perpendicular direction. In the EDR, electrons are accelerated in the X-
143
direction and ejected outward, forming a thin current layer (panel (c)). Electrons are also clearly
accelerated and heated by the reconnection electric field in the negative Y-direction (panel (d)).
The acute readers might find it puzzling that the electron distribution in the EDR at t=5.25 sec can
be fitted to a Maxwellian distribution as a function of energy, even though the velocity distributions
clearly demonstrate non-Maxwellian features. However, the key observation is that the velocity
distribution f (,vvx y ) is close to a half-Maxwellian distribution, namely
fv(,xy v 0)~(, f Mxy v v 0) and fv(,xy v 0)~0, where fM (,vvxy ) represents the Maxwellian
distribution.
(a) (c)
(b) (d)
Figure 6.6. Electron velocity distributions at the DF (a-b) and in the EDR (c-d) at t=5.25 sec.
144
When analyzing 25.5 keV-51 keV electrons shown in Figure 6.3 (panels (c-d) and (g-h)),
we found that the current sheet electrons contribute significantly to the energy flux at the DF, even
though they do not enter the diffusion region. We now discus the reason. We now discus the reason.
The densities of electrons and ions from the current sheet and background at Z=0 at t=5.25 sec are
shown in Figure 6.7. As expected, the background density peaks of both species are behind the DF
whereas the current sheet density peaks are ahead of the DF because of the sweeping and pileup
effect. However, two features are subtle and important. First, the shape of the electron background
density is similar to that of the DF, indicating the background electrons are convected with the DF and may be adiabatically accelerated as the DF intensifies. Second, the current sheet particles penetrate the DF. The electrons that penetrate into the slowly varying DF (on the ion time scale) are expected to be accelerated very efficiently because their first adiabatic invariant is expected to be conserved. The limited penetration is due to kinetic effect (finite-Larmor-radius effect and/or inertia effect), as suggested by the different penetration depths for electrons and ions. These pieces of evidence, together with the information that the electrons are accelerated predominated in the perpendicular direction, suggest that the acceleration mechanism operating at the DF is betatron acceleration. The betatron effect in the present study seems to be different from that discussed in
Chapter 2 (also see Pan et al. [2012]). There both the initial distributions and accelerated distributions were power law distributions, and the acceleration occurred in the region from the tail to the inner magnetosphere. The result of betatron acceleration was uniformly shifting the distribution to a higher-energy, without significantly changing the shape of distributions. In the present PIC simulation, Maxwellian initial distributions are applied. Observations have shown that the plasma sheet prior to reconnection has significant high-energy electrons [e.g. Deng et al., 2010].
Future studies of electron acceleration by magnetic reconnection should add high-energy electrons
145
to the initial distributions, in a same approach as for the source particles in the LSK simulations.
It is likely that the electron distributions as functions of energy will no longer be Maxwellian in
the EDR and that the difference between distributions in the EDR and at the DF is quantitative rather than qualitative. Another possible reason causing the difference in betatron effect is that some of the electrons immediately downstream of the outflow may encounter nonadiabatic orbits, as demonstrated by Hoshino et al. [2001]. Nonadiabatic and irregular orbits can prevent electrons from convecting with the magnetic field, leading to mixing of different populations during DF formation. Without electron orbit information, it is difficult to pinpoint the significance of the nonadiabatic effect. Nevertheless, with the evidence that electrons are predominantly accelerated in the perpendicular direction, that the background electrons are convected with the magnetic field,
and that the current sheet electrons penetrate into the DF, we feel betatron acceleration is a major
contributor to the energetic electrons in the outflow and at the DF in particular.
n and B (t = 5.25 sec) z 0.6 30 n e,c n i,c 0.5 n 25 e,b n i,b B 0.4 z 20
) 0.3 15 −3 (nT) z B n (cm 0.2 10
0.1 5
0 0
−0.1 −5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 x(Re)
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Figure 6.7. Densities of electrons and ions from the current sheet ( ne,c and ni,c ) and background
( ne,b and ni,b ) at the center of the current sheet (Z=0) at t=5.25 sec.
6.4. Conclusions
In this chapter, we used the implicit PIC code to simulate 2D magnetic reconnection with
physical parameters that are close to the observed values in the magnetotail. The simulation results
were validated against satellite measurements and previous explicit PIC simulations. Specifically,
the DF strength (~15-25 nT within X~1.03 Re) is comparable with observations. The associated
electric field (~30-60 mV/m) is much larger than the observed value (~10-20 mV/m) because of
lighter ion mass is applied in the simulation. We found that electrons are heated to TkeVe, ~2 4 in the parallel direction along the separatrix, similar to previous results. Electrons are heated to
TkeVe, ~2 4 in the perpendicular direction in the reconnection exhaust, with the temperature
maximum located behind the front. The energetic electrons (25.5 keV-51 keV) are generated in
the region where electrons are heated in the perpendicular direction. However, the peak flux of
energetic electrons occurs at the DF. The electron distribution function as a function of energy in
the EDR is Maxwellian. In contrast, in addition to a thermal Maxwellian component with
TkeVe ~ 2.86 , the electron distribution at the DF (~1.7 Re away from the X-line) at t=6.61 sec has
a substantial high-energy tail, with E14.7 keV and n ~5.15. These characteristic values are
comparable to those observed by THEMIS P2 (X 18.6RE ) in the reconnection outflow region with a DF passing in the February 07, 2009 substorm event, and those in the outflow region
(~20X RE ) inferred from the global MHD+LSK simulation of the March 11, 2008 substorm
147 event. The simulated power law distribution is also comparable with that from the Cluster observations reported by Imada et al. [2007]. Furthermore, the simulation reproduced the Cluster observed feature that electron spectrum at the DF in the downstream region is harder than that in the diffusion region. The power law tail in our simulation is caused by acceleration associated with the DF rather than by acceleration in the EDR because it is generated after the DF separates from the EDR. To our knowledge, there has been no simulation study which directly quantifies both the acceleration in the EDR and the acceleration associated with the DF. The energization mechanism operating in the EDR is the reconnection electric field acceleration, and that in the outflow and at
DF is likely betatron acceleration.
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CHAPTER 7
Conclusions and Problems for the Future
7.1. Conclusions
We have studied in this dissertation the problem of energization of particles (both electrons
and ions) to tens and hundreds of keV and the associated transport process in the magnetotail
during substorms.
We developed a simple analytical model to quantify nonlocal adiabatic acceleration effects
[Pan et al., 2012]. In this model, given the particle differential flux at the presumed source in the tail, the flux in the inner magnetosphere is calculated by assuming conservation of the first and second adiabatic invariants. The model is characterized by two parameters, the magnetic field compression factor accounting for the enhancement in the perpendicular flux due to betatron acceleration and the contraction factor accounting for the enhancement in the parallel flux due to
Fermi acceleration. The model shows that the adiabatic enhancement of flux strongly depends on the slope of the source flux. For example, at high energies, a large increase in perpendicular
(parallel) flux can be induced by a small compression (contraction) factor given a steep source
spectrum. When compared with THEMIS data, the model worked surprisingly well in predicting
flux enhancement of high-energy electrons from the tail to the inner magnetosphere, despite the
fact that betatron and Fermi acceleration were characterized by just two parameters. In retrospect,
the main reason for this was that high-speed flows were present in the events analyzed, which
confined the electrons to narrow flow channels and carried electrons of all energies earthward via
the charge- and energy-independent EB drift. As reviewed in Chapters 1-2, the theory of
149 adiabatic particle motion has been developed since 1940s [Alfvén, 1950; Northrop, 1963; Banõs,
1967] and has been extensively applied to the magnetosphere [e.g. Tverskoy, 1969; Sharber and
Heikkila, 1972; Kivelson et al., 1973; Lyons, 1984; Artemyev et al., 2011; Ashour-Abdalla et al.,
2011; Birn et al., 2012 and references therein]. In particular, it was demonstrated that by applying conservation of the first and second adiabatic invariants in an analytical model of the magnetic field, particle distributions at different locations can be mapped from the distribution at a given position [Tverskoy, 1969; Lyons, 1984; Artemyev et al., 2011; Birn et al., 2012]. The success of our two-parameter model, and the factor that differentiate it from previous models is that our model applies to cases in which the initial and final distributions (fluxes) are in the same high-speed flow channel.
The analytical model is rather restrictive because the particles are assumed to be uniformly accelerated. A more powerful approach is to combine a LSK simulation with a global MHD simulation driven by measured upstream solar wind. The approach is much less restrictive because in the LSK simulation trajectories of millions of particles are calculated in the realistically determined MHD fields and the results are statistical. The MHD+LSK method was applied to model electron acceleration and transport in the March 11, 2008 substorm event [Pan et al., 2014a].
We found that adding a high-energy tail of electrons obeying a power law distribution in the LSK electron source was crucial in obtaining consistency between the simulated energetic electron flux
and that measured by THEMIS P2 at X ~ 14.7RE , suggesting that the power law distributed electrons were produced from the source region, which was close to reconnection in the MHD simulation. Meanwhile, the LSK simulation reproduced the flux enhancement from P2 in the tail to P3/P4 in the inner magnetosphere. This flux increase was by about an order of magnitude for the 41-95 keV energy channel. The enhancement occurred in the dipolarization region where the
150
magnetic field was compressed by about a factor of 2. The enhancement was uniform for energies
greater than ~ 1 keV, suggesting that the nonlocal acceleration process was adiabatic. The adiabatic
acceleration mechanism was further confirmed by characteristics of electrons trajectories, which
showed that for typical tens of keV electrons the adiabaticity parameter ( ) was much greater than
unity ( 1) in the flow channels and the first adiabatic invariant was approximately conserved.
This study provided convincing quantitative evidence of local acceleration and nonlocal
acceleration in a substorm event. In addition, electron transport in the magnetotail was found to be
determined by high-speed flows generated by magnetic reconnection. The EB drift was
statistically dominant even for tens of keV electrons. Electrons gradient and curvature drifted towards the dawn side in the inner magnetosphere. Ashour-Abdalla et al. [2011] applied the
MHD+LSK simulations to study electron acceleration during a substorm event that occurred on
February 15, 2008. They found that nonlocal betatron acceleration was the major mechanism that
produced energetic electrons (approaching 100 keV) observed by THEMIS P4 in the inner magnetosphere in that substorm. In the February 15, 2008 event, there was no observation in the
tail that could be used to quantify the presence of energetic electrons closer to the reconnection
region. Ashour-Abdalla et al. [2011] used a Maxwellian source in the LSK simulation, which may
have underestimated the flux of energetic electrons resulting from reconnection. Our study
extended their work by utilizing simultaneous measurements in the tail and in the inner
magnetosphere and adding power law electrons to the source in the LSK simulation. As a result,
both local acceleration and nonlocal acceleration were quantified in a substorm event.
Having used the MHD+LSK simulations and THEMIS spacecraft measurements to study
electron energization in the magnetotail, we applied them to examine ion energization associated
with magnetic dipolarization during the weak February 07, 2009 substorm event [Pan et al.,
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2014b]. We found that the major high-energy ion flux enhancements observed in the inner magnetosphere were due to nonlocal acceleration by the dipolarizations and high-speed flows. Ions originating from the reconnection site underwent a two-stage energization process. Not far from the reconnection region, where the magnetic field was weak, the ions were nonadiabatically accelerated. Subsequently, they adiabatically gained energy as they caught up with and rode on the earthward propagating dipolarization structures. They could catch up with the dipolarizations because of plasma compression at the dipolarization regions. For ion transport, we found that in
the magnetotail, the high-speed flows controlled ion transport via the EB drift, whereas close to
the Earth, ions gradient and curvature drifted towards the dusk side. The ion acceleration scenario
is similar to the electron acceleration scenario presented by Pan et al. [2014a], but it is significantly different from the scenario of nonadiabatic acceleration in the wall region with ~1 [e.g. Ashour-
Abdalla et al. 1992b, 1992c, 2009; Zhou et al., 2011], even though in both scenarios, ions gain
energy from the perpendicular electric field in the Y-direction. Dipolarizations and high-speed
flows in narrow channels are critical for ion acceleration in the present study, while they are not
required for the acceleration in the wall region. Therefore, it will be interesting to examine ion
energization in events with different characteristics of flows and dipolarizations (see the discussion
in the next section). The acceleration of ions when they drift across high-speed flow channels was
also discussed by Birn et al. [2013]. However, compared to that study, the ion motion in our study
is much more adiabatic. Moreover, in our LSK simulation, ions originating in the flows closely
follow the flow channels in the tail while it appears that the ions simulated by Birn et al. [2013]
drift across the flow channels much faster (see Figure 3 therein). These differences probably stem
from the differences in the electric and magnetic fields. In our global MHD simulation driven by
realistic upstream solar wind conditions, the flow speed and width, the magnetic and electric fields,
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and the dynamics of the system are realistically determined and event-dependent, while their MHD simulation has generic characteristics of neutral line formation and dipolarizations in the magnetotail (see more discussion in the next section).
We extended the study of the February 07, 2009 event to electron energization and transport, and compared the global energization and transport mechanisms between electrons and ions [Pan et al., 2015a]. We found that thermal ions and electrons (a few keV) observed at the dipolarizations originated from a relatively wide region of the tail near the reconnection site and were convected to the inner magnetosphere. High-energy particles (tens of keV up to ~100 keV) were produced by the perpendicular electric fields associated with the dipolarizations and accompanying high- speed flows. The particle trajectories showed that electrons that originated from the reconnection site were adiabatically accelerated during earthward transport, and, surprisingly, ions were accelerated in a manner similar to that of electrons. However, the high-energy electron motion was adiabatic except in very limited regions near the reconnection while high-energy ion motion was marginally adiabatic in the dipolarization regions. Different from the aforementioned study by Birn et al. [2013], we showed that both electrons and ions closely follow the flow channels in the tail due to the dominant EB drift. To our knowledge, this was the first study that compared global electron and ion energization in the same substorm event.
To understand the power law electrons observed by THEMIS P2 in the tail and inferred from the MHD+LSK simulations, we applied the state-of-the-art implicit PIC code [Brackbill and
Forslund, 1982; Vu and Brackbill, 1992; Lapenta et al., 2006; Markidis et al., 2010 and references therein; Lapenta, 2012] to examine electron acceleration in the reconnection region [Pan et al.,
2015b]. We were able to extend previous simulations of magnetic reconnection by setting a relatively large simulation domain and using realistic physical parameters. The large simulation
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domain allowed us to examine electron acceleration by the reconnection electric field and
acceleration associated with dipolarization pulses in the outflow region [e.g. Sitnov et al., 2009;
Sitnov and Swisdak, 2011]. Previous studies of electron acceleration using explicit PIC
simulations typically chose the proton-to-electron mass ratio as mmie / ~ 100 , the plasma
frequency to electron gyro frequency ratio as pe/~2 ce , and the electron thermal speed in terms
of the speed of light as vce / ~ 0.2 [e.g. Hoshino et al., 2001; Drake et al., 2005; Pritchett, 2006a,
2006b]. In our simulations, these parameters were improved to mmie / 400 , pe/7.2 ce ,
vce / 0.04 . Note that the realistic values are mmie / 1836 , pe/~7.17 ce , vce / ~ 0.04. With
the larger mass ratio and other realistic parameters, the problem of interpreting electron energies
was significantly reduced, hence direct comparisons of simulation results with observations in the
same physical units were made possible. We found that in the reconnection region, electrons with
uniform initial temperature (TkeVe ~ 0.82 ) are heated to Te, ~2 4 keV in the parallel direction
along the separatrices by strong parallel electric fields. Electrons are heated to TkeVe, ~2 4 in
the perpendicular direction by the reconnection electric field in the EDR and by betatron
acceleration in the outflow region. The energy flux of 25.5-51 keV electrons peaks at the DF due
to betatron acceleration of electrons in the reconnection exhaust and penetration of current sheet
electrons into the DF. The electron distribution function in the EDR as a function of energy is
Maxwellian whereas at the DF it has a high-energy tail ( n ~4 6, EkeVmin ~10 20 ) in addition
to a thermal component (TkeVe ~ 2 4 ). These characteristic values are comparable to those
observed by THEMIS P2 (X 18.6RE ) in the reconnection outflow region with a DF passing in
the February 07, 2009 substorm event [Pan et al., 2015a], and those in the outflow region
(~20X RE ) inferred from the global MHD+LSK simulation of the March 11, 2008 substorm
154 event [Pan et al., 2014a]. The simulated power law distribution is also comparable with that from the Cluster observations reported by Imada et al. [2007]. As in the PIC simulation, Imada et al.
[2007] reported that Cluster observed a harder electron spectrum at the DF than in the diffusion region. The high-energy tail results from acceleration associated with the dipolarization pulse
(~1 2RE away from the X-line) rather than acceleration in the EDR because it is generated after the DF separates from the EDR. To our knowledge, there has been no simulation study which simultaneously quantifies the acceleration in the EDR and the acceleration associated with the DF.
To summarize, we have developed a complete model of particle multi-step energization on multiple scales in the magnetotail during substorms, including acceleration localized near the reconnection site and acceleration during plasma earthward transport. Figure 7.1 shows a schematic of the normalized particle distribution functions produced in this model:
(a) In the reconnection region, particles (electrons in particular) are heated to a few keV in
the parallel direction along the separatrices by reconnection parallel electric fields.
Electrons are heated to a few keV in the perpendicular direction by the reconnection
electric field in the EDR and by betatron acceleration in the outflow region. The
electron distribution function in the EDR as a function of energy is Maxwellian
(~24TkeVe ) whereas at the DF it has a high-energy tail ( n ~4 6 ,
EkeVmin ~10 20 ) in addition to a thermal component (TkeVe ~ 2 4 ) [Pan et al.,
21015b]. The DF and the high-energy tail are observed by a spacecraft close to the X-
line, e.g. THEMIS P2.
(b) The keV thermal particles and tens of keV power law distributed high-energy particles
produced by processes near the reconnection region are further nonlocally accelerated
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to tens of keV up to about a hundred keV by perpendicular electric fields associated
with the dipolarizations and reconnection jets as they propagate towards the Earth in
narrow channels [Ashour-Abdalla et al., 2011; Pan et al., 2014a, 2014b, 2015a; Liang
et al., 2014; Pan et al., 2015a]. The resultant energetic electrons are observed by a
spacecraft in the inner magnetosphere, e.g. THEMIS P4. The jet plasmas are hotter than
preexisting plasma sheet plasmas and their distributions are preferably increased at a
few tens of keV (see Figure 6 by Deng et al. [2010] and Figure 8 and Figure 9 by Birn
et al. [2014]), so a spacecraft would observe an increase of power law index (absolute
value) for energetic particles (e.g. EkeV 25 ) during a DF pass [Pan et al., 2012]. The
nonlocal acceleration is adiabatic for both electrons and ions, with a caveat that the
high-energy electron motion is adiabatic in a much wider region in the tail than that of
the ions [Pan et al., 2014b, 2015a]. The power law index for energetic particles under
adiabatic acceleration from the tail to the inner magnetosphere remains relatively
unchanged [Pan et al., 2012].
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P2 before DF (a) (b) EDR P2 at DF P2 at DF P4 at DF f(E) f(E)
Emin
E E0 E
Figure 7.1. Distribution functions produced in the model of particle multistep energization on
multiple scales in the magnetotail during substorms. (a) Particle (electron) acceleration near the
reconnection region. The electron distribution in the EDR is Maxwellian, whereas the electron
distribution at the DF in the immediate downstream of the outflow is hotter and has a high-energy
tail. (b) Particle acceleration during transport. Close to the X-line, the electron distribution in the
preexisting plasma sheet has a high-energy tail. The distribution at the DF is hotter and preferably
enhanced at a few tens of keV. The particles at the DF are adiabatically accelerated as the DF
propagates to the inner magnetosphere. The power law index above EkeV0 ~25 (absolute value)
increases as the DF passes a spacecraft, whereas at the DF it does not change significantly from the tail to the inner magnetosphere. Part (a) is derived from the THEMIS P2 measurements and the PIC simulations in Chapter 6, and part (b) is derived from the analytical model, the THEMIS measurements, and the MHD+LSK simulations presented in Chapters 2-5.
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The acceleration mechanism by early-stage dipolarizations ( ~ 1 2RE to the X-line) in our
PIC simulations is similar to that by late-stage dipolarizations (a few to more than 10RE to the X- line) in our global MHD+LSK simulations. However, the dipolarizations in the PIC simulation are different from those in the MHD simulations. The former dipolarizations are intensified to
Bz ~25nT within ~ 2RE from the X-line, and the dipolarizations are more pulse-like (a few to
tens of seconds long). Pulse-like dipolarizations are also produced in explicit PIC simulations
[Sitnov et al., 2009; Sitnov and Swisdak, 2011]. The dipolarizations in the MHD simulations,
however, are formed in the region close to the Earth. They are intensified as the plasmas carried
by the flows are compressed against the strong near-Earth magnetic field. They are more
sustainable (a few to tens of minutes long) and less pulse-like. In addition to the MHD simulations
in the two events presented in this dissertation, other global MHD simulations also reproduce
dipolarizations resulting from plasma compression and flow braking [Birn et al., 2004, 2011; El-
Alaoui et al., 2012, 2013]. In observations, multiple dipolarizations separated by tens of seconds
to a few minutes have been observed near the reconnection region [Fu et al., 2013] and in the inner
magnetosphere [Zhou et al., 2009]. In fact multiple dipolarization pulses proceeding a sustained
dipolarization were presented for the March 11, 2008 event in Chapter 3, see Figure 3.3-3.4. In
that event, the interval of the dipolarization pulses at P4 (X ~ 10.4RE ) was shorter that that at P2
(X ~ 14.7RE ), indicating the dipolarization pulse region was compressed as the flow speed was
slowed. Combining these pieces of evidence, it seems that dipolarization pulses are generated near
the reconnection region and pile up in the inner magnetosphere as they propagate towards the Earth.
The PIC simulations demonstrated the production of dipolarization pulses, whereas the MHD
simulations showed the plasma compression and the magnetic field pileup. This interpretation
seems to be consistent with a recent 2D multi-scale study of reconnection and dipolarizations by
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Ashour-Abdalla et al. [2015], which showed that a chain of dipolarization pulses are generated by
the unsteady reconnection at X ~ 32RE . The dipolarization pulses propagate earthward, and
merge atX ~ (15 20)RE . The dipolarizations are generated more frequently in the simulation
(one dipolarization in every two seconds) than in the aforementioned observations described in
Chapter 3, Zhou et al. [2009] and Fu et al. [2013]. Despite these indicative evidence, the possible connection between these two kinds of dipolarizations remains to be determined by future large- scale fully kinetic simulations. Therefore, connecting particle energization near the reconnection region demonstrated by the PIC simulations (part (a) of Figure 7.1) and nonlocal energization demonstrated by the global MHD+LSK simulations (part (b) of Figure 7.1) is not as straightforward as it appears.
7.2. Unsolved Problems and Future Work
The results that we have presented in this dissertation suggest directions of future work.
Having studied particle energization in a few substorm events, one can ask a natural question: how
do particle energization and transport depend on particular substorm events? Is the model of
particle multistep energization on multiple scales in the magnetotail still valid in large substorms?
This is an interesting question because of three reasons. First, the substorm events studied in this
dissertation are weak and moderate, reflecting the solar minimum effect during the THEMIS
mission. As we pointed out in Chapter 4 and commented above, a significant difference between
our study of the February 07, 2009 event and the study by Birn et al. [2013], who used idealized boundary conditions for their MHD simulation, is that the MHD fields are appreciably different.
The different MHD fields cause significant differences in ion trajectories. In our study, the ions
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follow the flow channels closely, whereas it appears that the ions simulated by Birn et al. [2013]
drift across the flow channels much faster. Moreover, the motion of ions of tens of keV energy is
marginally adiabatic in the dipolarization regions in our study, whereas it is much less adiabatic in
their study. The nonlocal adiabatic acceleration picture depends on the adiabaticity of particle
motion, which depends on the electromagnetic fields. Second, Liang et al. [2014] examined the
nonlocal acceleration mechanism in two substorm events that occurred on February 15, 2008 and
August 15, 2001. In these two events, the substorm magnitudes were approximately the same---
AE index peaks were ~300 nT and ~200 nT respectively (moderate substorms)---but the magnetotail configuration and electron acceleration were very different. During the February 15,
2008 event, the dayside reconnection occurred nearer the subsolar point, and the high-speed flows in narrow channels produced by azimuthally localized tail reconnection swept the electron sources.
The electrons were adiabatically accelerated, and the electron distribution was pancake-like
( f ()vfv () ) in the inner magnetosphere. In contrast, during the August 15, 2001 event, the
dayside reconnection occurred on the flanks of magnetopause. An X-line extending across the tail
was formed and the earthward flows were slow. The electrons were nonadiabatically accelerated
in the weak field region close to the X-line, resulting in cigar-like electron distributions
( f ()vfv ( )). Third, in Chapter 5 we suggested that characteristics of injections of energetic
particles during substorms are determined by dipolarizations and flow channels through three
factors: the locality of flow channels, the relative importance of EB drift compared to
gradient/curvature drifts, and the adiabatic acceleration efficiency. In larger substorms, the
dipolarizations are stronger and flow speeds are greater [e.g. Gabrielse et al., 2014]. Therefore, the convection electric fields are larger and the adiabatic acceleration is stronger. However, the dependence of dipolarization and flow characteristics (e.g. the width of the flow channels) on
160
geomagnetic activity level is not well established. Considering these three reasons, we feel that it
is important to examine our model in many more events. For example, by comparing weak
substorm events with strong substorm events, we can better understand the robustness of the nonlocal adiabatic acceleration mechanism (especially for ions), and the consequences on the
substorm injections.
Future studies should further investigate the differences and similarities between electron
and ion acceleration in the magnetotail. In this dissertation, we have presented a comparison study
of electron energization with ion energization for the February 07, 2009 substorm event. In this
substorm, ion and electron fluxes demonstrated similar features, e.g. the simultaneous increase of
high-energy fluxes and decrease of low-energy fluxes for both species upon the arrival of the
dipolarization. The ion energization was stronger than the electron’s for this event. However, there
are events in which electron energization is stronger, e.g. the February 15, 2008 event studied by
Zhou et al. [2009] and Ashour-Abdalla et al. [2011]. In both studies, ion fluxes were not shown
because no appreciable ion energization was associated with the dipolarization in the event. A six-
case study by Runov et al. [2011] indicated that both electron and ion high-energy fluxes increase
upon a DF arrival; ion high-energy flux increase is slower and less dramatic than that for electrons
(see Figure 5 therein). However, the limited number of events compromised this conclusion.
Previous statistical studies of many events investigated high-energy electron fluxes associated
dipolarizations by using data from Cluster [Fu et al., 2012c] and THEMIS [Wu et al., 2013], but did not investigate ion fluxes. Therefore, extending the statistical studies to ion fluxes and comparing them with electron fluxes are needed to quantify the similarities and differences between electron energization and ion energization. To better understand the differences and similarities across species, MHD+LSK simulations should be employed to study events in different
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categories, e.g. an event with stronger electron acceleration, an event with stronger ion acceleration,
and an event with approximately equal acceleration for electrons and ions.
Another intriguing problem concerns the pitch angle distributions. Recent studies of
electron distributions showed various features that were not expected. Using Cluster data, Fu et al.
[2012c] performed a statistical study and found that the distributions are pancake-like (maximum
at 90 pitch angle) in growing flux pileup regions (FPRs, i.e. dipolarization regions), characterized
by the Vx peaks lagging behind the Bz peaks. They are cigar-like (maximum at 0 and 180 ) in
decaying FPRs in which the Bz peaks lag behind the Vx peaks instead. This statistical study is
consistent with their earlier event study [Fu et al., 2011]. Their interpretation is that in growing
FPRs, the magnetic fields are compressed by higher-speed plasmas behind the DFs, so electrons
are betatron accelerated in the perpendicular direction. In the decaying FPRs, the expanding
magnetic flux tubes cool electrons in the perpendicular direction. The relationship between spatial
locations of FPRs and pitch angle distributions was not determined in their studies, but was
considered in a statistical study using THEMIS data by Wu et al. [2013]. They demonstrated that
the distributions are more pancake-like in the midtail (X 15RE ) than in the near-Earth region
(15REEXR 10 ). They interpreted this result as a consequence of stronger compression in
the midtail because of higher flow velocities there than in the near-Earth region. This observational feature and its interpretation were supported by a MHD plus test particle simulation [Birn et al.,
2014]. The simulation reproduced the observed spatial dependence of pitch angle distributions.
The simulation also reproduced the triple-peak structure (maximum at 0 , 90 and 180 ) in
electron distributions observed by THEMIS [Wu et al., 2013; Runov et al., 2013], although the
cause of the structure was not clearly explained. Future studies using MHD+LSK simulations may
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examine the pitch angle distributions in simulations and compare them with observations in
substorm events.
The study of particle energization using PIC simulations should be further improved in the
future. First, as pointed out in Chapter 6, the artificial mass ratio poses a nontrivial and open
question on interpreting the energetics of electrons because the electron acceleration process is
strongly coupled with both ion and electron dynamics [Hoshino et al., 2001]. In our simulations,
the mass ratio was mmie / 400 , and the electron energies were accurate within about a factor of
2. So the problem was reduced when compared to typical explicit PIC simulations [e.g. Hoshino
et al., 2001; Drake et al., 2005; Pritchett, 2006a, 2006b]. For an implicit PIC simulation of a
D/2 system of space-time dimension D , the computing time scales with mass ratio as mmie/ . It
is feasible to increase the mass ratio to its realistic value mmie / 1836 by increasing the
computing time by a factor of about 8, and the total computing time will increase to about half of
a million CPU hours. With the realistic mass ratio, the problem of interpreting electron energetics
given by the simulations will be completely eliminated. Second, the initial set-up in the PIC
simulations needs to be further adjusted according to in-situ measurements. An obvious
improvement would be to add high-energy particles to the initial distributions. In particular,
observations have shown that the preexisting plasma sheet has a significant population of high-
energy electrons [e.g. Deng et al., 2010]. It is likely that the electron distributions as functions of
energy will no longer be Maxwellian in the EDR if high-energy electrons are added to the initial
distributions. If so, the difference between distributions in the EDR and at the DF is quantitative
rather than qualitative. Another improvement would be to use different temperatures for the plasma
sheet and the lobe, e.g. a cold population for the lobe and a warm population for the plasma sheet
[Hoshino et al., 2001]. However, to truly resolve the issue of unrealistic initial distributions, we
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need to completely embrace measured distributions in the lobe and in the plasma sheet. The
ongoing Magnetospheric Multiscale (MMS) mission comprising four identically instrumented
spacecraft will study magnetic reconnection and particle acceleration in the magnetotail [Burch et
al., 2015]. Designed for measuring fields and particles on the electron scale, MMS will provide
unprecedented high quality data to constrain PIC simulations. Third, ion acceleration by magnetic reconnection needs to be examined and compared with electron acceleration. Recent observations
have overwhelmingly shown that electrons are accelerated to tens of keV to about a hundred keV
by processes close to the X-line [Øieroset et al., 2002; Imada et al., 2005, 2007; Chen et al., 2008;
Retinò et al., 2008; Wang et al., 2010; Imada et al., 2011; Huang et al., 2012a], while there are
very few corresponding observations of energetic ions. Previous PIC simulations showed that ion
distributions are non-Maxwellian near the X-line [e.g. Hoshino et al., 1998], and ions in the
outflow region are heated with effective thermal speeds on the order of Alfvén velocity based on
the reconnecting magnetic field [Drake et al., 2009], but no significant energetic ions are present.
With the high-quality MMS data, combined with the implicit PIC code and well-constrained
simulation set-up, significant progresses can be made in understanding energy partition and production of high-energy particles in the near-Earth reconnection region.
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APPENDIX 1
Particle Sources for LSK Simulations
For the particle sources in the LSK simulations, we used a Maxwellian distribution to
represent the thermal particles and a power law distribution to represent the high-energy tail.
Here is a full description of the combined distribution.
First, the general form of the power law distribution, f p , as a function of energy,
EEE[,min max ] is
n E()v fcp ()v (A1) Eth
where c is the normalization coefficient, Eth is a quantity with energy unit, n is the power law
index and Emin ( Emax ) are lower-energy (upper-energy) limits. Subscript p indicates a power
law distribution and will only appear below when necessary. The normalization condition is
vEEmax max / th 2EdE 4()fvdvvv2 1 4() f 1 (A2) vEEmin min / th mm 1 where Em v2 is used. By substituting (A1) into (A2), we have 2 n 3/2 a EEdE211 m 41cc (A3) b Ethmm42 E th a b
where n 3/2, aE min / Eth , and bE max / Eth . Hence, the normalized distribution takes the form,
nn3/2 Em11 E fc()v (A4) EEabEth42 th th
3/2 nn 3 v 3 vth mE 111 E ffv() ()v th (A5) vEabEabEth22 th th 2 th
165
This is an isotropic, three-dimensional velocity vector distribution rather than velocity magnitude
(scalar) distribution. The velocity magnitude distribution is
1n vv2 v 1 E ff()42 ()2 (A6) vvvabEth th th th Transforming the argument to energy, we have 11/2nn Evvvth11 th E E ff() ()2 (A7) EvvabvEabEth22 th th th Note that the selection of the arguments ( v or E ) of the distribution function matters when referring to the power law index. The differential flux as a function of energy is
1n EEvv11 E jf() () f () (A8) EEvvabEth th th2 th th
Second, the distribution function requires that the connection between the Maxwellian distribution and the power law distribution be continuous and smooth. This requirement gives the
exact relationship between the power law distribution lower-energy boundary, Emin and the power law index. The Maxwellian distribution is
2 v2 2 vv2 2vth feM () (A9) vvth th Transforming the argument to energy, we have
2 vv22 E 22 Evvvvth211 th 22vvth v th E 2 E th ffMM() () e e e (A10) Evvth2222 th vv th v th E th Thus 1 f () e /2 (A11) M 2
where EE/ th . Here Eth is one dimensional thermal energy of the Maxwellian distribution,
and we assign this meaning to the Eth in power law distribution, (A1). Now we calculate the
slope of Maxwellian distribution as a function of energy by taking the natural log of both sides,
166
1ln lnf ( ) ln (A12) M 22
Setting lnfM ( ) and ln , we have:
111 ed e ln (A13) 22d 2222
By requiring the same slope for the Maxwellian and power law distributions at Emin , the
relationship between the power law index and lower-energy boundary is
11 1Emin nn (A14) 22 2 2 2E EEmin min th
Third, we calculate the weights of the Maxwellian and power law distributions. The
accumulated flux of the Maxwellian distribution, I M is
2 vvvvvvvv/ th 2 IfderfMM() () ( ) exp() (A15) 0 2 vvvvvth th th2vth th2 th
EEEEE/ th 2 I ()fderfe () () /2 (A16) MM0 EEEth th th 2 We set the weight of the Maxwellian distribution to be x , and the weight of the power law distribution to be y . Neglecting the contribution by the Maxwellian distribution tail, which is
1( I EEmin /)th and very small, we have a set of equations due to normalization and the
requirement that the combined distribution be continuous,
I(/)EExymin th 1 (A17)
yfpthMth(/) Emin E xf (/) E min E (A18)
1/2n 11EEmin min Equation (A18) yields yxEE exp(min 2th ) . Substitute it ab Eth2 E th
into (A17) to obtain the weights. For example, for electrons in the March 11, 2008 event, we
167 have n 4.5 , Eth 1 keV , Emin 9 keV , Emax 450 keV , IEMth(min / E ) 0.97 , yx 0.04 , and x 1/1.01.
Both a Kappa distribution and the combined distribution derived here have two free parameters. For a Kappa distribution, they are the effective temperature and the Kappa value,
(e.g. equation (5) in Summers and Thorne [1991]). For the combined distribution, they are the
Maxwellian temperature and the power law index (or lower-energy boundary). The combined
distribution resembles one Kappa distribution with in the domain of EE min and another
Kappa distribution with 2 in the domain of EE min , because Maxwellian distribution corresponds the Kappa distribution with and Kappa distribution approximates to power law distribution at high energy.
168
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