The Microphysics of Collisionless Shocks
A DISSERTATION
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL
OF THE UNIVERSITY OF MINNESOTA
BY
Lynn Bruce Wilson III
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
Doctor Of Philosophy
September, 2010
c© Lynn Bruce Wilson III 2010
ALL RIGHTS RESERVED
Acknowledgements
The work presented in this thesis could not have been performed had I not been
given the chance to do the research involved. Thus, I am sincerely grateful an indebted
to Cindy Cattell for giving me the opportunity to prove myself.
I would like to thank the following people for use of instrument data: R. Lin for use
of the Wind/3DP data, K. Ogilvie for use of the Wind/SWE data, R. Lepping for use
of the Wind/MFI data, and J.L. Bougeret for use of the Wind/WAVES data.
I would like to thank the following people for help with the Wind/3DP software:
Peter Schroeder, Stuart Bale, Linghua Wang (for help in calibration of the SST Foil
data), Robert Lin, and most of all Marc Pulupa.
I would like to thank Adam Szabo for providing the high time resolution magnetic
field instrument data for the Wind spacecraft. It was an absolute necessity for a large
portion of my research. I would also like to thank Justin Kasper for his work in cal-
culating the shock normal vectors, without which I could not have done a significant
fraction of my work.
The following people were instrumental in my understanding of physics and therefore
cannot be ignored. So I would like to graciously thank the following people for useful
physics discussions and guidance, without which this paper would not have been possi-
ble: Cindy Cattell, John Wygant, Bob Lysak, Keith Goetz, Paul Kellogg, Tom Jones,
Jesse Woodroffe, Scott Thaller, Lei Dai, Adam Hupach, Kris Kersten, and Aaron Bren-
eman.
Of course, none of this work would have been accomplished without help from my
friends and family by supporting me in my efforts. Therefore, I would like to thank my
grandfather, Duane Helmer, for lighting the candle of curiosity in me at such a young
age. I must thank my grandmother, Gayle Helmer, for teaching me how to read. I
i
would like to thank my great uncle Terry Helmer for giving me a place to live for over
ten years. I would like to thank my sister, Steff, for helping me to keep my sense of
humor. I would like to thank both of my parents, Amy Helmer and Lynn Wilson Jr., for
their support and guidance. I espeically thank my father for teaching me self-discipline
and how to be logical and rational. I would like to thank Jennifer Lynn Schei for the
helpful and often humorous conversations about our experiences in graduate school. I
would like to thank Briggs Rolfsrud for keeping me honest and supporting my efforts in
the last few months of graduate school. I must not forget to thank all the people whom
I have lived with during my graduate school career, including: Stuart, Janine, Rachael,
and Hannah Helmer and Chris, Brianna, Gary, and Lynn Gadacz and Stephen Scherer.
I especially thank Chris and Brianna Gadacz for putting up with me the longest during
graduate school.
This research was supported by NESSF grant NNX07AU72H, grant NNX07AI05G,
and the Dr. Leonard Burlaga/Arctowski Medal Fellowship.
ii
Abstract
Shock waves in interplanetary (IP) space are of considerable interest due to their po-
tential to damage ground based electronic systems and their ability to energize charged
particles. The energization of charged particles at IP shocks has the obvious extrapola-
tion to supernova shock waves, which are thought to be a candidate for generating the
most energetic particles in the universe. The observations and theory behind collision-
less shock wave evolution suggest that IP shocks should, for the most part, be stable
structures which require energy dissipation. In a regular fluid, like our atmosphere,
energy dissipation is accomplished through binary particle collisions transferring the
loss of bulk flow kinetic energy to heat. Plasmas are mostly collisionless fluids, thus
requiring other means by which to dissipate energy.
The studies herein were performed using wave and particle data primarily from
the Wind spacecraft to investigate the microphysics of IP shock energy dissipation
mechanisms. Due to their lower Mach numbers, more simplified geometry, and quasi-
perpendicular nature, IP shock waves are an excellent laboratory to study wave-particle
related dissipation mechanisms. Utilization of multiple data sets from multiple high
time resolution instruments on board the Wind spacecraft, we have performed studies
on the transition region microphysics of IP shocks.
The work began with a statistical study of high frequency (&1 kHz) waveform cap-
ture data during 67 IP shocks with Mach numbers ranging from ∼1–6 found ion-acoustic
wave amplitudes correlated with the fast mode Mach number and shock strength. The
ion-acoustic waves (IAWs) were estimated to produce anomalous resistivities roughly
seven orders of magnitude above classical estimates.
Another study was an examination of low frequency waves (0.25 Hz < f < 10 Hz) at
five quasi-perpendicular IP shocks found the wave modes to be consistent with oblique
precusor whistler waves at four of the events. The strongest event in that study had low
frequency waves consistent with shocklets. The shocklets are seen simultaneously with
diffuse ion distributions. Both the shocklets and precursor whistlers are seen simulta-
neously with anisotropic electron distributions unstable to whistler anisotropy and heat
iii
flux instabilities. The IP shock with upstream shocklets showed much stronger electron
heating across the shock ramp than the four events without upstream shocklets.
Further investigation of the atypical IP shock found the strong heating to be associ-
ated with large amplitude (> 100 mV/m) solitary waves and electron Bernstein waves.
The observed heating and waveforms are likely due to instabilities driven by the free
energy provided by reflected ions at this supercritical IP shock, not the DC macroscopic
fields. The particle heating observed for the event with shocklets was observed to be
different from other events with similar shock parameters, suggesting a different dissi-
pation mechanism.
The work presented in this thesis has helped increase the understanding of the micro-
physics of IP shocks in addition to raising new questions regarding the energy dissipation
mechanisms dominating in the ramp regions. The initial work focused on a statistical
study of high frequency waveforms in IP shock ramps. The study results suggested a
re-evaluation of the relative importance of anomalous resistivity due to wave-particle
interactions. This assertion was further strengthened by the atypical particle heating
observed in the 04/06/2000 event which we claimed clearly showed a dependence on
the observed waveforms. Thus, the nearly ubiquitous observations of large amplitude
IAWs in the ramp regions of IP shocks raises doubts about ignoring these high frequency
fluctuations. In addition to these findings, we also observed a low frequency wave mode
which is only supposed to exist upstream of quasi-parallel shocks with small radii of
curvatures.
All of these findings have increased our knowledge of collisionless shock energy dis-
sipation, but they have raised many questions regarding our current theories. We have
raised doubts regarding the use of the solar wind electron distributions as one particle
population. We have showed evidence to support the energy dependence of wave-particle
interactions between low frequency whistler waves and ≤1 keV electrons. Thus, we con-
clude that in the analysis of IP shocks the microphysics can no longer be disregarded.
iv
Contents
Acknowledgements i
Abstract iii
List of Tables ix
List of Figures x
1 Introduction 1
1.1 The Sun and Solar Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Collisionless Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Basic Hydrodynamics and Shock Physics . . . . . . . . . . . . . . . . . . 4
1.2.1 Nonlinear Wave Steepening and Discontinuities . . . . . . . . . . 4
1.2.2 Introduction to Shock Physics . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Shock Parameter Definitions . . . . . . . . . . . . . . . . . . . . 7
1.2.4 Conservation Relations . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.5 Energy Dissipation Mechanisms in Collisionless Shocks . . . . . . 14
1.3 Shock Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.1 Bow Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.2 Coronal Mass Ejections . . . . . . . . . . . . . . . . . . . . . . . 22
1.4 Particle Distribution Observations in Solar Wind and at Shocks . . . . . 25
1.4.1 Distribution Function Moments . . . . . . . . . . . . . . . . . . . 25
1.4.2 Previous Observations and Examples of Electron Distributions . 26
1.4.3 Previous Observations and Examples of Ion Distributions . . . . 30
1.5 Wave Particle Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 35
v
1.5.1 The Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . 35
1.5.2 Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.5.3 Landau Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.5.4 Weak Growth Limit . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.5.5 Cyclotron Resonance . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.6 Introduction to the Microphysics of IP shocks . . . . . . . . . . . . . . . 47
1.7 Electromagnetic Waves: Background, Definitions, and Prior Work . . . 49
1.7.1 Langmuir Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.7.2 Ion-Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.7.3 Electrostatic Solitary Waves . . . . . . . . . . . . . . . . . . . . . 53
1.7.4 Whistler Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.7.5 Waves at or Near the Electron Cyclotron Frequency . . . . . . . 58
1.7.6 Lower Hybrid Waves . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.8 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2 Instrumentation, Measurements, and Analysis Techniques 68
2.1 Introduction to the Wind Spacecraft and instruments . . . . . . . . . . 68
2.1.1 Wind 3DP Particle Detector . . . . . . . . . . . . . . . . . . . . 69
2.1.2 WAVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.1.3 Magnetic Field Instrument . . . . . . . . . . . . . . . . . . . . . 77
2.2 Wind Data Analysis and Calibrations . . . . . . . . . . . . . . . . . . . 77
2.2.1 Magnetic Field Rotations . . . . . . . . . . . . . . . . . . . . . . 78
2.3 Particle Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3.1 Electron Distributions . . . . . . . . . . . . . . . . . . . . . . . . 78
2.3.2 Temperature Anisotropies . . . . . . . . . . . . . . . . . . . . . . 84
2.3.3 Ion Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.4 Waveform Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.4.1 Minimum Variance Analysis . . . . . . . . . . . . . . . . . . . . . 87
2.4.2 Bandpass Filters and Spectral Analysis . . . . . . . . . . . . . . 89
2.4.3 Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3 Previous Studies of Low Frequency Waves and Macroscopic Fields 99
3.1 Introduction to Low Frequency Waves at Collisionless Shocks . . . . . . 99
vi
3.1.1 ULF Foreshock Waves . . . . . . . . . . . . . . . . . . . . . . . . 100
3.2 Shocklets, SLAMS, and Whistler Foreshock Waves . . . . . . . . . . . . 102
3.2.1 Shocklets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.2.2 SLAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.2.3 1 Hz Whistlers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.2.4 ULF Foreshock Waves at IP shocks . . . . . . . . . . . . . . . . . 110
3.2.5 Simulations and Summary of ULF Foreshock Waves . . . . . . . 110
3.3 Introduction to Cross-Shock Potential . . . . . . . . . . . . . . . . . . . 113
3.3.1 Expected Observations . . . . . . . . . . . . . . . . . . . . . . . . 116
3.4 Electric and Magnetic Field Measurements . . . . . . . . . . . . . . . . 118
3.4.1 Electric Fields in Shock Ramps . . . . . . . . . . . . . . . . . . . 118
3.4.2 Magnetic Fields in Shock Ramps . . . . . . . . . . . . . . . . . . 121
3.5 Assumptions For de-Hoffmann-Teller Potential Calculations . . . . . . . 123
3.5.1 Calculation of the de Hoffmann-Teller Potential . . . . . . . . . . 125
4 Shocklets and Low Frequency Whistlers at an Interplanetary Shock 127
4.1 An Atypical IP Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.2 Comparison of the 04/06/2000 event to the four typical events . . . . . 133
4.3 Analysis of Particle Data for this Study . . . . . . . . . . . . . . . . . . 134
4.3.1 Comparison of Electron Heating . . . . . . . . . . . . . . . . . . 143
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5 High Frequency Waves at Interplanetary Shocks 153
5.1 Statistical Study of High Frequency Waves at Collisionless Shocks . . . 153
5.2 Analysis of Microphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.2.1 Data Sets and Analysis . . . . . . . . . . . . . . . . . . . . . . . 157
5.2.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.3 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6 Conclusion and Discussion 179
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
vii
References 184
Appendix A. Wind Measurements and Data Analysis Issues 216
A.1 The Pesa High Detector: The Glitch . . . . . . . . . . . . . . . . . . . . 216
A.2 SCET Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
A.3 Explicit Explanation of Actual Data Calculations . . . . . . . . . . . . . 221
A.3.1 Wind 3DP Software Calculations . . . . . . . . . . . . . . . . . . 221
A.3.2 Specific Software Implementations . . . . . . . . . . . . . . . . . 222
A.3.3 Pressure and Temperature Calculation . . . . . . . . . . . . . . . 224
Appendix B. Glossary and Acronyms 226
viii
List of Tables
2.1 Wind 3DP Instrument Specs . . . . . . . . . . . . . . . . . . . . . . . . 71
2.2 Wind WAVES TDS Specs . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.3 Wind WAVES TNR Specs . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.1 Wind 3DP Electron Stats on 04/06/2000 . . . . . . . . . . . . . . . . . 152
5.1 Statistical High Frequency Wave Occurrence . . . . . . . . . . . . . . . . 154
5.2 Wind 3DP ELB stats across the 04/06/2000 IP shock . . . . . . . . . . 162
5.3 Wind 3DP Global ELB stats across the five IP shocks . . . . . . . . . . 163
A.1 Pesa High Bad Data Bins . . . . . . . . . . . . . . . . . . . . . . . . . . 216
ix
List of Figures
1.1 Plasma Parameter Jumps Across Shock . . . . . . . . . . . . . . . . . . 8
1.2 Shock Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Shock Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Shock Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Foreshock, Bow Shock, and Magnetopause Structure . . . . . . . . . . . 21
1.6 Coronagraph of two CMEs . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7 CME Cartoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.8 Electron Distribution Example . . . . . . . . . . . . . . . . . . . . . . . 28
1.9 Ion Beam Distribution Example . . . . . . . . . . . . . . . . . . . . . . . 32
1.10 Intermediate Ion Distribution Example . . . . . . . . . . . . . . . . . . . 34
1.11 Intermediate and Diffuse Ions . . . . . . . . . . . . . . . . . . . . . . . . 36
1.12 Parallel Cyclotron Resonance Energy Example . . . . . . . . . . . . . . 46
1.13 Example Langmuir Wave . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.14 Example Ion Acoustic Wave . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.15 Example Solitary Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.16 Example Precursor Whistler Wave . . . . . . . . . . . . . . . . . . . . . 57
1.17 Example Electron Cyclotron Waves . . . . . . . . . . . . . . . . . . . . . 59
1.18 Example Lower Hybrid Wave . . . . . . . . . . . . . . . . . . . . . . . . 63
1.19 Example Lower Hybrid Wave: Low Frequency Filter . . . . . . . . . . . 64
1.20 Example Lower Hybrid Wave: High Frequency Filter . . . . . . . . . . . 65
2.1 Wind Spacecraft Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 73
2.2 Wind Spacecraft Rotations and B-field . . . . . . . . . . . . . . . . . . . 74
2.3 Wind TDS Rotation Example . . . . . . . . . . . . . . . . . . . . . . . . 77
2.4 Electron Core and Halo Distribution Fit Example . . . . . . . . . . . . . 80
x
2.5 Electron Distribution Fits . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.6 Power Spectrum Example for Upstream Shocklet . . . . . . . . . . . . . 90
2.7 Bandpass Filtered Magnetic Field Data . . . . . . . . . . . . . . . . . . 91
2.8 Low Frequency Zoom of 16:20:44 UT event on 04/06/2000 . . . . . . . . 92
2.9 Wavelet Spectral Analysis Example . . . . . . . . . . . . . . . . . . . . . 96
2.10 Example Sawtooth Wavelet versus FFT . . . . . . . . . . . . . . . . . . 98
3.1 Discrete Wave Packet with irregular low-frequency waves . . . . . . . . 103
3.2 Shocklets with and without Discrete Wave Packet . . . . . . . . . . . . . 104
3.3 Four Shocklet Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.4 Two Examples of SLAMS . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.5 Expected Heating Signatures . . . . . . . . . . . . . . . . . . . . . . . . 117
3.6 6 large amplitude TDSF Waves . . . . . . . . . . . . . . . . . . . . . . . 120
3.7 Comparison MFI at IP shock . . . . . . . . . . . . . . . . . . . . . . . . 122
4.1 Waves, Magnetic Field Magnitude, and Ion Density . . . . . . . . . . . . 128
4.2 Scaled Plot of Shocklets on 04/06/2000 . . . . . . . . . . . . . . . . . . 129
4.3 Low Frequency Zoom of 16:28:00 UT event on 04/06/2000 . . . . . . . . 131
4.4 Low Frequency Zoom of 16:31:00 UT event on 04/06/2000 . . . . . . . . 132
4.5 Typical Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.6 Histogram of Whistler Angles versus Shocklets . . . . . . . . . . . . . . 135
4.7 Comparison of 16:28:00 Shocklet to EL and EH PADs . . . . . . . . . . 137
4.8 Comparison of 16:31:00 Shocklet to EL PADs: 1st Half . . . . . . . . . . 139
4.9 Comparison of 16:31:00 Shocklet to EL PADs: 2nd Half . . . . . . . . . 140
4.10 Diffuse Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.11 Comparison of Temperature Anisotropies for Five Events . . . . . . . . 144
4.12 Comparison of Perpendicular Temperatures for Five Events . . . . . . . 146
4.13 Comparison of Parallel Temperatures for Five Events . . . . . . . . . . . 147
5.1 Wave Amplitude vs. Compression Ratio . . . . . . . . . . . . . . . . . . 155
5.2 Wave Amplitude vs. Mach Number . . . . . . . . . . . . . . . . . . . . . 156
5.3 HTR MFI TDS and DF Locations with DFs . . . . . . . . . . . . . . . . 160
5.4 Four ELB and PHB Distributions . . . . . . . . . . . . . . . . . . . . . . 161
5.5 Four Atypical Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.6 Wavelet Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 166
xi
5.7 Snapshot of Waveform A . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.8 Snapshot of Waveform B . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.9 Snapshot of Waveform C . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.10 Snapshot of Waveform D . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.11 SST Foil Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
A.1 Example Pesa High Distribution with Labels . . . . . . . . . . . . . . . 217
xii
Chapter 1
Introduction
1.1 The Sun and Solar Wind
The closest star to Earth, the sun, is a large ball of ionized gas called a plasma. The
sun has intense variable magnetic fields which are the drivers of most solar and geomag-
netic activity. The sun’s outer atmosphere, the corona, is hundreds of times hotter than
lower visible layer of the solar disk. The apparent violation of the second law of ther-
modynamics has provoked immense interest in the dynamics of the solar atmosphere.
The outer atmosphere is not bound to the sun and its expansion is observed as an ever
present flow called the solar wind [Lang , 2000]. The solar wind is composed of electrons,
protons, alpha particles, and heavy ions. The particle nature of the solar wind was first
measured by a Russian scientist named I. Gringauz [Gringauz et al., 1960]. However,
Gringauz et al. [1960] could only determine a relative ion flux and direction of flow. It
was not until 1962 that Marcia Neugebauer and Conway W. Snyder used more than
100 days of Mariner 2 data to show that charged particles were continuously emanating
from the sun [Neugebauer and Snyder , 1962].
The sun’s magnetic field has been found to flip roughly every 11 years going through
periods called solar maximum and solar minimum. The maximum and minimum peri-
ods are defined based upon the number of sun spots observed. Sun spots are seen as
dark localized regions on the visible disk of the sun and are now known to contain in-
tense magnetic fields (up to ∼0.3 Tesla, or ∼10,000 times the terrestrial magnetic field
strength). These localized regions of intense magnetic field are called active regions
1
2and are responsible for the solar storms which can excite a myriad of local terrestrial
phenomena (e.g. the aurora). The variation of the sun’s magnetic fields have impacts
on the expansion of the solar wind into the surrounding heliosphere.
It is now known that the sun expells roughly 109 kg of mass per second at speeds
ranging from roughly 300 km/s to over 800 km/s [Marsch, 2006]. The solar wind flow
is almost entirely in the radial direction and its speed has strong latitudinal variations
during solar minimum as discovered by the Ulysses spacecraft [McComas et al., 1998;
Phillips et al., 1995]. Later studies found that after removing radial gradients, the high
latitude regions showed little latitudinal variation [McComas et al., 2000]. During solar
maximum, the variations in solar parameters are much less well defined.
1.1.1 Collisionless Fluids
A fluid made of positively and negatively charged particles is called a plasma. Plas-
mas can be considered a neutral gas if their microscopic electric fields are screened out
over a distance known as the Debye length given by:
λDs =
√
εokBT s
nse2
(1.1a)
=
V Ts
ωps
(1.1b)
where εo is the permittivity of free space, kB is the Boltzmann constant, Ts is the
temperature, e2 is the charge squared, ns is the number density of particle species s,
VTs is the thermal speed of species s (=
√
kBT s/ms), and ωps is the plasma frequency
of species s defined as:
ωps ≡
√
nsqs2
msεo
(1.2)
The condition for defining a gas as a plasma is defined as:
nsλDs
3 ≫ 1 (1.3)
which states that the number of particles in a Debye sphere is large. Most plasmas
are considered collisionless, which means that the Coulomb collision time, τ c, is much
3larger than any relevant plasma time scale. In other words, the average mean free
path of a particle is often much larger than the gradient scale lengths of waves or
density fluctuations. The average mean free path of a charged particle in a fully ionized
plasma can be estimated by multiplying the average time between collisions, τ c, and
the average speed of the particles (assume VTs). A particle with speed VTs undergoes
a collision in unit time τ c with the particles that lie within a cylinder of volume πrc2
VTs, where rc is the limiting impact parameter for which the deflection of the incident
particle is sufficiently large to make the change in momentum comparable to the initial
momentum of the incident particle. Given that the particle number density is no, the
effective collision frequency is:
νc =
1
τ c
≈ πrc2V Tsno (1.4a)
≈ noe
4
32π(meεo)2V Ts3
(1.4b)
where we have assumed rc to be ≈ δo/(me VTs2) and δo = e2/(4πεo). We used me because
their light mass makes it easier to cause significant deflections of their trajectories,
which we define as a collision. A typical proton kinetic energy in the solar wind is
roughly 10 eV, which corresponds to a thermal speed VTp ∼ 31 km/s. The typical solar
wind particle density is roughly 107 particles per meter squared, thus a 10 eV proton
experiences roughly 4×10−7 collisions per second or τ c ∼ 2.5×106 seconds. So a typical
solar wind proton moving at 31 km/s travels, on average, 7.75×1010 meters in this time,
which is roughly half of an astronomical unit or AU (∼1.5×1011 m). Thus, it is likely
that the particles will interact with something else before colliding with another particle.
Since plasmas can act like a fluid by exhibiting bulk continuous motions and though
they are not dominated by collisions, they can produce discontinuities [Kellogg , 1962].
However, in a plasma, the particles are affected by not only the regular fluid parameters,
they are also affected by electric and magnetic fields due to their individual charges.
The consequences of the added forces result in new types of discontinuities and shock
parameters.
Discontinuities are changes in relevant parameters on scale lengths shorter than the
relevant communication scale lengths of the medium. For a regular fluid, the relevant
scale length is the mean free path for binary collisions. For a plasma, the relevant
scale lengths are the particle gyroradius, ρgs, and/or inertial lengths, c/ωps, where s
4represents the particle species (i.e. electrons or ions) and c is the speed of light. The
parameter, ρgs is defined by:
ρgs ≡ V ⊥
Ωcs
(1.5)
where V⊥ is the velocity of a particle perpendicular to the magnetic field (often assumed
to be the thermal speed of the particle), Ωcs is the particle gyrofrequency (= qs |B|/ms),
ns is the particle number density, qs is the particle charge, and ms is the particle mass.
For reference, a typical proton in the solar wind near 1 AU will have a gyrofrequency of
roughly 0.1 Hz (Bo ∼ 6.56 nT), a plasma frequency of roughly 600 Hz (ne ∼ 8.2 cm−3),
a thermal speed of roughly 30 km/s (Te ∼ 13 eV), a thermal gyroradius of roughly 300
km, and inertial length of roughly 80 km. For future reference, we define microscopic as
scales smaller than a proton gyroradius, ρgp, and macroscopic as scales larger than ρgp.
There is not a sharp distinction between the two other than to say that the phenom-
ena involved in macroscopic effects often can be treated as a fluid while microphysics
requires kinetic treatments.
1.2 Basic Hydrodynamics and Shock Physics
1.2.1 Nonlinear Wave Steepening and Discontinuities
Fluids are a ubiquitous substance found in every region of the universe. Fluids can
exist in a collision dominated or collisionless state. One of the characteristic properties
of this state of matter is a set of phenomena known as discontinuities. Discontinuities
can be described as a mathematical property defining a step-like or abrupt transition
between two points along a curve. Discontinuities can occur in fluids in different forms.
The simplest discontinuity is a density gradient with no flow across the sharp boundary,
known as a contact discontinuity. An example of a contact discontinuity in a regular
collisional fluid is a chemical explosion where a new source of hot gas is introduced to
a system.
Wave discontinuities can arise from a nonlinear wave, which has steepened to a
point where the flow, density, or temperature changes direction and/or magnitude in a
5distance shorter than the characteristic scale length of the medium. The characteristic
length is defined by the type of information transfer in the medium. In a regular fluid,
the characteristic scale length is the mean free path for binary particle collisions. The
mean free path of particles in our atmosphere is on the order of a micrometer and there
are roughly 1019 particles per centimeter cubed at standard temperature and pressure.
Thus, the number of binary collisions is easily sufficient to dissipate the free energy in
regular hydrodynamic discontinuities.
Discontinuities require the nonlinear growth a wave propagating in a medium, called
nonlinear wave steepening. The steepening can result from a dependence of the phase
velocity on the amplitude of the wave, causing the larger amplitude waves to catch up
with and overtake the smaller amplitude waves. For instance, acoustic waves having
a finite amplitude of any wave mode must always steepen, if some form of resistance
to steepening is ignored [Shu , 1992]. The equations of fluid mechanics are fundamen-
tally nonlinear. Almost all nonlinear waves have a tendency to steepen except for some
dispersive waves whose different Fourier components propagate at different speeds for
small amplitudes. These waves have special finite amplitude solutions, called solitons,
maintain their shape as they propagate because dispersion exactly balances steepening.
The process of wave steepening can be thought of as a series of small successive
step waves trailing each other. The leading waves alter their respective downstream
fluid speeds which define the propagation speeds of the trailing waves, allowing them
to overtake the leading waves. A compressional wave, for instance, increases the fluid
speed behind the compression which allows any compressional wave trailing the first
compressional wave to propagate faster than the first. Thus, the intrinsic nature of
compressional waves makes them prone to wave steepening [Kennel et al., 1985]. The
waves steepen until either some form of dissipation limits the steepening or the wave
reachs what is called a gradient catastrophe and breaks (e.g. water waves seen as white
caps on lakes). However, if there exists enough resistance to growth, the steepened wave
may reach a balance between steepening and the resistive forces. The resisting com-
ponent is energy dissipation due to non-conservative forces, much like friction between
two surfaces. Energy dissipation is the transfer of energy from one form (e.g. kinetic
energy) to another form (e.g. heat) through an irreversible process. The distinction of
irreversible is necessary because reversible forms of dissipation cannot, by themselves,
6produce specific types of discontinuities like shock waves.
1.2.2 Introduction to Shock Physics
Shock waves are type of propagating wave that occur obiquitously throughout the
universe as a discontinuous transition, called the ramp, between two regions defined
as upstream (unshocked) and downstream (shocked). Shock waves occur in collision
dominated media like our atmosphere and in collisionless media like interstellar space.
They are seen in our atmosphere around supersonic aircraft and in space as supernovae,
galactic/planetary/cometary bow shocks, and transient interplanetary (IP) events.
Across a shock wave, there is a discontinuous change in flow speed that results in a
bulk change in kinetic energy across the ramp. The finite difference in kinetic energy is
a source of free energy in the shock wave which must be dissipated in some way to con-
serve energy. It is well known that energy dissipation can be achieved easily in a regular
collision dominated fluid, like our atmosphere, through binary particle collisions. The
binary collisions transfer the free energy by randomly increasing their average kinetic
energy, also known as heating. The loss of information through the randomization of
particle kinetic energies produces an increase in entropy, thus the shock transition is
irreversible. Shock waves manage to heat particles in relatively short distances defined
by a characteristic scale size in the medium. The mean free path for binary parti-
cle collisions of particles in our atmosphere is on the order of a micrometer and there
are roughly 1019 particles per centimeter cubed at standard temperature and pressure.
However, as we discussed in Section 1.1.1, the average mean free path for a proton in
the solar wind is on the order of an AU and the typical number density is 10 cm−3.
Thus, shock waves which exist in the interplanetary or interstellar medium cannot rely
upon binary particle collisions to dissipate their energy.
Collisionless shock waves are capable of efficiently heating and/or accelerating charged
particles. For instance, the collisionless shocks involved in supernovae or binary star
collisions are thought to produce some of the highest energy cosmic rays
[Blandford and Eichler , 1987]. Cosmic rays are also produced locally, in our solar sys-
tem by transient IP events or at the heliospheric termination shock. These particles can
damage and/or destroy spacecraft or injure/kill astronauts. The transient IP events
7can also affect planetary magnetic fields by compressing the field, after resulting in
strong aurora that induce currents in long wires, pipelines, or power grids. The induced
DC currents are capable of destroying power stations Pirjola [1989]; Pirjola et al. [2000].
1.2.3 Shock Parameter Definitions
A shock is usually a relatively stable discontinuity, shown qualitatively in Figure 1.1.
Here, stable refers to the temporal structure of the discontinuity. This is in contrast
to the breaking water waves preferred by surfers, which are not stable. They are a
nonlinearly steepened wave which does not have enough energy dissipation to balance
the steepening. Shock waves often do have enough energy dissipation to balance the
steepening, thus they are seen as a time-stationary discontinuity.
In many plasma regimes, the magnetic field is the direction of reference, whereas in
a shock, the shock normal vector becomes the important direction. If we let nˆ ≡ shock
normal vector, θBn ≡ shock normal angle or angle between nˆ and upstream (subscript
1) magnetic field, B1, then we can define the tangential and normal components of any
given vector as:
Qn ≡ Q · nˆ (1.6a)
Qt ≡ (nˆ×Q)× nˆ (1.6b)
≡ Q− (Q · nˆ) nˆ (1.6c)
where Qn(t) refers to an arbitrary normal(tangential) vector with respect to the shock
normal vector.
Figure 1.1 is an illustrative cartoon used to show the change in various plasma
parameters one might measure in space across the shock. The parameters are shown in
the shock reference frame. In the figure, each panel corresponds to a different plasma
parameter(s) magnitude versus time. The panels are, from top to bottom: 1) the
magnitude of the normal, Un, and tangential, Ut, flow speeds (in the shock frame), 2)
the magnitude, |B|, and tangential component, Bt, of the magnetic field, 3) the particle
pressure, P (= no kB T), 4) the magnitude of the average temperature, T, and 5) the
particle number density, no. This figure is for illustrative purposes; real collisionless
shock crossings are much more complicated and far less laminar.
8U  , Un t
|B|, Bt
Pressure
Temperature
Density
Plasma Parameter Jumps across a Fast Mode Shock
Figure 1.1: An illustrative example of how the parameters change across a shock in the
shock reference frame. The top panel shows the normal, Un, and tangential, Ut, flow
speeds, the second panel is the same but for the magnetic field. The third through fifth
panels show the pressure, temperature, and density change across the shock. The image
is specifically for a fast mode shock in a collisionless plasma. The subscripts, n and t,
refer to normal and tangential directions respectively with respect to the shock normal
vector. For a regular fluid, simply ignore the panel that refers to the magnetic field.
For future purposes and definitions, an illustrative example of the shock geometry
can be seen in Figure 1.2. The blue line represents the shock normal vector, n, the red
B1(B2) represent the upstream(downstream) magnetic field vectors, Bt1(Bt2) represent
the tangential component of the upstream(downstream) magnetic field vectors, and θBn
is the shock normal angle. The shock normal angle is defined by:
θBn ≡ tan | Bt
Bn
| (1.7a)
= tan | (nˆ×B)× nˆ
B · nˆ | (1.7b)
where nˆ is the shock normal vector as defined above.
9Shock
θBn
Bt1
n
B1
Bt2
B2Upstream
Downstream
Figure 1.2: An illustrative example of the often assumed geometry of a collisionless
shock wave. The shock is assumed to have a planar surface with a well defined normal
vector, nˆ. The angle between the incident magnetic field, B1, and nˆ is θBn.
10
1.2.4 Conservation Relations
When observing collisionless discontinuities in space with satellites, it is necessary to
show that time series measurements of specific quantities satisfy a sequence of conditions
in order to verify the discontinuity as a shock wave [Vinas and Scudder , 1986]. In
regular fluid dynamics, a discontinuous transition between an upstream and downstream
state must conserve mass, momentum, and energy when considering a one-dimensional
time-stationary discontinuity. The conservation of mass results from integrating the
mass continuity equation over some volume which contains the shock transition. The
continuity equation is a solution to a moment of the Boltzmann equation given by:
∂f
∂t
+ v · ∇f + F
m
· ∇vf = δcf
δt
(1.8)
where f is the distribution function, f = f(v,x,t), F is some external force, ∇v is the
gradient operator in velocity space, and δc/δt is a collision operator. When δc/δt →
0, as in a collisionless plasma, the Boltzmann equation becomes the Vlasov equation.
We can define the number of particles, dN, in a phase space element, d3x d3v, in the
following way:
dN = f (v,x, t) d3xd3v (1.9)
where one assumes that the number of particles in d3x d3v is sufficiently large such
that f(v,x,t) can be regarded as a continuous function of v and x. The total number of
particles can be found by integrating over all of phase space. The following quantities
will be used later, so we define:
ns =
∫
V
f s (v,x, t) d
3v (1.10a)
Us =
1
ns
∫
V
vf s (v,x, t) d
3v (1.10b)
W s =
ms
2
∫
V
v2f s (v,x, t) d3v (1.10c)
←→
P s = ms
∫
V
(v−Us) (v−Us) f s (v,x, t) d3v (1.10d)
where ns is the number density, Us is the average velocity, Ws is the kinetic energy den-
sity, ms is the mass, and
←→
P s is the pressure tensor of species s [Gurnett and Bhattacharjee ,
2005].
11
Moment equations are calculated by multiplying the Vlasov equation by powers of
the velocity then integrating over velocity space. As mentioned before, the continuity
equation is a solution to a moment equation, specifically the zeroth moment given by:
∂ρm
∂t
+∇ · (ρmU) = 0 (1.11)
where ρm is the mass density, given by Equation 1.10a multiplied by the particle mass,
and U is the bulk flow velocity, given by Equation 1.10b. The right-hand side of this
equation is zero because no sources/losses of mass inside the discontinuity are assumed
to be present; mass is neither created nor destroyed in the discontinuity. Momentum
conservation requires a little more work because the Vlasov equation is multipled by v
before integration is performed. After a great deal of arithmetic and a few assumptions,
the momentum equation can be shown to be:
msns
[
∂Us
∂t
+ (Us · ∇)Us
]
= nsF−∇ ·←→P s (1.12)
where F = qs(E + v × B), is the Lorentz force here.
In a magnetized fluid, known as magnetohydrodynamics or MHD, the conservation of
each moment is altered slightly to accomodate the electric and magnetic forces involved.
The conservation of momentum and energy can be written as:
∂
∂t
(ρmUs) +∇ ·
[
ρmUsUs +
(
P +
B2
2µo
)←→
I − BB
µo
]
= 0 (1.13a)
∂
∂t
(
1
2
ρmU s
2 +
P s
γ − 1 +
B2
2µo
)
+∇ ·
[
1
2
ρmU s
2Us +
γP s
γ − 1Us +
E×B
µo
]
= 0 (1.13b)
where P is some scalar isotropic pressure, γ is the statistical mechanics parameter which
depends on the number of degrees of freedom, = (d + 2)/d with d being the degrees of
freedom (also the ratio of specific heats). For closure, the pressure is typically defined
by the adiabatic equation of state, PVγ = constant. For future reference, a one dimen-
sional(degree of freedom) gas has γ = 3, an isothermal gas has γ = 1, a three dimensional
gas has γ = 5/3, and an incompressible gas has γ = ∞ [Gurnett and Bhattacharjee ,
2005].
If we let {} define the difference between downstream and upstream quantities across
a discontinuity, then the set of relationships defined by these conservation laws are
12
known as the Rankine-Hugoniot relations, given by Gurnett and Bhattacharjee [2005]:
{ρmUn} = 0 (1.14a){
(ρmUn)Ut − BnBt
µo
}
= 0 (1.14b){
ρmUn2 + P +
Bt2
2µo
}
= 0 (1.14c){(
1
2
ρmU2 +
γP
γ − 1 +
B2
µo
)
Un − (U ·B) Bn
µo
}
= 0 (1.14d)
{Un (nˆ×Bt) + (Ut × nˆ)Bn} = 0 (1.14e)
{nˆ ·B} = 0 (1.14f)
(1.14g)
where Equation 1.14a corresponds to mass continuity, Equation 1.14b is the tangential
momentum, Equation 1.14c is the normal momentum, and Equation 1.14d is energy,
while the last two equations are specific to MHD and Maxwell’s equations. Equation
1.14e results from an assumption made in MHD where the plasma has an infinite con-
ductivity. Using a simplified version of Ohm’s law and assuming an ideal MHD plasma,
then E = - U × B. Using this assumption, we assume the discontinuity is a sharp
boundary and so there can be no sudden time variation of B within the layer. This
results in Faraday’s law changing to ∇ × {E} = 0, which after linearizing (discussed in
the next chapter) goes to nˆ × {E} = 0, where nˆ is the shock normal vector. Equation
1.14f results from linearizing Maxwell’s equation which claims no existence of magnetic
charges(monopoles). Using this set of equations, one can determine the relevant shock
parameters as described by Vinas and Scudder [1986]. Using the adiabatic equation of
state we can replace the scalar pressure by recalling that the speed of sound in a fluid,
Cs, is given by ∂P/∂ρm and using Pρm−γ = constant, we see:
Cs2 =
∂P
∂ρm
(1.15a)
∂P
∂ρm
= (constant)γρmγ−1 (1.15b)
=
γP
ρm
(1.15c)
13
which results in Equations 1.14c and 1.14d going to:{
ρm
(
Un2 +
Cs2
γ
)
+
Bt2
2µo
}
= 0 (1.16a){
ρm
(
1
2
U2 +
ρmCs2
γ − 1 +
B2
ρmµo
)
Un − (U ·B) Bn
µo
}
= 0 . (1.16b)
(1.16c)
As mentioned previously, when an object/wave propagates through a medium faster
than the relevant speed of information transfer, a shock wave may form. In a collision-
less plasma, the relevant speed is known as the fast mode or magnetosonic speed. A
magnetosonic wave is a compressive mode where the density fluctuations are in phase
with the magnetic fluctuations leading to a wave whose phase speed depends on both
the sound speed and Alfve´n speed, given by:
2Cf
2 =
(
Cs
2 + V A
2
)
+
√
(Cs2 − V A2)2 + 4Cs2V A2 sin2 θBn (1.17a)
=
(
Cs
2 + V A
2
)
+
√
(Cs2 + V A2)
2 − 4Cs2V A2 cos2 θBn (1.17b)
where VA = B/
√
µoρm is the Alfve´n speed. There are three different relevant speeds in
a plasma, Cf , Cslow, and VA corresponding to the three MHD wave modes. Therefore,
one might expect there to be three different Mach numbers given by:
M f =
Un
Cf
(1.18a)
M slow =
Un
Cslow
(1.18b)
MA =
Un
CA
. (1.18c)
Though the transition across the shock is an abrupt change, allowing little time
for heat to flow, it would be incorrect to use only the adiabatic equation of state to
describe the shock. The reason is that the irreversible nature of the shock implies that
the fluid is not in a succession of equilibrium states [Gurnett and Bhattacharjee , 2005;
Shu, 1992]. The Rankine-Hugoniot relations only refer to the asymptotic values far from
the shock transition region, not within the dissipative scale lengths [Kennel et al., 1985].
As discussed previously, the bulk kinetic energy across the shock is not conserved, thus
it is transferred to some other form in the shock transition region. The transformation
14
of the energy into some other form (i.e. heat) results in the irreversibility of the shock.
One can show that the entropy increases across the shock, proving that the transition
is irreversible, but we will not discuss that here [Gurnett and Bhattacharjee , 2005; Shu ,
1992].
In this thesis, I utilize shock parameters determined by Kasper [2007] using the
conservation relations discussed in this section.
1.2.5 Energy Dissipation Mechanisms in Collisionless Shocks
Collisionless shock waves are known to be efficient mechanisms by which charged par-
ticles can be heated and/or accelerated. The collisionless nature of these shocks prevents
binary particle collisions from dominating the transfer of bulk kinetic energy to thermal
energy. Thus, energy dissipation mechanisms have been a focus for research [Mellott ,
1984] since the prediction of collisionless shock waves [Kellogg , 1962]. Three possible en-
ergy dissipation mechanisms discussed here are wave dispersion [Mellott and Greenstadt ,
1984], anomalous resistivity due to wave-particle interactions [Gary , 1981], and particle
reflection [Edmiston and Kennel , 1984; Kennel , 1987].
The type of energy dissipation important in a collisionless shock wave depends
strongly on the magnetosonic or fast mode Mach number, Mf , shock normal angle, θBn,
and the ratio of particle to magnetic pressures called the plasma beta, β [Mellott , 1984].
At low Mach number shocks, the energy dissipation has been shown to include wave dis-
persion, thermal conduction, and anomalous resistivity [Mellott and Greenstadt , 1984;
Mellott , 1985; Wilson III et al., 2007]. As the Mach number increases, one can show
that there exists a Mach number where resistivity cannot limit the wave steepening. For
a true shock transition, the form of energy dissipation, within a few wavelengths, must
act to increase upstream perturbations (i.e. whistler waves) approaching the shock and
decrease downstream perturbations leaving the shock [Kennel et al., 1985]. However,
when the downstream flow speed exceeds the downstream sound speed, stationary point
analysis shows that perturbations downstream will grow. This implies that the shock
is unstable. Typically when one includes dispersion, conduction, resistivity, or viscosity
into the Rankine-Hugoniot relations, the mechanisms producing these terms are not
considered. In other words, the inclusion of these terms into the conservation relations
across a discontinuity are devoid of any physically significant content. Since the analysis
15
is done on the asymptotic values conserved across the ramp, the specifics of these effects
can be ignored and treated simply as sources or losses of energy. The type of dissipation
also affects the scale length of the shock transition region. In the following paragraphs,
we will discuss the types of energy dissipation focused on in this thesis.
There are three basic scale lengths in a two-fluid model of a plasma, c/ωpe, c/ωpi,
and λDe. These scale lengths serve as deterministic limiting factors for the time scales
over which currents or charge densities can change. So as a nonlinear compressional
wave steepens to dispersive scale lengths (larger than resistive scales), the dispersion
can limit steepening. The dissipation is accomplished by the radiation of a nonlinear
wave which carries the short-wavelength energy in the dispersive range away from the
compressive nonlinearly steepened wave. The nonlinear wave, originally radiated by the
steepened front, is later dissipated by wave damping. Whether these short-wavelength
waves propagate faster or slower than the steepened front determines whether they lead
or trail the front itself. If the front forms into a shock and dispersion is limiting the
steepening of the shock, then the radiated nonlinear waves must damp out asymptoti-
cally away from the shock. In the simplest case, the entire wave train is time-stationary
in the shock frame, which implies that the nonlinear waves are phase standing with
respect to the shock. In other words, their wave vector is aligned with the shock wave
vector and their phase speeds match. To carry energy away from the shock, the wave
group velocity must be greater than(less than) the upstream(downstream) flow speed if
it phase stands upstream(downstream) [Kennel et al., 1985].
At short wavelengths, the fast mode converts to an elliptically polarized whistler
wave whose phase speed exceeds the fast mode phase speed. As the wavelength con-
tinues to decrease, the fast mode phase speed increases which implies a leading non-
linear wave train for shocks propagating obliquely to the magnetic field. It should be
noted that only low Mach number shocks exhibit a leading wave train and that all
dispersive scale lengths are larger than resistive dissipative scale lengths. Wave disper-
sion has been well investigated at low Mach number (.2) collisionless shock waves by
Mellott and Greenstadt [1984] and Mellott [1984, 1985].
Ion-acoustic waves (IAWs) have long been believed to provide anomalous resistivity
in collisionless shock waves [Sagdeev , 1966] because the current threshold is low when
Te > Ti and the Debye length is small compared to electron inertial scales. In the
16
solar wind, ωpe ≫ Ωce which causes the IAWs to become decoupled from the mag-
netic field. This allows the waves to interact with the bulk of the electron distribution,
which can lead to energy dissipation through particle heating [Dum et al., 1974; Gary ,
1981]. In the case of collisionless shock waves, IAWs are thought to be driven by
a relative drift between electrons and ions across the magnetic field, or a cross-field
current. Anomalous resistance results from radiated waves, due to a relative drift be-
tween electrons and ions, scattering the drifting particles in such a way as to reduce
the drift between the two species. Thus, the waves act to mediate a transfer of mo-
mentum from the electrons to the ions [Kennel et al., 1985]. It is important to note
that there are number of possible wave modes/instabilities which have been suggested
to provide anomalous resistivity at quasi-perpendicular shock waves besides current-
driven IAWs [Gary and Sanderson , 1970; Gary , 1970a, 1981; Lemons and Gary , 1978],
including lower-hybrid waves [Lemons and Gary , 1978], lower-hybrid drift instability
[Davidson and Gladd , 1975], two-stream or modified two-stream instabilities [Kellogg ,
1965; Lemons and Gary , 1978], electron cyclotron drift instability [Forslund et al., 1970,
1971, 1972; Lampe et al., 1971a,b, 1972a; Wong , 1970], and the Buneman instability
[Buneman , 1959; Kellogg , 1964].
Only a few studies have been done on resistive dissipation [Wilson III et al., 2007],
and its significance with respect to the other possible mechanisms is not well under-
stood. It is thought that wave-particle interactions produce an anomalous resistivity
that limits the cross-field currents in the shock ramp that produce the abrupt change
in magnetic field [Gary , 1981]. Though some studies suggest that wave-particle in-
teractions are not necessary to heat the downstream plasma [Hull et al., 1998, 2000;
Hull and Scudder , 2000; Hull et al., 2001], recent observations [Wilson III et al., 2007]
and simulation studies using realistic mass ratios [Petkaki et al., 2006] suggest other-
wise.
In classical fluid theory, there is a Mach number called the critical Mach num-
ber, Mcr, above which resistive energy dissipation can no longer limit wave steepening
[Edmiston and Kennel , 1984; Kennel et al., 1985; Kennel , 1987]. To determine this
parameter, one assumes that the downstream normal flow speed is equal to the down-
stream sound speed. Thus, any acoustic waves which would normally propagate with
the shock in the ramp would be convected into the downstream making them no longer
17
Time (HH:MM:SS)
16:39:00 16:40:00 16:41:00 16:42:00
Wind 3s B-Field:  10/21/2001
|B
| 
(n
T
)
20
15
10
0
-10
-20
10
20
B
 [
G
S
E
] 
(n
T
) X
Y
Z
Ramp
Foot
Overshoot
Figure 1.3: An example of a supercritical quasi-perpendicular IP shock observed by
Wind on 10/21/2001 illustrating the different regions of a collisionless shock. The foot
of the shock is highlighted by the blue region, the ramp by green and overshoot by red.
18
a source of energy dissipation in the ramp. The often quoted M cr ≈ 2.7 is valid only
for a perpendicular shock (shock normal angle, θBn = 90◦) propagating into a cold
plasma. For more realistic ranges of θBn and temperature, typical solar wind conditions
will actually yield M cr ∼1–2 [Edmiston and Kennel , 1984; Kennel , 1987], values often
observed at interplanetary (IP) shocks which suggested particle reflection may occur
even at low Mach number shocks.
In the simplest treatment one can ignore the specifics of the reflection process and
treat the reflected particle population as an extra energy loss term in the Rankine-
Hugoniot relations. However, the consequences of the microphysics behind ion reflection
leads to a number of instabilities which help dissipate energy. For instance, if ions reflect
and stream upstream of the shock they can excite ion-ion instabilities [Gary et al., 1981].
The reflected ions can also interact with the incident electrons producing a modified two-
stream instability, which can act to slow the incident plasma [Matsukiyo and Scholer ,
2003, 2006a]. The deceleration of the incident flow compresses the plasma producing
a shock foot. The relative fraction of reflected ions could potentially be reduced by
cross-field current driven instabilities in the shock ramp [Forslund et al., 1984].
Figure 1.3 highlights the different regions of a quasi-perpendicular shock observed
by the Wind spacecraft on 10/21/2001. The structure of collisionless shocks is often
defined by the magnitude of the magnetic field. The shock itself requires a discontinuous
transition from a high speed upstream to a low speed thermalized downstream, referred
to as the ramp (see green region of Figure 1.3). The ramp of a magnetized shock can be
defined as the duration from the point of lowest magnetic field immediately preceding
the discontinuous transition to the point of highest magnetic field immediately following
the discontinuous transition [Farris et al., 1993].
Particle reflection/energization at collisionless shocks has been extensively stud-
ied [Paschmann et al., 1980, 1981, 1982; Meziane and D’Uston, 1998; Meziane et al.,
2004a,b]. Furthermore, ISEE 1 and 2 observations of ion reflection at a subcritical
terrestrial bow shock have found evidence to suggest that the transition between sub-
critical and supercritical may not be abrupt [Greenstadt and Mellott , 1987]. Thus, ion
reflection is likely an ever-present dissipation mechanism for quasi-perpendicular shocks,
but it does not become a dominant mechanism until higher Mach numbers.
The magnetic foot, a region immediately preceeding the ramp with thickness of
19
roughly an ion inertial length (c/ωpe), is created by reflected gyrating ions (blue region
of Figure 1.3) [Leroy et al., 1982; Livesey et al., 1984; Paschmann et al., 1982]. The gy-
rating ions are produced by nearly specular reflection, where roughly 20% of the incident
ions are reflected [Leroy et al., 1982; Paschmann et al., 1980; Thomsen et al., 1985a].
The foot region itself supports a large fraction of the energy dissipation in supercritical
shock waves by slowing and pre-heating the incident plasma [Leroy et al., 1982]. The
ions skip along the shock front but the electrons remain constrained to the magnetic field
and the relative drift between the two produces a current. It is this current that causes
the small increase in magnetic field strength which defines the foot. Such gyrating ions
are sources of free energy for multiple instabilities [Matsukiyo and Scholer , 2006a] and
they can produce instabilities upstream of the shock which alter the Rankine-Hugoniot
conditions [Scholer and Belcher , 1971].
Giacalone et al. [1991] examined the effect of the shock overshoot on reflection co-
efficients of ions at collisionless shocks. They found that an increase in the overshoot
amplitude increased the number of reflected particles and the loss of energy by the
transmitted particles. Leroy [1983] found that a portion of the incident ion population
is directly transmitted and accumulates in the immediate post-ramp region developing
an overshoot as a natural consequence of the Rankine-Hugoniot conservation relations.
Mellott and Livesey [1987] found that magnetic overshoots resulted from a combination
of reflected ions and turbulence in supercritical shocks. Paschmann et al. [1982] ob-
served gyrating ions at a nearly perpendicular bow shock crossing. The gyrating ions
were clearly distinct from the specularly reflected ions. Such gyrating ions are thought
to influence if not produce the shock overshoot.
Figure 1.4 is four IP shock examples of 2.5 minutes magnetic field data taken from
the Wind spacecraft centered on the shock ramps. The two panels on the left are exam-
ples of quasi-parallel shocks and the two on the right are quasi-perpendicular. The top
left panel is an approximately laminar quasi-parallel shock and the turbulent counter-
part (directly below) shows the chaotic transition region of a supercritical quasi-parallel
shock. However, for both quasi-perpendicular examples, the fields are much more lam-
inar when viewed on these time scales. The figure is useful in illustrating the range of
turbulence one can observe in IP shock waves.
20
|B
| 
(n
T
)
8
6
10
B
 [G
S
E
] (
nT
)
0
-5
5
10
HTR Wind B-Field:  02/11/2000 (2.5 Min. Window)
Time (HH:MM:SS)
Time (ms)
02:34:22
0.2485
02:34:52
0.1485
02:35:22
0.0495
02:35:51
0.9495
02:36:21
0.8495
Turbulent Quasi-Parallel
Time (HH:MM:SS)
Time (ms)
06:39:39
0.9315
06:40:09
0.8325
06:40:39
0.8245
06:41:09
0.7245
06:41:39
0.6245
25
20
15
10
|B
| 
(n
T
)
B
 [G
S
E
] (
nT
)
10
0
-10
20
HTR Wind B-Field:  08/26/1998 (2.5 Min. Window)
Laminar Quasi-Perpendicular
(Supercritical)(Supercritical)
Time (HH:MM:SS)
Time (ms)
17:28:23
0.9945
17:28:53
0.8935
17:28:23
0.7935
17:29:53
0.6945
17:30:23
0.5945
B
 [G
S
E
] (
nT
) 2
-2
0
-4
-6
4
|B
| 
(n
T
)
6
7
5
4
HTR Wind B-Field:  02/27/1997 (2.5 Min. Window)
Laminar Quasi-Parallel
(Subcritical)
Time (HH:MM:SS)
Time (ms)
11:17:24
0.9715
11:17:54
0.9165
11:18:24
0.9085
11:18:54
0.8555
11:19:24
0.8005
|B
| 
(n
T
)
B
 [G
S
E
] (
nT
)
-5
0
-10
-15
5
16
12
14
18
8
20
10
HTR Wind B-Field:  10/24/1997 (2.5 Min. Window)
Turbulent Quasi-Perpendicular
(Subcritical)
XYZ XYZ
XYZ XYZ
Figure 1.4: Four examples of a super and subcritical quasi-perpendicular and quasi-
parallel IP shocks observed by Wind. The two panels on the left are examples of sub
and supercritical quasi-parallel IP shocks. The two panels on the right are the same for
quasi-perpendicular. Notice that the term laminar is a relative term for each type of
shock wave.
1.3 Shock Types
1.3.1 Bow Shocks
Bow shocks form when an object obstructs a supersonic flow in some medium. Ex-
amples would be a planet or comet in the solar wind. The solar wind is highly supersonic,
thus an object roughly at rest with respect to the sun would appear to the solar wind as
a ballistic object moving at supersonic speeds. They were predicted to exist upstream of
planetary magnetospheres by Kellogg [1962] and first measured by Sonett et al. [1964].
An illustrative cartoon of the Earth’s foreshock, Figure 1.5, defines the different
regions of a bow shock. There are two main types of collisionless shock geometries,
quasi-parallel and quasi-perpendicular. Quasi-parallel shocks occur when the angle
between the shock normal vector (the blue nˆ) and the interplanetary magnetic field
(IMF), called the shock normal angle or θBn, is ≤ 45◦. A quasi-perpendicular shock
21
Magnetopause
Ion
Foreshock
IM
F Foreshock
Electron
Quasi-Parallel
Quasi-Perpendicular
n^
n^
Earth
M
ag
ne
to
sh
ea
th
Bow Shock
Sun
Figure 1.5: This figure is an illustrative example of the Earth’s foreshock, bow shock, and
magnetopause. The different regions are labeled appropriately along with illustrative
examples of quasi-parallel versus quasi-perpendicular shock definitions. The yellow dots
are supposed to represent electrons while the red protons. Due to the higher speeds of
electrons, on average, one can see the electron foreshock extends further upstream than
the ion foreshock. Also note, the quasi-parallel section of the bow shock appears much
more turbulent than the quasi-perpendicular due to the reflected ions. This image is
adapted from Plate 1 in [Tsurutani and Rodriguez , 1981].
22
occurs when θBn > 45◦. One of the more important differences between a collisionless
shock and a shock wave in a regular fluid is the capacity for collisionless shock waves to
communicate with the upstream medium. This is accomplished by reflecting particles
off the shock front that propagate away from the shock [Edmiston and Kennel , 1984;
Greenstadt and Mellott , 1987; Kennel , 1987]. The region upstream of a shock that can
communicate with the shock is called the foreshock region. As a consequence of their
much smaller mass, electrons can often travel further upstream than ions. This can be
seen as the yellow region illustrated in Figure 1.5, called the electron foreshock. The
ion foreshock corresponds to the red region of Figure 1.5.
1.3.2 Coronal Mass Ejections
Coronal Mass Ejections (CMEs) are large transient eruptions of plasma from the
sun’s surface. Once a CME leaves the Sun, it is usually called an interplanetary CME
(ICME). They are the result of the largest explosions in the solar system releasing up
to ∼ 1034 ergs (or ∼ 1027 joules) This is roughly the equivalent of the energy released
by detonating 20 million 100 Megatons nuclear bombs [Lang , 2000]. CMEs were first
discovered using coronagraphs on the OSO-7 and Skylab spacecraft during the early
1970’s [Kahler , 1992]. The CME is observed in white light as a bright ’umbrella-like’
structure on the leading edge of a dark cavity. The light observed is caused by Thom-
sen scattering of photons by free electrons. Thus the intensity depends on the density
of electrons, not their temperature. The bright leading edge is a high density plasma
known as the sheath while the dark trailing cavity is a low particle density high mag-
netic field region, called a magnetic cloud. Upwards of 5-50 billion tons of plasma can
be ejected at speeds over 2000 kilometers per second in some of the strongest events
[Lang , 2000]. The eruptions can take anywhere from a few minutes to hours to leave
the solar surface [Schwenn , 2006].
Figure 1.6 is an example of coronagraph images of two different erupting CMEs
taken from the High Altitude Observatory MacQueen et al. [1980]. The images are
white light images taken on April 14th, 1980 and October 24, 1989. The leading edge of
the CME, seen as a bright light bulb-shaped emission, is the sheath region of the shock
wave formed by the erupting CMEs.
23
CMEs are typically assumed to result from large scale changes in the magnetic field
topology of the sun. Solar flares were previously thought to cause many of the phe-
nomena now known to be associated with CMEs [Bougeret , 1985; Gosling et al., 1968;
Schwenn , 2006]. CMEs can cause large geomagnetic storms either by compressing the
magnetosphere due to a high ram pressure or by having a large and continuously south-
ward magnetic field component allowing for reconnection at the Earth’s magnetopause
[Luhmann et al., 1998; Schwenn, 2006]. CMEs are also known to accelerate high energy
particles towards Earth which can, in extreme cases, cause lethal radiation doses to
spacecraft and astronauts [Schwenn , 2006].
Figure 1.7 is a cartoon which attempts to show a possible 2D magnetic field topology
Figure 1.6: The evolution of two CMEs in white light from the High Altitude Obser-
vatory MacQueen et al. [1980]. The white light seen is due to Thomsen scattering of
electrons in the high density sheath of the CMEs.
and various regions in and around a CME. The upstream region is the area farther away
from the sun than the red bow-shaped line (marking the shock ramp) while the down-
stream is closer to the sun. The black(green) lines correspond to open sunward(anti-
sunward) magnetic field lines while the blue lines correspond to closed field lines. As with
the bow shock example, the quasi-parallel section of the shock is accompanied by low
24
frequency electromagnetic waves which are less easily excited at the quasi-perpendicular
sections.
As a CME leaves the surface of the sun, it accelerates and compresses the plasma on
Magnetic
Cloud
Shock
n^
n^
Quasi-Parallel
Quasi-Perpendicular
Low Frequency
Waves
Figure 1.7: A cartoon of a CME with magnetic cloud propagating away from the sun.
The black(green) lines correspond to open sunward(anti-sunward) magnetic field lines
while the blue lines correspond to closed field lines. The cartoon illustrates the different
regions of the shock.
the leading edge of a magnetic cloud (seen in Figure 1.7). If the CME accelerates at a
sufficient rate and reaches a high enough speed, the wave forming on the bright leading
edge (seen in Figure 1.6) steepens. Since compressional waves have the fundamnetal
property that they steepen, the compressed plasma at the leading of the CME steepens
unless sufficient energy is removed from the wave. When a CME reaches a speed faster
than the local magnetosonic speed (Cf or fast mode speed), a shock wave can form. The
magnetic cloud of a CME, or ejecta, is often separated from the leading plasma. The
25
shock front is on the leading edge of a plasma sheath which results from compression,
deflection, and heating of the solar wind. The ejecta are separated from the sheath by a
tangential discontinuity [Schwenn , 2006]. A tangential discontinuity is a special type of
contact discontinuity where no mass or magnetic flux cross the boundary. An example
of a tangential discontinuity is the Earth’s magnetopause in regions where magnetic
reconnection is not occurring. CMEs are known to expand nearly as fast as they prop-
agate radially outward [Siscoe and Schwenn , 2006; Siscoe et al., 2006]. Thus, by the
time they reach 1 AU, they are often in excess of 1 AU in diameter. Since IP shocks
are extremely large compared to the Earth’s bow shock, one can typically assume a
planar geometry. This assumption simplifies the shock normal vector calculation rather
significantly. CME-driven shock waves are also more likely to be quasi-perpendicular
than quasi-parallel at 1 AU which has implications on dissipation mechanisms [Gary ,
1981]. In this thesis, I will focus on observations of waves and particle distributions
at IP shocks.
1.4 Particle Distribution Observations in Solar Wind and
at Shocks
1.4.1 Distribution Function Moments
Particle distributions functions are useful because they are solutions to the equations
of motion described by the Vlasov and/or Boltzmann equations, thus they allow for a
statistical description of a fluid.
For instance, the heat flux is an important quantity because it describes the flow of
thermal energy into or out of some volume. In the solar wind, electrons carry the flow
of energy due to their lighter mass than that of the much heavier ions. The electron
heat flux is due to a relative drift between the higher energy and lower energy elec-
trons. Since the electron heat flux is responsible for the expansion of the solar corona
[Feldman et al., 1975], it is important to study and understand how the heat flux is
generated, maintained, and altered in the solar wind.
The heat flux, also called the kinetic energy flux in the solar wind reference frame,
is derived from the third moment of the distribution function. Though in its general
26
mathematical form the heat flux is simply a third moment integral resulting in an asym-
metric rank-3 tensor. Typically the heat flux is seen as a tensor in the second moment
integral. However, one usually assumes certain symmetries to reduce the calculation to
a rank-2 tensor, which can be treated as a matrix. In practical applications, such as
calculating the heat flux for a real particle distribution function, the heat flux is derived
from the third moment. In general, the third moment is known as the skewness, or the
measure of the degree of asymmetry of a distribution function. Further assumptions
can reduce the calculation to a rank-1 tensor, which can also be a simple vector. The
vector form of the heat flux can be shown as:
~q =
ms
2
∫
V
d3v~vv2f s(~x, ~v, t) . (1.19)
The heat flux vector, ~q, is the form most often used in plasma physics because it is
more physically tangible and easier to work with.
In this thesis, I examine evidence of wave-particle interactions observed as scattering
of the electron heat flux vector in Chapter 4.
1.4.2 Previous Observations and Examples of Electron Distributions
Electron velocity distributions have been measured in the tenuous solar wind for
over 30 years [Feldman et al., 1975]. A number of very distinct properties were quickly
distinguished, starting with the observation that the solar wind electrons are often com-
posed of a cool dense Maxwellian core (typically < 60 eV) and a more tenuous hot halo
component consistent with a power-law [Feldman et al., 1975]. The energy at which the
core component no longer dominated the distribution was defined as the break energy.
Later a very prominent field-aligned signature was noticed, now called the strahl (Ger-
man for bright or beam), in the high energy halo electrons [Pilipp et al., 1987a] and
lastly a high energy super halo [Lin et al., 1996]. The differences in functional form of
the two electron components is important because they can be used to infer their pro-
genitor/source [Lin et al., 1996]. For instance, any distribution can be shown to relax to
a Maxwellian if it is in a collision dominated medium [Gurnett and Bhattacharjee , 2005]
and a power-law can illuminate the possible mechanisms which led to the energization
of these particles [Lin et al., 1996].
27
Due to their much lighter mass than hydrogen and helium, electrons carry the bulk
of thermal energy flow away from the sun. The electron heat flux, or more appropri-
ately, the kinetic energy flux in the plasma rest frame, is the dominant component of
energy flux in the solar wind. This heat flow is also thought to play a critical role in the
expansion of the solar corona, thus studying electron moments in the solar wind have
been motivated by many factors [Feldman et al., 1975].
The halo electrons were originally named due to their often isotropic shape in phase
space, much like a halo, and nonthermal properties [Feldman et al., 1975]. The halo
electrons were also observed to drift relative to the cold dense thermal core of the distri-
bution, producing a net kinetic energy flux. This portion of the distribution drifting rela-
tive to the cold core is the part responsible for the finite electron heat flux observed in the
solar wind streaming away from the sun along the magnetic field [Feldman et al., 1973].
It was later observed that the electron heat flux in the solar wind would occasionally
”disappear.” The effect is referred to as a heat flux dropout (HFD), which is currently
assumed to result from a magnetic disconnection from the sun [Crooker et al., 2003;
Pagel et al., 2005]. de Koning et al. [2006, 2007] examined the width of the suprather-
mal electron pitch-angle distributions (PADs) during solar electron burst events with
the ACE spacecraft. Simulations and theory have been pursued to explain the discrep-
ancies between the expected heat flux values and the observed values [Vocks and Mann,
2003; Vocks et al., 2005]. A number of instability mechanisms driven by electron heat
flux in the solar wind have been theoretically examined in detail by Gary et al. [1975,
1994, 1999].
Many different distribution types observed in the terrestrial magnetosphere have
been observed near collisionless shocks or in the solar wind. For instance, Larson et al.
[1996] examined loss-cone like distributions at the bow shock with the Wind spacecraft,
Gosling et al. [1989a] examined suprathermal electrons in the bow shock and magne-
tosheath with ISEE 1 and 2, and temperature anisotropies were examined in the solar
wind between 0.3 and 0.8 AU with Helios 1 and 2 [Pilipp et al., 1987b] and at IP shocks
[Wilson III et al., 2009] with the Wind spacecraft.
Figure 1.8 shows an example of an electron distribution seen in the solar wind ob-
served by the Wind spacecraft. This figure is representative of many of the particle dis-
tribution function figures shown later in this thesis. The top panel is a two dimensional
28
V
 p
e
rp
e
n
d
ic
u
la
r 
(1
0
0
0
 k
m
/s
)
-20
-10
0
10
20
V parallel (1000 km/s)
-20 -10 0 10 20
Velocity (1000 km/s)
-20 -10 0 10 20
10
-16
10
-14
10
-10
10
-12
3
3
3
d
f 
(s
e
c
  
/k
m
  
/c
m
  
)
--- : Parallel Cut
- - - : Perpendicular Cut
Wind 3D Plasma Eesa Low
2000-04-06/16:28:57 - 16:29:00
Vsw
Qf
nsh SunDirection
Figure 1.8: An example of an electron particle distribution. The top panel, the contour
plot, is 2D projection of the particle distribution function where the horizontal axis is the
velocity parallel(anti-parallel) to the magnetic field and the vertical axis is perpendicular
to the magnetic field. Four vectors are projected onto the contour plot: heat flux (Qf),
solar wind velocity (Vsw), shock normal vector (nsh), and the sun direction. The bottom
panel is a plot of the parallel and perpendicular cuts of the distribution function.
29
contour projection of the three dimensional distribution function where the horizontal
axis is the parallel/anti-parallel (with respect to the magnetic field) velocity direction
and the vertical axis is perpendicular velocity. The axes range from -20,000 km/s to
+20,000 km/s. The distribution function was constructed assuming gyrotropy, or az-
imuthal symmetry with respect to the magnetic field. Although this is not a necessary
process, it often makes analysis easier. On the top panel, there are four straight lines
corresponding to the direction of the electron heat flux (red line), solar wind velocity
(black line), shock normal vector (blue line), and sun direction (purple line). Each line
is projected into the same coordinate system as the distribution contours. The bottom
panel shows the parallel (black line) and perpendicular (blue line) cuts of the distribu-
tion function shown in the contour plot. The vertical axis corresponds to the magnitude
of the phase space density (s3km−3cm−3), or units of the distribution function, while
the horizontal axis is velocity (with the same limits as the contour plot).
The transfer of energy from a nonthermal source to heat is a topic of considerable
interest in collisionless plasmas because energy transfers to heat tend to occur through
binary collisions in a regular fluid or free-free Bremsstrahlung interactions in the solar
atmosphere [Aschwanden , 2005]. The lack of sufficient binary collisions in a plasma
requires other mechanisms by which to transfer energy [Gary et al., 1975]. Electron
heating has been examined at the bow shock using ISEE 2 data [Feldman et al., 1982],
compared to ion heating at the bow shock using ISEE 1 and 2 data [Thomsen et al.,
1985a], compared to the de Hoffmann-Teller frame potential across IP and bow shocks
with ISEE spacecraft [Schwartz et al., 1988], used as a proxy for the de Hoffmann-Teller
frame potential across collisionless shocks [Hull et al., 2000; Hull and Scudder , 2000],
and in association with atypical wave modes at IP shocks [Wilson III et al., 2009].
Electrons have been theorized to drive and/or damp electromagnetic fluctuations in
many regions of space. Early predictions of electron-ion energy/momentum exchanges,
thermal conductivities, and heat flux originally suggested electron(ion) temperatures
which were higher(lower) than the observed values in the solar wind [Gary et al., 1975;
Spitzer and Ha¨rm , 1953]. Thermal conductivities, produced by binary particle colli-
sions, were originally predicted to produce an electron heat flux proportional to a con-
stant multiplied by the electron temperature gradient [Gary et al., 1975]. The proposed
solutions included a variety of plausible explanations, but the most likely is heat flux
30
generated instabilities affecting the solar wind plasma [Gary et al., 1975, 1994, 1999;
Gurnett et al., 1979a; Saito and Gary , 2007; Wilson III et al., 2007, 2009].
Electron beams are known to be a source of free energy for instabilities and have been
examined with ISEE 1 data [Fitzenreiter et al., 1984], in the ionosphere with sounding
rockets [Kellogg et al., 1986], in the foreshock with the Wind spacecraft
[Fitzenreiter et al., 1996], in association with Type III radio bursts and impulsive elec-
tron events using Wind [Ergun et al., 1998a], and to map the structure of the source
region for Type II radio bursts in IP shocks observed by Wind [Pulupa and Bale , 2008].
In this thesis, I utilize electron distribution measurements to examine particle heat-
ing signatures at IP shocks and to look for evidence of wave-particle interactions.
1.4.3 Previous Observations and Examples of Ion Distributions
The irregular turbulence seen in magnetic field measurements upstream of plane-
tary bow shocks for over 40 years [Fairfield , 1969] often had frequencies in the range
of the ion cyclotron frequency. Thus, the examination of ion particle distributions in
association with the waves seemed like the natural thing to do. Three ion popula-
tions, reflected, intermediate, and diffuse, are commonly found in planetary foreshocks.
The distinction between different distributions became important when it was discov-
ered there was a strong correlation between the distribution type and location within
the foreshock [Hoppe et al., 1981; Paschmann et al., 1979, 1981]. Later studies found
there to be what appeared to be boundaries between different regions of the foreshock
that strongly depended on the angle between the interplanetary magnetic field and the
X-GSE direction [Meziane and D’Uston, 1998]. Since, the first observations of ions up-
stream of collisionless shocks, they have been a topic of intense interest due to their
capacity to carry energy away from the shock and interact with waves.
Reflected Ions
Reflected ions have a beam-like feature in their distribution with bulk speeds on the
order of 1-5 times the solar wind speed. They occur predominantly near shocks with
shock normal angles, θBn, between 30◦-75◦ (see Figure 1.9) [Bonifazi and Moreno, 1981;
Paschmann et al., 1981; Schwartz et al., 1983]. They have broad thermal spreads and
are often anisotropic with T⊥b/T‖b > 1 [Paschmann et al., 1981]. The beams typically
31
have nb/no ∼ 0.3-13%, where nb is the number density of the beam and no is the den-
sity of the solar wind [Bonifazi and Moreno, 1981; Paschmann et al., 1981]. Reflected
ions carry most of their energy in the form of kinetic energy and not thermal energy
[Bonifazi and Moreno, 1981].
Figure 1.9 is an example of a reflected ion distribution function observed by the
Wind spacecraft. The top panel plots contours of constant phase space density versus
particle velocity with the horizontal axis being the velocity parallel to the background
magnetic field and the vertical axis perpendicular to magnetic field in the plane defined
by the magnetic field and solar wind velocity (black line projected onto contour). The
bottom panel plots the parallel and perpendicular cuts of the distribution function. The
wind ion measurements shown range from -2500 km/s to +2500 km/s. The color con-
tours range from 10−12 to 10−10 s3 km−3 cm−3.
Reflected ions streaming away from the shock along the magnetic field can inter-
act with the incident solar wind through ion-ion instabilities producing large amplitude
magnetic fluctuations both parallel and oblique to the magnetic field [Akimoto et al.,
1993; Gary et al., 1981] and low frequency oblique whistler waves [Akimoto and Winske,
1989]. These hydromagnetic waves can pitch-angle scatter the reflected ion beams as
they convect back towards the bow shock because their group and phase velocities are
smaller than the solar wind bulk flow (typically). They are usually transverse at this
stage with very small compressive components. As a consequence of the scattering, the
ion beams broaden (in pitch-angle) into crescent-shaped ion distributions called interme-
diate ions, which are usually seen deeper (i.e. further from the sun) into the foreshock
region. Eventually, it is thought, these wave-particle interactions scatter the ions to
the point of suprathermal spherical shells in phase space called diffuse ion distributions
[Gary et al., 1981].
Intermediate and Gyrating Ions
Intermediate ion distributions represent a transition between reflected and diffuse
ion distributions (see panel A in Figures 1.11 and 1.10) and appear as a crescent-shaped
distribution with centers of curvature near the solar wind velocity. They have been ob-
served in association with ULF foreshock waves and show a distinct boundary with
32
3
3
3
d
f 
(s
e
c
  
/k
m
  
/c
m
  
)
10
-10
10
-12
10
-11
10
-9
10
-8
10
-7
-2000
-1000
0
1000
2000
Vp
e
rp
 
[km
/s
]
-2000 -1000 0 1000 2000
Vpara [km/s]
-2000 -1000 0 1000 2000
Velocity [km/s]
Wind 3DP Pesa High Burst
2000-04-10/15:22:15-15:22:18
Solar Wind Direction
+++ : Parallel Cut
* * * : Perpendicular Cut
Ion Beam
Figure 1.9: Here is an example of an ion beam seen upstream of the Earth’s bow
shock by the Wind Pesa High instrument immediately after (i.e. closer to the sun) the
intermediate distribution in Figure 1.10. The contour ranges are the same for both of
these plots.
33
gyrating and diffuse ions [Meziane and D’Uston, 1998; Meziane et al., 2001]. Interme-
diate ions often show total energy densities that are nearly the same as the reflected
ions observed adjacent to them (spatially) [Bonifazi and Moreno, 1981; Gary , 1985].
Figure 1.10 is an example of an intermediate ion distribution function. The format
is similar to Figure 1.9. The intermediate ions can be seen as the crescent-shaped peak
in the contour plot outlined by the red oval and in the parallel cut of the distribution
by the red box. The main difference from a reflected ion distribution is that the inter-
mediate ions are much broader in the direction perpendicular to the magnetic field in
phase space. They are thought to result from the wave-particle scattering of reflected
ion beams [Mazelle et al., 2003].
Gyrating ion distributions show signatures of gyromotion about the magnetic field
and can be gyrophase-bunched (nongyrotropic) or nearly gyrotropic [Meziane et al.,
2001]. The gyrotropic distributions are typically observed closer to the shock than
the nongyrotropic ions. Gyrating ions are commonly observed in association with the
magnetic foot and overshoot of quasi-perpendicular supercritical shocks due to specular
reflection [Paschmann et al., 1980, 1982; Sckopke et al., 1983; Thomsen et al., 1985b].
Such distributions are thought to play an integral part in the processes which heat
the ion distributions downstream of the shock [Sckopke et al., 1983]. The gyrophase-
bunched ions can be created/formed by reflection at a collisionless shock or the disrup-
tion of an ion beam by waves generated from a beam-plasma instability [Akimoto et al.,
1991; Meziane et al., 2001; Thomsen et al., 1985b]. The finite thermal velocity of the
gyrophase-bunched ions should lead to gyrophase mixing of the distributions, which
would result in a more gyrotropic distribution [Burgess and Schwartz , 1984]. Thus the
observations of nongyrotropic distributions far upstream of the bow shock suggest local
production mechanisms [Mazelle et al., 2003; Thomsen et al., 1985b].
Diffuse Ions
Diffuse ion distributions are a highly nonthermal, relatively isotropic spherical shell-
like distribution in phase space extending up to ∼40 keV (see panel B in Figure
1.11). Diffuse ions often show anisotropies with pitch-angle distributions peaking at
90◦ [Gosling et al., 1984; Paschmann et al., 1981; Sentman et al., 1981] and have aver-
age thermal spreads ∼5-7 times greater than reflected ions [Bonifazi and Moreno, 1981].
34
3
3
3
d
f 
(s
e
c
  
/k
m
  
/c
m
  
)
10
-10
10
-12
10
-11
10
-9
10
-8
10
-7
-2000
-1000
0
1000
2000
Vp
e
rp
 
[km
/s
]
-2000 -1000 0 1000 2000
Vpara [km/s]
-2000 -1000 0 1000 2000
Velocity [km/s]
Wind 3DP Pesa High Burst
2000-04-10/15:22:12-15:22:15
+++ : Parallel Cut
* * * : Perpendicular Cut
Solar Wind Direction
Intermediate Ions
Figure 1.10: Here is an example of intermediate ions seen upstream of the Earth’s bow
shock by the Wind Pesa High instrument. The color contours range from 10−12 to 10−10
s3 km−3 cm−3 and the UV contamination effects have been reduced.
35
They are likely the result of wave-particle interactions that pitch-angle scatter and heat
an ion beam into a nearly isotropic shell, thus a shock generated source [Gosling et al.,
1984, 1989b; Winske and Leroy , 1984].
Figure 1.11 is two examples of an diffuse ion distributions upstream of an IP shock.
The contour plots have the same format as those in Figures 1.9 and 1.10. The diffuse
ions are seen as a nonthermal halo or shell-like distribution between 800-2500 km/s in
both parallel and perpendicular directions. The theories behind their production and
their importance will be discussed below.
Many theories, in conjunction with observations, suggest a spatial correlation with
observations of diffuse ions and their production results from the interaction of low
frequency hydromagnetic waves interacting with the foreshock ions [Bame et al., 1980;
Gary et al., 1981; Gosling et al., 1984; Hoppe et al., 1981; Mazelle et al., 2003]. Diffuse
ions can dramatically alter the index of refraction of the local medium [Hada et al., 1987;
Omidi and Winske, 1990] because they can carry a significant fraction of the total en-
ergy density of the ions [Bonifazi and Moreno, 1981]. As a consequence of their influence
on the medium, a class of highly steepened magnetic structures referred to as shock-
lets and/or SLAMS are always seen in association with diffuse ions (see Chapter 3 for
further discussion) [Hoppe et al., 1981, 1982; Hoppe and Russell , 1983; Thomsen et al.,
1985b, 1990; Wilson III et al., 2009].
In this thesis, I utilize ion distribution measurements to determine solar wind density
and velocity, and examine evidence for ion reflection and acceleration.
1.5 Wave Particle Interactions
1.5.1 The Dispersion Relation
In order to discuss the interactions of waves and particles, one must begin by dis-
cussing a property called dispersion. Dispersion is a property of a medium where the
frequency, ω, depends on the wave number, k. A simple example of a dispersive medium
would be a prism. If one sends multi-wavelength (e.g. white light) into the prism, one
notices a rainbow when the light exits the opposite side. The incident light is undergo-
ing a process called refraction where the light in the first medium with a specific wave
number, k1, enters a second medium that alters its wave number to k2. The manner
36
-2000
-1000
0
1000
2000
Vp
e
rp
 
[km
/s
]
-2000 -1000 0 1000 2000
Vpara [km/s]
Wind 3DP Pesa High Burst
2000-04-06/16:32:03-16:32:06
UV contamination (not real)
Thermal core saturation
Di!use Ions
A
Wind 3DP Pesa High Burst
2000-04-06/16:31:38-16:31:41
-2000
-1000
0
1000
2000
Vp
e
rp
 
[km
/s
]
-2000 -1000 0 1000 2000
Vpara [km/s]
Intermediate Ions
Solar Wind Direction
Shock Normal Direction
B
Figure 1.11: An example of intermediate and diffuse ion distributions seen upstream of a
IP shock on 04/06/2000. Panel A shows an example of an intermediate ion distribution
mixed with a diffuse ion distribution while panel B is purely diffuse ions.
37
in which a wave propagates through any particular medium can be characterized by a
property called the index of refraction, n2 = k2c2/ω2, where c is the speed of light in
vacuum. Thus one can see, n2 = 1 for vacuum.
The simplest dispersion relation in a plasma is for a homogeneous, source free plasma
with only small linear fluctuations/perturbations in any given quantity defined by their
Fourier transform in the following manner:
δA(k, ω) =
∑
k
Ake
i(k·r−ωt) (1.20)
where δA(k,ω) can represent any of the following relevant quantities: particle density
(ns), velocity (Vs), electric or magnetic fields (E or B), etc. where the subscript s
represents the particle species. If one assumes that the zeroth order functions have
neither temporal nor spatial variations, then the operators ∂/∂t and ∇ go to -iω and
+ik respectively. This allows one to reduce Faraday’s and Ampere’s laws, to first order,
to the following two equations:
−ik× δE = (−iω)δB (1.21a)
ik× δB = −iω
c2
K˜ · δE (1.21b)
where K˜ is the dielectric tensor. Eliminate δB between the two equations and remember
that n = ck/ω to find:
n× (n× δE) + K˜ · δE = 0 (1.22)
which can be rewritten in a short form as D˜ · δE = 0, where setting the determinant of
the tensor D˜ equal to zero yields a solution for δE. One should note that the determinant
of D˜ is not, by definition, zero. When it is equal to zero, the solution is referred to as
the dispersion relation, D(ω,k). Thus, the eigenmodes of the tensor, D˜, are the modes
of the allowed waves in the given system. Or in other words, the solution(s) to the
dispersion equation relate ω to k which describe the normal modes of the system [Gary ,
1993].
38
1.5.2 Instability
In a dispersive plasma, waves can grow or damp depending on driving mechanisms
and interactions with the medium itself. Plasmas are incredibly unstable and they
are rarely, if ever, in thermodynamic equilibrium. Departure from thermodynamic
equilibrium, i.e. non-Maxwellian features in a velocity distribution, can be a source
of free energy which needs to be dissipated [Gary , 1993]. In order to dissipate the
energy provided by a non-Maxwellian distribution, a plasma must transform the free
energy into some other form which will allow the non-Maxwellian features to relax to a
Maxwellian. This is done through normal modes of a system called instabilities.
An instability is a normal mode that can grow in space and time with well defined
relationships between ω and k [Gary , 1993]. Each wave mode resulting from a source
of free energy has its own dispersive properties much like any given medium has its
own dispersive properties. If an instability grows in time or space it can couple to
stable wave modes of the system (i.e. the instability uses radiation to dissipate the
free energy). Thus, an instability is effectively the transfer mechanism by which the
free energy can be dissipated. Given that instabilities depend on both ω and k, one
can assume that there are going to be different ranges or scales for any given system.
For small wavelength(large k), we refer to these instabilities as microinstabilities and
for large wavelength(small k), we refer to these instabilities as macroinstabilities. More
quantitatively, a microinstability is any instability that satisfies kρi & 1, where ρi is the
ion gyroradius. In this thesis, we will focus on microinstabilities but also look at some
macroinstabilities.
In order for wave damping/growth to occur, the wave modes must have only an
imaginary part to their frequency, since purely growing modes (i.e. real part of frequency
is zero) can exist. Assuming a non-zero real part to the frequency, let us define the
frequency of the wave as:
ω(k) = ωr(k) + iγ(ωr,k) (1.23)
where the real (ωr) and imaginary (γ) parts of the frequency are generally functions
of k. However, it is often safe to assume that γ = γ(k) or even just a constant. By
this definition, if γ >(<) 0 the wave amplitude grows(decays) exponentially in time.
One should note, however, that a solution to the dispersion relation of γ > 0 is not, in
39
itself, sufficient to produce an instability. Instabilities require a source of free energy,
otherwise the growth implied in the result is an unphysical solution.
The intrinsically unstable nature of a plasma allows for a number of different in-
stabilities, depending on the free energy source and the local plasma parameters of
the medium. For instance, ion-acoustic waves can be excited by the following sources of
free energy: cross-field currents [Lemons and Gary , 1978], ion velocity ring distributions
[Akimoto et al., 1985; Akimoto and Winske, 1985], electron heat flux [Dum et al., 1980;
Gurnett et al., 1979b], cold ion beam streaming through a warm electron background
across the magnetic field [Moses et al., 1985, 1988], etc. Often times the instabilities
are named after the source of free energy or the wave mode they excite which leads to a
confusing misinterpretation of the physical differences between the free energy source,
the instability, and the wave produced by the instability.
1.5.3 Landau Interactions
The collisionless nature of plasmas initially raised questions regarding the processes
of energy dissipation. In 1946, Lev Davidovich Landau proposed a theory of collisionless
dissipation where particles with velocities near the phase velocity of a plasma wave, ω/k,
interact resonantly with the wave fields [Landau , 1946]. The theory was slightly altered
and corrected a few years later by Jackson [1960]. It is now known as Landau damping
and was first shown to be real in experiments by Malmberg and Wharton [1964, 1966].
The physical interpretation of Landau damping was first explained by
Bohm and Gross [1949a,b]. It can be thought of as an analog to elastic particle colli-
sions, where one of the particles is the wave. One can think of particles moving in a
frame of reference where the wave is at rest and consider the wave as a series of potential
hills and troughs. In this frame, the particles with speeds greater(less) than the phase
speed of the wave can transfer energy to(from) the wave, thus causing growth(damping)
of the wave and the particles lose(gain) energy on average. The analogy often used is to
consider a surfer riding water waves. If the surfer can speed up enough to just slightly
less than the phase speed of the wave, they will be able to ride the wave, gain kinetic
energy and thus surf. If they do not move, the wave will pass by them and they will
rise and fall without any real energy transfer occuring between wave and surfer. If the
surfer goes slightly too fast, they will transfer energy to the wave and never pass over
40
the wave only reaching just below the peak.
Landau interactions can also lead to phase trapping and mixing. In the wave frame,
we consider a periodic wave to be a series of electric potential hills and troughs. In this
frame, the particles with speeds greater than the phase speed of the wave can escape the
troughs of the wave while those with speeds slightly less than the phase speed cannot.
The particles with higher speeds than the wave will suffer small periodic fluctuations
in their velocities, but nothing more. The particles which cannot escape the troughs
will oscillate back and forth in the trough and are trapped. Note that the oscillation
frequency, also called the bounce frequency, ωb, depends on how nearly matched their
velocity is to the resonant velocity. The difference in oscillation frequencies of the dif-
ferent particles leads to a phenomena called phase mixing. Phase mixing can also give
rise to the dissipation implicitly implied in Landau damping.
It is important to note that the above discussion was only assuming linear Landau
damping. During the linear phase, one can show that the total nonrelativistic energy
of the particles is given by:
WKE =
1
2
mV ‖res
2 + qΦ (1.24)
where V‖res is the resonant velocity of the particles, q their charge, m their mass, and
Φ the electrostatic potential of the wave in the wave frame [Gurnett and Bhattacharjee ,
2005]. During the linear phase, the total energy of the system is conserved. Therefore,
if the kinetic energy of the particles increases, the potential energy of the wave must
decrease. To determine whether the particle kinetic energy increases or decreases, one
must consider the slope of zeroth order velocity distribution function, Fo(v), at the
resonant velocity. If the slope is given by:
∂F o(v)
∂v
|V res < 0 (1.25)
where Vres is the parallel resonant velocity, then the number of particles initially moving
slower than V‖res is higher than the number initially moving faster. Then the wave
transfers energy to the particles in a similar fashion to how a surfer gains kinetic energy
by riding a wave. The wave is initially faster than the particles and thus effectively runs
into them, transfering momentum. The end result is particles gain kinetic energy and
the wave loses amplitude due to a loss of its potential energy, thus the name Landau
damping.
41
After some time, the interaction described above must change. The reason is that
the bounce period of the trapped particles is often much greater than the period of the
wave. Thus in the linear phase, the damping of the wave happens faster than than
the period of the trapped particles. During this time, one should also note that some
energy was given to particles which weren’t trapped. This is important to mention
because eventually the trapped particles begin to give energy back to the wave. Since
some energy was given to untrapped particles, not all the energy can be transferred back
to the wave. The trapped particles, by this time, have also begun to phase mix which
reduces the effective transfer of energy back to the wave. Though the particles will not
cause the wave amplitude to grow in this phase, they do slow the damping predicted
by linear Landau damping. Eventually the damping ceases after many bounce periods
and after the trapped particles become thoroughly phase mixed. Note, unless some
dissipative/irreversible process occurs, the phase mixed particles are not truly randomly
mixed. Their phase information can still be recovered.
In this thesis, I examine evidence for Landau damping of low frequency waves in
Chapter 4.
1.5.4 Weak Growth Limit
Since nonlinear treatments of plasmas are often impossible to deal with analytically,
linear approximations are considered. A specific example for linear waves is the weak
growth approximation which assumes | γ | ≪ | ωr |. The following derivations can be
found in more detail in Gurnett and Bhattacharjee [2005].
One can then expand the dispersion relation in a Taylor series to obtain:
D(ω,k) = D(ωr,k) + iγ
∂D(ωr,k)
∂ωr
= 0 (1.26)
where each part, D(ωr,k) and ∂D(ωr,k)/∂ωr, contains real and imaginary parts. Each
term must be evaluated in the limit γ → 0 to find solutions to the dispersion relation in
the weak growth approximation [Gurnett and Bhattacharjee , 2005]. Note that for the
second term in this limit, γ ∂D(ωr,k)/∂ωr, is negligible. Thus, only the real part is
kept. The first term simplifies to:
D(ωr,k) = Dr(ωr,k) + iDi(ωr,k) (1.27)
42
where the subscripts, r and i, refer to the real and imaginary parts respectively. Since
we are interested in solutions of D(ωr,k) = 0, we solve the real and imaginary parts
separately to give:
Dr = 0 (1.28a)
γ
∂Dr(ωr,k)
∂ωr
+Di = 0 (1.28b)
In the process of evaluating Equations 1.28a and 1.28b individually, one finds that
D(ωr,k) has an integral form of:
D(ωr,k) = 1 + lim
γ→0
ωpe2
k2
∫
C
∂F o/∂V ‖
V ‖ − ωr/k − iγ/kdV ‖ (1.29)
which requires the use of the Plemelj relation given by:
lim
ǫ→0
∫ ∞
−∞
f(x)
x− (xo ± iǫ)dx = P
∫ ∞
−∞
f(x)
x− xodx± iπf (xo) (1.30)
where ǫ > 0 and the P refers to the principal value integral defined by:
P
∫ ∞
−∞
...dx = lim
δ→0
[
∫ xo−δ
−∞
...dx +
∫ ∞
xo+δ
...dx] . (1.31)
After applying these mathematical techniques, one finds the relationships for the real
and imaginary parts of the dispersion relation reduce to:
Dr = 1− ωpe
2
k2
P
∫ ∞
−∞
∂F o/∂V ‖
V ‖ − ωr/k (1.32a)
Di = −π k| k |
ωpe2
k2
∂F o
∂V ‖
|V ‖=ω/k . (1.32b)
The growth/damping rate, γ, can be found from Equations 1.28b and 1.32b, given by:
γ ∝
(
∂Dr
∂ωr
)−1 ∂F o
∂V ‖
|V ‖=ω/k (1.33)
which is the classic result of Landau damping where γ depends on the slope of the
distribution function at the phase velocity of the wave.
43
1.5.5 Cyclotron Resonance
Cyclotron interactions between waves and particles are similar to Landau interac-
tions in some respects, but differ in one very important quality. The damping rate
of cyclotron interactions, in the weak damping limit for an isotropic distribution, is
directly proportional to the distribution function evaluated at a parallel resonance ve-
locity, V‖res. Recall in the weak damping limit for Landau interactions, the damping
rate was proportional to the slope at the resonance velocity. Thus, a greater number of
particles can contribute to the damping in cyclotron interactions than in Landau damp-
ing because all the particles near V‖res interact with the wave not just the particles with
velocities just below V‖res. However, when the distributions become anisotropic, cy-
clotron damping suddenly depends on the slope of the distribution. In the weak growth
limit, the growth rate goes as:
γ =
π
∂Dr/∂ω
∑
s
(ωps
ω
)2 [
− ω| k‖ |
∫ ∞
0
dv⊥ 2πv⊥F so
]
− π
∂Dr/∂ω
∑
s
(ωps
ω
)2 [ k‖
| k‖ |
∫ ∞
0
dv⊥ πv⊥
2
(
v‖
∂F so
∂v⊥
− v⊥∂F so
∂v‖
)]
|V ‖,res (1.34)
where Fso (= fso/nso) is the normalized zero-order distribution function of species s
and ωps is the plasma frequency of species s. For an isotropic distribution, the second
integrand in Equation 1.34 goes to zero because Fso = Fso(v), thus one can use v = (v‖
2
+ v⊥2)1/2 and the commutation property of partial derivatives to show:
∂F so
∂v⊥,‖
=
∂v
∂v
∂F so
∂v⊥,‖
(1.35a)
=
∂v
∂v⊥,‖
∂F so
∂v
(1.35b)
=
v⊥,‖
v
∂F so
∂v⊥,‖
. (1.35c)
If the distribution is anisotropic, then Fso 6= Fso(v) and the second integrand in Equation
1.34 does not go to zero (see Gurnett and Bhattacharjee [2005] for more detail). It was
shown in detail by Kennel and Petscheck [1966] that if the anisotropy of the particles
is great enough, the distribution can cause a cyclotron resonance that leads to growth.
44
The criteria for growth is given by:
As >
1
Ωce/ω − 1 (1.36)
where As is the anisotropy given by:
As =
∂F so
∂v⊥
+
k‖
ω
(
v⊥
∂F so
∂v‖
− v‖ ∂F so
∂v⊥
)
(1.37)
which reduces to (T⊥,s/T‖,s - 1) for a bi-Maxwellian distribution. Anisotropic electron
distributions are an important source of free energy for whistler waves [Gary et al., 1994,
1999].
In the solar wind, the plasma almost always satisfies the condition ωpe ≫ Ωce. In the
high density limit and the cold plasma approximation, which assumes the particles are
initially at rest with no thermal motions (i.e. Te = Ti = 0), one can estimate the index
of refraction for a RH-mode whistler wave. If the wave has frequency ω2 ≪ ωpe2 and
propagates at an oblique angle to the magnetic field, θkB, then the index of refraction
for a RH-mode whistler wave is:
n2 =
ωpe2
ω(ΩcecosθkB − ω) . (1.38)
The angle at which n2 goes to infinity, called resonance, is known as the resonance cone
angle, θkB → θres ≡ cos−1 (ω/Ωce). An important aspect of a whistler mode wave is
that their frequencies always satisfy ω < Ωce. Replacing ωpe2 with ζ2Ωce2, where ζ2 =
(c2µo me ne)/Bo2 and after some manipulation, the index of refraction becomes:
n2 = ζ2
Ωce
ω(cosθkB − ω/Ωce) . (1.39)
For particles to undergo resonance with a wave, their velocities parallel to the back-
ground magnetic field, V‖res, must satisfy the following condition:
V ‖res = −ω +mΩce
k‖
(1.40)
where the parallel wave number, k‖, is given by |k| cosθkB and m = 0, −1, or +1 (though
higher integer values can exist, we only consider these for now). The different values
of m represent different resonances or different types of interactions. The first possible
45
value, 0, is for Landau resonance which is discussed in Section 1.5.3. The following two
are two different types of cyclotron resonance called normal and anomalous, respectively.
Normal cyclotron resonance occurs between an electron and a RH-mode wave. In
the case of the whistler mode, the electron cyclotron frequency always exceeds the
whistler frequency. For normal cyclotron resonance to occur, the electron’s gyromotion
must match the rotation frequency of the fields of the whistler mode. Thus, the elec-
tron’s guiding center velocity along the zero order magnetic field must Doppler shift
the whistler’s frequency up to the electron cyclotron frequency (i.e. ω’ = ω - k‖V‖ =
Ωce). This implies that k‖V‖ < 0 and whistler instabilities must be driven by positive
anisotropies in the electron velocity distributions. In other words, the average kinetic
energy perpendicular to the magnetic field should be larger than the average kinetic
energy parallel to the magnetic field.
Anomalous cyclotron resonance applies a similar principle to the normal resonance,
but with ions replacing electrons. Now the wave polarization in the particle’s guiding
center rest frame must appear to be a LH-mode and the Doppler shifted frequency must
match the ion cyclotron frequency. Thus, k‖V‖ > 0 and V‖ > ω/k‖. One should note,
however, that such resonance conditions are rarely met since V‖ is often very high.
To estimate the resonant energies of particles with the whistler mode in the cold
plasma approximation, we must take our arguments from above a little further. Recall
that n2 = k2c2/ω2, which allows us to solve for k2. After some algebra, we find:
k2 =
(
µomene
Bo2
)[
ωΩce
(cosθkB − ω/Ωce)
]
. (1.41)
Now we replace k‖ in Equation 1.40 by |k| cosθkB, square V‖res, and replace |k|2 with
the result from Equation 1.41 to get:
V ‖res
2 =
(ω +mΩce)2
k2cos2θkB
(1.42a)
=
( Bo2
µomene
Ωce
ωcos2θkB
)[
cosθkB − ω
Ωce
][
m+
ω
Ωce
]2
. (1.42b)
In the nonrelativistic limit, the kinetic energy of a resonant particle is just the typical
kinetic energy or 1/2 me V‖res
2, which gives:
E‖res =
( Bo2
2µone
)( Ωce
ωcos2θkB
)[
cosθkB − ω
Ωce
][
m+
ω
Ωce
]2
(1.43)
46
θ    (  )
kB
o
10 20 30 40 50 60 70
Ω
  
 /
ω
ce
100.0
10.0
5.0
2.0
1.0
V   /c  =  0.013AeΩ    =  357.15  (Hz)ce
E 
   
 (k
e
V
)
re
s
10.0
1.00
0.10
0.01
Figure 1.12: An example calculation of the parallel cyclotron resonance energy for
normal (m = -1) cyclotron resonance. The plot shows E‖res plotted versus Ωce/ω and
θkB. The energy ranges from 10 eV to 10 keV and the values of Ωce and VAe/c are shown
at the top of the plot.
where the first factor, (Bo2/2µone) is 1/2 me VAe2 where VAe is the electron Alfve´n
speed. Though we should note that there is no electron Alfve´n wave that propagates at
this speed.
An example calculation of Equation 1.43 is shown in Figure 1.12. E‖res is plotted
versus ω/Ωce and θkB. The frequency dependence is weaker than the angle dependence
at very oblique angles (&60◦), but becomes dominant at lower angles. As one can see,
at lower wave frequencies (high values of ω/Ωce) higher energy electrons are required
to resonate with whistler waves through a normal cyclotron (m = -1) resonance. The
reason is that the electrons must move faster to Doppler shift the low frequencies up to
the electron cyclotron frequency in the guiding center reference frame of the electrons.
47
Applications of Equation 1.43 have been used in research by Lengyel-Frey et al. [1994]
and Wilson III et al. [2009] to estimate the energy range of electrons which would be
resonant with the whistler waves they observed.
In this thesis, I examine evidence for cyclotron interactions with low frequency waves
in Chapter 4 and higher frequency waves in Chapter 5.
1.6 Introduction to the Microphysics of IP shocks
Anomalous resistivity, through wave-particle interactions, has been thought to act as
an energy dissipation mechanism in collisionless shock waves [Kellogg , 1965; Sagdeev ,
1966]. It is thought to provide much of the dissipation at low Mach number shock
waves [Gary , 1981], along with dispersive [Mellott and Greenstadt , 1984] and conductive
[Kennel et al., 1985] effects. However, at higher Mach numbers other mechanisms be-
come important, such as particle reflection [Edmiston and Kennel , 1984; Kennel , 1987].
Due to their generally lower Mach numbers at 1 AU and their larger radii of curvature,
IP shocks provide an excellent opportunity to study the role of wave dissipation at low
Mach number shocks.
The importance of wave-particle interactions in the total energy dissipation budget of
collisionless shocks is not well known. To zeroth order and ignoring wave-particle effects,
electrons conserve their first adiabatic invariants which states that the magnetic mo-
ment, µ, is given by = Bo/(0.5 me Ve,⊥2) = constant. If the electrons conserve their mag-
netic moments, then one expects Ve,⊥2 to increase(decrease) as Bo increases(decreases).
Wave-particle interactions can ”de-magnetize” the electrons, or cause µ 6= constant,
which has a very important implication, namely irreversibility due to an increase in
entropy. Recent observations of IP shocks [Wilson III et al., 2007, 2009, 2010] and
simulation studies with a realistic mass ratio (Mi/me ∼ 1836) [Petkaki et al., 2006;
Petkaki and Freeman , 2008; Yoon and Lui , 2006, 2007] have found evidence to sug-
gest that wave-particle interactions may be more important than previously thought.
However, some previous studies of collisionless shocks have suggested that the electron
heating can be adequately explained by an ES electric field in the shock ramp called the
cross-shock potential [Scudder et al., 1986a]. For a more detailed discussion of macro-
scopic fields, see Section 3.3.
48
Quasi-linear theory estimates the effective resistivity produced by wave particle in-
teractions:
ηIA =
νIA
εoωpe2
(1.44)
where νIA is the effective collision frequence between the wave electric fields and electrons
given by:
νIA = ωpe
εo | δE |2
2nekBT e
(1.45)
where ne is the electron number density, Te is the electron temperature, and | δE| is
the amplitude of the fluctuating electric field due to the wave. Vlasov simulations using
realistic mass ratios have found that the results of Equation 1.44 can be up to 2-3
orders of magnitude smaller than the momentum transfers observed in the simulations
[Petkaki et al., 2006; Petkaki and Freeman , 2008; Yoon and Lui , 2006, 2007]. Thus, the
quasi-linear estimates of wave-particle collision rates can serve as a lower bound when
estimating resistivities in observations.
The instabilities expected to occur in the ramps of collisionless shock waves depend
on the shock geometry and Mach number. The main source of free energy in a low
Mach number (Mf ≤ 3) quasi-perpendicular (θBn > 45◦) collisionless shock wave is
thought to be due to the relative drift between electrons and ions (either field-aligned
or cross-field), a current [Lemons and Gary , 1978]. As a consequence of the free energy
source, instabilities are thought to arise and contribute to resistive energy dissipation
in the transition region of quasi-perpendicular collisionless shocks. Some commonly
expected instabilities thought to be important in collisionless shocks are: ES IAWs,
bipolar ES structures with Debye length scales parallel to the background magnetic
field, called solitary waves or phase space holes, modified two stream instability (MTSI),
lower-hybrid-drift instability (LHDI), and electron cyclotron drift instability (ECDI)
[Wu et al., 1984].
49
1.7 Electromagnetic Waves: Background, Definitions, and
Prior Work
1.7.1 Langmuir Waves
Langmuir waves have been studied extensively in the terrestrial foreshock and some-
what at IP shocks [Bale et al., 1997; Fitzenreiter et al., 2003;Kellogg et al., 1999]. Lang-
muir waves are usually linearly polarized parallel to the ambient magnetic field with
narrow frequency peaks near fpe. They are capable of pitch-angle scattering electrons
and peturb the background density levels [Soucek et al., 2005]. Langmuir waves are also
thought to be the progenetors of solar radio emissions, specifically Type II and Type
III radio bursts [Bale et al., 1999; Kellogg , 2003; Pulupa and Bale, 2008; Pulupa et al.,
2010]. Though they are not directly related to collisionless shocks, the shock structure
is related to the source of free energy for the Langmuir waves, electron beams [Kellogg ,
2003]. Langmuir waves are also thought to scatter off of density perturbations, which
can be a useful tool in describing source regions of radio bursts or shock structure
[Krasnoselskikh et al., 2007].
Figure 1.13 is an illustrative example of a linearly polarized Langmuir wave seen
upstream of an IP shock on 04/06/2000 at 16:31:54.951 UT by the Wind spacecraft’s
TDS detector [Bougeret et al., 1995]. The figure is a ∼17 ms waveform capture that
has been rotated into field-aligned coordinates defined by the XY-GSE projection of the
magnetic field (due to only two component observations). The top two panels are the
parallel (E‖ in red) and perpendicular (E⊥ in blue) projections of the electric field while
the bottom two are the Morlet wavelet transforms. As one can see, Langmuir waves
are high frequency (typically ∼10-50 kHz in the solar wind), linearly polarized, large
amplitude (>10 mV/m) waves. They are observed as both ES and electromagnetic in
the solar wind [Bale et al., 1999; Pulupa and Bale, 2008; Wilson III et al., 2007].
Figure 1.13 is an example of a waveform capture of an electric field measurement
of a Langmuir wave observed upstream of an IP shock. The electric fields have been
rotated into field-aligned coordinates where the top panel (red electric field) is the par-
allel component, E‖, and the second panel (blue electric field) is the perpendicular, E⊥.
In the top two panels are vertical black arrows that define the relative amplitudes of
50
0.01
6240
0.86
73.0
W
av
e
le
t 
P
o
w
e
r
E
1.0
10.0
Fr
e
q
u
e
n
cy
 (k
H
z)
Rotated Electric Fields for 16:31:54.951 UT on 04/06/2000
1.0
10.0
Fr
e
q
u
e
n
cy
 (k
H
z)
E
0.01
6240
0.86
73.0
W
av
e
le
t 
P
o
w
e
r
~ 80 mV/m
Parallel
Perpendicular
~ 80 mV/m
-40
 0
40
E
-40  0 40
E
Figure 1.13: An example Langmuir wave observed upstream of the shock ramp of the
04/06/2000 event of Wilson III et al. [2009]. The top two panels are the parallel (E‖ in
red) and perpendicular (E⊥ in blue) projections of the electric field while the bottom
two are the Morlet wavelet transforms. The wavelet transforms are labeled for each
relative component. The wavelets are plotted from 100 Hz to 60 kHz. The hodogram
to the right plots E⊥ versus E‖ in the region outlined by the magenta box with the solid
green line representing XY-GSE projection of the shock normal vector in FACs.
51
each component of the electric field measured. Below the two waveforms are the cor-
responding frequency spectrum calculated using wavelet analysis (see Section 2.4.3 for
more information). To the right is a hodogram plotting E⊥ versus E‖ for the time range
outlined by the magenta box in the top two panels to the left. The solid green line in
the hodogram represents the XY-GSE projection of the shock normal vector in FACs.
As one can see, this particular Langmuir wave appears to be linearly polarized roughly
parallel to the magnetic field.
1.7.2 Ion-Acoustic Waves
A number of authors [Gurnett et al., 1979b,a; Hess et al., 1998; Thomsen et al.,
1985a] have concluded that IAWs are important in dissipating energy in lower Mach
number shocks. Wave amplitudes in previous studies were found to be correlated with
the electron to ion temperature ratio, Te/Ti [Gurnett et al., 1979a]. They tend to be
broadband bursty waves with Doppler shifted frequencies between 1-10 kHz (typically
fpi < f < fpe) in the solar wind with a maximum intensity around 3 kHz [Gurnett et al.,
1979b,a; Hess et al., 1998]. They are usually linearly polarized close to parallel or
oblique to the ambient magnetic field Akimoto et al. [1985]; Akimoto and Winske [1985];
Fuselier and Gurnett [1984]. In a shock, the instability is thought to be driven by a rela-
tive drift between electrons and ions [Mellott , 1985], with threshold drifts increasing for
small Te/Ti. A number of studies have concluded that IAWs are likely to be dominant
in the terrestrial bow shock despite questions about high damping effects due to small
Te/Ti [Akimoto et al., 1985; Akimoto and Winske, 1985; Fuselier and Gurnett , 1984].
Theoretical studies suggest temperature gradients and oblique propagation of the waves
can reduce damping when Te ∼ Ti.
Figure 1.14 is an example of an IAW observed in the shock ramp of an IP shock on
1996-04-08. The format is the same as Figure 1.13. As one can see, the wave appears to
be roughly linearly polarized parallel to the magnetic field in the plane of measurement,
though their polarization can occasionally be quite oblique and more eliptical. They
often exhibit a relatively narrow and well defined frequency peak, as illustrated in the
wavelet transform plots below the waveforms.
52
Rotated Electric Fields for 02:41:08.830 UT on 04/08/1996
E
E
~ 20 mV/m
~ 20 mV/m
2 4 6 8 10 12 14 16
Time (ms)
0.01
1000
0.46
21.5
W
av
e
le
t 
P
o
w
e
r
1.0
10.0
Fr
e
q
u
e
n
cy
 (k
H
z)
1.0
10.0
Fr
e
q
u
e
n
cy
 (k
H
z)
0.01
1000
0.46
21.5
W
av
e
le
t 
P
o
w
e
r
Perpendicular
Parallel
-10
 0
10
E
-10  0 10
E
Figure 1.14: An example IAW observed in the shock ramp of the 04/08/1996 event of
Wilson III et al. [2009]. The format is similar to that of Figure 1.13, except the wavelets
are plotted from 100 Hz to 10 kHz.
53
1.7.3 Electrostatic Solitary Waves
ES solitary waves (ESWs) are characterized as nonlinear ES Debye-scale bipolar
electric field signatures parallel to the ambient magnetic field [Cattell et al., 2002a,b,
2005; Ergun et al., 1998b; Franz et al., 2005; Pickett et al., 2004], often associated with
electron beams [Cattell et al., 2005; Ergun et al., 1998b; Franz et al., 2005]. Phase
space holes were first thought to be a nonlinear mode consistent with a BGK mode
by [Matsumoto et al., 1994]. Thus, the component of the electric field parallel to
the magnetic field is seen as a bipolar pulse while the components perpendicular are
monopolar, both are derivatives of a Gaussian. Solitary waves have been observed
at the Earth’s bow shock [Bale et al., 1998, 2002], and at an IP shock near ∼8.7
AU [Williams et al., 2005], as well as within the magnetosphere at many boundaries
[Cattell et al., 2002a,b, 2005] possibly providing energy dissipation. Simulations have
shown them to form in and around the ramp regions of high Mach number collisionless
shock waves [Matsukiyo and Scholer , 2006a; Shimada and Hoshino, 2000].
ESWs act like clumps of positive charge, if electron holes. In the frame of the electron
hole, the ions can be incident on the hole at very large speeds relative to their thermal
speed. The relative speed is large because electron holes travel at roughly the electron
drift velocity or fractions of electron beam speeds which is much much larger than the
ion thermal speed [Behlke et al., 2004; Cattell et al., 2002a, 2003, 2005; Ergun et al.,
1998a,b; Franz et al., 1998]. Since the structures are on the order of an electron Debye
length, λDe, the transit times of incident ions will be relatively small compared to the
local ion gyroperiod. Thus, the ions can become demagnetized if scattered. To visualize
this, assume an ion is incident on an electron hole with an impact parameter of b. Then
the perpendicular impulse of the ion in response to the electron hole’s electric field can
be shown as:
M i∆V i,⊥ (b) = e
∫ ∞
−∞
dt E⊥ [b, z (t)] (1.46)
where Mi is the ion mass, ∆Vi,⊥ is the change in ion velocity, E⊥ is the electric field
perpendicular to the magnetic field, and z(t) is the position along the direction parallel
to the magnetic field at time t. If we assume that the incident ion velocity in the electron
hole frame is Vehole and that the electron hole will not recoil upon impacting a single ion,
then we can also estimate the parallel impulse of the ions as ∆Vi,‖ ≃ -∆Vi,⊥2/(2 Vehole).
54
The net result is an exchange of momentum between the ions and the electrons both
parallel and perpendicular to the magnetic field. If there is a relative drift between the
two species, this momentum exchange can act to reduce the relative drift, thus reduce
a current. In this way, electron holes can be effective waves for inducing anomalous
resistivity. The end result is strong perpendicular ion heating (i.e. random perpendicu-
lar ion acceleration) and a significant amount of parallel momentum imparted upon the
electrons from the ions. Observations have shown that ∆Ti,⊥ across a train of electron
holes can be a significant fraction of the ion thermal energy [Ergun et al., 1998b].
Since the holes act like clumps of positive charge, they have positive potentials and
thus act to trap incident electrons [Dyrud and Oppenheim, 2006; Lu et al., 2008]. The
particles that are trapped are the ones with energies below that of the max potential of
the solitary structure. The accumulation of electrons trapped in the solitary structure
acts to damp/saturate the instability driving the holes [Lu et al., 2008]. In addition
to trapping electrons, the electron holes can create double-peaked electron distribu-
tions at low energies which are unstable to other wave modes [Berthomier et al., 2008;
Matsukiyo and Scholer , 2006a].
The last, somewhat indirect, way in which solitary waves can heat/scatter particles
is by coupling to other wave modes. Solitary waves can either couple to or directly cause
IAWs [Dyrud and Oppenheim , 2006], whistler mode waves [Lu et al., 2008], and electron
acoustic waves [Matsukiyo and Scholer , 2006a]. IAWs are known to heat electrons par-
allel to their fluctuating electric fields (typically along the magnetic field) [Dum et al.,
1974]. Whistler waves are known to cause a perpendicular pitch-angle diffusion and
heating of electrons [Brice , 1964; Kennel and Petscheck , 1966]. Electron acoustic waves
are thought to produce strong parallel (with respect to the magnetic field) electron
heating [Matsukiyo and Scholer , 2006a], but to the best of our knowledge, these modes
have not been observed.
Figure 1.15 is an example of an ESW observed by the Wind spacecraft in the ter-
restrial magnetosphere. The format is the same as Figures 1.13 and 1.14. Though these
wave modes are observed at collisionless shocks, they are typically much larger ampli-
tude in the magnetosphere making the characteristic bipolar signature more obvious.
Notice that the defining characteristic, parallel component (red) is bipolar while the
perpendicular is monopolar (blue), is very obvious in this example.
55
E
Rotated Electric Fields for 14:45:55.464 UT on 10/03/2001
1.0
10.0
Fr
e
q
u
e
n
cy
 (k
H
z)
0.01
83000
2.02
410
W
av
e
le
t 
P
o
w
e
r
Parallel
1.0
10.0
Fr
e
q
u
e
n
cy
 (k
H
z)
0.01
83000
2.02
410
W
av
e
le
t 
P
o
w
e
r
E
Perpendicular
-100  0 100
E
-100
 0
100
E
~ 200 mV/m
~ 200 mV/m
Figure 1.15: An example solitary wave observed in the terrestrial magnetosphere. The
format is similar to that of Figures 1.13 and 1.14. The parallel and perpendicular
component wavelet transforms are labeled respectively.
56
1.7.4 Whistler Waves
Whistler waves were first discovered by Barkhausen [1919] while listening to signals
from an antenna connected to a simple vacuum tube amplifier. The signals were heard
to decrease in frequency with increasing time. Over thirty years later, Storey [1953]
managed to explain these strange signals as being the result of lightning strikes. It is now
known that whistler waves can exist as a RH electromagntic mode [Kennel and Petscheck ,
1966] or a slightly electrostatic mode and interact with both ions [Hoppe et al., 1982;
Stasiewicz et al., 2003] or electrons [Brice, 1964], [Kennel and Petscheck , 1966], and
[Lyons et al., 1972]. Due to their capacity to resonantly and nonresonantly interact
with particles, whistler waves are a topic of extreme interest in collisionless shock dissi-
pation topics.
A specific class of whistler wave is often observed immediately upstream of a quasi-
perpendicular collisionless shock wave in magnetic field data, called precursor whistler
waves or just precursor waves. Their existence was theorized as a necessary part of
the shock structure since collisionless shock waves were first predicted [Kellogg , 1962].
The necessity of their existence at low Mach number collisionless shock waves was more
rigorously shown byMorton [1964] and Stringer [1963]. These waves are low enough fre-
quency that they couple to the magnetosonic wave responsible for the shock ramp. Thus,
they can provide energy dissipation in shock waves through dispersive effects and wave-
particle interactions upstream of the shock ramp [Gary and Mellott , 1985]. Occasionally
precursor whistlers phase stand with respect to the shock front [Fairfield and Feldman ,
1975], which means their phase velocity matches the shock front phase velocity. There-
fore, in the shock frame of reference, the waves appear as standing wave modes. This
condition is satisfied at the bow shock when the phase speed of the precursor whistler
matches the shock wave phase speed and the group velocity exceeds the incident solar
wind speed [Greenstadt et al., 1975].
Figure 1.16 shows an example of a quasi-perpendicular, low Mach number IP shock
with a precursor whistler wave (highlighted by purple box). The top panel shows the
magnetic field magnitude and the bottom panel shows the three GSE components of the
magnetic field. The fluctuations in the magnitude of the magnetic field illustrate the
compressive and oblique nature of this class of whistler wave. This particular whistler
was observed to propagate at an oblique angle to the magnetic field and has spatially
57
|B| (nT)
B [GSE] (nT)
XYZ
Precursor
Whistler
Figure 1.16: An example of an IP shock with an upstream precursor whistler wave.
The top panel is the magnitude of the magnetic field and the bottom panel is the GSE
components. The region with the precursor is outlined by the transluscent blue box.
dispersive properties.
Fairfield and Feldman [1975] initially identified precursor whistler waves at the quasi-
perpendicular bow shock using magnetometer data from Explorer 43 as phase standing.
Later studies, using OGO 5 [Greenstadt et al., 1975] and ISEE 1 and 2
[Mellott and Greenstadt , 1984] magnetometer data, found many of the upstream waves
to be inconsistent with phase standing whistlers. Mellott and Greenstadt [1984] found
two different types of precursor whistler waves, a phase standing whistler wave prop-
agating parallel to the shock normal and another whistler propagating parallel to the
magnetic field. In the SC frame, the precursors propagating parallel to the magnetic
field had higher frequencies (∼1 Hz) than the phase standing precursor whistlers (∼0.1
Hz). It is important to note that the ∼1 Hz waves studied by Hoppe et al. [1982] had
relatively large θkB values, while the precursors of Mellott and Greenstadt [1984] were
propagating parallel to the magnetic field, thus θkB ∼ 0◦. Mellott and Greenstadt [1984]
proposed that the parallel propagating precursors were products of the phase standing
precursors. The precursors propagating parallel to the shock normal (the phase standing
58
precursors) were found to have higher rest frame frequencies than the precursors prop-
agating parallel to the magnetic field. The difference was due to their propagation with
respect to the magnetic field. The Doppler effects on the parallel propagating precursors
were negligible because the magnetic field was primarily directed in Y-GSE direction,
roughly perpendicular to the solar wind velocity. Both the parallel propagating and
phase standing precursors are characterized by a high degree of RH polarization and
nearly monochromatic frequency spectrum. A more recent study by Farris et al. [1993]
found that the comparison between observed and predicted wavelengths for phase stand-
ing precursor whistlers to be consistent with the results ofMellott and Greenstadt [1984].
However, the estimates by Farris et al. [1993] of the ratio of precursor whistler wave-
length to shock thickness differed from those of Mellott and Greenstadt [1984]. Thus,
Farris et al. [1993] concluded that the thickness of the shock ramp was not strictly
dependent/related to the wavelength of the precursor whistler.
1.7.5 Waves at or Near the Electron Cyclotron Frequency
Electron cyclotron harmonic, electron Bernstein, (n + 1/2), or ”totem pole” waves
have been observed throughout planetary magnetospheres by Barbosa et al. [1990] and
Usui et al. [1999]. These emissions can be both broad or narrow in frequency range
[Hubbard and Birmingham , 1978]. They are typically driven unstable by loss-cone or
anisotropic electron distributions in the high energy hot halo in planetary magneto-
spheres. Usui et al. [1999], in a study near the terrestrial magnetopause, found the
emissions to be associated with increases in the ratio of hot halo to cold core electron
densities, nh/nc. To the best of our knowledge, these emissions have not been observed
previously in the solar wind.
Figure 1.17 shows two examples of electron cyclotron harmonic waves observed down-
stream of an IP shock by the Wind spacecraft. The top(bottom) four panels correspond
to the waveform observed at 16:32:25.358 UT(16:32:25.428 UT) on 04/06/2000. The
top row of panels for each waveform with the red(blue) lines correspond to the E‖(E⊥)
component of the wave electric field. The bottom two panels of each waveform show
the power spectra (mV/m2/Hz) versus frequency (kHz) plots corresponding to the time
range defined by the orange box in the top two panels. The verticle green(magenta)
lines overplotted on the power spectra correspond to integer(half-integer) harmonics of
59
fcen Peaks
fce(n + 1/2) Peaks
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
2 4 6 8 10 12 14 16
Time (ms)
2 4 6 8 10 12 14 16
Time (ms)
Electric Field vs. Time on:  04-06-2000 at:  16:32:25.428 UT
W
av
e
 P
o
w
e
r
Frame: 729
E
Frame: 729
E
E
21 3
E
21 3
Frame: 729
~20 mV/m~20 mV/m
Frame: 729
10
-410
-310
-210
-110
 0
10
-5
10
-6
9/2 13/2 17/2 21/2
11/2 19/215/2
9/2 13/2 17/2
11/2 15/2
fcen Peaks
fce(n + 1/2) Peaks
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
2 4 6 8 10 12 14 16
Time (ms)
2 4 6 8 10 12 14 16
Time (ms)
Electric Field vs. Time on:  04-06-2000 at:  16:32:25.358 UT
W
av
e
 P
o
w
e
r
Frame: 442
E
Frame: 442
E
~40 mV/m
E
Frame: 442
2 4 951 3 876 1110
10
-410
-310
-210
-110
 0
10
-5
10
-6
E
2 4 951 3 876 1110
Frame: 442
~40 mV/m
Figure 1.17: Two examples of electron cyclotron harmonic waves observed downstream
of an IP shock. The top waveform was observed at 16:32:25.358 UT and the bottom at
16:32:25.428 UT. The top row contains E‖ (red) and E⊥ (blue). Below the waveforms
are the power spectra (mV/m2/Hz) versus frequency (kHz) plots corresponding to the
time range defined by the orange box. The verticle green(magenta) lines overplotted on
the power spectra correspond to integer(half-integer) harmonics of fce. The 16:32:25.428
UT has a similar format.
60
fce. Note that the 16:32:25.358 UT waveform primarily shows enhanced power at inte-
ger harmonics of fce while the 16:32:25.428 UT waveform shows mixtures of integer and
half-integer harmonics of fce. The wave power enhancements shift dynamically in time
throughout the waveform, thus why only small windows of time were used to calculate
the power spectra. These waveforms are over two orders of magnitude above the back-
ground levels (∼0.1 mV/m at 1 AU).
Simulations have that found bipolar ES phase space holes form in and around the
ramp regions of high Mach number collisionless shock waves [Dyrud and Oppenheim ,
2006; Matsukiyo and Scholer , 2006a]. Due to their ability to efficiently exchange mo-
mentum between electrons and ions, the holes can heat and scatter particles. Sim-
ulations also show that the holes can also couple with other wave modes like IAWs
and lower hybrid waves, providing resistive dissipation [Dyrud and Oppenheim , 2006;
Matsukiyo and Scholer , 2006a]. Matsukiyo and Scholer [2006a] examined microinsta-
bilities in the foot of supercritical collisionless shocks using a two dimensional PIC sim-
ulation with a realistic mass ratio (Mi/me ∼ 1836). They observed six different types
of instabilities excited in less than an ion gyroperiod with the dominant modes includ-
ing ECDI, whistler instability, electron acoustic instability, and two different modified
two-stream instabilities (MTSIs); MTSI-2 excited by relative drifts between incident
electrons and reflected ions and MTSI-1 due to the relative drift between electrons and
incident ions. Reflected ions cause the incident solar wind ions to decelerate in the
shock foot, which locally decelerate the electrons to maintain current continuity in the
shock normal direction. These instabilities give rise to waves which scatter and heat
the plasma, thus dissipating energy.
In this thesis, we will discuss a type of cyclotron wave/instability in detail in Chap-
ter 5 and their effect on local electron distributions.
1.7.6 Lower Hybrid Waves
Lower hybrid waves (LHWs) are typically an electrostatic (ES) mode propagating
perpendicular to the magnetic field with a frequency given by:
f lh
2 =
f cef ci
1 + (f cef ci)/f pi2
(1.47)
61
where fcs is the cyclotron frequency of species s and fpi is the ion plasma frequency. In
the high density limit (fpi2 ≫ fce fci), typical of the solar wind, flh ∼ (fce fci)1/2. LHWs
are capable of resonating with the bulk of the ion distribution, thus they can serve
as an effective ion heating mechanism [Davidson and Gladd , 1975; Lemons and Gary ,
1978; Mellott , 1985]. In the ES limit and large propagation angles, the waves can
couple the parallel motion of the electrons to the perpendicular motion of the ions
[Marsch and Chang , 1983]. LHWs can be driven unstable by cross-field currents
[Lemons and Gary , 1978], electron heat flux in the solar wind [Marsch and Chang ,
1982], and the lower-hybrid drift instability (LHDI) [Cairns and McMillan, 2005]. They
are an attractive candidate for resistive energy dissipation in collisionless shocks be-
cause their critical drift speeds, Vdcr ≈ VTi, are much lower than that of an IAW,
Vdcr ≈ VTe. LHWs can couple to other wave modes like drift waves, modified two-
stream instabilities (MTSIs), etc., all of which cause significant wave-particle inter-
actions [Lemons and Gary , 1978]. LHWs can heat the ions transverse to the magnetic
field producing anisotropic ion distributions [Marsch and Chang , 1982]. Because LHWs
have ω/k‖ ≫ ω/k⊥, they tend to interact with the higher energy electrons producing
broadened high energy tails [Cairns and McMillan, 2005].
Electromagnetic lower hybrid waves (EMLHWs), or hybrid whistler waves, prop-
agate nearly perpendicular to the ambient magnetic field and can appear to have a
broadband frequency spectrum. One should note that in the limit of large k⊥, LHWs
are on the same branch of the dispersion relation as whistler waves. As with the ES
LHWs, they are thought to be driven unstable by the solar wind electron heat flux
[Marsch and Chang , 1983]. They can, much like ES LHWs, heat the ions perpendicular
to the magnetic field. For these modes, Vi,res,⊥ ≈ ω/k⊥ ≪ ω/k‖ ≈ Ve,res,‖, which means
the Landau interactions are perpendicular for the ions and parallel for the electrons, with
respect to the magnetic field. They have frequencies of fci ≪ f≪ fce [Marsch and Chang ,
1983]. These waves dissipate their wave energy through Landau interaction with the
ions producing perpendicular ion heating. They propagate very obliquely to the field
within a cone defined by k‖/k⊥ ≤ 1/5 and k‖/k⊥ ≥ VTi,⊥/VTe,‖ [Marsch and Chang ,
1983]. Zhang and Matsumoto [1998] observed EMLHWs at an IP shock using Geotail
and showed that the waves propagate nearly perpendicular to the magnetic field.
Another related wave mode is the LHDI, which in the presence of strong plasma
62
gradients, acts like a fluid instability excited through the coupling of a LHW and a drift
wave [Davidson and Gladd , 1975; Huba et al., 1978]. When the gradients are weak,
the LHDI is a kinetic instability driven by a resonance between ions and a drift wave.
When in the presence of a finite plasma β, the LHDI exists as an ES and electromag-
netic mode [Davidson and Gladd , 1975; Huba et al., 1978]. The growth rate of the LHDI
peaks at kρe ≈ 1, for a broad range of frequencies near flh [Davidson and Gladd , 1975;
Cairns and McMillan, 2005]. The mode is strongly unstable when the magnetic field
gradient scale lenght, LB, is comparable to ρi. The LHDI produces strong anomalous
resistivity due to the wave’s electric fields, δE⊥, perpendicular to the ambient magnetic
field, Bo, which create (δE⊥ × Bo)-drifts that transport particles across Bo. Thus, the
LHDI causes cross-field diffusion which is an increase in entropy, thus irreversible and
important for energy dissipation [Coroniti , 1985].
Figure 1.18 is a TDSS sample observed downstream of the shock ramp of an IP
shock observed by the Wind spacecraft on 02/11/2000. The waveform is an example
of an EMLHW. The left-hand side of the plot shows the three components of the mag-
netic field measured by the TDSS detector search coils on Wind in minimum variance
coordinates (see Section 2.4.1 for more details). The right-hand side of the plot shows
the corresponding wavelet transforms (see Section 2.4.3 for more details). To the right
of the wavelets are the labels of the relevant frequencies for this wave, where fce is the
electron cyclotron frequency and flh is the lower hybrid resonance frequency. In the
left-hand panel, one can see the angle of propagation with respect to the magnetic field
is θkB ∼ 85◦, roughly perpendicular to the magnetic field. Note that at multiple points
in the waveform, there appear to be low and high frequency signals intermixed. At
roughly 400 ms, a higher frequency (at roughly 100 Hz), is superposed on the lower
frequency (∼10 Hz or ∼flh) signal.
Figure 1.19 shows the result of filtering the signal for 7 Hz < f < 20 Hz (see Sec-
tion 2.4.2 for discussion of frequency filtering). Note that the mid-to-min eigenvalue
ratio, λ2/λ3, is much higher now at ∼12 compared to ∼3 in Figure 1.18. Also, θkB has
increased to ∼90◦, consistent with previous observations [Zhang and Matsumoto , 1998]
and theory [Marsch and Chang , 1983]. The amplitude of the low frequency component
is ∼1 nT.
Figure 1.20 shows the result of filtering the signal for 60 Hz < f < 200 Hz. The
63
B    (nT)min
λ  / λ   =  3.310
2 3
λ  / λ   =  1.7841 2
θ   =   85.11  +/-  1.05
kB
o o
fce
Time (ms)
200 400 600 800 1000
k     = < +0.714,+0.376,-0.590 >  +/-  < 0.040, 0.091, 0.106 >
GSE
Rotated Magnetic Fields for 23:33:58.351 UT on 02/11/2000
flh10 Hz
100 Hz
0.01 1000.22 4.64
Wavelet Power
Time (ms)
200 400 600 800 1000
10 Hz
10 Hz
100 Hz
100 Hz
B    (nT)mid
B    (nT)max
~ 2 nT
fce
flh
fce
flh
~ 2 nT
~ 2 nT
Figure 1.18: An example TDSS sample observed downstream of the shock ramp of an
IP shock observed by the Wind spacecraft on 02/11/2000. The left-hand side of the
plot shows the three components of the magnetic field measured by the TDSS detector
search coils on Wind in minimum variance coordinates. The right-hand side of the plot
shows the corresponding wavelet transforms. To the right of the wavelets are the labels
of the relevant frequencies for this wave, where fce is the electron cyclotron frequency
and flh is the lower hybrid resonance frequency. Other relevant information is given in
the plot including the wave vector, k, in GSE coordinates, the angle of propagation
with respect to the magnetic field, θkB, and the eigenvalue ratios, λ1/λ2 and λ2/λ3.
64
B    (nT)min
~ 2 nT
θ   =   89.81  +/-  0.42
kB
o o
λ  / λ   =  12.082 3
λ  / λ   =  1.9091 2
B    (nT)mid
B    (nT)max
m
ax
mid
m
ax
min
min
m
id
~ 2 nT
7 Hz  <  f  <  20 Hz
Time (ms)
200 400 600 800 1000
k     = < +0.714,+0.376,-0.590 >  +/-  < 0.040, 0.091, 0.106 >
GSE
Rotated Magnetic Fields for 23:33:58.351 UT on 02/11/2000
Figure 1.19: The low frequency filter (7 Hz < f < 20 Hz) of the TDSS sample observed
downstream of the 02/11/2000 IP shock ramp in Figure 1.18. The left-hand side of the
plot shows the three components of the magnetic field measured by the TDSS detector
search coils on Wind in minimum variance coordinates. The right-hand side of the plot
shows the corresponding hodograms.
65
B    (nT)min
θ   =   10.51  +/-  0.42
kB
o o
λ  / λ   =  12.232 3
λ  / λ   =  1.8461 2
B    (nT)mid
B    (nT)max
m
ax
mid
m
ax
min
min
m
id
60 Hz  <  f  <  200 Hz
Time (ms)
200 400 600 800 1000
k     = < -0.585,+0.810,+0.045 >  +/-  < 0.054, 0.039, 0.006 >
GSE
Rotated Magnetic Fields for 23:33:58.351 UT on 02/11/2000
~ 0.4 nT ~ 0.4 nT
Figure 1.20: The high frequency filter (60 Hz < f < 200 Hz) of the TDSS sample
observed downstream of the 02/11/2000 IP shock ramp in Figure 1.18. The left-hand
side of the plot shows the three components of the magnetic field measured by the TDSS
detector search coils on Wind in minimum variance coordinates. The right-hand side of
the plot shows the corresponding hodograms.
66
format is the same as in Figure 1.19, but the amplitude of the wave is smaller and it is
right-hand polarized with respect to the magnetic field, consistent with an electromag-
netic whistler wave. The wave is propagating at a slightly oblique angle with respect to
the magnetic field, which is roughly anti-parallel to the wave vector.
In this thesis, I utilize waveform captures of each type of wave reviewed in this sec-
tion to look for evidence of wave-particle interactions observed as wave induced heating,
pitch-angle scattering and/or diffusion, and particle acceleration.
1.8 Thesis Overview
Ever since the prediction of the existence of collisionless shock waves [Kellogg ,
1962], intense interest has been focused on possible dissipation mechanisms [Sagdeev ,
1966]. As discussed above, the energy dissipation must be irreversible for a true shock
transition to occur. Possible energy dissipation mechanisms include wave dispersion
[Mellott and Greenstadt , 1984], particle reflection [Edmiston and Kennel , 1984; Kennel ,
1987], macroscopic field effects [Bale and Mozer , 2007; Hull et al., 2001; Walker et al.,
2004; Wygant et al., 1987], and anomalous resistivity due to wave-particle interactions
[Gary , 1981]. The relative significance of each type of energy dissipation can depend
upon multiple factors, but they are not well understood. It is thought that particle
reflection is more important at higher Mach number shocks, while lower Mach number
shocks depend upon wave dispersion and anomalous resistivity [Mellott and Greenstadt ,
1984]. Yet it should be noted that particle reflection alone cannot produce a complete
shock transition [Kennel et al., 1985]. In space, the terrestrial bow shock has been ex-
tensively studied [Bale et al., 1997, 1998, 2000, 2002; Hull et al., 2006; Kellogg et al.,
1999], but IP shocks are less well examined [Fitzenreiter et al., 2003; Gurnett et al.,
1979b; Hess et al., 1998; Thejappa and MacDowall , 2000; Wilson III et al., 2007]. Due
to their lower Mach numbers, it is expected that IP shocks rely upon wave dispersion
and anomalous resistivity for energy dissipation [Mellott and Greenstadt , 1984]. Thus,
examining the waves in IP shocks is a necessary requirement for understanding the evo-
lution of IP shocks as they propagate away from the sun.
This thesis examines energy dissipation mechanisms, shock structure, and particle
heating in IP shocks using data from the Wind spacecraft. The thesis is organized as
67
follows:
1. Chapter 2 outlines and summarizes the Wind spacecraft instrumentation and the
analysis techniques used herein.
2. Chapter 3 outlines the important low frequency waves seen in and around colli-
sionless shock waves. This chapter also discusses some of the macroscopic fields
observed in shock ramps and their effects on particle distributions.
3. Chapter 4 reports on the study done by Wilson III et al. [2009] on low frequency
(0.25 Hz < f < 10 Hz) and summarizes their results and conclusions.
4. Chapter 5 reports on the studies done by Wilson III et al. [2007] and
Wilson III et al. [2010] on high frequency (& 400 Hz) waves in and around IP
shocks.
5. Chapter 6 summarizes the conclusions of this thesis and outlines some future work.
This thesis includes material from the following papers: Wilson III et al. [2007],
Wilson III et al. [2009], and Wilson III et al. [2010].
Chapter 2
Instrumentation, Measurements,
and Analysis Techniques
In this chapter, the measurement and data analysis techniques needed for studies of
IP shocks are described. Potential error sources are discussed, although some details are
deferred to Appendix A. Illustrative examples from Wind measurements are presented.
2.1 Introduction to the Wind Spacecraft and instruments
The Wind spacecraft was launched on November 1st, 1994 by a Delta II rocket
from Cape Canaveral Air Force Station in Merritt Island, FL. For the first two years
of the mission WIND was in a highly elliptical orbit on the sunward side of the Earth
with an apogee of 250 Earth radii (RE) and a perigee of at least 5 RE. Wind was the
first of NASA’s Global Geospace Science (GGS) program, which was part of the In-
ternational Solar-Terrestrial Physics (ISTP) Science Initiative, a collaboration between
several countries in Europe, Asia, and North America. The aim of ISTP was to under-
stand the behavior of the solar-terrestrial plasma environment in order to predict how
the Earth’s magnetosphere responds to changes in solar wind conditions. WIND’s ob-
jective is to measure the properties of the solar wind before it reaches the Earth [Desch,
2005].
The Wind spacecraft has an array of instruments including: Konus [Aptekar et al.,
1995], the Wind Magnetic Field Investigation (MFI) [Lepping et al., 1995], the Solar
68
69
Wind and Suprathermal Ion Composition Experiment (SMS) [Gloeckler et al., 1995],
the Solar Wind Experiment (SWE) [Ogilvie et al., 1995], a Three-Dimensional Plasma
and Energetic Particle Investigation (3DP) [Lin et al., 1995], the Transient Gamma-Ray
Spectrometer (TGRS) [Owens et al., 1995], and the Radio and Plasma Wave Investiga-
tion (WAVES) [Bougeret et al., 1995]. The Konus and TGRS instruments are primarily
for gamma-ray and high energy photon observations of solar flares or gamma-ray bursts.
The SMS experiment measures the mass and mass-to-charge ratios of heavy ions. The
SWE and 3DP experiments were designed to measure the lower energy (below 10 MeV)
solar wind protons and electrons. The WAVES and MFI experiments were designed to
measure the electric and magnetic fields observed in the solar wind. All together, the
Wind suite of instruments allows for a complete description of plasma phenomena in
the solar wind in the ecliptic plane [Desch, 2005].
2.1.1 Wind 3DP Particle Detector
The Wind/3DP instrument was designed to make full three-dimensional measure-
ments of the distributions of suprathermal electrons and ions in the solar wind. The in-
strument includes three arrays, each consisting of a pair of double-ended semi-conductor
telescopes each with two or three closely sandwiched passivated ion implanted silicon
detectors, which measure electrons and ions above ∼20 keV. The instrument also has
top-hat symmetrical spherical section electrostatic analyzers with microchannel plate de-
tectors (MCPs) are used to measure ions and electrons from ∼3 eV to 30 keV [Lin et al.,
1995].
The two types of detectors have energy resolutions ranging from ∆E/E ≈ 0.3 for the
solid state telescopes (SST) and ∆E/E ≈ 0.2 for the top-hat electrostatic (ES) analyz-
ers. The angular resolutions are 22.5◦ × 36◦ for the SST and 5.6◦ (near the ecliptic) to
22.5◦ for the top-hat ES analyzers. The particle detectors can obtain a full 4π steradian
coverage in one full(half) spin (∼3 s) for the SST(top-hat ES analyzers).
ES Analyzers
The arrays of detectors are mounted on two opposing booms, each 0.5 m in length.
The top-hat ES analyzers are composed of four separate detectors, each with different
geometric factors to cover different ranges of energies. The electron detectors, EESA,
70
and ion detectors, PESA, are each separated into low (L) and high (H) energy detectors.
The H and L analyzers contain 24 and 16 discrete anodes, respectively. The anode layout
provides a 5.6◦ angular resolution within ±22.5◦ of the ecliptic plane (increases to 22.5◦
at normal incidence to ecliptic plane). The analyzers are swept logarithmically in energy
and counters sample at 1024 samples/spin (∼3 ms sample period). Thus the analyzers
can be set to sample 64 energy samples per sweep at 16 sweeps per spin or 32 energy
samples per sweep at 32 sweeps per spin,etc. The detectors are defined as follows:
1. EESA-L (EL): covers electrons from ∼3 eV to ∼1 keV1 with a 11.25◦ spin
phase resolution. EL has a total geometric factor of 1.3 × 10−2 E cm2-sr (where
E is energy in eV) with a nearly identical 180◦ field of view (FOV), radial to the
spacecraft, to that of PESA-L.
2. EESA-H (EH): covers electrons from ∼200 eV to ∼30 keV (though typical
values vary from a minimum of ∼137 eV to a maximum of ∼28 keV) in a 32
sample energy sweep each 11.25◦ of spacecraft spin. EH has a total geometric
factor of 2.0 × 10−1 E cm2-sr, MCP efficiency of about 70% and grid transmission
of about 73%. EH has a 360◦ planar FOV tangent to the spacecraft surface which
can be electrostatically deflected into a cone up to ±45◦ out of its normal plane.
3. PESA-L (PL): covers ions with a 14 sample energy sweep2 from ∼100 eV to
∼10 keV (often energies range from ∼700 eV to ∼6 keV) each 5.6◦ of spacecraft
spin. PL has a total geometric factor of only 1.6 × 10−4 E cm2-sr but an identical
energy-angle response to that of PESA-H. While in the solar wind, PL reorients
itself along the bulk flow direction to capture the solar wind flow which results in
a narrow range of pitch-angle coverage.
4. PESA-H (PH): covers ions with a 15 sample energy sweep from as low as ∼80
eV to as high as ∼30 keV (typical energy range is ∼500 eV to ∼28 keV) each
11.25◦ of spacecraft spin3 . PH has a total geometric factor of 1.5 × 10−2 E
1 These values vary from moment structure to moment structure depending on duration of data
sampling, spacecraft potential, and whether in burst or survey mode. The typical range is ∼5 eV to
∼1.11 keV.
2 Note that in survey mode the data structures typically take 25 data points at 14 different energies
while in burst mode they take 64 data points at 14 different energies.
3 Note that PH has multiple data modes where the number of data points per energy bin can be
any of the following: 121, 97, 88, 65, or 56.
71
cm2-sr with a MCP efficiency of about 50% and grid entrance post transmission
of about 75%.
Characteristics of the entire instrument suite are shown in Table 2.1 from Lin et al.
[1995].
Table 2.1: Wind 3DP Instrument Specs
Detector Energy Geometric Factor FOV Dynamic Range3
Range (cm2-sr) (◦)
EH/FPC1 100eV - 30keV 0.1 E 360 × 90 ∼100-108
EL1 3eV - 30keV 0.013 E 180 × 14 ∼102-109
PH2 3eV - 30keV 0.015 E 360 × 14 ∼101-109
PL2 3eV - 30keV 0.00016 E 180 × 14 ∼104-1011
Foil1 25-400keV 1.7 E 180 × 20 ∼10−1-106
Open2 20keV - 6MeV 1.7 E 180 × 20 ∼10−1-106
Solid-State Telescopes
The SST detectors consist of three arrays of double-ended telescopes, each of which
is composed of either a pair or triplet of closely-sandwiched semi-conductor detectors.
The center detector (Thick or T) of the triplet is 1.5 cm2 in area, 500 µm thick, while
the other detectors, foil (F) and open (O), are the same area but only 300 µm thick. One
direction of the telescopes is covered in a thin lexan foil4 (SST-Foil) where the thickness
was chosen to stop protons up to the energy of electrons (∼400 keV). Electrons are
essentially unaffected by the foil. On the opposite side (SST-Open), a common broom
magnet is used to refuse electrons below ∼400 keV from entering but leaves the ions
essentially unaffected. Thus, if no higher energy particles penetrate the detector walls,
the SST-Foil should only measure electrons and the SST-Open only ions. Each double-
ended telescope has two 36◦ × 20◦ FWHM FOV, thus each end of the five telescopes
can cover a 180◦ × 20◦ piece of space. Telescope 6 views the same angle to spin axis
as telescope 2, but both ends of telescope 2 have a drilled tantalum cover to reduce the
geometric factor by a factor of 10 to measure the most intense fluxes. The SST-Foil data
4 The foil has ∼1500 A˚ of aluminum evaporated on each side to completely eliminate sunlight.
72
structures typically have 7 energy bins each with 48 data points while the SST-Open
has 9 energy bins each with 48 data points. Both detectors have energy resolutions of
∆E/E ≈ 30%.
2.1.2 WAVES
The electric field detectors of the Wind WAVES instrument [Bougeret et al., 1995]
are composed of three orthogonal electric field dipole antenna, two in the spin plane
(roughly the plane of the ecliptic) of the spacecraft and one along the spin axis. The
complete WAVES suite of instruments includes five total receivers including: Low Fre-
quency FFT receiver called FFT (0.3 Hz to 11 kHz), Thermal Noise Receiver called
TNR (4-256 kHz), Radio receiver band 1 called RAD1 (20-1040 kHz), Radio receiver
band 2 called RAD2 (1.075-13.825 MHz), and the Time Domain Sampler called TDS
(≤7.5 kHz in slow mode and ≤120 kHz in fast mode). The longer of the two spin plane
antenna, defined as Ex, is 100 m tip-to-tip while the shorter, defined as Ey, is 15 m
tip-to-tip. The spin axis dipole, defined as Ez, is roughly 12 m tip-to-tip. When ac-
counting for spacecraft potential, these antenna lengths are adjusted to ∼41.1 m, ∼3.79
m, and ∼2.17 m, respectively (P.J. Kellogg, Personal Communication, 2007). The mag-
netic field detectors of the Wind WAVES instrument are composed of three orthogonal
search coil magnetometers (designed and built by the University of Iowa). The XY
search coils are oriented to be parallel to the XY dipole antenna. Thus, in Figure 2.1,
the X-component search coil is at an angle θ from the X-GSE direction. The search
coils allow for high frequency magnetic field measurements (defined as Bx, By, and Bz).
The WAVES Z-Axis is anti-parallel to Z-GSE direction. Thus any rotations can be
done about the Z-Axis in the normal Eulerian sense followed by a change of sign in the
Z-component of any GSE vector rotated into WAVES coordinates.
Figure 2.1 is a schematic used to illustrate the relationship between the various field
instruments on the Wind spacecraft and the GSE-coordinate system. The point of view
is from the negative Z-GSE direction (i.e. below the plane of the ecliptic) looking to-
ward the positive Z-GSE direction (i.e. above the plane of the ecliptic). The spacecraft
rotates in a counterclockwise direction in this coordinate system. The house keeping
information which informs a user where each boom is with respect to the sun direction
is returned as an eight bit integer, providing ∼1.5◦ accuracy. The angle θ is relevant to
73
X-GSE
X-Antenna
Y-Antenna
ψ
MFI-Boom
{@ 0/256 - Tick, MFI-Boom pointed at Sun}
{@ 32/256 - Tick, pointed at Sun}
-Ey
+Ex
+Ey
-Ex
{@ 224/256 - Tick, pointed at Sun}
θ
θ = −90
o
θ = +90
o
Y-GSE
45
o
ψ = Rotation Direction 
of Wind Satellite
θ = Angle between +X-Antenna
and X-GSE direction
θ = −180  , +180
o o
Figure 2.1: This is a schematic used to illustrate the relationship between the various
field instruments on the Wind spacecraft and the GSE-coordinate system. The relative
angles are determined by an eight bit integer. As seen in the image, when this integer
equals zero, the magnetic field boom (the x-framed boom along the X-GSE axis) is
pointed towards the sun (X-GSE). When the integer reads 224, the positive X-antenna
(electric field, Ex) is pointed towards the sun. The magnetic field boom is roughly 45◦
from either X (red line) or Y-antenna (blue line). As indicated by the figure, the satellite
rotates in a counter-clockwise direction in this coordinate system.
74
the software which retrieves TDS samples from the VAX/ALPHA systems (see Section
2.1.2).
Time Domain Sampler Receiver
φ
ζ
B
X-Antenna
Y-Antenna
X-GSE
Y-GSE
Z-GSE
GSE
φ
R(φ).BGSE BWAVES=
=BGSERotate into WAVESCoordinates
Rotate GSE
Basis by φ
Cos(φ) Sin(φ) 0
0
0 0
R(φ) = -Sin(φ)Cos(pi) Cos(φ)Cos(pi)
Cos(pi)
Z-Antenna
Figure 2.2: This is a schematic cartoon used to illustrate the rotation from a GSE basis
to the WAVES antenna basis. The Z-component of the magnetic field must change a
sign after rotation about the Z-GSE axis. The figure shows a rotation about the Z-Axis
by an angle φ (blue) which results in the X-antenna aligning with the new X’-Axis. The
X’Y’-projection of the magnetic field in this basis is the rotated field one uses when
plotting hodograms of the electric field from WAVES.
Electric (and magnetic) field waveform captures can be obtained from the Time
Domain Sampler (TDS) receiver [Bougeret et al., 1995]. In the highest sampling rates,
the TDS samples are ∼17 ms waveform capture of 2048 points (120 kHz) for the Fast
(TDSF) sampler and 7.5 kHz for the Slow (TDSS) sampler. The TDS receiver has four
possible modes (see Table 2.2). In the solar wind, TDSF is usually set to sample in the
120 kHz mode but was changed to the 7.5 or 1.9 kHz modes during the petal orbits
75
through the Earth’s magnetosphere. TDSF samples (see Figure 2.3) are two components
of the electric field (typically in the XY-GSE plane), defined as Ex and Ey (occasionally
chosen to be Ez prior to 1996). For TDSF samples, we define |Exy|=
√
E2x + E
2
y as
the peak-to-peak (pk-pk) amplitude of the waveform. When operating at 120 kHz, the
TDSF receiver has little to no gain below ∼120 Hz, thus the data is cutoff below 150 Hz
when performing gain corrections on ground (P.J. Kellogg, Personal Communication,
2007). Nearly all of the data presented in this thesis is from the TDSF receiver.
Figure 2.2 is a schematic cartoon used to illustrate an example rotation from the
GSE basis to the WAVES antenna basis. The blue lines represent the WAVES basis
at rotated about the negative Z-GSE axis by an angle φ. The angle ζ represents the
angle between the projection of the magnetic field in GSE coordinates (BGSE) onto the
XY-GSE plane and the X-GSE axis. The rotation matrix given in the figure will rotate
BGSE into WAVES coordinates resulting in, BWAV ES. This is calculation necessary for
analysis of wave polarizations. Also, for lower sampling rates (e.g. ≤7.5 kHz), this
rotation should be done on each data point since the angle φ can change by up to
∼120◦ during on TDS sample when sampling at 1875 Hz.
The TDSS samples return 4 field vectors, either three electric and one magnetic
Table 2.2: Wind WAVES TDS Specs
Speed Fast Sampler Slow Sampler
(sps) (sps)
A 120,000
B 30,000
C 7,500 7,500
D 1,875 1,875
E 468
F 117
or one magnetic and three electric field measurements. The TDSS receiver also has
four possible sample rates, as shown in Table 2.2 from Bougeret et al. [1995]. The gain
for the TDSS search coils rolls off below 3.3 Hz. When returning three electric fields,
the TDSS samples are often contaminated by spin effects due to different levels of
photoelectron currents on the antenna in sunlight versus shadow (i.e. only affects the X
76
and Y components) and a glitch associated with the Z-antenna (P.J. Kellogg, Personal
Communication, 2007). However, if the electric fields are large enough and Wind is in
the shadow of the Earth, these effects are negligible. When returning three magnetic
and one electric field, the TDSS receiver can return well defined waveforms with only
small noise and glitch effects. The TDSS receiver is rarely used because the triggering
mechanism did not work correctly. Later in its operation, the TDSS receiver was set to
trigger off of the TDSF receiver (P.J. Kellogg, Personal Communication, 2010). This
can result in a failure to observe large amplitude wave modes. On occasion, the TDSS
receiver does return large amplitude wave modes which are of interest because they are
often below the low frequency cutoff of TDSF (see Figures 1.18 through 1.20).
Radio Receivers
The TNR measures ∼4-256 kHz electric fields in up to 5 logarithmically-spaced fre-
quency bands, though typically only set at 3 bands (K. Goetz, Personal Communication,
2007), from 32 or 16 channels per band, with a 7 nV/
√
Hz sensitivity, 400 Hz to 6.4
kHz bandwidth, and total dynamic range in excess of 100 dB [Bougeret et al., 1995].
The data are taken by two multi-channel receivers which nominally sample for 20 ms
at a 1 MHz sampling rate (see Table 2.3 for more information). The TNR is often used
to determine the local plasma density by observing the plasma line, an emission at the
local plasma frequency due to a thermal noise response of the wire dipole antenna. One
should note that observation of the plasma line requires the dipole antenna to be longer
than the local λDe [Meyer-Vernet and Perche, 1989]. For typical conditions in the solar
wind, the wire dipole antenna on Wind easily satisfy this condition.
Table 2.3: Wind WAVES TNR Specs
Band Range Sampling Rate Measurement Time
(kHz) (kHz) (ms)
A 4-16 64.1 320
B 8-32 126.5 160
C 16-64 255.7 80
D 32-128 528.5 40
E 64-256 1000.0 20
77
2.1.3 Magnetic Field Instrument
The magnetic field instrument (MFI) on board Wind [Lepping et al., 1995] is com-
posed of dual triaxial fluxgate magnetometers. The MFI has a dynamic range of ±4 nT
to ±65,536 nT, digital resolution ranging from ±0.001 nT to ±16 nT, sensor noise level
of < 0.006 nT (r.m.s.) for 0-10 Hz signals, and sample rates varying from 44 samples
per second (sps) in snapshot memory to 10.87 sps in standard mode. The data are also
available in averages at 3 seconds, 1 minute, and 1 hour. The data sampled at higher
rates (i.e. >10 sps) will be referred to as High Time Resolution (HTR) data from here
on.
2.2 Wind Data Analysis and Calibrations
E
  
(m
V
/m
)
y
-100
-50
 0
50
100
Angle of X-Antenna from Sun Direction:  -43.96   to   -45.94
o o
2 4 6 8 10 12 14 16
Time (ms)
E
  
(m
V
/m
)
x
-40
-20
 0
20
40
04/06/2000 at 16:32:12.498 UT, Event #:  27126375
E
  
(m
V
/m
)
y
-50
 0
50
-50  0 50
E  (mV/m)x
Rotated B-Field:  <12.54, 14.14, 11.51> (nT, 10 Hz)
GSE B-Field:  <-1.12, -18.86, -11.51> (nT, 10 Hz)
Angle of XY-Projection (B-field) from X-GSE Direction:  ~-93.4
o
γ
γ ~ 61.4
o
φ
φ ~ 48.4
o
Figure 2.3: This is an example of a TDS sample from the Wind/WAVES instrument.
The two panels on the left are the X and Y-antenna measurements of the electric field
in the WAVES coordinate system. The hodogram on the right is a plot of Ex vs. Ey
with the associated magnetic field rotated into the proper coordinate system. Notice
the angle between the X-antenna and the sun-direction (θ from Figures 2.1 and 2.2) is
roughly -43.96◦ ≤ θ ≤ -45.94◦ over the duration of the TDS sample. The angle of the
XY-GSE projection of the magnetic field from the X-GSE direction is roughly -93.4◦.
78
2.2.1 Magnetic Field Rotations
The magnetic field data is typically obtained in GSE coordinates and thus we need to
rotate the data into WAVES coordinates, as explained in Section 2.1.2, for polarization
analysis. Figure 2.3 shows an example of TDSF sample in 120 kHz sample rate. The
event, shown in WAVES coordinates, is an example of a large amplitude solitary wave
observed just downstream of an IP shock observed on 2000-04-06. On the left, the
top(bottom) panel is a plot of Ex(Ey). To the right is the hodogram, plot of Ex vs.
Ey, for the region outlined by the red box in the left two panels. Overplotted on the
hodogram, with a red line, is the XY-projection of the background DC magnetic field.
The angle of the X-antenna from the sun direction varies from -43.96◦ to -45.94◦ over
the course of the 17 ms TDS sample. In GSE coordinates, the XY-GSE projection of
the DC magnetic field is roughly -93.4◦ away from the sun direction, thus varies away
from the X-antenna by roughly 44.8◦ (shown in red in the hodogram). Notice that
the bipolar signature of the solitary wave is roughly aligned with the magnetic field,
consistent with magnetospheric observations [Ergun et al., 1998a,b; Franz et al., 2000].
2.3 Particle Data Analysis
2.3.1 Electron Distributions
Due to their much lighter mass than hydrogen and helium, electrons carry the bulk
of thermal energy flow away from the sun. Non-Maxwellian aspects of electron distribu-
tion functions (eDFs) have been theorized to produce a number of plasma instabilities
[Gary et al., 1975; Gary , 1981] and supported by observations [Gurnett et al., 1979b;
Thomsen et al., 1983a, 1987; Wilson III et al., 2009]. Thus, it is important to examine
the multi-component characteristics of the solar wind eDFs.
Due to the existence of multiple components of electrons in the solar wind, there
have been many approaches to modeling the observed eDFs. Feldman et al. [1983a]
modeled the parallel and perpendicular (with respect to the magnetic field) cuts of the
electron distribution functions as one dimensional functions (see Section 1.4.2 for more
details). Fitting of each component individually tends to be much easier than attempt-
ing to simultaneously fit both. The problem with one dimensional fits of the eDFs is
79
that the amplitudes are inseparable functions of temperature (if anisotropic) as shown
by Thomsen et al. [1983a]. Therefore, estimation of temperature components (parallel
or perpendicular to the magnetic field) become subject to scrutiny. However, one can
still estimate an anisotropy for the electrons by using the ratio of the thermal speeds:∥∥∥∥vTs,⊥vTs,‖
∥∥∥∥
2
∼ T s,⊥
T s,‖
(2.1)
where vTs,‖(vTs,⊥) is the corresponding parallel(perpendicular) thermal speed of species
s (e.g. core) and Ts,‖(Ts,⊥) is the effective parallel(perpendicular) temperatures. The
reason that one would like an estimate for the temperature anisotropy of the halo (or
core) electrons is that it is a potential free energy source for instability [Gary et al.,
1994, 1999].
Figure 2.4 shows an example of an electron distribution seen by the Wind spacecraft
in the solar wind with fit lines superposed. The actual data is shown as the asterisks
while the fits for the core (blue) and halo (red) are solid lines. The different components
of the distribution are outlined by color-coded boxes. The cold core (outlined by blue
box in the bottom panel) is modeled as a bi-Maxwellian (Equations 2.3a through 2.4).
The hot halo (outlined by the green and gray boxes in the bottom panel) is modeled
as a modified Lorentzian (Equations 2.5 and 2.6). In addition to the power-law tail of
the halo electrons, Figure 2.4 shows an additional higher energy component of the solar
wind electrons known as the super halo [Lin et al., 1996]. Note that the perpendicular
cut appears far more symmetric about zero velocity than the parallel cut. Though this
observation can be a real phenomena, the symmetry in these cuts is due to an assumed
gyrotropy in the creation/calculation of the distribution functions. Regardless, it is
generally true that the perpendicular components are more symmetric than the par-
allel component in the solar wind due to the strahl component (outlined by the green
box in the top panel). As previously mentioned in Section 1.4.2, each component of
the distributions are thought to provide information regarding their origin/progenitors
[Feldman et al., 1975; Lin et al., 1996].
In the analysis done in this thesis, the energies used to estimate the halo and core
electron temperatures for all the EL distributions were determined by fitting the core to
a Maxwellian velocity distribution and the higher energy halo to a modified Lorentzian
[Thomsen et al., 1983a]. The point where the Lorentzian begins to dominate the overall
80
3
6
d
f 
(s
e
c
  
/m
  
)
10
-15
10
-14
10
-20
10
-18
10
-17
10
-16
10
-13
10
-19
10
-15
10
-14
10
-20
10
-18
10
-17
10
-16
10
-13
10
-19
V parallel (1000 km/s)
-20 -10 0 10 20
V perpendicular (1000 km/s)
-20 -10 0 10 20
Phase Space Density
3
6
d
f 
(s
e
c
  
/m
  
)
Core Fit Line
Halo Fit Line
Halo Electrons
Super Halo
Core Electrons
Maxwellian Distribution
Power Law
Power Law
(smaller exponent)
Strahl
Figure 2.4: An example of the different components of a solar wind electron distribu-
tion. The top plot is the parallel cut of the distribution and the bottom plot is the
perpendicular cut.
81
distribution is defined as the break energy, used as the upper bound on the core electrons
and the lower bound on the halo electrons. The moments were then calculated directly
from the full 3D electron distributions using my modified version of the Wind/3DP SSL
IDL Software package (original version provided by R.P. Lin). The strahl component
introduces a highly anisotropic peak in the parallel cuts of distribution functions which
can increase the difficulty fitting a function to the halo electron distribution. Thus,
the strahl electrons were removed in the halo electron fits. The relevant parameters
are then calculated from the original electron distributions using the energy bins below
the break energy for the core and the energy bins above the break energy for the halo
electrons. One should note that the use of energy bin cutoffs instead of the fit functions
can lead to increased uncertainty in the estimates of core and halo parameters. The
core and halo components overlap in energy, thus one may have core(halo) electrons in
their halo(core) moment calculations. These details will be discussed later in Section
2.3.2.
This approach, often used in solar wind studies, is used to find temperatures for
each respective component [Feldman et al., 1983a; Thomsen et al., 1983a]. The mea-
sured electron moments are used to estimate temperatures, velocities, fluxes, etc.. The
core and halo components were determined by fitting the data to a Maxwellian (for
the core) and a modified Lorentzian or kappa distribution (halo). The distributions
are estimated by separately fitting the core and the halo components then adding the
resultant distributions together as:
f e(v) = fL(v) + fM(v) (2.2)
where fL(fM) represent the modified Lorentzian(Maxwellian) components of the distri-
bution. For the Maxwellian core, we model the parallel and perpendicular components
separately [Feldman et al., 1983a; Thomsen et al., 1983a] as individual one dimensional
functions in the following manner:
f ‖M(v) = A‖Me
−
„
v‖−vo
vT‖
«2
(2.3a)
f⊥M(v) = A⊥Me
−
“
v⊥
vT⊥
”2
(2.3b)
where A‖M(A⊥M) is the parallel(perpendicular) amplitude of the one dimensional
Maxwellian, v‖(v⊥) is the parallel velocity, vo is the parallel drift velocity (typically
82
zero because one transforms into the solar wind rest frame), and vT‖(vT⊥) is the par-
allel(perpendicular) thermal speed. It would be more appropriate to model both the
parallel and perpendicular components as one bi-Maxwellian with:
fM(v) = NM
(
me
2πT⊥M
)√√√√( me
2πT ‖M
)
e
− me
2kB
((v‖−vo)2/T ‖M+v⊥2/T⊥M) (2.4)
where NM is the number density, me is the electron mass, T‖M(T⊥M) is the paral-
lel(perpendicular) electron temperature, v‖(v⊥) is the parallel(perpendicular) velocity
as in Equations 2.3a and 2.3b, and vo is the parallel drift velocity is the same as in Equa-
tion 2.3a. This is not done in our analysis because we are interested in determining the
break energies for the parallel and perpendicular components. Attempting to fit the
multi-component anisotropic solar wind electron distributions to a bi-Maxwellian poses
a number of problems due to beam-like features and the solar wind strahl component.
Thus, we used one dimensional functions to fit the parallel and perpendicular compo-
nents to find the break energies after fitting the halo components, discussed below.
The hot halo electrons were modeled as a one dimensional modified Lorentzian func-
tion [Feldman et al., 1983a]:
fL(v) = AL
[
1 +
(
v⊥,‖
v(⊥,‖),L
)2m]−(m+1)/m
(2.5)
where v⊥,‖,L are speeds which characterize the break points (i.e. points at which a
Maxwellian no longer dominates the distribution) in the distribution perpendicular and
parallel to the ambient magnetic field. The exponent, m, characterizes the sharpness of
the break and the hardness of the high energy power law tail. Of course, one should
model these in two dimensions, as Thomsen et al. [1983a] did, to examine flat-topped
distributions and/or just the halo component of the electron distributions which is shown
as:
fL(v) = AL
[
1 +
(
v⊥
v⊥,L
)2m
+
(
v‖
v‖,L
)2m]−(m+1)/m
(2.6)
where again v⊥,‖,L are speeds which characterize the break points in the distribution
perpendicular and parallel to the ambient magnetic field and AL is the amplitude defined
by:
AL =
NLm sin (π/(2m))
π2v⊥,L2v‖,L
. (2.7)
83
One can also use a kappa distribution in two dimensions, as seen here:
f(v⊥
2, v‖) =
(π
κ
)(−3/2) v⊥2j
θ⊥2(j+1)θ‖
Γ(κ+ j + 1)
κjj!Γ(κ − 1/2)
(
1 +
v⊥2
κθ⊥2
+
v‖
2
θ‖2
)−(κ+j+1)
(2.8)
where κ is some index, j is an index which indicates the strength of a loss-cone (j =
0 indicates no loss-cone), Γ() are gamma functions. One should note that as κ → ∞,
f(v⊥2,v‖) → Dory-Guest-Harris loss-cone bi-Maxwellian distribution [Mace, 1998]. The
parameters θ⊥,‖ are effective thermal speeds given by:
θ⊥ =
(
2κ− 3
κ
)1/2√ kBT⊥
(j + 1)me
(2.9a)
θ‖ =
(
2κ− 3
κ
)1/2√kBT ‖
me
(2.9b)
where T⊥,‖ are the respective temperatures perpendicular and parallel to the ambient
magnetic field [Mace, 1998]. An electron (kappa dependent) plasma beta is defined
when using this distribution by:
βe =
θ‖
2ωpe2
c2Ωce2
(2.10a)
=
(κe − 3/2)
κe
neT ‖,e
Bo2/(2µo)
(2.10b)
where Bo is the average magnetic field amplitude. This reduces to the usual βe in the
limit as κe → ∞.
As mentioned earlier, there is a certain type of electron distribution often seen down-
stream of strong quasi-perpendicular shocks called a flat-topped (or flattop) distribution
[Feldman et al., 1983a; Thomsen et al., 1983a]. These distributions can be modeled with
the modified Lorentzian (shown in Equation 2.6) or with a modified exponential given
by:
f(v) = Afe
−(|v⊥/vTh,⊥ |5+|v‖/vTh,‖ |5) . (2.11)
This specific distribution is thought to be produced by interactions with strong current-
driven ion acoustic waves [Dum et al., 1974; Dum, 1975; Feldman et al., 1983a] or due
to the cross-shock potential [Scudder et al., 1986a]. The distribution is referred to as a
self-similar distribution and the exponent in its general form should be proportional to
84
(v⊥/vTh,⊥)x where x → 5 in the quasi-linear limit [Dum et al., 1974].
In most cases presented in this thesis, the electron distributions upstream of IP
shocks are modeled with one dimensional functions of a bi-Maxwellian core and bi-
Lorenztian (typically m = 2 to 3) halo. Most of the flattop distributions were modeled
with a self-similar distribution (x = 4) for the core and bi-Lorenztian halo. An auto-
mated fit routine was used to gain a zeroth order estimate of the relevant fit parameters
(e.g. T⊥,‖ etc.) and then corrections were made by hand if necessary. Typically the core
and halo fits were adjusted to correct errors caused by the strahl component or beam-
like features. Once the functional forms were determined for the respective components,
the break energy was determined. The break energy defined the separation between the
core and halo electrons in 3DP data structures. Thus, we defined the relevant energy
bins for each component, computed the moments of the distribution function, and an-
alyzed the results. Since we were often interested in the respective temperatures, or
average kinetic energy per particle, the moment calculations done by the Wind/3DP
software was an appropriate approximation to actual temperatures because it directly
calculates the average kinetic energy from the second moment. The software does not
ignore the strahl nor does it assume any functional form for the data. Calculation of
the moments directly from the data yielded more accurate results than fitting cuts of
the distribution function to model functions like a bi-Maxwellian.
2.3.2 Temperature Anisotropies
Temperature anisotropies are known to be sources of free energy for instabilities
in magnetized plasmas [Gary et al., 1975]. They have been examined between 0.3 and
0.8 AU using Helios 1 and 2 data [Pilipp et al., 1987b], in the near Earth solar wind
using ISEE data [Phillips et al., 1989a,b], and at IP shocks [Wilson III et al., 2009].
Therefore, estimating an effective temperature anisotropy can allow one to examine the
potential to excite waves.
Figure 2.5 presents an example of Wind data used to show the comparison between
the modeled distribution function cuts (solid blue lines) and the actual data (* for par-
allel cut and ♦ for perpendicular cut) for the distribution shown in Figure 1.8. To argue
for the use of my modified Wind/3DP SSL IDL Software package in calculating the mo-
ments of the distribution function, I compared the temperature anisotropies calculated
85
(seen in bottom panel) by the software with the values calculated (seen in top panel)
from the modeled functions. In every case examined, the modeled distribution functions
show a stronger anisotropy (i.e. T⊥/T‖ is larger) in both the core and halo components
than the program estimates. As previously discussed, the larger anisotropies in the fit
function estimates result from the neglect of the strahl component when fitting the data.
The fit estimates for the distribution in Figure 2.5 are Tc,⊥/Tc,‖ ∼ 1.43 and Th,⊥/Th,‖
∼ 0.99. The computed estimates of the temperature anisotropies from the actual data
are Tc,⊥/Tc,‖ ∼ 0.99 and Th,⊥/Th,‖ ∼ 0.84.
The consistent difference between modeled fit function and program estimates in
temperature anisotropies due to the solar wind strahl (see Figure 2.4) component, skews
the parallel temperature in such a way as to reduce the halo (and often core) temper-
ature anisotropy, Th,⊥/Th,‖. As a consequence, the strahl component is either removed
or not included in the modeled fits used in this thesis due to its highly non-Maxwellian
behavior and intermittent occurrence. Since instability estimates often ignore the strahl
component when examining the solar wind heat flux [Gary et al., 1994, 1999], the fit
estimates in Figure 2.5 are relevant to at least the threshold estimates of Gary et al.
[1994] and Gary et al. [1999]. One should also note that the average temperature esti-
mates for the core electrons tend to be higher for the 3DP software estimates than the
model fits because of the strahl contamination.
As previously discussed, the strahl electrons can often skew the anisotropies away
from distributions which may be unstable to the whistler heat flux instability [Gary et al.,
1994, 1999]. Thus, when calculating the temperatures of both the core and halo compo-
nents using the 3DP software, one finds that Th,⊥/Th,‖ is lower than from fit estimates.
As a consequence, the 3DP software estimates were used as a lower bound on the temper-
ature anisotropy estimates byWilson III et al. [2009] when comparing to the instability
thresholds of Gary et al. [1994].
2.3.3 Ion Distributions
The Pesa Low detector is a spherical top-hat ES particle detector which covers en-
ergy ranges typically between ∼700 eV to ∼6 keV when in the solar wind. The detector
is designed to detect the solar wind beam and thus has a narrow field of view. It is also
86
Wind 3D Plasma Eesa Low
2000-04-06/16:28:57 - 16:29:00
Velocity (1000 km/s)
-20 -10 0 10 20
3
6
d
f 
(s
e
c
  
/m
  
)
10
-15
10
-14
10
-20
10
-18
10
-17
10
-16
10
-13
10
-19
6
d
f 
(s
e
c
  
/m
  
)
10
-15
10
-14
10
-20
10
-18
10
-17
10
-16
10
-13
10
-19
Fit Line
Actual Data
T     / T     : 0.99 (program)ce ce
T     / T     : 0.84 (program)he he
T     / T     :  1.43 (from Þts)ce ce
(V     /V     )   :  0.99 (from Þts)he he
2
Figure 2.5: An example of the parallel and perpendicular cuts of the distribution func-
tion. The asterisks in the top panel and diamonds in the bottom panel are the real
parallel and perpendicular cuts from the bottom panel of Figure 1.8. The solid lines are
the composite fit examples. The top panel shows the solar wind strahl as the asymmetry
anti-parallel to the magnetic field.
87
prone to saturation and innaccurate measurements if the solar wind parameters change
abruptly, like in IP shocks. If the ions get too hot, the detector can saturate as well.
However, for most of the duration of Wind’s experience in the solar wind the detector
measures nearly all of the solar wind beam.
The Pesa High detector is a spherical top-hat ES particle detector which covers
energy ranges typically between ∼500 eV to ∼28 keV when in the solar wind. The
detector has a relatively high threshold at low energies making it difficult to detect
background ions below ∼500 km/s unless there is a well defined peak. In other words,
below ∼500 km/s the one-count levels of the detector, which scales as E−2, are too high
to observe the particles often found in this energy range. At the lowest energies, below
∼200 km/s, one can see the thermal core (for an example, see the bottom two panels of
Figure A.1). The thermal core often saturates the detector and is often contaminated
with UV light. The detector is not, however, by any means useless. It offers a full
4π steradian coverage every ∼3 seconds in burst mode and has been used to detect
gyrating ion distributions upstream of the terrestrial bow shock and their association
with low frequency waves [Meziane et al., 2001]. However, with any instrument there
are idiosyncracies which must be regarded before interpreting data. As was previously
mentioned, the low energies of the Pesa High detector have relatively high one-count
levels. Thus, if the particles in that low energy population are shifted to higher energies
by say, a convectional electric field, they can appear as an asymmetric particle distri-
bution function even though the distribution may really be symmetric.
2.4 Waveform Analysis
2.4.1 Minimum Variance Analysis
Minimum variance (MV) analysis utilizes a property of plane polarized linear elec-
tromagnetic waves which allows one to assume that fluctuations in the electric (δE) and
magnetic (δB) fields are are in a plane orthogonal to the direction of propagation (kˆ)
[Khrabrov and Sonnerup, 1998]. Using a plane wave assumption for ∇ · B = 0 results
in the linearized equation kˆ · δB = 0. The analysis is performed by minimizing the
88
variance matrix of the magnetic field given by:
Spq =
〈(
Bp −
〈
Bp
〉)(
Bq −
〈
Bq
〉)〉
(2.12)
where <Bp > is the average of the pth component of the magnetic field. We assume
Spq to be a non-degenerate matrix with three distinct eigenvalues, λ3 < λ2 < λ1, and
three corresponding eigenvectors, e3, e2, e1. Thus the minimum variance eigenvalue
and eigenvector are λ3 and e3. The propagation direction is assumed to be along eˆ3 if
one assumes small isotropic noise and the condition λ2/λ3 ≥ 10 is satisfied. Then the
uncertainty in this direction is given by Kawano and Higuchi [1995]:
δkˆ = ±
(
eˆ1
√
δλ3
λ1 − λ3 + eˆ2
√
δλ3
λ2 − λ3
)
(2.13)
where K is the number of vectors used and δλ3, the uncertainty in the λ3 eigenvalue, is
given by:
δλ3 = ±λ3
√
2
(K − 1) . (2.14)
In general, the uncertainty of δλi is given by:
δλi = ±
√
2λ3(2λi − λ3)
(K − 1) . (2.15)
Another useful quantity to know is the angle between the local ambient magnetic field
and the propagation direction, θkB. This can be calculated in the typical manner,
θkB ≡ cos−1
(
kˆ · bˆ
)
, with associated uncertainties of:
δθkB = ±
√
λ3λ2
(K − 1)(λ2 − λ3)2 . (2.16)
MV analysis can be done on specific time ranges to determine the wave vector, k,
and the polarization. The use of bandpass filters can increase the accuracy of MV
analysis by removing low frequency compressional components (typical of magnetosonic
whistler waves in the solar wind). Using the wave vector from MV analysis, the angle
of propagation for each wave with respect to the shock normal vector, θkn, upstream
averaged solar wind velocity, θkV , and the magnetic field, θkB, were examined. These
angles are determined by the following equation:
θkj ≡ cos−1
(
k · jˆ
| j |
)
(2.17)
89
where j can be b, n, or Vsw.
In our use of MV analysis, we define the eigenvalues of the spectral matrix, from
minimum to maximum, as λ3, λ2, and λ1. As a general rule for determining whether the
MV analysis has yielded a well determined plane circularly polarized wave, we require
λ2/λ3 ≥ 10.0 and λ1/λ2 ∼ 1.0 if less than 50 field vectors were used in the analysis.
For the case where λ2/λ3 ≥ 10.0 but 1.0 < λ1/λ2 ≪ λ2/λ3, the wave is elliptically
polarized. If λ1/λ2 ≫ λ2/λ3 ∼ 1.0 then the wave is linearly polarized and the k-vector
cannot be trusted. These assumptions hold for data with small isotropic Gaussian
noise [Khrabrov and Sonnerup, 1998]. Single satellite measurements introduce another
complication. Though the plane orthogonal to the k-vector may be well determined, the
sign of the vector cannot be known without at least one component of the electric field
or another satellite measurement [Hoppe et al., 1981, 1982; Hoppe and Russell , 1983].
2.4.2 Bandpass Filters and Spectral Analysis
High time resolution (HTR) magnetic field data were obtained from the Wind space-
craft within ±1 hour of the IP shocks analyzed. The HTR MFI data used in this the-
sis were sampled at two different rates: ∼22 samples/s (for 04/03/1996, 04/08/1996,
12/02/1996, 01/10/1997, and 10/24/1997) and∼11 samples/s (02/27/1997, 12/10/1997,
04/23/1998, 04/30/1998, 05/15/1998, 08/26/1998, 02/11/2000, 02/20/2000, and
04/06/2000). HTR magnetic field data, was used to define the ramp region, or tran-
sition region, of the IP shock (see Figure 1.3 for visual reference). The wave vector
and other wave properties were determined using Minimum Variance (MV) analysis
[Khrabrov and Sonnerup, 1998], as described in detail in Section 2.4.1. The magnetic
field fluctuations were identified and analyzed using a bandpass filter, as shown in Fig-
ure 2.7.
The frequency ranges for each bandpass filter was determined from spectral analysis.
Spectral analysis is simply an examination of the data in Fourier space as seen in Figure
2.6, which can illustrate where the greatest power is located with respect to frequency.
Finding a frequency peak, as seen in the black box of Figure 2.6, can help one identify
the particular wave mode being examined or other processes occuring in the plasma.
Spectral analysis was used to determine an approximate range of frequencies to use for
the initial guess of the bandpass filter frequencies then refined by maximizing the ratio
90
0.01 0.10 1.00
Frequency (Hz)
n
T 
  H
z
2
-1
10
3
10
-5
10
-4
10
-3
10
-2
10
-1
100
10
1
102
Power Spectrum for 16:28:00 UT Shocklet on 04/06/2000
Figure 2.6: A plot of the magnetic field power spectrum as a function of frequency for
16:27:12-16:28:45 UT on 04/06/2000. The three colors represent the X (red), Y (green),
and Z (blue) GSE components of the magnetic field. Black box illustrates the range of
frequencies used for the bandpass filter in panel A of Figure 4.3.
of intermediate to maximum eigenvalues, λ2/λ3. The data is plotted and adjustments
are made to the frequency range of the filter until the wave envelope resembles some-
thing similar to the three higher frequency filters seen in Figure 2.7.
Figure 2.7 is an example of four different bandpass filters being applied to the mag-
netic field data shown in the top panels measured by the Wind spacecraft upstream of
an IP shock. The example shown is a plot of a shocklet (see Section 3.2.1) observed
roughly five minutes upstream of the 2000-04-06 event discussed in detail in Chapter 4.
All six panels plot the same time range and the upper left is a plot of the magnetic field
magnitude and upper right are the XYZ-GSE components. For this event, the MFI
data was sampled at ∼10 Hz. The bottom two rows are plots of the resulting XYZ-GSE
components after filtering the data. The lower frequency components (upper left panel
91
|B
| 
(n
T
)
6
7
8
9
10
B
 [G
S
E
] (
nT
)
0
-5
5
-10
HTR Wind B-Field:  04/06/2000 16:27:43 - 16:29:46 UT
190 mHz < f < 600 mHz 450 mHz < f < 1.0 Hz
500 mHz < f < 1.0 Hz 600 mHz < f < 3.0 Hz
0
-2
2
0.0
-1.0
1.0
1.5
-1.5
-0.5
0.5
0
-1
1
0.0
-1.0
1.0
1.5
-1.5
-0.5
0.5
~ 2 nT ~ 3 nT
~ 3 nT
~ 4 nT
16:29:0016:28:00 16:29:0016:28:00
B
 [G
S
E
] (
nT
)
B
 [G
S
E
] (
nT
)
Bx By Bz
Figure 2.7: An example plot of bandpass filtered HTRMFI data on 04/06/2000. The top
two panels have not been filtered and represent the magnitude and vector components
near a shocklet upstream of the 04/06/2000 IP shock event. In every vector plot,
red is the X-GSE direction, green the Y-GSE, and blue the Z-GSE directions. The
frequencies at which each of the bottom four panels were filtered is labeled in each plot.
The relative magnitude of the fluctuations are also labeled with the verticle black arrows
in each filtered plot.
of filtered signals) often have larger amplitudes and in the case of dispersive waves like
shocklets, have different spatial locations than their high frequency counterparts. The
spatial dispersion can be seen by comparing the upper left filtered panel to the three
other filtered data panels. Notice that the high frequency wave envelope occurs roughly
30 seconds ahead of the lower frequency amplitude peak in the upper left filtered data
plot.
Figure 2.8 is an example of MV analysis done on low frequency magnetosonic whistler
waves upstream of an IP shock. The left hand set of wave events plot the GSE (gray
scale) components, the middle panels plot the MV (color scale) components, and the
right hand set of panels show the hodograms, By vs. Bx, Bz vs. Bx, and Bz vs. By. The
time for each wave event in Figure 2.8 is defined by the color-coded boxes outlined in
92
Time (HH:MM:SS)
16:20:44 - 16:25:24 UT on 04/06/2000
B
 [G
S
E
] (
nT
)
0
-2
4
-4
2
-6
-8
|B
| 
(n
T
)
7.5
8.0
9.0
6.5
7.0
8.5
16:22:00 16:24:00
SCET :  16:22:31.1145 - 16:22:34.3345
Date:  04/06/2000
-0.5
0.0
0.5
k
GSE
^ = <-0.900,0.384,0.205>
-0.5
0.0
0.5
-0.5
0.0
0.5
SCET :  16:22:36.0825 - 16:22:38.2905
Date:  04/06/2000
B
  
(n
T
)
x
0.0
-0.2
0.2
0.4
-0.4
B
  
(n
T
)
y
0.0
-0.2
0.2
0.4
-0.4
B
  
(n
T
)
z
0.0
-0.2
0.2
0.4
-0.4
k
GSE
^ = <0.728,-0.622,-0.289>
SCET :  16:22:31.4825 - 16:22:33.9665
Date:  04/06/2000
-0.5
0.0
0.5
-0.5
0.0
0.5
-0.5
0.0
0.5
k
GSE
^ = <-0.900,0.382,0.209> SCET :  16:24:07.0705 - 16:24:15.9945
Date:  04/06/2000
B
  
(n
T
)
y
0.0
-1.0
1.0
2.0
-2.0
B
  
(n
T
)
x
0.0
-1.0
1.0
2.0
-2.0
B
  
(n
T
)
z
0.0
-1.0
1.0
2.0
-2.0
k
GSE
^ = <0.626,-0.716,-0.309>
C
D
A
B
Y Y Y Y
Y YY Y
Y YY Y
Y Y Y Y
Z ZZ Z
Z Z Z Z
Z Z
Z ZZ Z
Z Z
X X
X X
X X
X XX X
X X
X X
X X
X Y Z
B
  
(n
T
)
y
B
  
(n
T
)
x
B
  
(n
T
)
z
B
  
(n
T
)
y
B
  
(n
T
)
x
B
  
(n
T
)
z
Figure 2.8: An example of MV analysis on the leading whistler waves of a low frequency
magnetosonic whistler wave, called a shocklet (discussed later). The frequency ranges
and angles of propagation are: f > 0.6 Hz and θkB = 41◦(139◦) for A, 0.25 Hz < f <
0.45 Hz and θkB = 38◦(142◦) B, f > 1.0 Hz and θkB = 41◦(139◦) for C, and 0.6 Hz <
f < 3.0 Hz and θkB = 27◦(153◦) for D. The eigenvalue ratios from the MV analysis are
also shown with the MV estimate of the k-vector direction in GSE coordinates above
each hodogram. The purple arrows indicate the direction of rotation for each respective
plot.
93
the center plot of the magnetic field magnitude and GSE components. The difference in
polarization, seen in the hodograms, between wave events A and C in Figure 2.8 can be
explained by projection effects due to single satellite measurements using only magnetic
field measurements [Hoppe et al., 1981]. In the spacecraft frame, wave events A and C
in Figure 2.8 are LH polarized with respect to the propagation direction, but all wave
events in Figure 2.8 show a RH sense with respect to the magnetic field, characteristic
of whistler modes. Wave events B and D in Figure 4.3 are RH polarized both with
respect to the wave vector and the magnetic field.
2.4.3 Wavelet Analysis
Wavelet transforms are square integrable basis functions used as a mathematical
technique to analyze waveform data in both space and scale [Farge , 1992; Hudgins et al.,
1993; Lau and Weng , 1995; Torrence and Compo, 1998a]. Wavelets are basis functions
based on group theory and are used for multiple purposes including signal analysis,
image coding, and renormalization theory in quantum electrodynamics [Farge , 1992].
Traditionally Fast Fourier Transforms or FFTs have been used to analyze electric and
magnetic field data in space physics. Wavelets have only recently become a popular
analysis tool, and a great deal of confusion and apprehension surround this mathemat-
ical tool.
Let us first consider the traditional waveform analysis technique, the Fourier trans-
form. Fourier transforms are a mathematical technique where one attempts to re-
construct an input signal using the superposition of a semi-infinite or infinite set of
trigonometric functions. There are advantages and disadvantages to using trigonomet-
ric functions in signal decomposition. In particular, a disadvantage is that the basis
functions oscillate forever. Thus any localized signal, which we’ll define as a data glitch
of non-continuous behavior, in space will be spread out over all Fourier coefficients. One
should also note that any signal which has an abrupt change in amplitude with respect
to time, such as a solitary wave, would show a similar result on a Fourier transform.
This results in an inability to filter out the glitch [Farge , 1992]. For reference, we de-
fine local as the subdomain of time about which the transform is being applied. For
instance, let us assume we have some input signal, f(t), which depends on time and has
an equal time spacing of δt. The generalized Fourier series of f(t) is given by:
94
f(t) =
1
2
ao +
∞∑
n=1
an cosnt+
∞∑
n=1
bn sinnt (2.18)
where n = 0, 1, ..., N-1 is the time index and:
ao =
1
π
∫ π
−π
f(t)dt (2.19a)
an =
1
π
∫ π
−π
f(t) cosntdt (2.19b)
bn =
1
π
∫ π
−π
f(t) sinntdt (2.19c)
where an and bn are the Fourier coefficients of f(t). The Fourier transform of f(t) is
given by:
F (ω) =
√
1
2
∫ ∞
−∞
f(t)e−iωtdt (2.20)
where F(ω) is said to be the Fourier transform of the function f(t). We return to the
example of a glitch in a finite time series. One can see that there is no local information
stored in the Fourier coefficients defined in Equations 2.19a through 2.19c. The main
consequence is that the glitch is found in the phase of every Fourier coefficient. In
contrast, wavelet transforms intrinsically retain localized information allowing one to
isolate and remove data glitches or more clearly analyze real signals that are localized.
In 1980 a French researcher, named Jean Morlet, working on seismic data collabo-
rated with a theoretical physicist, named Grossmann, to develop a geometrical trans-
lation and dilation [Farge , 1992]. They were interested in decomposing seismic signals
into both space and scale, an analysis technique of extremely useful potential. The
wavelet transform removes the behavior of a signal at infinity, which an FFT cannot
do. The wavelet coefficients retain the locality of the input signal allowing for a recon-
struction of the original input signal locally. Meaning, if there is a glitch at one point in
an otherwise smooth/continuous input signal, the amplitude of the wavelet coefficients
will increase near the glitch in time and frequency, but be small elsewhere. This allows
one to reconstruct the wavelet locally without requiring the entire transform.
I will focus my efforts on the most commonly used wavelet transform in this thesis,
95
the un-normalized Morlet wavelet transform [Morlet et al., 1982; Morlet , 1982] defined
by:
ψo(η) = π
−1/4eiωoηe−η
2/2 (2.21)
where ωo is a non-dimensional frequency (taken to be 6 to satisfy the admissibility
condition set by the uncertainty principle) and η is a non-dimensional time parameter.
When we scale and translate ψo(η), the wavelet becomes a function of scale, s, and time,
nδt. The Fourier transform of the scaled and translated wavelet is given by:
ψˆ(sωk) =
(
2πs
δt
)1/2
ψˆo(sωk) (2.22)
where s is the wavelet scale, δt is the sample period, and ωk is defined by:
ωk =
{
2πk
Nδt for k ≤ N/2,
− 2πkNδt for k > N/2.
Every unscaled ψo must have unit energy to satisfy the admissibility condition seen as:∫ ∞
−∞
| ψˆo(ω′) |2 dω′ = 1 (2.23)
which results in there being a factor N in the normalization shown by:
N−1∑
k=0
| ψˆ(sωk) |2= N (2.24)
where N is the number of points in the input array. The Fourier transform of ψo is
defined as:
ψˆo(sω) = π
−1/4H(ω)e−(sω−ωo)
2/2 (2.25)
where H(ω) is the Heaviside step function, H(ω) = 1 for ω > 0, H(ω) = 0 otherwise.
Thus, if one has an input time series xn, with a uniform sample rate of 1/δt, and time
index n = 0, 1, ..., N-1. We define the Fourier transform of xn, xˆk, as:
xˆk =
1
N
N−1∑
n=0
xne−2πikn/N (2.26)
where k = 0, 1, ..., N-1 is the frequency index. From this, we can construct the wavelet
transform as a function of wavelet scale, s, to be:
W n(s) =
N−1∑
k=0
xˆkψˆ
∗(sωk)e
iωknδt (2.27)
96
where ωk is defined as in Equation 2.4.3 [Torrence and Compo, 1998a].
Figure 2.9 is an example image of a Morlet wavelet transform. The figure is pro-
Morlet Wavelet Transform
F
re
q
u
e
n
cy
Time
fce2
fce
Cyclotron
Harmonics
Level
Power Spectral Density
Figure 2.9: An example of a wavelet transform plot. The plot represents a wave ampli-
tude (color scale) as a function of frequency versus time. This figure is illustrative of
later images.
duced from a real TDS event, but it is used here to illustrate the format of the wavelet
transforms presented later. The color scale is a relative intensity of the wave mode as
a function of frequency versus time. The plot was produced using software produced
by Torrence and Compo [1998b]. There are two significance tests shown in Figure 2.9:
the cone of influence and the 95% confidence level. The cone of influence is the bowl-
shaped line that outlines the region of the plot where edge effects become important due
to the finite sample time. The 95% confidence level is a statistical test of significance
comparing the relative amplitudes of the wave mode at different frequencies and times.
The 95% confidence level is seen as the contour outlining the most intense part of the
wavelet transform. It represents the region of the data composed of the top 5% of all
the amplitudes in the wave. Meaning, inside the contour for a particular time, all the
amplitudes are larger than 95% of the other amplitudes at any other frequency outside
of the contour. Finally, the two horizontal lines at the bottom of the wavelet transform
represent the electron cyclotron frequency and its first harmonic.
Figure 2.10 is an illustrative comparison between a Morlet wavelet transform and
97
three different size windowed FFTs. The windowed FFT was originally implemented
to force a locality into the time/frequency decomposition of time series analysis prior
to wavelet transforms [Hudgins et al., 1993]. All four spectral power density plots are
plotted on the same scale. The three different sized windows are shown as the magenta
colored boxes on the left-hand side of the image. The sizes are labeled at the top of
each windowed FFT panel as a fraction of the total time range of the input signal. One
should note that there is no common standard for defining a useful window width for
a windowed FFT. Given the arbitrary choice in window width, one can see that the
interpretation of an input signal can vary drastically depending on the choice in size
of FFT window. For instance, the 1/64th window is so narrow, it fails to resolve the
actual frequency of the input sawtooth signal. As with the wavelet transform, there is a
balance between the number of frequency bins and time steps for a windowed FFT. The
ultimate lower bound is set by the uncertainty principle [Torrence and Compo, 1998a].
Figure 2.10 illustrates how an arbitrary choice in window size could lead to a dra-
matically different interpretation of the waveform. The Morlet transform, however,
very distinctly resolves the fundamental frequency of the sawtooth waveform. Due to
the advantages listed above and the clear difference in the resolving power of the Morlet
wavelet over a windowed FFT (seen in Figure 2.10), I will primarily rely on Morlet
wavelets throughout this thesis for dynamic waveform analysis.
98
Morlet Wavelet TransformF
re
q
u
e
n
cy
Actual Frequency
1/16th Window
1/32nd Window
1/64th Window P
o
w
e
r 
S
p
e
ct
ra
l 
D
e
n
si
ty
F
re
q
u
e
n
cy
F
re
q
u
e
n
cy
F
re
q
u
e
n
cy
Sawtooth Wave Example
Wavelet and Fourier Transforms
Time
Figure 2.10: An example of a sawtooth wave analyzed with a Morlet wavelet trans-
form and three different windowed FFTs. The relative floating FFT window sizes are
shown on the left-hand side by the magenta colored boxes. The actual frequency of
the sawtooth wave is shown as the white line in the top wavelet transform panel (and
subsequent FFT panels).
Chapter 3
Previous Studies of Low
Frequency Waves and
Macroscopic Fields
This chapter presents an overview of previous work done on low frequency (<20
Hz) magnetic fluctuations and macroscopic (scales > ρgi) electromagnetic fields in the
solar wind and planetary foreshocks. The purpose of this chapter is to provide some
background for arguments and analyses made in Chapters 4 and 5. The discussion of
low frequency waves presents a history of observations with detailed figures to serve
as illustrative examples. The discussion regarding macroscopic fields in shock waves
highlights the work of previous studies and presents arguments for considering higher
frequency fluctuations when examining the shocked particle distribution functions.
3.1 Introduction to Low Frequency Waves at Collisionless
Shocks
Irregular turbulence upstream of planetary bow shocks has been observed for over
40 years [Fairfield , 1969]. These waves are thought to play an intrinsic role in particle
acceleration, heating, and energy dissipation in collisionless shock waves by allowing the
shock to communicate with the upstream plasma [Hada et al., 1987; Stasiewicz et al.,
99
100
2003]. The frequencies of this turbulence were often observed to fall in the range of the
ion cyclotron frequency. Thus, the magnetic turbulence upstream of the Earth’s bow
shock was initially examined in association with ion particle data [Paschmann et al.,
1981; Sentman et al., 1981]. Waves in the frequency range around the ion cyclotron
frequency are important dynamically for shock formation and stability. By radiating
energy away from the shock itself, both transverse Alfve´nic and compressional magne-
tosonic modes allow a collisionless shock to communicate with the upstream plasma by
pre-heating or decelerating the incoming plasma, thus altering the Rankine-Hugoniot
conditions [Scholer and Belcher , 1971]. The waves in this frequency regime upstream of
collisionless shocks are thought to play an intrinsic role in particle acceleration, heating,
and energy dissipation [Hada et al., 1987; Stasiewicz et al., 2003].
3.1.1 ULF Foreshock Waves
Fairfield [1974] separated the magnetic turbulence observed upstream of the terres-
trial bow shock into two categories using Imp 6 magnetometer data, due to possible
generation mechanisms and polarization: low frequency (0.01-0.05 Hz) and high fre-
quency (0.5-4.0 Hz), which will be discussed in the next section. We will briefly discuss
the 0.01-0.05 Hz class of waves in this section but this thesis does not focus on them.
However, they are an important aspect of collisionless shock dynamics so we cannot
ignore their possible impacts on the energy dissipation occurring in the shock.
The 0.01-0.05 Hz class of waves discussed by Fairfield [1974] has been extensively
studied. They were observed to be mostly transverse, with δB ∼ several nT, and a left-
hand polarization in the spacecraft frame. These waves were associated with the bow
shock since they are only observed on magnetic field lines connected to the bow shock.
Hoppe and Russell [1983] studied ultra-low frequency (ULF) waves using the ISEE 1 and
2 spacecraft. The waves were observed to be associated with intermediate and diffuse
ions, and Hoppe and Russell [1983] classified them as mixtures of transverse Alfve´nic
and compressional magnetosonic modes with rest frame frequencies ∼0.1 Ωcp (∼0.005-
0.01 Hz in solar wind) and wavelengths ∼6000 km. More recently, Eastwood et al.
[2005a,b] examined a specific class of foreshock ULF waves they called the 30 sec-
ond quasi-monochromatic wave using multi-spacecraft observations from Cluster. They
101
found, through the use of cross-correlation and MVA that the waves had wavelengths
∼600 km, right(left)-hand polarized in plasma(spacecraft) frame, and θkB = 21◦ ± 14◦.
They also observed that when the angle between the interplanetary magnetic field (IMF)
and the X-GSE direction were large, the wave vectors were confined to the plane defined
by the IMF and X-GSE vectors. They could not conclude why the waves were observed
to propagate obliquely to the magnetic field, however the waves were typically observed
simultaneously with intermediate or gyrophase bunched ion distributions. The 30 sec-
ond ULF waves are thought to be driven unstable by the interaction of backstreaming
(with respect to the solar wind flow) ions and incident solar wind ions through the right-
hand ion-ion cyclotron beam instability [Gary , 1993]. From all of these observations,
Eastwood et al. [2005a,b] concluded that the 30 second quasi-monochromatic foreshock
ULF wave is actually a kinetic fast magnetosonic ULF wave. More recent studies with
higher resolution particle instruments have found that the ULF wave boundary of the
foreshock coincides with an inner (i.e. closer to Earth) boundary of field-aligned ion
beams [Meziane et al., 2004a], consistent with theory [Gary , 1993].
Such low frequency waves have been observed upstream of the terrestrial bow shock
by Archer et al. [2005], Balikhin et al. [1997a,b, 2003], Bavassano-Cattaneo et al. [1983],
Behlke et al. [2003, 2004], Eastwood et al. [2005a,b], Hoppe et al. [1981, 1982],
Hoppe and Russell [1980, 1983], Russell et al. [1971], and Schwartz et al. [1992], the
Mercurian bow shock by Orlowski et al. [1990], the Venusian bow shock by
Orlowski and Russell [1991] and Orlowski et al. [1994], the Martian bow shock by
Brain et al. [2002], the Jovian bow shock by Bavassano-Cattaneo et al. [1987],
Bertucci et al. [2005], and Sonnerup et al. [1981], the Saturnian bow shock by
Bertucci et al. [2007] and Orlowski et al. [1992], the Uranian bow shock by Smith et al.
[1989, 1991], and upstream of comets by Brinca and Tsurutani [1989], Galeev et al.
[1989], Kennel et al. [1986], Klimov et al. [1986], Le et al. [1989], Tsurutani et al. [1987,
1989a,b]. Their ubiquitous nature, potential to alter shock parameters, and contribu-
tion to energy dissipation make these waves of extreme interest in collisionless shock
studies. We will not focus on these low frequencies in this thesis, but we acknowledge
their importance in shock wave dynamics.
102
3.2 Shocklets, SLAMS, and Whistler Foreshock Waves
In the previous section we highlighted the studies done on some of the lowest fre-
quency waves of interest connected to energy dissipation and wave-particle interactions
in the foreshock region of collisionless shocks. This section will examine some of the
higher frequency (still <10 Hz) foreshock waves often observed in the foreshocks of bow
shocks and IP shocks. Initial observations found these waves to be composed of multi-
ple wave modes and properties making them more difficult to analyze than their lower
frequency counterparts [Fairfield , 1974]. Here we will highlight some previous work
and examine the importance of each wave mode in the 0.5-4.0 Hz frequency range of
foreshock waves.
Fairfield [1974] initially found that the 0.5-4.0 Hz waves had a left-hand polarization
below 2.5 Hz and right-hand above 2.5 Hz. This led them to conclude that the waves
were all right-hand polarized waves in the plasma frame, but the waves with frequencies
below 2.5 Hz were being Doppler shifted causing a reversal of polarization. The Doppler
effect is observed as a reversal of polarization in the spacecraft frame because the waves
with phase velocities slower than the solar wind are convected back over the spacecraft.
The wave frequencies in the plasma frame, ωo, are observed as ω’ (= ωo + k · Vsw) in
the spacecraft frame. A reversal of polarization occurs when ω’ < 0, assuming ωo > 0,
which implies that k · Vsw < 0 and | k · Vsw | > ωo. Since whistler waves are known
to have higher phase velocities at higher frequencies in this frequency range, the cor-
responding right-hand polarization of the higher frequency waves observed by Fairfield
[1974] led him to conclude that the observed modes were in fact Doppler shifted whistler
waves. Fairfield [1974] determined, using cold plasma dispersion, that the waves were
propagating at angles between 20◦ ≤ θkB ≤ 40◦ and their wavelengths were ∼100 km.
Later studies began to find that the 0.5-4.0 Hz waves were actually composed of
two different waves with slightly different characteristics. In the 0.5-4.0 Hz frequency
range, the lower frequencies were found to have a larger amplitude and more compres-
sive component than the higher frequency end of this frequency range. Both modes were
observed to have a left-hand (LH) polarization in the spacecraft (SC) frame, but a RH
polarization in the plasma rest frame [Hoppe et al., 1981, 1982]. The higher frequency
modes were also shown to be a whistler wave, but were observed farther upstream of the
103
bow shock in association with ion beams [Hoppe et al., 1981, 1982] (see Section 1.7.4).
There has been some confusion regarding the terminology surrounding the lower fre-
9
-1
-6
4
-1
9
0
10
20
|B| (nT)
Bx
By
Bz
S
p
a
c
e
c
ra
ft
 C
o
o
rd
in
a
te
s
Time (HH:MM:SS UT)
08:57:00 08:58:00 08:59:00 09:00:00
 Fig. 3.  Discrete wave packets occurring in association with irregular low-frequency waves
on March 10, 1968, while the satellite was at 19.1 R   and at a sun-earth-satellite angle of 
67.5  .
E
o
Russell et. al. [1971]
Discrete Wave Packets
Figure 3.1: The irregular low-frequency fluctations in this image are examples of the
multi-frequency shocklets observed by Russell et al. [1971] using the OGO 5 spacecraft.
This is adapted from Figure 3 in Russell et al. [1971].
quency larger amplitude mode in this frequency range. The waves were first described
by Russell et al. [1971] and called a discrete wave packet. Figure 3.1 is an adapted
image from Russell et al. [1971] showing a few examples of what they referred to as
discrete wave packets observed by OGO 5 upstream of the terrestrial bow shock. The
top line corresponds to the magnitude of the magnetic field and the next three lines
are the spacecraft coordinate components of the magnetic field. The image is roughly
three minutes of data sampled at 56 samples per second. The waves are compressive
and propagating at large θkB’s (e.g. >30◦).
Later, Hoppe et al. [1981]; Stasiewicz et al. [2003]; Thomsen et al. [1990] all referred
to these same magnetic fluctuations as shocklets. Figure 3.2 is an adapted image from
Hoppe et al. [1981] showing four examples of shocklets. All four panels show the three
spacecraft components of the magnetic field followed by the magnetude of the magnetic
104
    Fig. 26.   Individual examples of the low-frequency structures of November 3, 1977, seen plotted on our standard scale
in Figure 21, are presented on expanded time scales.  The plots present the three vector components (in spacecraft coordi-
nates) and magnitude of the magnetic field measured at ISEE 1.
02:17:30 02:18:00 02:18:300
4
8
0
-4
4
0
-4
4
0
4
Bx
By
Bz
|B| (nT)
S
p
a
c
e
c
ra
ft
 C
o
o
rd
in
a
te
s
Time (HH:MM:SS UT)
November 3, 1977
02:09:00 02:09:30 02:10:00
4
8
0
-4
4
0
-4
4
0
4
Bx
By
Bz
|B| (nT)
S
p
a
c
e
c
ra
ft
 C
o
o
rd
in
a
te
s
Time (HH:MM:SS UT)
November 3, 1977
|B| (nT)
Bx
By
Bz
S
p
a
c
e
c
ra
ft
 C
o
o
rd
in
a
te
s
02:03:30 02:04:00 02:04:30
0
-4
0
4
8
4
0
-4
4
0
4
Time (HH:MM:SS UT)
November 3, 1977
Time (HH:MM:SS UT)
02:20:35 02:21:00 02:21:30
4
8
0
-4
4
0
-4
4
0
4
Bx
By
Bz
|B| (nT)
S
p
a
c
e
c
ra
ft
 C
o
o
rd
in
a
te
s
November 3, 1977
A
B
C
D
Figure 3.2: Examples of shocklets with (D) and without (A-C) leading high frequency
wave packets. This is adapted from Figure 26 in Hoppe et al. [1981].
field (nT). The different panels show examples of shocklets with (D) and without (A-
C) leading whistler wave trains. As one can see, the wave modes can exhibit multiple
characteristics which led to confusion in the literature (e.g. Lucek and Balogh [1997]
referred to similar fluctations as discrete wave packets while Schwartz et al. [1992] called
them ULF Waves).
Further examination of foreshock waves led Schwartz and Burgess [1991] and
Schwartz et al. [1992] to identify another type of compressive wave they defined as short
large-amplitude magnetic structures (SLAMS). They readily admit that the only iden-
tifiable difference (in their data set) between shocklets (what they call foreshock ULF
waves) and SLAMS is the amplitude of SLAMS tends to be much greater than that
105
of shocklets. Regardless, a cursory view of the various examples used in the literature
suggests strong similarities between these phenomena, though they have been given
multiple names. We will discuss, in detail, all of these foreshock modes in detail in the
following sections.
3.2.1 Shocklets
Hoppe et al. [1981], using the ISEE 1 and 2 satellites, identified the structures they
called shocklets as ULF magnetosonic waves associated with a leading magnetosonic
whistler wave train, though they have been observed without a whistler wave train
[Hoppe et al., 1981; Le et al., 1989]. The whistler wave train is thought to result from
wave spreading due to the difference in phase velocities between lower and higher fre-
quency whistler waves. Thus, the higher frequency waves can out-run the lower fre-
quency waves leading to spatially dependent frequency spectrum within the shocklets
[Omidi and Winske, 1990]. Observations found that the high frequency whistler wave
train on the leading edge of the shocklets have rest frame frequencies of 0.1 < ω/Ωcp <
40 (∼0.001-0.4 Hz in solar wind), wavelengths of 30 km ≤ λ ≤ 2100 km and propagation
angles of θkB ∼ 20◦-30◦ [Russell et al., 1971; Hoppe et al., 1981].
Shocklets appear to be a commonly observed magnetic fluctuation upstream of quasi-
parallel bow shocks and very rarely observed upstream of IP shocks. They have been
observed upstream of the quasi-parallel terrestrial bow shock [Hoppe et al., 1981], the
Saturnian and Jovian bow shock [Bertucci et al., 2005], cometary foreshocks [Le et al.,
1989; Tsurutani et al., 1987, 1990a,b], one observation at a quasi-parallel IP shock
[Lucek and Balogh , 1997], and 12 shocklets were observed upstream of one supercrit-
ical quasi-perpendicular IP shock [Wilson III et al., 2009] (see Chapter 4). Planetary
foreshock studies [Hoppe et al., 1981, 1982; Hoppe and Russell , 1983; Thomsen et al.,
1985b, 1990] and one IP shock study [Wilson III et al., 2009] have shown shocklets to
occur in association with diffuse ion distributions, suggesting such ion distributions may
be a necessary component for shocklet formation.
Figure 3.3 shows four examples of shocklets observed upstream of a quasi-
perpendicular IP shock on 2000-04-06. In each panel, the top plot is the magnitude of
the magnetic field and the bottom plot shows the three GSE components of the mag-
netic field (X-Red, Y-Green, and Z-Blue). All plots are shown in units of nanotesla.
106
Time (HH:MM:SS)
16:31:30 16:31:40 16:31:50 16:32:00
16:31:30 - 16:32:06 UT on 04/06/2000
|B
| 
(n
T
)
6
7
8
9
5
B
 [G
S
E
] (
nT
)
0
-2
2
4
-4
-6
-8
Bx
By
Bz
D
High Frequency
Wave Packet
16:22:00 16:24:00
16:20:44 - 16:25:24 UT on 04/06/2000
|B
| 
(n
T
) 8.5
9.0
8.0
7.5
7.0
6.5
B
 [G
S
E
] (
nT
)
0
-2
2
4
-4
-6
-8
Bx
By
Bz
Steepened edge
(or leading edge)
Long Period
Magnetosonic Wave
Long Period
Magnetosonic Wave
Long Period
Magnetosonic Wave
B
16:26:04 - 16:31:32 UT on 04/06/2000
|B
| 
(n
T
)
6
7
8
9
10
B
 [G
S
E
] (
nT
)
0
-5
5
-10
High Frequency
Wave Packet
Long Period
Magnetosonic Wave
16:28:00 16:30:00
Bx
By
Bz
C
15:59:46 - 16:03:04 UT on 04/06/2000
16:00:00 16:01:00 16:02:00 16:03:00
|B
| 
(n
T
)
6.0
5.5
7.0
6.5
B
 [G
S
E
] (
nT
)
0
-2
2
4
-4
-6
Bx
By
Bz
A
Long Period
Magnetosonic Wave
Low Frequency
Wave Packet
Figure 3.3: Four examples of shocklets upstream of an IP shock on 04/06/2000. The
IP shock was observed by Wind at 16:32:09.237 UT. The shocklets are labeled with the
various substructures observed simultaneously. The first example in panel A is the only
example here without the leading high frequency magnetosonic whistler.
These plots highlight the various forms shocklets may exhibit. For instance, shocklet
A is an example without a leading high frequency (&1 Hz) whistler. It is observed
further upstream from the three other events, but under similar plasma densities and
magnetic field strengths. The increase in magnetic field strength in all four shocklets
is observed concurrently with an increase in plasma density (see Figure 4.1), consistent
with magnetosonic waves. Shocklets C and D are examined in greater detail in Chapter
4.
Shocklets have been examined in association with high frequency electric fields
107
finding the downstream region (earthward side) of the shocklet to contain strong ion-
acoustic-like emissions [Stasiewicz et al., 2003]. In the foot (sunward side) of the shock-
lets, high frequency plasma emissions were observed. Both wave types are capable of
heating the plasma, thus possible dissipation mechanisms for the entire shocklet struc-
ture. In Chapter 4, I will discuss the detailed analysis of 12 shocklets observed upstream
of a quasi-perpendicular shock and their effects on particle distributions.
3.2.2 SLAMS
In 1991, Schwartz and Burgess [1991] proposed a new framework for explaining the
complex and diverse phenomena observed upstream of the quasi-parallel bow shock, in
which they coined the acronym SLAMS. They proposed that ULF waves grow out of
background fluctuations and steepen into the structures refered to as SLAMS. SLAMS
have a short duration, ∼10 seconds, are generally embedded in the other foreshock
waves though they can be isolated, have wavelengths ∼0.5 RE (or half that of the up-
stream waves observed with them) along the bulk flow direction, generally share the
polarization characteristics of the ULF waves surrounding them (RH-polarized in SC
frame but LH in rest frame), have phase speeds which increase(decrease) with ampli-
tude in the rest(SC) frame, and amplitudes which are generally larger than what is
expected from adiabatic compression (i.e. ratio of magnetic field strengths exceeds
four) [Schwartz and Burgess , 1991; Schwartz et al., 1992].
Figure 3.4 shows two examples of SLAMS observed upstream of the terrestrial bow
shock observed by the Wind spacecraft. The top panel plots the magnitude of the
magnetic field and the bottom panel plots the three GSE components. Red boxes are
used to outline the two SLAMS in both panels. Notice that the difference between the
maximum value of the magnetic field in the SLAMS and the average value upstream
of the SLAMS exceeds 15 nT in both cases. To contrast, the same difference for the
shocklets observed in Figure 3.3 rarely exceeds 5 nT.
Since the introduction of the term SLAMS, a great deal of effort has been put forth
to distinguish between characteristics of structures given different names for a phenom-
ena which apear to result from the same source. Lucek et al. [2002], using data from
the four Cluster spacecraft, concluded that SLAMS have shorter scale lengths than
108
|B
| 
(n
T
)
B
 [G
S
E
] (
nT
)
0
-20
20
-40
40
10
20
30
40
50
16:55:30 16:56:00 16:56:30 16:57:00 16:57:30
16:55:05 - 16:57:35 UT on 12/02/1996
A
B
SLAMS
XYZ
Bow Shock Crossing
at: 16:56:20.015 UT
Figure 3.4: The figure above is an example of a bow shock crossing by Wind on
12/02/1996 at 16:56:20.015 UT. Two examples of SLAMS are seen upstream of the
bow shock. Note the magnitude of the change of the magnetic field from the average
upstream values to the maximum value in SLAMS A, B, and the shock itself are ∼15
nT, ∼19 nT, and >40 nT respectively. The ratio of the maximum field magnitude to
the upstream averages are ∼2.3, ∼2.7, and >4 for the same three events.
shocklets and that the two structures had many similarities. Others have simply an-
alyzed the data to determine relevant and critical properties useful for identification
and/or differentiation. Behlke et al. [2003], using the four Cluster spacecraft, examined
the convective electric fields associated with SLAMS at the terrestrial bow shock. The
convective electric showed that the SLAMS were moving with the solar wind plasma
and they also observed a downstream wake of decreased density. Behlke et al. [2004]
observed electric field signatures inside of SLAMS consistent with BGK phase space
holes moving above the ion thermal speed with negative potential structures but were
unable to determine their impact on the structure of the SLAMS.
109
No experimental study, however, has shown any truly distinct characteristics that
distinguish SLAMS from shocklets (or discrete wave packets). Only their scale lengths
have been reported to be different [Lucek et al., 2002], and this is not a truly distin-
guishing characteristic in a dispersive medium. One characteristic, which seems to be
prevelent in the observational studies, is the change in magnitude of the magnetic field
across each structure. As one can see from Figures 3.1 through 3.3, the maximum
amplitude acheived by any particular shocklet rarely exceeds twice that of the average
upstream field magnitude. In almost every example figure presented by Schwartz et al.
[1992], the average upstream field magnitude is ∼5-10 nT and the peak amplitude in the
SLAMS vary from as low as 25 nT to as high as >60 nT. The ratio of amplitudes, peak
in SLAMS to average upstream, is often in excess of a factor of five, well beyond what
is expected for adiabatic compression. The two examples in Figure 3.4 have magnetic
compression ratios greater than two.
3.2.3 1 Hz Whistlers
The higher frequency (&1 Hz) waves of Fairfield’s high frequency category (0.5-4.0
Hz) have been identified as whistler waves [Fairfield , 1974; Hoppe et al., 1982]. A nearly
monochromatic whistler wave was discovered by Hoppe et al. [1981] far upstream (>1
RE) of the quasi-parallel bow shock in association with reflected ion beam distributions.
Hoppe et al. [1982] showed that the rest frame frequencies were 20 < ω/Ωcp < 100
(∼0.2-10 Hz in the solar wind), wavelengths were ∼100 km, and propagation angles
with respect to the magnetic field θkB ∼ 20◦-45◦. Oblique whistler waves with f ∼1 Hz
observed by Farris et al. [1993] upstream of low β quasi-perpendicular subcritical and
marginally supercritical bow shocks were consistent with the ∼1 Hz whistlers reported
by Russell et al. [1971] and Hoppe et al. [1981, 1982]. Whistler waves observed upstream
of the Mercurian, Venusian, and Saturnian bow shocks were found to have characteristics
similar to the ”one-Hz” whistler waves observed by Hoppe et al. [1981, 1982].
110
3.2.4 ULF Foreshock Waves at IP shocks
There are fewer observations of shocklets, SLAMS, or foreshock whistlers observed
upstream of IP shocks. This may be due to the fact that IP shocks are unlikely to pro-
duce conditions conducive to the production of these waves due to their massive scales,
larger radius of curvature at 1 AU, tendency to have a quasi-perpendicular geometry,
and typically lower Mach numbers. Not only are quasi-perpendicular shocks less likely
to produce the diffuse ion distributions thought to be necessary for shocklet genera-
tion and growth, they are not intrinsically unstable to reformation when sub-critical
[Farris et al., 1993]. Both Hada et al. [1987] and Omidi and Winske [1990] suggested
shocklet generation mechanisms which require conditions more likely to occur in the
foreshocks of planetary or cometary bow shocks connected to the quasi-parallel section.
This is not to say that ULF foreshock waves are not observed upstream of IP shocks,
rather that the phenomena are far more rarely observed. Using multi-satellite measure-
ments upstream from quasi-perpendicular and quasi-parallel IP shocks, Russell et al.
[1983] observed two distinct ULF foreshock wave types, a whistler precursor near the
ramp and a 30 second wave they called irregular turbulence farther upstream which had
a nearly featureless frequency spectrum. Tsurutani et al. [1983], in a study of ∼100
quasi-parallel (defined by the authors as θBn < 65◦) IP shocks, found low frequency
waves (∼0.05 Hz) to propagate within 15◦ of the ambient magnetic field.
3.2.5 Simulations and Summary of ULF Foreshock Waves
Simulations have attempted to clarify the distinguishing characteristics between fore-
shock ULF waves, shocklets, and SLAMS. Omidi and Winske [1990] used an electro-
magnetic hybrid code to study shocklets and they showed that the leading wave train
of the shocklets were a consequence of wave spreading due to dispersive effects. They
initialized an elliptically polarized ULF wave in a nonperiodic box and allowed it to in-
teract with a proton beam with a spatially dependent density. The interaction between
their artificial solar wind and the gradient in beam density caused the ULF waves to
steepen and form kinetic magnetosonic waves. As the magnetosonic waves steepened,
magnetosonic whistler waves grew just downstream of the steepened portions and be-
gan to propagate upstream away from the steepened edge of the magnetosonic waves.
111
Omidi and Winske [1990] found that the beam provides the free energy for wave growth
and steepening and that the propagation of the wave results in the wave train due to
wave spreading. Thus, in their simulation the beam is a necessary source of free energy
for the formation of the steepened magnetosonic waves with leading whistler wave train.
Akimoto et al. [1991] performed a 1-D electromagnetic hybrid simulation to investi-
gate the steepening of parallel propagating hydromagnetic waves (ULF foreshock waves)
into magnetic pulsations driven by ion beams. The waves were found to strongly in-
teract with an ion beam, trapping it, and then gyrophase-bunching it which locally
generates more waves. The waves could also resonantly interact with the shorter wave-
length waves, adding to the total wave energy present [Akimoto et al., 1991].
Scholer [1993], also using an electromagnetic hybrid code, examined ULF waves,
excited by ion beams, steepening upstream of a quasi-parallel shock as they convect
into regions of increasing diffuse ion density. The diffuse ions were shown to have a
significant impact on the index of refraction for the whistler and magnetosonic waves.
As the ULF waves steepen, they dispersively radiate a whistler wave. Scholer [1993]
also found that the leading whistler train was phase standing with respect to the shock
front, which is different from the results of Omidi and Winske [1990]. Scholer [1993]
found that the existence of diffuse ions is a necessary component for the formation of
the steepened magnetosonic waves with leading whistler wave train.
Two more recent 1-D PIC simulation studies focused on the evolution of SLAMS
excited by diffuse ion distributions [Scholer et al., 2003; Tsubouchi and Lembe`ge, 2004].
Both studies found SLAMS to result from steepened ULF foreshock waves and the lead-
ing whistler train to result from the radiation of the dispersive waves by the steepened
edge of the SLAMS. Some of the first studies referred to the phenomena as shock-
lets [Omidi and Winske, 1990; Scholer , 1993] while later studies called them SLAMS
[Scholer et al., 2003; Tsubouchi and Lembe`ge, 2004]. Regardless of the terminology they
used, Scholer [1993]; Scholer et al. [2003]; Tsubouchi and Lembe`ge [2004] all concluded
that ULF foreshock waves were the source of the shocklets/SLAMS and that they were
all the same phenomena, just at different stages in their evolution, consistent with the
predictions of Schwartz and Burgess [1991].
To summarize, the evolution of the low frequency foreshock waves ocurrs in the
following manner:
112
1. Initially low frequency ULF waves, both magnetosonic and Alfve´nic in nature, are
excited by field-aligned proton beams in cyclotron resonance with the waves far
upstream of the bow shock (i.e. greater than an ion gyroradius) near the ion fore-
shock boundary [Hoppe et al., 1982; Meziane et al., 1999, 2004b; Mazelle et al.,
2003]
2. The foreshock ULF waves have a phase velocity less than that of the solar wind,
which causes the waves to be convected back towards the bow shock where they in-
teract with changing ion distributions [Hoppe et al., 1981, 1982;Hoppe and Russell ,
1983; Meziane and D’Uston, 1998]. As the ULF waves convect and interact with
the ion beams, they pitch-angle scatter the beam producing intermediate and/or
gyrating ion distributions [Mazelle et al., 2003].
3. These anisotropic and nongyrotropic ion distributions alter the index of refraction
of the waves leading to wave growth and/or steepening [Akimoto et al., 1991;
Schwartz and Burgess , 1991; Scholer , 1993].
4. The steepened waves convect still further into the foreshock and the ion distribu-
tions become broader to the point of becoming diffuse in phase space.
5. When the steepened waves convect into a region of diffuse ions (see Section 1.4.3),
the index of refraction becomes even more strongly dependent on these suprather-
mal diffuse ions [Omidi and Winske, 1990; Scholer , 1993] causing the waves to
steepen more still to the point of breaking. The breaking waves are the result of
insufficient energy dissipation needed to limit the steepening.
6. As the waves break, they disperively radiate whistler waves upstream of the steep-
ened edge because the whistlers in this frequency range have larger phase veloc-
ities than the lower frequency magnetosonic waves. The frequency dependence
of the phase velocity leads to the leading whistler wave train, with the highest
frequencies observed at the greatest spatial distance from the steepened edge of
the magnetosonic wave. This process leads to the shocklets of the type seen in
Figure 3.3.
7. As the shocklets continue to convect towards the bow shock, the wave ampli-
tude increases still more due to pressure gradients [Giacalone et al., 1993] and the
113
structures eventually grow to become SLAMS, extracting energy from suprather-
mal ions [de Wit et al., 1999], and diffuse ions [Scholer , 1993; Scholer et al., 2003;
Tsubouchi and Lembe`ge, 2004]. These structures are then thought to play an im-
portant role in quasi-parallel shock reformation and energy dissipation
[Scholer et al., 2003; Tsubouchi and Lembe`ge, 2004].
3.3 Introduction to Cross-Shock Potential
The phenomena responsible for changing the electron distribution functions (eDFs)
across collisionless shock waves has been a topic of considerable debate since the predic-
tion of the existence of collisionless shock waves. Some theories invoke microscopic wave-
particle interactions to explain the observed heating [Dum et al., 1974; Forslund et al.,
1970; Fredricks et al., 1965; Gary , 1970b; Lampe et al., 1972b; Scarf et al., 1965; Wong ,
1970] while others claim that only macroscopic effects are necessary to explain the heat-
ing [Hull et al., 1998, 2000; Hull and Scudder , 2000; Hull et al., 2001; Kuncic et al.,
2002; Scudder et al., 1986a]. Early observations of weak IP shocks found that electrons
were preferentially heated perpendicular to the magnetic field, consistent with adiabatic
compression and conservation of the first electron adiabatic invariant [Feldman et al.,
1983b]. Observations of strong shocks, however, showed a preference for parallel heat-
ing with downstream distributions resembling an inflated Maxwellian with a flattened
characteristic at low energies in the parallel cut of the eDFs, called a flattop distribution
[Feldman et al., 1983b]. Similar eDFs features were observed near the terrestrial bow
shock [Feldman et al., 1983a], which is often a much higher Mach number shock than
IP shocks.
To produce a flattop eDF, some suggested that a macroscopic (& scale size of
shock thickness) quasi-static electric field could produce the observed distributions
[Feldman et al., 1982, 1983a,b; Scudder et al., 1986a; Thomsen et al., 1987].
Scudder et al. [1986a] showed, using ISEE observations, that this electric field was frame
dependent and that the most important reference frame for the electrons was the de
Hoffmann-Teller frame of reference. The de Hoffmann-Teller frame is the frame of refer-
ence where the incident flow is parallel to the magnetic field, thus no upstream convective
electric field exists in this frame. The electric potential due to the quasi-static electric
114
field in the shock ramp is called the de Hoffmann-Teller potential or ∆ΦdHT . Numerous
studies have shown evidence to suggest that ∆ΦdHT plays an important role in shap-
ing the eDFs downstream of collisionless shock waves [Feldman et al., 1982, 1983a,b;
Scudder et al., 1986b,c,a; Thomsen et al., 1987].
To zeroth order, ∆ΦdHT can accelerate the incident electrons through the shock re-
sulting in a peak (e.g. drifting Maxwellian) in the eDFs offset parallel to the magnetic
field. In the absence of any wave-particle interactions, this peak in the eDF would re-
main constant through the shock and into the downstream. However, the combination
of conservation of magnetic moment, µ, shocked electrons attempting to move back
upstream, and wave-particle scattering/heating are thought to give rise to the observed
downstream eDFs [Scudder et al., 1986a; Thomsen et al., 1987]. Some have shown ev-
idence to suggest that wave-particle interactions are not necessary to fill the voids in
phase space resulting from acceleration of an entire distribution by a quasi-static electric
potential [Hull et al., 1998, 2000; Hull and Scudder , 2000; Hull et al., 2001] assuming
the maximal trapping approximation developed by Morse [1965]. If one assumes a con-
stant phase space density along a phase space trajectory, then one requires that the
downstream phase space density cannot exceed the maximum upstream phase space
density along that same trajectory. Maximal trapping is an assumption whereby one
assumes that changes in background field parameters are slow enough to allow a suffi-
cient number of electrons to fill the voids in phase space produced by the quasi-static
electric potential in the shock ramp resulting in a phase space density that approaches
the maximum upstream phase space density [Morse , 1965]. This assumption is thought
to be satisfied because shocked electrons in the downstream region have thermal speeds
much greater than the shock speed and solar wind speed, thus a fraction of the thermal
electrons can attempt to move back upstream. The electric field produced by ∆ΦdHT
acts to accelerate upstream electrons into the downstream, so the shocked electrons
attempting to return upstream are decelerated by this same potential and in some cases
trapped in the downstream. Also, time-dependent magnetic fields and electric poten-
tials in the shock ramp can act to increase the number of trapped electrons [Morse ,
1965]. In summary, the process by which ∆ΦdHT can lead to heating and flattop eDFs
is:
115
1. Incident electrons are accelerated parallel to the magnetic field by ∆ΦdHT pro-
ducing a peak in the eDF offset along the magnetic field
2. This peak in phase space density decreases in amplitude as the spacecraft tran-
sits the shock [Feldman et al., 1983b], suggesting the accelerated particles are
relaxing toward an equilibrium state which is thought to occur through either
wave-particle interactions [Thomsen et al., 1983a] and/or maximal trapping ef-
fects [Morse , 1965]
3. Shocked electrons in the downstream region attempt to return upstream and are
subject to wave-particle effects Thomsen et al. [1983a] and/or trapping in the ES
fields due to ∆ΦdHT [Scudder et al., 1986a], which has been observed near the
terrestrial bow shock [Larson et al., 1996]
4. These combined effects can result in the parallel flattop eDF.
The perpendicular heating observed in collisionless shocks often can be explained by a
conservation of the first adiabatic invariant (or magnetic moment) for thermal electrons,
thus Te,⊥ ∝ Bo. However, conservation of magnetic moment can lead to mirroring which
should result in phase space voids inaccessible in the downstream eDFs. To explain this,
Hull et al. [2000] and Scudder et al. [1986a] suggest that wave-particle interactions only
act to fill the voids in phase space and smooth the downstream eDFs. Other studies have
gone further and shown evidence to suggest that macroscopic fields due to ∆ΦdHT and
gradients in the magnetic field magnitude can explain the downstream eDFs without
any wave-particle interactions [Hull et al., 1998; Hull and Scudder , 2000; Hull et al.,
2001]. In the following sections we will explain why many of the assumptions made
when calculating ∆ΦdHT are not valid for most collisionless shock waves. However,
we cannot definitively show that wave-particle interactions are absolutely necessary to
explain the downstream eDFs. In other words, the invalidation of the assumptions made
by Hull et al. [1998, 2000]; Hull and Scudder [2000]; Hull et al. [2001]; Scudder et al.
[1986a] may not invalidate their theory. A corollary is observing whistler waves in finite
temperature plasmas yet their characteristics conform to the cold plasma dispersion
relation [Fairfield , 1974].
116
3.3.1 Expected Observations
Flattop parallel eDFs observed downstream of collisionless shocks present an argu-
ment in favor of quasi-static electric field driven electron heating [Scudder et al., 1986a].
However, these same eDFs have been shown to be produce by IAWs in the quasi-linear
limit [Dum et al., 1974]. The question we address in this section is, which mechanism
dominates the downstream eDF structure, the quasi-static macroscopic fields or the
high frequency microscopic fields?
Figure 3.5 is an example of a typical electron heating signature observed at a super-
critical quasi-perpendicular IP shock. The IP shock, observed by the Wind spacecraft
on 09/24/1998 at 23:20:37 UT, had Mf = 2.87 ± 0.08, θBn = 78.6◦ ± 2.4◦, N2/N1 =
2.17 ± 0.38, and β1 = 0.215 ± 0.214 [Kasper , 2007]. Each eDF has a similar format
to that of Figure 1.8 except that the parallel cuts are red, perpendicular are blue and
the one-count levels are included. The top panels are two-dimensional contours of con-
stant phase space density in the plane defined by the solar wind velocity and magnetic
field, showing the projections of the shock normal vector (blue dotted line), nˆsh, the
solar wind velocity (solid black line), Vsw, and the electron heat flux vector for these
specific distributions (solid red line), Qf . The data ranges are ±20,000 km/s for the
parallel (horizontal), V‖, and perpendicular (vertical), V⊥, axes. The bottom panels
contain the parallel (red line) and perpendicular (blue line) cuts of the eDFs along with
the one-count (green line) levels for uncertainty estimates. In between the eDFs is a
vertical line used as an illustration to distinguish between the upstream eDF on the left
and downstream eDF on the right. The upstream magnetic field for this IP shock was
mostly in the positive Y-GSE and negative X-GSE directions. Thus, the deviation in
the parallel cut of the eDF on the left from isotropy is likely due to a combination of
the solar wind strahl and mirrored electrons.
As illustrated in the parallel cut (red line) of the eDF on the right in Figure 3.5,
the downstream electron distribution can have a flattened characteristic at low energies,
known as a flattop distribution. The specific example shown also has a beam-like feature
on top of the flattop distribution, a feature often seen downstream of the terrestrial bow
shock [Thomsen et al., 1983a]. The large strahl component observed in the parallel cut
in the left eDF (23:20:31 UT) skews the estimate of the electron temperature anisotropy
(T⊥/T‖) for the core, halo, and entire distribution. In the eDF on the right (23:22:11
117
Wind 3D Plasma Eesa Low
1998-09-24/23:20:31 - 23:20:34
V
 p
e
rp
e
n
d
ic
u
la
r 
(1
0
0
0
 k
m
/s
)
-20
-10
0
10
20
V parallel (1000 km/s)
-20 -10 0 10 20
Vsw
Qf
nsh
Velocity (1000 km/s)
-20 -10 0 10 20
P
e
rp
e
n
d
ic
u
la
r 
C
u
t
P
ar
al
le
l C
u
t
O
n
e
 C
o
u
n
t 
Le
ve
l
3
3
3
d
f 
(s
e
c
  
/k
m
  
/c
m
  
)
10
-14
10
-15
10
-16
10
-12
10
-13
10
-11
P
e
rp
e
n
d
ic
u
la
r 
C
u
t
P
ar
al
le
l C
u
t
O
n
e
 C
o
u
n
t 
Le
ve
l
Wind 3D Plasma Eesa Low
1998-09-24/23:22:11 - 23:22:14
Vsw
Qf
nsh
Velocity (1000 km/s)
-20 -10 0 10 20
V parallel (1000 km/s)
-20 -10 0 10 20
Shock Crossing
~23:20:37 UT
Upstream Downstream
Figure 3.5: The plot shows two EL eDFs upstream and downstream of a supercrit-
ical quasi-perpendicular IP shock observed by the Wind spacecraft on 09/24/1998 at
23:20:37 UT. Each eDF has a similar format to that of Figure 1.8 except that the parallel
cuts are red, perpendicular are blue and the one-count levels are included. In between
the eDFs is a vertical line used as an illustration to distinguish between the upstream
eDF on the left and downstream eDF on the right. The upstream magnetic field for
this IP shock was mostly in the positive Y-GSE and negative X-GSE directions.
118
UT), one can see that at very low energies (. 45 eV or ∼4000 km/s), the electrons
are more anisotropic parallel to the magnetic field but at slightly higher energies (∼70
eV or ∼5000 km/s) the eDF starts to shift towards T⊥ > T‖. The halo electrons (&
100 eV or ∼6000 km/s) become more and more perpendicularly anisotropic with in-
creasing energy. The anisotropy values for the core (subscript c) and halo (subscript
h) for both distributions are T⊥,c/T‖,c ∼ 0.82, 0.93 and T⊥,h/T‖,h ∼ 0.81, 1.77. Note
that for the eDF on the right (23:22:11 UT), the halo is almost twice as anisotropic
as the core. Though the low energy parallel flattop is consistent with previous bow
shock observations, many supercritical IP shocks show very anisotropic halo electrons
and nearly isotropic core electrons near the downstream edge of the shock and the halo
electrons relax to more isotropic distributions further downstream. For instance, the
eDF at 1998-09-25/00:08:53 UT (not shown) had T⊥,c/T‖,c ∼ 0.94 and T⊥,h/T‖,h ∼
0.97. We will show an example of an IP shock where the halo and core electrons do not
isotropize within an hour of crossing the shock ramp in Chapters 4 and 5, suggesting
local processes pitch-angle scattering the electrons in the downstream.
3.4 Electric and Magnetic Field Measurements
3.4.1 Electric Fields in Shock Ramps
This section will highlight some of the direct measurements of electric fields which
can exist in the shock transition region [Bale and Mozer , 2007]. The illustrations will
show that the electric fields within a shock layer consist of a spectrum of frequencies
which accumulate to make up the cross-shock potential. Thus, we will argue that one
cannot use only the quasi-static fields in the shock ramp to describe the fundamental
physics of energy dissipation. For future reference, we will define the electrons as being
demagntized if ~E + ~ve × ~B 6= 0.
Wygant et al. [1987] measured low frequency (.32 Hz) electric fields within the
shock ramp using the ISEE-1 instrument. They presented the first measurements of
high time resolution (32 samples/s) 1-D electric field data at a collisionless shock sup-
plimented with low time resolution (3 second averages) 2-D electric field data. The
low frequency measurements were the first measurements of the cross-shock quasi-static
119
electric field, measured in both the normal-incidence and de-Hoffmann-Teller frame of
reference. The quasi-static electric fields were on the order of 4-10 mV/m resulting in
potentials >300 Volts while the higher frequency data showed intense spikes with ampli-
tudes ranging from 10-100 mV/m. The lower limit on the potential of the largest spike
was estimated to be ∼8 Volts (assuming 80 m wavelength). When they assumed the
waveform was driven by the lower hybrid instability, for which the wavelength should
be roughly an electron gyroradius, ∼rge, or roughly 1 km, the resulting potential drop
was roughly 50 Volts. Though this value is smaller than the quasi-static potentials, it
is not insignificant and should be considered when estimating the electric fields in the
ramp of collisionless shocks.
Walker et al. [2004] measured the quasi-static electric field perpendicular to the
magnetic field, E⊥, finding the amplitudes to exceed 70 mV/m and the scale sizes to be
as small as an electron inertial length, c/ωpe. They also found that as the shock nor-
mal angle increased to 90◦, the scale length of these electric field structures decreased.
The two dimensional measurements were made at 25 samples per second with the four
Cluster spacecraft, slightly lower than the highest frequency 1-D measurements made
by Wygant et al. [1987]. The sharp gradients observed by Walker et al. [2004] could
potentially demagntize the electrons, changing the shock dynamics.
Hull et al. [2006] measured 3-D electric fields in the bow shock using the Polar space-
craft, finding ion-acoustic-like waves in the ramp with amplitudes exceeding 80 mV/m.
The data was sampled at up to 8000 Hz (occasionally at 1600 Hz), but the frequencies of
the ion-acoustic-like waves extended beyond 100 Hz, above the sampling frequencies of
Wygant et al. [1987] and Walker et al. [2004]. It may be that the measurements made
by Wygant et al. [1987] and Walker et al. [2004] were attenuated by aliasing implying
their electric field measurements underestimated the electric fields in the shock ramp.
Bale and Mozer [2007] measured parallel and perpendicular electric fields, E‖ and
E⊥, at 1600 Hz using the 3-D electric field instrument on the Polar spacecraft. They
reported on the first direct measurements of E‖ at a collisionless shock. At nearly per-
pendicular shocks, E⊥ will act to slow the incident solar wind while the existence of E‖ in
a shock ramp is significant since it can directly demagnetize electrons. Bale and Mozer
[2007] observe E‖ ≤ 100 mV/m and E⊥ ≤ 600 mV/m corresponding to localized po-
tential wells of 10’s of volts and 100-1000’s of volts, respectively. The scale lengths of
120
Ex Ey
A A
B B
C C
D D
E E
~17 ms
TDSF Waveform Captures
~ 100 mV/m
~ 60 mV/m ~ 200 mV/m
~ 400 mV/m
~ 60 mV/m ~ 300 mV/m
~ 30 mV/m ~ 100 mV/m
~ 40 mV/m ~ 300 mV/m
~ 60 mV/m ~ 100 mV/m
~17 ms
12-02-1996 from 16:56:22.628 - 16:57:05.662
F F X:  ~1.18 < f /f   < ~5.88ce
Y:  ~1.18 < f /f   < ~23.54ce
X:  ~1.56 < f /f   < ~4.15ce
Y:  ~1.56 < f /f   < ~7.26ce
X:  ~1.22 < f /f   < ~8.56ce
Y:  ~1.22 < f /f   < ~12.23ce
X:  ~1.66 < f /f   < ~16.60ce
Y:  ~1.66 < f /f   < ~16.60ce
X:  ~3.68 < f /f   < ~18.42ce
Y:  ~3.68 < f /f   < ~24.56ce
X:  ~3.59 < f /f   < ~11.99ce
Y:  ~4.19 < f /f   < ~23.98ce
f   ~ 903.56 Hzce
f   ~ 849.73 Hzce
f   ~ 833.86 Hzce
f   ~ 814.31 Hzce
f   ~ 817.52 Hzce
f   ~ 964.25 Hzce
Figure 3.6: The plot shows six different TDS samples from the Wind/WAVES instru-
ment observed during a bow shock crossing on 12/02/1996 at roughly 16:56:20 UT (see
Figure 3.4). The left hand column (red waveforms) correspond to the X-antenna and
the right hand column (blue waveforms) correspond to the Y-antenna. Each waveform
is labeled with a respective letter and amplitude illustrated by the vertical black line
on the left hand of each panel. Notice that at least one component of each waveform
exceeds 100 mV/m peak-to-peak. On the right hand side are lists of the cyclotron fre-
quency, fce, and the range of frequencies for the peak power of the wave normalized by
fce. All the waveforms are oblique linearly polarized waves (hodograms not shown). The
sample rate was ∼120 kHz.
121
the observed E‖ were as small as fractions of c/ωpe while E⊥ scale lengths were on the
same order. Sharp gradients in the electric field could demagnetize electrons even in
the absence of large E‖.
An illustrative example of higher frequency waveforms can be seen in Figure 3.6. The
waves have frequency peaks between 1-10 kHz, are oblique linearly polarized waves con-
sistent with Doppler shifted ion-acoustic waves [Gurnett et al., 1979b,a;Wilson III et al.,
2007]. Wilson III et al. [2007] calculated quasi-linear estimates of the anomalous re-
sistivities produced by large amplitude ion-acoustic waves in the ramps of IP shocks
finding they ranged from ∼1–856 Ω ·m (∼107 times greater than classical Spitzer par-
ticle collision estimates). However, recent Vlasov simulations have compared quasi-
linear estimates of anomalous resistivities produced by ion-acoustic waves with the
momentum exchange calculated from their simulation results. They found the simu-
lation results can be 2-3 orders of magnitude greater than the quasi-linear estimates
[Dyrud and Oppenheim, 2006; Petkaki et al., 2003, 2006; Petkaki and Freeman , 2008;
Watt et al., 2002], suggesting microscopic wave-particle interactions may be more im-
portant than previously thought.
In this section we have presented multiple examples from previous studies and our
own observations of electric fields within the ramp region of collisionless shocks. Exactly
how the cross-shock potential arises is still poorly understood and how it scales with up-
stream parameters is even less so. Regardless, the examples presented herein show that
the electric fields can produce both large potential drops and localized potential wells
which can affect the local particle distributions. Thus, observed changes in the eDFs
downstream of collisionless shocks may not be entirely due to macroscopic quasi-static
fields.
3.4.2 Magnetic Fields in Shock Ramps
This section highlights some of the measurements of magnetic fields within shock
ramps. These illustrative observations will show that the assumptions of laminar,
steady-state changes used in calculation of ∆ΦdHT may not be valid for many IP shocks.
An illustrative example will be used to explain why the use of DC measurements in
shock analysis can be misleading. The quantitative implications for theories relying
upon monotonic shock transitions, however, will not be addressed here.
122
A cursory examination of most IP or bow shock crossings with magnetic field data
XYZ
XYZ
|B
| 
(n
T
)
B
 [
G
S
E
] 
(n
T
)
0
-10
-15
-5
14
10
18
12
16
14
10
18
12
16
-10
-12
-2
-4
-6
-8
-14
|B
| 
(n
T
)
B
 [
G
S
E
] 
(n
T
)
11:17:50 11:18:00 11:18:10 11:18:20 11:18:30
Comparison of  0.33 Hz and ~22 Hz
Magnetic Field Data for 10/24/1997
~22 Hz
~0.33 Hz
Figure 3.7: The plot shows the magnitude and the GSE coordinates of the magnetic field
for the IP shock on 10/24/1997 in 3 second (top two panels) and 22 samples/second
(bottom two panels) data. The upstream flow speed in the shock frame is roughly
117 km/s which means that roughly each sample for the two different sample rates
corresponds to length scales of roughly 351 km and 5.32 km, respectively. For reference,
the average upstream electron inertial length is roughly 10 km.
sampling at rates higher than 1 Hz will show that fluctuations on the same order as
the amplitude of the shock ramp can occur, even at weak IP shocks (see example in
Wilson III et al. [2009]). Figure 3.7 is a perfect example of how a shock crossing may
look very different when examining at different sample rates. It is also clear from the
figure that the amplitude of the high frequency (∼1-3 Hz) magnetic fluctuations can
be similar to that of the shock itself, which immediately throws into question any of
123
the small amplitude fluctuation arguments (i.e. monotonic magnetic field profile, time-
stationary, and planar shock arguments). The 10/24/1997 event had a Mach number
of ∼1.7, shock normal angle ∼68◦, compression ratio ∼2.5, average upstream electron
inertial length ∼10 km, and normal thickness of roughly 35 km (∼3.5 <c/ωpe,1> or
average upstream electron inertial lengths). Also, the 10/24/1997 event had an up-
stream normal flow speed of roughly 117 km/s. This implies that when sampling at 22
samples/second, each sample corresponds to roughly 5.32 km (∼1/2 <c/ωpe,1>) of solar
wind plasma flowing past the spacecraft. For example, in one cycle of a 3 Hz wave, ∼4
<c/ωpe,1> of solar wind plasma has traversed the satellite (a similar number is found
when estimating the whistler wavelength in the Alfve´n Mach number limit [Mellott ,
1984]). Often the assumption is made that all fluctuations and/or relevant scale lengths
in a shock are much greater than the electron inertial length which allows one to drop
electron inertial terms in certain calculations. However, in this case the relevant scale
lengths are on the same order as <c/ωpe,1>.
In addition, not all electrons will interact in the same manner with a given wave,
as seen in Sections 1.5.3 through 1.5.5. Saito and Gary [2007] found that higher en-
ergy electrons were more efficient at interacting with whistler waves, reinforcing the
treatment of the solar wind as a multi-component kinetic plasma, not a single elec-
tron fluid. The magnetic fluctuations shown in Figure 3.7 combined with the electric
fields observed in the multiple previous studies highlighted in Section 3.4.1 raise doubts
about whether one can assume electrons stay magnetized everywhere within a shock.
The magnetic field measurements in Figure 3.7 bring into question whether the shock
is time-stationary. Regardless, the purpose of these illustrations are to exhibit why the
detailed microphysics of collisionless shock waves is important.
3.5 Assumptions For de-Hoffmann-Teller Potential Calcu-
lations
If one transforms into the de-Hoffmann-Teller frame of reference, theory shows that a
DC electric field should exist across the shock, producing the de-Hoffmann-Teller poten-
tial, ∆ΦdHT . In this frame of reference, the convection electric field (E = -V × B) of the
124
solar wind disappears in the upstream region. The electrons are assumed to be magne-
tized everywhere in this frame of reference. Numerous studies have calculated ∆ΦdHT ,
given specific assumptions, many of which may not be physically realistic for many
shocks [Hull et al., 1998, 2000; Hull and Scudder , 2000; Hull et al., 2001; Kuncic et al.,
2002; Scudder et al., 1986a]. However, it is difficult to determine the relative importance
and validity of each assumption without doing thorough simulations and comparisons
with data. The following is a list of assumptions that have been made in order to
estimate ∆ΦdHT :
1. electrons are magnetized everywhere (i.e. changes in magnetic field should be
larger than the electron inertial length, c/ωpe)
2. neglect electron heat flux, assume isotropic pressure, and adiabatic gas law
3. field-aligned flow for electrons
4. shock is time-stationary and planar
5. shock has a monotonic jump in the magnetic field magnitude
6. flat-topped electron distributions downstream of the shock (at the same time as
assuming the weak shock limit)
7. ∆ΦdHT ∝ ∆B
8. the total energy of the electrons in the de-Hoffmann-Teller frame is a constant
9. steady-state Vlasov fluid equations
In Section 3.4.1, previous observations which contradict Assumption 1 (thus also As-
sumption 3) were presented, including the existence of sharp electric field gradients and
E‖. Assumption 2 can be problematic when dealing with IP shocks since there are
rarely events without a finite electron heat flux near 1 AU in the solar wind. Further
more, the isotropic pressure assumption can be significantly inaccurate as observations
of strong anisotropies in the electron distributions have been presented [Pilipp et al.,
1987b; Thomsen et al., 1985a; Wilson III et al., 2009]. A cursory examination of most
IP or bow shock crossings with magnetic field data sampling at rates higher than 1
125
Hz will show that most events show Assumptions 3 through 5 to be inaccurate, where
Assumptions 4 and 5 can break down even for some weak shocks (see examples in
Wilson III et al. [2009]). For most IP shocks, Assumption 6 is not accurate, yet studies
often make this assumption even when invoking the weak shock limit. Demagnetiza-
tion of electrons due to stochastic wave-particle interactions can affect the validity of
Assumptions 8 and 9.
In summary, all of the assumptions used to calculate ∆ΦdHT are often innaccurate
for IP shocks and planetary bow shocks. Whether the innaccuracy of these assump-
tions is great enough to invalidate the calculations is not known. Scudder et al. [1986a]
presented one example where the calculations matched their observations within uncer-
tainties. Thus, we believe that the downstream eDFs result from some combination of
quasi-static macroscopic and high frequency microscopic fields affecting the particles in
the shock ramp.
3.5.1 Calculation of the de Hoffmann-Teller Potential
The de Hoffmann-Teller potential, ∆ΦdHT , was first shown to be the relevant cross-
shock potential when considering eDFs by Scudder et al. [1986a]. They estimated
∆ΦdHT by integrating the measured electron pressure across the shock ramp, given
by:
∆ΦdHT =
∫ x2
x1
dx
∇ ·P
nee
(3.1)
where x is a measure of distance across the shock in the shock normal direction and
∆ represents the total difference between upstream and downstream. The discussion
in Appendices A.3.2 and A.3.3 is a specific example of the moment calculations done
using particle distribution function measurements. The purpose of referencing those
sections is to highlight the fact that moment software often uses the assumption <Te>
= Tr[Pi,j]/ne to calculate the components of the ”temperature” directly from the particle
distribution function measurements. If one makes this assumption, then all calculations
using the electron pressure should be considered nearly the same as calculations done
using the electron temperature. However, some studies calculate ∆ΦdHT using the
electron pressure measurements and then correlate ∆ΦdHT to Te, which we do not feel
as being physically significant. This is not to say that ∆ΦdHT is insignificant nor do
126
we dispute the theory. We simply do not agree with the arguments that suggest the
observed downstream eDFs can be entirely explained by the macroscopic DC fields in the
shock ramp [Hull et al., 1998; Hull and Scudder , 2000; Hull et al., 2001] nor do we agree
with the interpretation of the correlation observed between ∆ΦdHT and Te reported by
Hull et al. [2000]. In summary, we do not believe that changes in the eDFs across
collisionless shock ramps are adequately explained using only macroscopic DC fields.
In this thesis, we will examine in detail the eDFs and higher frequency waveforms and
argue that our observations provide evidence to support the theory that wave-particle
interactions are important in collisionless shock waves.
Chapter 4
Shocklets and Low Frequency
Whistlers at an Interplanetary
Shock
4.1 An Atypical IP Shock
[Work presented in Wilson III et al. [2009, 2010]]
In the examination of low frequency waves using high time resolution magnetometer
data from the Wind spacecraft at ten IP shocks, five quasi-perpendicular events had
foreshock or precursor waves which were examined in detail by Wilson III et al. [2009].
They observed unusual waveforms upstream of one IP shock on 04/06/2000, finding the
waves to be consistent with shocklets. This event was the strongest shock examined
in this study with Mf ∼ 4, θBn ∼ 68◦, and Ni2/Ni1 ∼ 4 [Kasper , 2007]. The shock
also showed strong electron and ion heating that far exceeded that of the other four
more typical IP shocks. Within ±5 seconds of the shock crossing, four large amplitude
(> 15 mV/m peak-to-peak) waveforms were observed, two of which are consistent with
solitary waves. The unusual upstream shocklets and strong electron and ion heating led
to another study [Wilson III et al., 2010] which examined the microphysics of the shock
ramp region.
Figure 4.1 shows the shock on 04/06/2000 with the 12 observed shocklets labeled
127
128
|B
| [
G
SE
,n
T
]
10
15
20
30
25
TN
R
 F
re
q
u
e
n
cy
 (k
H
z)
10
100
10
15
20
30
25
N
   
(c
m
   
)
i
-3
In
te
n
sity (d
B
)
[R
e
lative
 to
 B
ackg
ro
u
n
d
]1.5
2.5
4.2
12.0
7.1
Shocklets
WAVES TNR, Magnetic Field Magnitude, and Ion Density
06 April, 2000 from 15:35:00 - 17:30:00 UT
Shock Arival:16:32:09.237 UT
16:30 17:3016:00 17:00
Figure 4.1: A plot of the electric field intensity as a function of frequency and time, the
magnetic field magnitude (3 second), and the ion density from PL on 04/06/2000. The
vertical blue lines indicate the location of the 12 shocklets observed upstream of this
event. One can see that the magnetic field magnitude and density/thermal line are in
phase, consistent with magnetosonic waves.
with blue lines, showing the relationship between magnetic field magnitude and ion
density. The top panel shows the electric field power from the WAVES TNR detector.
Upstream of the shock one can easily see the plasma line which is proportional to the
square root of the plasma density. Note that the TNR data shown is one minute averages
and on a log scale, thus the relative changes in phase with the magnetic field (panel
directly below) are not always obvious. The bottom panel of Figure 4.1 plots the ion
density from the Pesa Low detector on the Wind 3DP instrument [Lin et al., 1995].
One can clearly see that the magnetic field magnitude and ion density are in phase at
the main shock front and at each shocklet, consistent with magnetosonic waves.
Figure 4.2 shows the same event as Figure 4.1, on a shorter time scale, with the
magnetic field magnitude scaled to emphasize the shocklets seen upstream (indicated
by the blue arrows). Wind crossed the main shock ramp at 16:32:09.237 UT (i.e. the
far right-hand side of the top panel of Figure 4.2 or roughly 3 seconds after shocklet B).
129
15:40:00 15:50:00 16:00:00 16:10:00 16:20:00 16:30:00
B
 [G
S
E
] (
nT
)
|B
| 
(n
T
)
6
10
9
8
7
0
-10
5
-5
15:39:00 - 16:36:00 UT on 04/06/2000
Shocklets
1 2 3 4 5 6 7 8 9 10 11 12
Bx
By
Bz
Time (HH:MM:SS)
16:31:30 16:31:40 16:31:50 16:32:00
16:31:30 - 16:32:06 UT on 04/06/2000
|B
| 
(n
T
)
6
7
8
9
5
B
 [G
S
E
] (
nT
)
0
-2
2
4
-4
-6
-8
Bx
By
Bz
Bx
By
Bz
Time (HH:MM:SS)
16:28:00 16:28:30 16:29:00 16:29:30 16:30:00
16:28:00 - 16:30:00 UT on 04/06/2000
|B
| 
(n
T
)
6
7
8
9
10
B
 [G
S
E
] (
nT
)
0
-5
5
-10
A B
Steepened edge
(or leading edge)
High Frequency
Wave Packet
Long Period
Magnetosonic Wave
Upstream
Figure 4.2: A plot of the magnetic fields for the shock on 04/06/2000. The shock
arrival time was 16:32:09.237 UT. The top panel is the magnetic field strength, |B|
(nT), followed by the field components in GSE coordinates from 15:39:00 - 16:36:00
UT. In the top panel, the location of each shocklet observed for this event is labeled
with a blue number and an arrow. The two shocklets which will be analyzed in detail
in this paper are labeled A and B (Figures 4.3 and 4.4 show a more detailed picture of
each shocklet). The top two panels have been scaled down to show the shocklets more
clearly because the IP shock itself jumps to over 30 nT downstream (only a few seconds
after the shocklet B).
130
The bottom two panels are examples of two shocklets with waves on the leading edge
consistent with a RH-whistler mode. The typical structure of the shocklets is labeled
in panel B. Shocklets 1, 3-5, 7, and 9-12 all had clearly formed waves on their leading
edges. These waves had RH polarizations with respect to the magnetic field and an
increase in ion density coincident with the increase in |B|, consistent with magnetosonic
whistlers and magnetosonic waves.
Figures 4.3 and 4.4 show examples of the analysis done on the leading whistler
wave of each shocklet. The left hand set of wave events plot the GSE (gray scale)
components, the middle panels plot the MV (color scale) components, and the right
hand set of panels show the hodograms, By vs. Bx, Bz vs. Bx, and Bz vs. By. The time
ranges for the selected subintervals seen in wave events A-C, were chosen to maximize
the intermediate to minimum eigenvalue ratio, seen in red in each wave event. In every
wave analysis presented and every wave examined (125 different analyses for all five IP
shocks), λ2/λ3 ≥ 10.0 and 56 cases had λ2/λ3 ≥ 50.0.
The use of multiple band pass filters on the shocklets revealed that their wave vectors
remained relatively unchanged (within ∼10◦-15◦) between the different frequency bands
chosen for our filters for similar time intervals. This is illustrated clearly in events A
and C of Figure 4.3 and events A, C, and D of Figure 4.4. The time interval length
for each event are similar, but each event was filtered over a different frequency bin.
The wave vector is the same for each event within uncertainties (see Equations 2.13
through 2.17 for uncertainty calculation). We also observed a clear dependence of the
frequency at peak power on the distance from the steepened edge of the shocklets,
with the higher frequency waves being seen first, followed by the lower frequency waves
as the shocklets convected over the satellite. This is consistent with the frequency
dependence of magnetosonic whistlers, whose group velocities increase with increasing
frequency. This result is also seen in simulations [Omidi and Winske, 1990; Scholer ,
1993; Scholer et al., 2003; Tsubouchi and Lembe`ge, 2004].
The difference in polarization between wave events A, C, and D in Figure 4.4 can be
explained by projection effects due to single satellite measurements using only magnetic
field measurements. In the spacecraft frame, wave events A, B, and D in Figure 4.3 are
LH polarized with respect to the propagation direction, but all wave events in Figures
4.3 and 4.4 show a RH sense with respect to the magnetic field, characteristic of whistler
131
16:28:00 - 16:30:00 UT on 04/06/2000
|B
| 
(n
T
)
6
7
8
9
10
B
 [G
S
E
] (
nT
)
0
-5
5
-10
Time (HH:MM:SS)
16:28:00 16:28:30 16:29:00 16:29:30 16:30:00
Bx
Bz
By
SCET :  16:28:19.8865 - 16:28:30.5585
Date:  04/06/2000
B
  
 (
n
T
)
x
-1.0
0.0
1.0
-2.0
2.0
B
  
 (
n
T
)
y
-1.0
0.0
1.0
-2.0
2.0
B
  
 (
n
T
)
z
-1.0
0.0
1.0
-2.0
2.0
C
k
GSE
^ = <-0.604,0.620,-0.502>
Y Y
Y
Y
Z
Z
Z
Z
X
X
X
X
D
SCET :  16:28:40.9545 - 16:28:44.8185
Date:  04/06/2000
k
GSE
^ = <0.681,-0.685,-0.260>
B
  
 (
n
T
)
x
-0.5
0.0
0.5
-1.0
1.0
B
  
 (
n
T
)
y
-0.5
0.0
0.5
-1.0
1.0
B
  
 (
n
T
)
z
-0.5
0.0
0.5
-1.0
1.0
Y
Y
Z
Z
Y
Y
Z
Z
X
X
X
X
SCET :  16:28:31.1105 - 16:28:34.3305
Date:  04/06/2000
B
  
 (
n
T
)
x
-1.0
0.0
1.0
-2.0
2.0
B
  
 (
n
T
)
y
-1.0
0.0
1.0
-2.0
2.0
B
  
 (
n
T
)
z
-1.0
0.0
1.0
-2.0
2.0
k
GSE
^ = <-0.512,0.835,-0.202>
B
Y
X
Y
X
ZZ
Z Z
Y
Y
XX
SCET :  16:28:26.5105 - 16:28:30.0985
Date:  04/06/2000
B
  
 (
n
T
)
x
-1.0
0.0
1.0
-2.0
2.0
B
  
 (
n
T
)
y
-1.0
0.0
1.0
-2.0
2.0
B
  
 (
n
T
)
z
-1.0
0.0
1.0
-2.0
2.0
k
GSE
^ = <-0.657,0.584,-0.478>
A
Y
Y
X
X
Z
Z
Y
Y
X
X
Z
Z
Figure 4.3: An example of MV analysis on the leading whistler waves of a shocklet
shown in wave event A, Figure 4.2. The figure has the same format as Figure 2.8.
The frequency ranges and angles of propagation are: 0.5 Hz < f < 1.0 Hz and θkB =
35◦(145◦) for A, 0.5 Hz < f < 1.0 Hz and θkB = 14◦(166◦) B, 0.6 Hz < f < 3.0 Hz
and θkB = 41◦(139◦) for C, and 0.6 Hz < f < 3.0 Hz and θkB = 18◦(162◦) for D. The
eigenvalue ratios from the MV analysis are also shown with the MV estimate of the k-
vector direction in GSE coordinates above each hodogram. The purple arrows indicate
the direction of rotation for each respective plot.
132
Time (HH:MM:SS)
16:31:30 16:31:40 16:31:50 16:32:00
16:31:30 - 16:32:06 UT on 04/06/2000
|B
| 
(n
T
)
6
7
8
9
5
B
 [G
S
E
] (
nT
)
0
-2
2
4
-4
-6
-8
Bx
Bz
By
SCET :  16:31:41.5505 - 16:31:44.3105
Date:  04/06/2000
B
  
 (
n
T
)
x
-1.0
0.0
1.0
-2.0
2.0
B
  
 (
n
T
)
y
-1.0
0.0
1.0
-2.0
2.0
B
  
 (
n
T
)
z
-1.0
0.0
1.0
-2.0
2.0
k
GSE
^ = <-0.685,0.728,-0.024>
B
Z
Z
Y
Y
X
XX
X
Y
Y
Z
Z
SCET :  16:31:36.6745 - 16:31:39.2505
Date:  04/06/2000
A
B
  
(n
T
)
x
0.0
-1.0
1.0
k
GSE
^ = <-0.679,0.693,0.241>
B
  
(n
T
)
y
0.0
-1.0
1.0
B
  
(n
T
)
z
0.0
-1.0
1.0
Z
Z
Y
Y
X
X
X
X
Y
Y
Z
Z
B
  
(>
1
 H
z
)
x
-1.0
0.0
1.0
B
  
(>
1
 H
z
)
y
-1.0
0.0
1.0
B
  
(>
1
 H
z
)
z
-1.0
0.0
1.0
SCET :  16:31:36.0305 - 16:31:39.3425
Date:  04/06/2000
k
GSE
^ = <-0.686,0.689,0.236>
D
Y
Z
Z
Y
X
XX
X
Y
Y
Z
Z
k
GSE
^ = <-0.688,0.687,0.235>SCET :  16:31:35.5705 - 16:31:39.2505
Date:  04/06/2000
B
  
(>
6
0
0
 m
H
z
)
x -1.0
0.0
1.0
B
  
(>
6
0
0
 m
H
z
)
y -1.0
0.0
1.0
B
  
(>
6
0
0
 m
H
z
)
z -1.0
0.0
1.0
C
Z
Y
X
X
Z
Z
Y
Y
X
Z
Y
X
Figure 4.4: Another example of MV analysis on the leading whistler waves of a shocklet
(See panel B in Figure 4.2). The format of this figure matches that of Figure 4.3 but
with different frequency ranges. The frequency ranges and angles of propagation are:
0.6 Hz < f < 3.0 Hz and θkB = 27◦(153◦) for A, 0.45 Hz < f < 1.0 Hz and θkB =
49◦(131◦) B, f > 0.6 Hz and θkB = 25◦(155◦) for C, and f > 1.0 Hz and θkB = 26◦(154◦)
for D.
133
modes. Wave event C in Figure 4.3 and wave events A and B in Figure 4.4 are RH
polarized both with respect to the wave vector and the magnetic field.
4.2 Comparison of the 04/06/2000 event to the four typ-
ical events
As mentioned above, four of the IP shocks had waves with characteristics consistent
with previous shock studies. Figure 4.5 shows the magnetic field magnitude and GSE
components for the four IP shocks with typical characteristics. The four events with
precursor waves are much lower Mach number shocks (Mf ≤ 2.3) than the unusual
event (Mf ∼ 4), consistent with theory [Morton , 1964; Stringer , 1963] and previous
observation [Mellott and Greenstadt , 1984].
The events shown in the upper panels of Figure 4.5 are examples of lower Mach
|B
| 
(n
T
)
B
 [G
S
E
] (
nT
)
-5
0
-10
-15
5
16
12
14
18
8
20
10
|B
| 
(n
T
)
8
6
7
9
5
10
B
 [G
S
E
] (
nT
)
2
-2
0
-4
-6
4
6
|B
| 
(n
T
)
4
6
7
5
B
 [G
S
E
] (
nT
)
2
-2
0
4
6
|B
| 
(n
T
)
B
 [G
S
E
] (
nT
)
-5
0
-10
-15
5
16
12
14
18
8
10
6
HTR Wind B-Field:  10/24/1997 (30 Sec. Window)
11:17:55.010 - 11:18:24.956 UT
HTR Wind B-Field:  04/03/1996 (15 Sec. Window)
09:47:09.676 - 09:47:24.626 UT
HTR Wind B-Field:  04/08/1996 (15 Sec. Window)
02:41:02.244 - 02:41:17.196 UT
HTR Wind B-Field:  12/10/1997 (30 Sec. Window)
04:32:59.765 - 04:33:29.297 UT
X-GSE
Y-GSE
Z-GSE
X-GSE
Y-GSE
Z-GSE
X-GSE
Y-GSE
Z-GSE
X-GSE
Y-GSE
Z-GSE
M   =  1.59      0.02f +
θ    = 75.7     1.4Bn
o o
+
+N  / N   =  1.49    0.052 1
M   =  1.73      0.06f +
θ    = 68.3     4.5Bn
o o
+
+N  / N   =  2.47    0.452 1
M   =  2.26      0.10f +
θ    = 70.9     1.6Bn
o o
+
+N  / N   =  2.49    0.352 1
M   =  2.08      0.03f +
θ    = 73.3     1.1Bn
o o
+
+N  / N   =  1.68    0.032 1
Figure 4.5: Four IP shocks with upstream precursor whistler waves, typical of subcrit-
ical quasi-perpendicular shocks. The top two panels (04/03/1996 and 04/08/1996) are
examples of laminar quasi-perpendicular shocks with leading precursor whistler waves.
The bottom two panels (10/24/1997 and 12/10/1997) show a far more turbulent tran-
sition from upstream to downstream.
134
number quasi-perpendicular shocks with a leading wave train and a relatively stable
transition from up to downstream. The events in the lower panels are examples of
higher Mach number shocks with a much more turbulent transition. The relevant shock
parameters are given in the green box. The waves in Figure 4.5 are similar in frequency
and their propagation angle with respect to the shock normal, θkn to the precursor
whistler waves observed by Russell et al. [1983]. Since θkn is not small it is not likely
that these waves are phase standing with respect to the shock. The precursors had nearly
circular RH polarization with respect to the magnetic field, and propagate obliquely to
the field with > 95% having propagation angles θkB > 20◦. However, Russell et al.
[1983] found that 75% of the precursors had propagation angles θkB < 20◦.
The whistler precursor waves and shocklets shared some characteristics. The range of
θkB values can be seen in Figure 4.6. There is an obvious difference between the whistlers
upstream of the four typical events (bottom panel) and the 12 shocklets (top panel)
observed upstream of the unusual event. The shocklets have a much broader range of
θkB and tend to be more oblique than the precursor whistler waves. If the shocklets are in
fact magnetosonic whistlers, the higher values of θkB would be consistent with their more
compressive nature than that of the precursor whistlers. Almost 80% of the shocklets
observed for the 04/06/2000 event had SC frame frequencies f > 0.45 Hz had θkB ≤ 45◦,
consistent with bow shock observations [Hoppe et al., 1981, 1982; Hoppe and Russell ,
1983; Russell et al., 1971] and cometary foreshock observations [Le et al., 1989]. Over
90% of the whistler observed for the 4 IP shocks without shocklets had θkB ≤ 45◦,
consistent with theory [Gary et al., 1994, 1999] and observations of whistler precursor
waves at IP shocks [Russell et al., 1983]. There were no distinguishing characteristics
in θkn or θkV between the shocklets and precursor whistlers.
4.3 Analysis of Particle Data for this Study
Estimates of the electron temperature anisotropies in both the cold dense core (sub-
script c) and the hotter more tenuous halo (subscript h) can be obtained from full 3D
electron distributions (see Chapter 2 for more details). For both EL and EH distribu-
tions, average temperatures, parallel (subscript ‖), and perpendicular (subscript ⊥) to
the magnetic field are computed. Temperature anisotropies, T⊥j/T‖j (j = c or h), were
135
Total # of Events:  66 (f > 450 mHz)
Shocklet Whistlers on 04/06/2000
N
u
m
b
e
r 
o
f 
Ev
e
n
ts
10
15
5
θ
kB
 (deg)  
10.0-17.1 17.1-24.3 24.3-31.4 31.4-38.6 38.6-45.7 45.7-52.9 52.9-60.0
Total # of Events:  44 (f > 600 mHz)
IP Shock Precursor Whistlers
N
u
m
b
e
r 
o
f 
Ev
e
n
ts
10
15
5
Figure 4.6: Histograms comparing the angle of propagation with respect to the magnetic
field for the IP shock precursor whistler waves and the shocklet whistlers. The top panel
shows the range of angles for all bandpass frequency bins greater than 0.45 Hz but only
for the 12 shocklets (each shocklet has multiple waves as seen in Figures 4.3 and 4.4)
observed on 04/06/2000. The bottom panel shows the range of angles for all bandpass
frequency bins greater than 0.6 Hz for the 4 IP shocks without shocklets. The horizontal
and vertical axes are on the same scales for both panels. Multiple frequency ranges were
chosen for each IP shock.
136
computed for each PAD and compared to threshold conditions for whistler heat flux
and anisotropy instabilities of Gary et al. [1994] and Gary et al. [1999].
The wave events of Figures 4.3 and 4.4 show a clear relationship with the low to
mid energy (∼few eV to 9 keV) electron distributions. Previous studies of whistler
waves at shocks suggested a relationship between whistler mode generation and elec-
trons [Tokar et al., 1984; Tsurutani et al., 1983]. Gary et al. [1994] determined the
threshold conditions for whistler heat flux and whistler anisotropy instabilities for typi-
cal solar wind conditions. They found that the instabilities were strongly dependent on
the core parallel plasma beta, β‖c, the ratio of parallel halo temperature to parallel core
temperature, T‖h/T‖c, and the temperature anisotropy of the halo, T⊥h/T‖h. Using lin-
ear Vlasov theory, Gary et al. [1999] showed that the halo temperature anisotropy has
a larger effect on the heat flux instability than the core temperature anisotropy. They
also found that the heat flux-driven whistler mode was always unstable for T⊥h/T‖h >
1.01 and always stable for β‖c ≤ 0.25. In the cases where T⊥h/T‖h > 1.01 but T‖h/T‖c is
small, Gary et al. [1994] suggested that a whistler anisotropy instability may be excited
even in the absence of a relative drift between the core and halo electrons. Thus if
the halo electrons initially meet this criteria, the whistler anisotropy instability would
dominate over the whistler heat flux instability. The whistler anisotropy instability
acts to reduce the relative drift between the halo and core electrons (if present) and
isotropize the halo temperatures. T⊥h/T‖h would reduce faster than the halo/core drift
(and T‖h/T‖c) could reduce causing the electron distributions to become unstable to
a whistler heat flux instability. In the case of large T‖h/T‖c and small T⊥h/T‖h, the
whistler heat flux instability would initially dominate over the whistler anisotropy in-
stability. This instability would increase T⊥h/T‖h slower than it could reduce T‖h/T‖c
[Gary et al., 1994].
The electron pitch-angle distributions (PADs) for three wave events are plotted in
Figures 4.7, 4.8, and 4.9. The primary influence of whistler heat flux instability is to
pitch-angle scatter the halo electrons through a cyclotron resonance. There is a clear
increase in the halo electron temperature anisotropy, T⊥h/T‖h (see Table 4.1), as one
crosses each wave, consistent with normal cyclotron resonance increasing the transverse
energy of the electrons. T⊥c/T‖c follows the same pattern, but the increase is not
137
SCET :  16:28:19.8865 - 16:28:30.5585
Date:  04/06/2000
B
  
 (
n
T
)
x
-1.0
0.0
1.0
-2.0
2.0
B
  
 (
n
T
)
y
-1.0
0.0
1.0
-2.0
2.0
B
  
 (
n
T
)
z
-1.0
0.0
1.0
-2.0
2.0
k
GSE
^ = <-0.604,0.620,-0.502>
k
GSE
^
Y
Y Y
Y
Z
Z
Z
Z
X
X
X
X
107
108
106
105
103
104
105
106
102
689.2 eV
426.8 eV
264.8 eV
169.0 eV
103.3 eV
65.2 eV
41.8 eV
27.2 eV
1113.0 eV
689.2 eV
426.8 eV
264.8 eV
169.0 eV
103.3 eV
65.2 eV
41.8 eV
27.2 eV
1113.0 eV
4161.3 eV
2849.2 eV
1339.4 eV
920.3 eV
634.4 eV
432.7 eV
200.1 eV
136.8 eV
8874.9 eV
1952.3 eV
6076.5 eV
292.1 eV
106
107
105
104
103
104
105
100
102
101
90 1800
Pitch Angle  (deg)
16:27:18 - 16:27:21 UT 
Eesa Low PAD
Eesa High PAD Eesa High PAD Eesa High PAD
16:30:36 - 16:30:39 UT 
Eesa Low PAD
16:28:57 - 16:29:00 UT 
Eesa Low PAD
90 1800
Pitch Angle  (deg) 90 1800 Pitch Angle  (deg)
N
u
m
be
r 
Fl
u
x
En
e
rg
y 
Fl
u
x
N
u
m
be
r 
Fl
u
x
En
e
rg
y 
Fl
u
x
4161.3 eV
2849.2 eV
1339.4 eV
920.3 eV
634.4 eV
432.7 eV
200.1 eV
136.8 eV
8874.9 eV
1952.3 eV
6076.5 eV
292.1 eVθkB ~ 40  (140  )
o o
θ
kB
~ 40  (140  )
o o θ
kB
~ 40  (140  )
o o
Figure 4.7: Comparison of wave polarization and electron distributions (wave event D
from Figure 4.3) with six pitch-angle distributions (PADs) from the Eesa Low (EL) and
High (EH) instruments. Each PAD is plotted in number (# s−1sr−1cm−2eV−1) and
energy (eV s−1sr−1cm−2eV−1) flux. The energies plotted range from 27 eV - 1113 eV
for EL and 137 eV - 8875 eV for EH. The frequency range is 0.6 Hz < f < 3 Hz for the
bandpass filter used on the MFI data and θkB = 40◦(140◦). The vertical lines on the
PADs represent an average estimate of the propagation angle, θkB, for the wave shown.
The electron temperature anisotropies and other parameters can be found in Table 4.1.
138
as dramatic. The threshold conditions for a whistler heat flux or anisotropy instabil-
ity (Figures 7 and 8 of Gary et al. [1994]) are met by most of the EL PADs up and
downstream of the waves in these three figures. There are, however, differences in our
estimates of nhe/ne, distributions used to model the halo electrons, and definition of
heat flux from those of Gary et al. [1994]. Our estimates of nhe/ne were often a factor
of 10 or more smaller than the estimates used by Gary et al. [1994] (∼0.05) for the
PADs shown herein (see Table 4.1). Gary et al. [1994] used bi-Maxwellian distributions
to model both core and halo electrons whereas we fit the halo distributions to modified
Lorentzian distributions. Gary et al. [1994] used a simplified version of the heat flux
from Feldman et al. [1975] whereas we calculated the full heat flux tensor, assuming
it symmetric, and derived a vector from that tensor. The main difference is a loss of
the vector direction associated with the heat flux, since Gary et al. [1994] assumed it
parallel to the ambient magnetic field. The magnitude estimates should not be affected
significantly.
One can see that T⊥h/T‖h increases across the waves in Figures 4.8 and 4.9 (from
0.55 for PAD 16:31:32-16:31:35 UT to 1.02 for PAD 16:31:44-16:31:47 UT shown in Table
4.1, an increase of ∼85%). Gary et al. [1994] found that the whistler heat flux instability
reduced T‖h/T‖c but increased T⊥h/T‖h at a faster relative rate. We observed T‖h/T‖c
to decrease across the waves (from 13.4 for PAD 16:31:32-16:31:35 UT to 11.4 for PAD
16:31:44-16:31:47 UT, a decrease of ∼15%). The same change occurred across the wave
in Figure 4.7. T⊥h/T‖h increases across the wave (0.65 for PAD 16:27:18-16:27:21 to 0.91
for PAD 16:30:36-16:30:39, an increase of ∼40%) and T‖h/T‖c decreases (13.7 for PAD
16:27:18-16:27:21 to 10.9 for PAD 16:30:36-16:30:39, a decrease of ∼20%). Notice in
both cases T⊥h/T‖h increases at a faster relative rate than T‖h/T‖c decreases, consistent
with the simulation results by Gary et al. [1994].
If we assume that the rest frame frequencies of the observed waves are consistent
with Hoppe et al. [1982] and use our measured θkB , then the resonant energies for nor-
mal cyclotron resonance can be calculated from Equation 1.43. The resonant energies
are 250 eV ≤ Eres ≤ 4 keV for the event in Figure 4.7, and 200 eV ≤ Eres ≤ 3 keV for the
wave in Figure 4.8. In Figure 4.7, most of the energy bins (65 eV - 689 eV for EL and
136 eV - 3 keV for EH) of the PADs which undergo the greatest change across the wave
are within the estimated resonant energy range. The PADs in Figures 4.8 and 4.9 are
139
B
  
(>
1
 H
z
)
x
-1.0
0.0
1.0
B
  
(>
1
 H
z
)
y
-1.0
0.0
1.0
B
  
(>
1
 H
z
)
z
-1.0
0.0
1.0
k
GSE
^ = <-0.686,0.689,0.236>
SCET :  16:31:36.0305 - 16:31:39.3425
Date:  04/06/2000
Y
X
X
Z
Z
Y
X
X
Y
Y
Z
Z
689.2 eV
426.8 eV
264.8 eV
169.0 eV
103.3 eV
65.2 eV
41.8 eV
27.2 eV
1113.0 eV
689.2 eV
426.8 eV
264.8 eV
169.0 eV
103.3 eV
65.2 eV
41.8 eV
27.2 eV
1113.0 eV
16:31:32 - 16:31:35 UT 
Eesa Low Burst PAD
16:31:35 - 16:31:38 UT 
Eesa Low Burst PAD
16:31:38 - 16:31:41 UT 
Eesa Low Burst PAD
90 1800
Pitch Angle  (deg) 90 1800 Pitch Angle  (deg)90 1800 Pitch Angle  (deg)
107
108
106
105
103
104
105
106
102
N
u
m
be
r 
Fl
u
x
En
e
rg
y 
Fl
u
x
Figure 4.8: This figure has the same format as Figure 4.7. The frequency range is f > 1.0
Hz and θkB = 25◦(155◦). The electron temperature anisotropies and other parameters
can be found in Table 4.1.
140
SCET :  16:31:41.5505 - 16:31:44.3105
Date:  04/06/2000
B
  
 (
n
T
)
x
-1.0
0.0
1.0
-2.0
2.0
B
  
 (
n
T
)
y
-1.0
0.0
1.0
-2.0
2.0
B
  
 (
n
T
)
z
-1.0
0.0
1.0
-2.0
2.0
k
GSE
^ = <-0.685,0.728,-0.024>
X
Y
Z
Y
X
Z Z
Z
X X
YY
689.2 eV
426.8 eV
264.8 eV
169.0 eV
103.3 eV
65.2 eV
41.8 eV
27.2 eV
1113.0 eV
689.2 eV
426.8 eV
264.8 eV
169.0 eV
103.3 eV
65.2 eV
41.8 eV
27.2 eV
1113.0 eV
90 1800
Pitch Angle  (deg) 90 1800 Pitch Angle  (deg)90 1800 Pitch Angle  (deg)
16:31:38 - 16:31:41 UT 
Eesa Low Burst PAD
16:31:41 - 16:31:44 UT 
Eesa Low Burst PAD
16:31:44 - 16:31:47 UT 
Eesa Low Burst PAD
107
108
106
105
103
104
105
106
102
N
u
m
be
r 
Fl
u
x
En
e
rg
y 
Fl
u
x
Figure 4.9: This figure has the same format as Figure 4.8 except that the wave occurs
roughly 2.5 seconds later. The frequency range is 0.45 Hz < f < 1.0 Hz and θkB =
49◦(131◦). The electron temperature anisotropies and other parameters can be found
in Table 4.1.
141
also consistent with the resonant energies showing the greatest changes from the PAD at
16:31:32 UT in Figure 4.8 to the PAD at 16:31:44 UT in Figure 4.9. The average increase
in pitch-angle of the electrons in this energy range across the waves would be consistent
with pitch-angle diffusion were we observing the same distribution in time. It is difficult
to say whether the strong anisotropies in the electron PADs downstream of the shock-
lets are a consequence of their traversal of the shocklets or if their downstream location
with respect to the shocklet results in the isotropization through leakage. Leakage is
the process whereby the downstream electrons with large pitch-angles are restricted to
the downstream by the shocklet’s magnetic fields, while the lower pitch-angle electrons
can move freely upstream [Larson et al., 1996; Thomsen et al., 1983b].
The electron heat flux, or more appropriately, the kinetic energy flux in the plasma
rest frame, was calculated to more thoroughly examine the instability thresholds dis-
cussed by Gary et al. [1994, 1999]. Each 3DP electron distribution was first transformed
into the solar wind frame, including the effects of the spacecraft potential, and then the
first four moments of the distribution function were calculated. The heat flux, cal-
culated from Equation 1.19, changed both in angle with respect to magnetic field and
wave vector and in magnitude across each shocklet. The angle between the wave vectors
and heat flux vectors calculated for each PAD change from ∼20◦ to ∼27◦ for 16:31:32-
16:31:44 UT. Also, the magnitude of the heat flux changes from ∼49 to ∼85 keV cm−3
km/s for 16:31:32-16:31:44 UT. Thus, the heat flux magnitude, angle between wave
vector and heat flux, and angle between heat flux vector and magnetic field all peak in
the 16:31:44-16:31:47 UT PAD which is just downstream of the peak amplitude of the
shocklet in panel B of Figure 4.2.
Most studies of shocklets at the terrestrial bow shock focused on ion distributions.
Shocklets were observed to have a location dependence in the terrestrial ion foreshock
with respect to the sunward edge (boundary between red and yellow regions in Figure
1.5). The location dependence is due to the evolution of the observed ion distributions
in the foreshock. Shocklets are observed in association with diffuse ion distributions,
a characteristic distribution seen in deeper regions (i.e. farther from the sun) of the
foreshock [Hoppe et al., 1981, 1982; Hoppe and Russell , 1983]. Figure 4.10 shows exam-
ples of diffuse ions seen simultaneously with the shocklet in panel B of Figure 4.2. The
plots are Pesa High Burst (PHB) distribution functions plotted with the horizontal axis
142
---  :  Solar Wind Direction
- - -  :  Shock Normal Direction
-2000
-1000
0
1000
2000
Vp
e
rp
 
[km
/s
]
-2000 -1000 0 1000 2000
Vpara [km/s]
Wind 3D-Plasma Pesa High Burst
2000-04-06/16:31:29-16:31:32
Wind 3D-Plasma Pesa High Burst
2000-04-06/16:31:38-16:31:41
-2000
-1000
0
1000
2000
Vp
e
rp
 
[km
/s
]
Wind 3D-Plasma Pesa High Burst
2000-04-06/16:31:35-16:31:38
Wind 3D-Plasma Pesa High Burst
2000-04-06/16:31:32-16:31:35
Wind 3D-Plasma Pesa High Burst
2000-04-06/16:31:44-16:31:47
Wind 3D-Plasma Pesa High Burst
2000-04-06/16:31:41-16:31:44
-2000 -1000 0 1000 2000
Vpara [km/s]-2000 -1000 0 1000 2000Vpara [km/s]
ED F
A B C
Figure 4.10: This figure is a series of contour plots of the ion distribution function
from PHB from 16:31:29 UT to 16:31:47 UT. The high energy diffuse ions are seen as
the semi-isotropic ring between 1000 and 2000 km/s. These distributions are observed
simultaneously with the shocklet in panel B of Figure 4.2.
143
parallel to the ambient magnetic field in the plane created by the magnetic field and
solar wind velocity. In each plot, the solar wind direction (black line) and shock normal
direction (red line) are projected onto the distributions for reference.
4.3.1 Comparison of Electron Heating
The electron distributions at the unusual event show strong heating in the down-
stream region. The downstream region of the 04/06/2000 event had broad flattop
distributions downstream, thought to result from strong current-driven ion-acoustic
waves [Thomsen et al., 1983a]. The 12/10/1997 event showed weak flattop distribu-
tions downstream for a few seconds, followed by a Maxwellian that was hotter than the
upstream distributions. The flattop distributions lasted for over an hour downstream
of the 04/06/2000 IP shock.
Figure 4.11 shows the temperature anisotropies for both the halo and core electrons
within ±1 hour of each of the five IP shocks examined. Each panel is labeled with a
capital letter differentiate each event, where A is for 1996-04-03, B is for 1996-04-08, C
is for 1997-10-24, D is for 1997-12-10, and E is for 2000-04-06. The level of unity (i.e.
isotropy) is marked by a horizontal blue line. The red vertical bars in the right hand
side of each panel denotes the magnitude of ∼0.2 for quick comparison between each
panel. The important thing to note is that the red bar is the smallest in both panels
of the 2000-04-06 event, which shows that the changes in temperature anisotropies are
the largest for that event. The 2000-04-06 event is also the only event to show a global
increase in T⊥h/T‖h.
The halo electrons have strong anisotropies in the downstream region of the
04/06/2000 event. The halo electrons show strong heating perpendicular to the magnetic
field with a remarkably low change in the parallel halo temperature. The preferential
perpendicular heating of the halo electrons may be explained by the efficiency of the
pitch-angle scattering discussed by Saito and Gary [2007] which showed a preferential
efficiency with higher kinetic energy electrons. The reason for the low heating in the
parallel halo electrons is not known at this time.
The fact that shocklets were observed upstream at only one of the five IP shocks
raises the question of what characteristics of the 04/06/2000 event might lead to their
generation. It is highly likely that the 04/06/2000 event is a supercritical shock (i.e.
144
09:00 09:30 10:00 10:30
08:45:00 - 10:45:00 UT on 04/03/1996
09:00 09:30 10:00 10:30
02:00 02:30 03:00 03:30
01:41:00 - 03:41:00 UT on 04/08/199601:41:00 - 03:41:00 UT on 04/08/1996
02:00 02:30 03:00 03:30
10:14:00 - 12:14:00 UT on 10/24/1997
11:00 11:30 12:0010:30
10:14:00 - 12:14:00 UT on 10/24/1997
11:00 11:30 12:0010:30
04:00 04:30 05:00 05:30
03:33:00 - 05:33:00 UT on 12/10/199703:33:00 - 05:33:00 UT on 12/10/1997
04:00 04:30 05:00 05:30
16:00 16:30 17:00 17:3015:30
15:00:00 - 18:00:00 UT on 04/06/200015:00:00 - 18:00:00 UT on 04/06/2000
16:00 16:30 17:00 17:3015:30
~0.2
08:45:00 - 10:45:00 UT on 04/03/1996
TceTce / TheThe /
1.10
1.05
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1.00
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~0.2
~0.2
~0.2
~0.2
~0.2
~0.2
~0.2
~0.2
~0.2
AA
BB
CC
DD
EE
Figure 4.11: A summary plot of temperature anisotropies for both the halo and core
electrons for the five events of interest in Wilson III et al. [2009]. Each event is labeled
with a capital letter. The left column is for the core electrons and the right column is
the halo electrons. The blue horizontal line marks the 1.0 level and the red bar is used
to show the relative amplitude (equal to ∼0.2) of any change between each plot.
145
requires particle reflection for energy dissipation), which may explain why shocklets
are observed upstream of this event and none of the others. Particle reflection could
explain the difference in heating between the 04/06/2000 event and the other four.
Recent observations [L.B. Wilson III et. al., Large amplitude electrostatic waves ob-
served at a supercritical interplanetary shock, 2009] have shown evidence of the mod-
ified two stream instabilities discussed by Matsukiyo and Scholer [2006a] which were
shown to strongly heat the electrons and ions. The core electrons show significant
heating (Tc2/Tc1 & 3.5) which one would expect from an interaction with electro-
static waves [Thomsen et al., 1985a] and/or electromagnetic modes like the modified
two stream instability [Matsukiyo and Scholer , 2006a]. Thus, ion reflection is likely
playing a more significant role in energy dissipation than at the other lower Mach num-
ber events [Thomsen et al., 1985a].
Figure 4.12 shows the perpendicular temperatures for both the halo and core elec-
trons within ±1 hour of each of the five IP shocks examined. The format is same as in
Figure 4.11, except the vertical red bars denoting the relative magnitudes are different
between halo (∼30.0) and core (∼3.0). Again, the relative magnitude is the largest for
the 2000-04-06 event compared to the other four events.
Figure 4.13 shows the parallel temperatures for both the halo and core electrons
within ±1 hour of each of the five IP shocks examined. The format is same as in Figure
4.12. Here the vertical red bars denoting the relative magnitudes for the halo and core
are ∼30.0 and ∼3.0, respectively. Again, the relative magnitude is the largest for the
2000-04-06 event compared to the other four events. Also notice that the 2000-04-06
event is the only event to show a global decrease in T‖h as seen in the right hand panel
E.
One can see from examining Figures 4.11 through 4.13 that there is a preference for
perpendicular heating in the 2000-04-06 event in both the core and halo electrons. The
change in core temperatures for the 2000-04-06 event is also well beyond the adiabatic
limit, the only one of the five events shown in these figures (see Section 5.2 for more
details). The differences in anisotropic heating is well illustrated in Figure 4.11.
146
09:00 09:30 10:00 10:30
08:45:00 - 10:45:00 UT on 04/03/1996
09:00 09:30 10:00 10:30
02:00 02:30 03:00 03:30
01:41:00 - 03:41:00 UT on 04/08/199601:41:00 - 03:41:00 UT on 04/08/1996
02:00 02:30 03:00 03:30
10:14:00 - 12:14:00 UT on 10/24/1997
11:00 11:30 12:0010:30
10:14:00 - 12:14:00 UT on 10/24/1997
11:00 11:30 12:0010:30
04:00 04:30 05:00 05:30
03:33:00 - 05:33:00 UT on 12/10/199703:33:00 - 05:33:00 UT on 12/10/1997
04:00 04:30 05:00 05:30
16:00 16:30 17:00 17:3015:30
15:00:00 - 18:00:00 UT on 04/06/200015:00:00 - 18:00:00 UT on 04/06/2000
16:00 16:30 17:00 17:3015:30
~3.0
TheTce
08:45:00 - 10:45:00 UT on 04/03/1996
8
9
7
10
40
50
60
70
80
40
50
60
70
8
9
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11
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90
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120
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160
180
200
10
15
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35
~30.0
~30.0
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~80.0
~35.0~7.00
~10.0
~11.0
~14.0
~17.0
~15.0
~10.0
~35.0
~120.0
~200.0
~70.0
~35.0
~105.0
~50.0
~115.0
~85.0
~7.50
~9.00
AA
BB
CC
DD
EE
Figure 4.12: A summary plot of perpendicular temperatures for both the halo and core
electrons for the five events of interest in Wilson III et al. [2009]. The format is the
same as in Figure 4.11 except the vertical red bars are different in magnitude between
the halo and core.
147
09:00 09:30 10:00 10:30
08:45:00 - 10:45:00 UT on 04/03/1996
09:00 09:30 10:00 10:30
02:00 02:30 03:00 03:30
01:41:00 - 03:41:00 UT on 04/08/199601:41:00 - 03:41:00 UT on 04/08/1996
02:00 02:30 03:00 03:30
10:14:00 - 12:14:00 UT on 10/24/1997
11:00 11:30 12:0010:30
10:14:00 - 12:14:00 UT on 10/24/1997
11:00 11:30 12:0010:30
04:00 04:30 05:00 05:30
03:33:00 - 05:33:00 UT on 12/10/199703:33:00 - 05:33:00 UT on 12/10/1997
04:00 04:30 05:00 05:30
16:00 16:30 17:00 17:3015:30
15:00:00 - 18:00:00 UT on 04/06/200015:00:00 - 18:00:00 UT on 04/06/2000
16:00 16:30 17:00 17:3015:30
~3.0
08:45:00 - 10:45:00 UT on 04/03/1996
~30.0
Tce The
15
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~3.0
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~80.0
~35.0
~80.0
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~115.0
~70.0
~145.0
~95.0
~160.0
~180.0
~7.00
~11.5
~9.00
~13.0
~16.0
~35.0
~13.0
~14.0
~6.50
~9.50 AA
BB
CC
DD
EE
Figure 4.13: A summary plot of parallel temperatures for both the halo and core elec-
trons for the five events of interest in Wilson III et al. [2009]. The format is the same
as in Figure 4.12.
148
4.4 Conclusions
The strongest event in our low frequency upstream waves study [Wilson III et al.,
2009] was the most unusual of the five because it was the only event with shocklets.
We observed 12 shocklets upstream (≤ 1 hour of shock ramp) of the shock. Al-
most 80% of the shocklets had θkB ≤ 45◦, consistent with the cometary bow shock
study by Le et al. [1989] and terrestrial bow shock studies [Hoppe et al., 1981, 1982;
Hoppe and Russell , 1983; Russell et al., 1971]. It is likely that shocklets only occurred
at the 04/06/2000 event because of its unusually high Mach number (Mf ∼ 4). The
high Mach number and quasi-perpendicular nature of the shock suggest that it is a
supercritical shock, thus requiring ion reflection to dissipate energy. Ion reflection
has been shown to be an important aspect of ULF wave generation in observations
[Hoppe and Russell , 1983; Thomsen et al., 1983b;Meziane and D’Uston, 1998] and sim-
ulations [Omidi and Winske, 1990; Scholer et al., 2003; Tsubouchi and Lembe`ge, 2004].
Shocklets are often seen in association with diffuse ion distributions [Hoppe et al., 1981;
Hoppe and Russell , 1983] and the ULF waves observed by Hoppe and Russell [1983]
were seen to steepen into shocklets with associated leading magnetosonic whistler wave
packets when in regions of diffuse ions. Simulation studies have supported these observa-
tions and suggest that diffuse ions may be a necessary factor for ULF waves steepening
into shocklets [Omidi and Winske, 1990; Scholer et al., 2003; Tsubouchi and Lembe`ge,
2004]. Thus, the simultaneous observation of the diffuse ion distributions in Figure 4.10
and the wave in panel B of Figure 4.2 supports our hypothesis that these magnetic
structures are in fact shocklets.
The major differences in electron moments between the 04/06/2000 event and the
other four occurred primarily in the halo electrons. The only IP shock to show a strong
global increase (∼ a factor of 2) in T⊥h/T‖h from upstream to downstream was the
04/06/2000 event. The relative heating between the electron halo and core (T‖h/T‖c)
components is more dramatically affected in the 04/06/2000 event than any other stud-
ied decreasing by a factor of & 4 across the IP shock (other events increase by ≤ 2). The
decrease is due to the large increase in T‖c and slight decrease in T‖h across the shock.
The β‖c is almost always ≤ 1.0 upstream (within an hour of the ramp) of the 04/06/2000
event, while of the four other IP shocks examined, only the 10/24/1997 event has β‖c ≤
149
1.0 anywhere upstream (for ∼24 minutes immediately upstream of the ramp). The low
β‖c may be a necessary condition for the excitation of shocklets. Recent observations
[Wilson III et al., 2010] have found evidence to suggest that the stronger heating of the
core electrons at the 04/06/2000 event may have resulted from the microinstabilities of
Matsukiyo and Scholer [2006a]. The instabilities are excited by reflected ions, known
to exist upstream of supercritical quasi-perpendicular shocks, interacting with the inci-
dent solar wind. Thus the likely supercritical nature of the 04/06/2000 event makes it
a prime candidate for these instabilities. To summarize, the the 04/06/2000 event was
the most unusual in the study for the following reasons: 1) it had the lowest upstream
βe, 2) the largest change in Ti and Te across the shock, 3) the highest Mf , n2/n1, and
Un, 4) only event to strong anisotropic preferential heating of the electrons, and 5) large
amplitude atypical electric field waveforms observed in the ramp regions.
The electron distributions showed a clear relationship between the core/halo temper-
ature anisotropy and shocklets. Regardless of how the halo electrons became anisotropic
T⊥h/T‖h downstream of the shocklets, it exceeded the threshold estimated byGary et al.
[1994, 1999] for excitation of the whistler heat flux instability. One should also note
T⊥h/T‖h increases more dramatically than T⊥c/T‖c across almost every shocklet ob-
served with a leading magnetosonic whistler, consistent with the simulation results of
Gary et al. [1994, 1999]; Saito and Gary [2007]. Thus it appears that the higher en-
ergy halo electrons may have experienced a more efficient pitch-angle diffusion than the
lower energy core. This suggests that the anisotropies often observed in the the ambient
solar wind electron distributions may be more unstable than previous estimates which
used the entire distribution to estimate T⊥/T‖. Another interesting observation is that
nearly every distribution within 30 seconds of each IP shock ramp exceeded the T‖h/T‖c
threshold for whistler heat flux instability estimated by [Gary et al., 1994].
In addition to particle distributions unstable to the whistler heat flux instability, an
electron heat flux was observed. The heat flux itself is the source of the free energy for
the instability while the halo electron anisotropies change the threshold and growth rate
of the instability. The the heat flux magnitude, angle between wave vector and heat flux,
and angle between heat flux vector and magnetic field all peak in the 16:31:44-16:31:47
UT PAD which is just downstream of the peak amplitude of the shocklet in panel B of
Figure 4.2. Also, the angle between the wave vectors and heat flux vectors calculated
150
for each PAD change from ∼20◦ to ∼27◦ for 16:31:32-16:31:44 UT. The absolute angle
between the magnetic field and heat flux vector changes from ∼9◦ to ∼24◦ for the same
PADs, consistent with the magnetosonic whistler pitch-angle scattering the heat flux
carrying electrons [Gary et al., 1994].
The PADs downstream of the shocklets are suggestive of perpendicular heating and
pitch-angle diffusion. However, it is unclear whether the unstable electron distributions
seen in the downstream region play any role in the shocklet formation. If the unstable
distributions do excite a whistler heat flux instability, the resultant waves could poten-
tially propagate upstream of the steepened edge of the shocklets producing the observed
wave train. The waves could prolong or increase the perpendicular electron heating, thus
producing more unstable distributions. This would lead to a cyclical behavior of wave
formation, propagation, and damping that would be self reinforcing. However, it is also
possible that electrons with smaller pitch-angles were able to return upstream of the
shocklets while the higher pitch-angle electrons could not [Larson et al., 1996]. This
could also explain the strong parallel anisotropies in the electron PADs observed up-
stream of the shocklets.
The other four IP shocks discussed herein showed characteristics typical of sub-
critical to marginally-critical quasi-perpendicular shocks [Farris et al., 1993]. They each
had precursor whistler waves upstream of the ramp and in the ramp. Almost all the
precursor whistlers observed had 20◦ ≤ θkB ≤ 45◦, consistent with theory [Gary et al.,
1994, 1999]. The precursor waves observed in this study actually show more similarities
to the ∼1 Hz whistlers of Hoppe et al. [1982]; Sentman et al. [1983]. The waves are
far more oblique than the previous observations of precursors at IP shocks observed by
Russell et al. [1983] or the upstream whistlers at IP shocks observed by Tsurutani et al.
[1983]. More than 90% of the waves had 29◦ ≤ θkn ≤ 75◦, thus it is not likely that they
are phase standing. The waves are also seen in association with anisotropic electron dis-
tributions, though less oblique than previous observations of whistlers associated with
anisotropic electrons [Sentman et al., 1983]. We found that the anisotropic electron
distributions exceed the thresholds for the whistler heat flux instability estimated by
Gary et al. [1994, 1999]. The threshold conditions for a whistler heat flux instability
[Gary et al., 1994, 1999] are met by almost all electron distributions within 30 seconds
of every IP shock in this study.
151
In this chapter we presented observations on two classes of low frequency (0.25 Hz
< f < 10 Hz) waves at five quasi-perpendicular IP shocks [Wilson III et al., 2009]. The
first class of low frequency waves is a non-phase standing precursor whistler observed
just upstream of four of the IP shocks examined in this study. The second class is a
steepened magnetosonic wave, with a leading magnetosonic whistler wave train, called
a shocklet. The shocklets and precursor whistlers are observed in association with elec-
tron distributions unstable to whistler heat flux and/or whistler anisotropy instabilities.
152
Table 4.1: Wind 3DP Electron Stats on 04/06/2000
Start-End Time Tce (eV) The (eV) T⊥c/T‖c T⊥h/T‖h nce (cm−3) nhe (cm−3) T‖h/T‖c β‖c
Eesa Low
16:27:18-16:27:21 10.80 125.34 0.86 0.66 5.95 0.030 13.69 0.86
16:28:57-16:29:00 10.56 125.52 0.99 0.84 9.95 0.039 13.19 0.71
16:30:36-16:30:39 11.14 117.85 0.95 0.91 9.03 0.051 10.86 0.70
Eesa Low Burst
16:31:32-16:31:35 10.02 117.29 0.70 0.55 5.02 0.026 13.42 1.16
16:31:35-16:31:38 10.40 120.33 0.73 0.59 4.74 0.026 13.07 1.02
16:31:38-16:31:41 10.28 127.57 0.73 0.55 5.16 0.023 14.53 1.14
16:31:41-16:31:44 11.54 129.47 0.94 0.84 6.08 0.039 12.03 0.79
16:31:44-16:31:47 11.16 126.61 1.02 1.02 10.42 0.050 11.37 0.75
Chapter 5
High Frequency Waves at
Interplanetary Shocks
5.1 Statistical Study of High Frequency Waves at Colli-
sionless Shocks
In a study of high frequency (f & 600 Hz) waves at 67 IP shocks with 1 < Mf < 6,
Wilson III et al. [2007] observed three electrostatic (ES) wave modes in the ramp region
of the IP shocks: Langmuir waves, IAWs, and solitary waves. Thirty-three of the 67
IP shocks had at least one waveform capture in the ramp region. The average number
of TDS samples in the 33 IP shock ramps analyzed was ∼3. The IAWs were observed
to have Doppler shifted frequencies ranging from fpi . f . 10 kHz, where fpi is the ion
plasma frequency (∼500 Hz in solar wind at 1 AU). The Langmuir waves are always
above fpe, but typically the shift away from fpe due to thermal effects is small.
Table 5.1 summarizes the statistics on the waves in the upstream, ramp, and down-
stream regions for the 33 IP shocks. The last three rows were normalized by the average
duration for their respective regions: up/downstream τ¯u = τ¯ d ≃3596 s, and ramp τ¯ r ≃8
s. Though it is known that the shock ramp is much shorter than 8 seconds, the use of 3s
magnetic field data restricted our estimation of the ramp durations. To normalize our
data, we take the number of events observed in each region, nregion, and divide by the
duration of each region to get an expected observation frequency, ηregion. Each ηregion
153
154
Table 5.1: Wave Occurrence by Regions for the 33 IP Shocks with Ramp TDS Samples
All Waves Large(≥5 mV/m)
Lang. IAW Solitary Lang. IAW Solitary
♯ Up 52 102 20 33 28 1
♯ in Ramps 5 78 7 5 71 5
♯ Down 8 91 12 6 29 3
Normalized Probability of Occurrence
Up 2.3% 0.3% 0.6% 1.4% 0.1% 0.0%
Ramp 97.4% 99.5% 99.0% 98.3% 99.8% 99.8%
Down 0.3% 0.3% 0.4% 0.3% 0.1% 0.1%
is summed to give a total, ηT , which is then used to get a percentage by: ηregion/ηT ×
100%. Thus, we have a normalized percentage of occurrence per region. One can easily
see the ramp region has the highest probability of wave occurrence. Roughly 87% of all
the waves seen in the 33 IP shock ramps were IAWs and 90% of these were large (≥5
mV/m). The dominant wave mode upstream is large amplitude Langmuir waves (∼53%
of all the large waves upstream for the 33 shocks with ramp waves), consistent with other
observations at IP shocks [Fitzenreiter et al., 2003; Thejappa and MacDowall , 2000] and
the terrestrial bow shock [Bale et al., 1997]. The downstream region is dominated by
IAWs (∼82% of all the downstream waves for the 33 IP shocks); however, ∼68% of those
were small IAWs. The dominant wave mode (for fpi < f < fpe) in the ramp region is large
IAWs. The most striking observation is that essentially 100% of the large amplitude
IAWs occur in the ramp when normalized by time, supporting theories on dissipation
in low Mach number shocks [Matsukiyo and Scholer , 2006b,a; Shimada and Hoshino,
2000].
Wilson III et al. [2007] also showed that the amplitude of large amplitude (≥ 5
mV/m peak-to-peak) IAWs increased with increasing Mach number and the shock
strength, consistent with larger shock strengths causing larger cross-field currents which
may provide free energy for wave generation, consistent with theory [Gary , 1981].
Because of the high probability of occurrence for large amplitude IAWs in the ramp
regions, the dependence of wave amplitude (|Exy |) on various shock parameters includ-
ing: θBn, n2/n1, MA, Mf , Mcs, and Te/Ti were tested for only the largest IAWs in each
155
Figure 5.1: Correlation plot for the largest IAWs in the ramp regions of 33 IP shocks
relating wave amplitude to the shock compression ratio (n2/n1).
shock ramp. The largest correlation is between |Exy | and n2/n1 (seen in Figures 5.1 and
5.2), consistent with larger shock strengths resulting in larger cross-field currents which
may provide free energy for wave generation. A slightly weaker correlation is seen be-
tween |Exy | (mV/m) and the fast mode Mach number (Mf ). Similar correlations were
seen for the other Mach numbers. No correlation between Te/Ti and wave amplitude was
observed. Polarization vectors for IAWs are consistent with a cross–field current source.
Wilson III et al. [2007] found the waves to produce resistivities of ∼1–856 Ω·m, using
Equation 1.44, which is roughly 107 times greater than the classical Spitzer electron-
ion collision estimates. One should also note that numerous simulation studies have
found Equations 1.44 and 1.45 to under estimate the actual resistivity due to IAWs
by up to 3 orders of magnitude [Dyrud and Oppenheim, 2006; Hellinger et al., 2004;
Petkaki et al., 2003, 2006; Petkaki and Freeman , 2008; Watt et al., 2002; Yadav et al.,
2009; Yoon and Lui , 2007]. These results are consistent with the theory that wave-
particle interactions are important for dissipation at subcritical shocks
Matsukiyo and Scholer [2006b,a].
156
Figure 5.2: Correlation plot for the largest IAWs in the ramp regions of 33 IP shocks
relating wave amplitude to the fast mode Mach number (Mf).
5.2 Analysis of Microphysics
[Work presented in Wilson III et al. [2010]]
This study is motivated by the examination of an unusual supercritical IP shock
observed by the Wind spacecraft on 04/06/2000 [Wilson III et al., 2009]. The previ-
ous study found unusual waveforms, called shocklets, upstream of the shock and strong
electron heating across the shock ramp. This section will focus on the unusually high
electron heating observed in the ramp region. Strong perpendicular electron heating
could be evidence of resistive heating due to wave-particle interactions [Thomsen et al.,
1985a]. Within ±5 seconds, corresponding to ±14 c/ωpi (upstream ion inertial lengths
where c is the speed of light and ωpi is the average upstream ion plasma frequency) of
the shock crossing, four waveform captures were obtained by the Wind spacecraft, two
of which have frequency spectra consistent with the ECDI and two with solitary wave
157
signatures. The results of this section were presented by Wilson III et al. [2010].
5.2.1 Data Sets and Analysis
Waveform captures were obtained from the Wind/WAVES instrument
[Bougeret et al., 1995], through the time domain sampler (TDS) receiver, which provides
a ∼17 ms waveform capture of 2048 points (from here on, a waveform capture is called
a TDS sample). TDS samples utilized herein provide two components of the electric
field in the XY-GSE plane, called Ex and Ey. The spin axis (roughly the Z-GSE com-
ponent) electric field was not sampled for these TDS samples. The TDS buffer stores
and evaluates waveforms based upon their amplitude. Thus if small amplitude waves
are observed, they will not be stored and transmitted to the ground if larger amplitude
waves fill the buffer. We define |Exy | =
√
Ex2 + Ey2 as the maximum peak-to-peak
(pk-pk) amplitude of the TDS samples. We only measure two components of the electric
field vector, roughly in the XY-GSE plane. This limits our ability to measure electric
fields parallel(perpendicular) to the magnetic field direction. Thus, when rotating the
electric field components into magnetic field-aligned coordinates (FACs), we rotate the
fields by the angle between the positive X-antenna and the XY-GSE projection of the
magnetic field vector. In FACs, we define the subscript ‖(⊥) as the direction paral-
lel(perpendicular) to the XY-GSE projection of the magnetic field direction. However,
if a significant fraction of the magnetic field is in the Z-GSE direction, the measured
E‖ may be significantly different from the total electric field along the magnetic field.
The converse applies to the measured values of E⊥. The lack of full 3D electric field
measurements can influence the determination of polarization as well.
To analyze the waveforms dynamically in time and frequency, we computed the Mor-
let wavelet transform [Torrence and Compo, 1998b] for the four waveforms examined in
this study. The wavelet transform has a number of distinct advantages over windowed
FFTs, but the two most important for our analysis are the resolved dynamic power spec-
tra at low frequencies (∼200 to 1000 Hz) and the analysis of non-stationary power inten-
sifications at many different frequencies simultaneously [Torrence and Compo, 1998a].
Windowed FFTs have imposed periodicity and have a single time scale while wavelets
do not. However, two sanity checks are performed to expose noise or numerical artifacts
158
due to the interpolation routines. The cone of influence is a calculation done to deter-
mine the time-frequency region of the wavelet transform which is subject to edge effects.
Thus, values above (with respect to frequency and time) this line can be trusted to not
have artificial effects due to the finite time range of the data. The second calculation
determined the 95% confidence level (also called ”significant at the 5% level”). The
95% confidence level outlines regions of the wavelet transform which enclose intensities
above 95% of the data. The 95% confidence level can be calculated from a theoretical
red or white noise spectrum, but here we used the actual waves since they are clearly
well above the background noise level of the solar wind (.0.1 mV/m at 1 AU). Both
significance tests are used to increase our ability to make quantitative analysis using
the wavelet transforms.
The magnetic field instrument on board Wind [Lepping et al., 1995] is composed of
dual triaxial fluxgate magnetometers. High time resolution (HTR) magnetic field data,
sampled at ∼11 samples/s, were used to define the ramp region, or transition region,
of the IP shock, as well as to identify low frequency magnetic fluctuations upstream of
the ramp [Wilson III et al., 2009]. The ramp is defined as the interval from the point
of lowest magnetic field immediately preceding the discontinuity (in magnetic field am-
plitude) to the point of highest magnetic field immediately following the discontinuity
[Farris et al., 1993].
Low energy (<30 keV) electron and ion distributions were obtained from the Wind
3DP EESA and PESA particle detectors [Lin et al., 1995]. The EESA Low (EL and
ELB in burst mode) instrument can measure electrons at 15 different energies from a
few eV to a little more than a keV. The PESA Low (PL and PLB in burst mode) instru-
ment measures ions at 14 different energies from as low as 100 eV to as high as 10 keV
(typical range in the solar wind is 700 eV to 6 keV). The instrument is used primarily
for bulk solar wind properties like ion velocity, density, and temperature. The PESA
High (PH and PHB in burst mode) instrument measures ions at 15 different energies
from as low as 80 eV to as high as 30 keV (typical range in the solar wind is 500 eV to
28 keV). The time resolution of each instrument depends on whether the instruments
are in burst mode or not. In burst mode, both EESA and PESA instruments return full
three dimensional particle distributions every three seconds (∼1 spin period). Electron
and ion distributions were examined for possible wave free energy sources and evidence
159
of heating. Estimates of the electron temperature anisotropies in both the cold dense
core (subscript c) and the hotter more tenuous halo (subscript h) can be obtained from
full 3D electron distributions. The method for determining the break energy between
halo and core electrons is outlined by Wilson III et al. [2009].
High energy (>30 keV) electron and proton measurements were obtained from three
pairs of double-ended solid state telescopes (SSTs), each with a pair or triplet of closely
stacked silicon semiconductor detectors [Lin et al., 1995]. The SSTs provide a full 4π
steradian coverage with a 22.5◦ × 36◦ angular resolution and ∆E/E ≈ 0.2 energy res-
olution. One side of each detector is covered in a thin lexan foil (SST Foil) to stop
protons up to ∼400 keV while leaving electrons relatively unaffected. The opposite end
of the detector is left open (SST Open) using a common broom magnet to sweep away
electrons below ∼400 keV while leaving the protons relatively unaffected. Thus, in the
absence of any higher energy (penetrating) particles, the SST Foil counts only electrons
and the SST Open counts only ions.
The relevant shock parameters, determined by Kasper [2007], are the shock normal
angle, θBn, fast mode Mach number, Mf , shock normal vector, nˆ, upstream normal flow
velocity in the shock frame, Un, upstream solar wind velocity, Vsw, and shock strength,
Ni2/Ni1. The values for the 04/06/2000 event are: Mf ∼ 4, θBn ∼ 68◦, Un ∼ 278
km/s, Vsw ∼ <-370.,-18.,-22.> km/s (GSE coordinates), nˆ ∼ <-0.98,-0.08,-0.16> (GSE
coordinates), and Ni2/Ni1 ∼ 4.
5.2.2 Observations
Figure 5.3 shows a plot of the magnitude (first panel) and the GSE components of
the magnetic field (labeled with color coded component letters, second panel) for the
04/06/2000 event between 16:32:03 and 16:32:15 UT. The shock arrival time, or middle
of the magnetic ramp, is 16:32:09.2 UT. The shaded regions correspond to the time
ranges of each particle distribution found in Figure 5.4. The vertical color-coded lines
labeled with capital letters indicate the locations of the TDS samples shown in Figure
5.5.
Figure 5.4 shows four ion (top row) and electron (bottom row) distribution functions
plotted with the horizontal axes corresponding to the direction parallel to the magnetic
field. The plots are projected into the plane produced by the solar wind velocity and the
160
Time (HH:MM:SS)
16:32:08 16:32:10 16:32:12 16:32:1416:32:04 16:32:06
A
B
C
D
|B
| 
(n
T
)
20
15
10
B
 [
G
S
E
] 
(n
T
)
0
-10
5
-15
-5
HTR Wind B-Field:  04/06/2000 16:32:02-16:32:16  UT
XYZ
Figure 5.3: The plot shows the magnitude and the GSE coordinates of the magnetic
field for the IP shock on 04/06/2000. The shaded regions correspond to the time ranges
of each particle distribution found in Figure 5.4. The vertical color-coded lines indicate
the locations of the TDS samples shown in Figure 5.5 labeled with the respective capital
letters. The time range of the plot is 16:32:02-16:32:16 UT and the shock arrival time
is 16:32:09.2 UT.
local magnetic field. The electron and ion velocity ranges in the plots are ±20,000 km/s
and ±2,500 km/s, respectively. The phase space density color scales for each instrument
are the same for the four different distributions in each row.
A summary of the relative changes in moments for the four electron distributions
shown in Figure 5.4 can be found in Table 5.2. The relative change in any given
moment is defined as ∆Qs = ((Qf - Qi)/Qi)s × 100%, where the subscripts f and
i represent the final and initial state, respectively and the subscript s represents the
particle species (e.g. core). The final and initial state for the calculations in Table 5.2
are specific only to the four distributions shown in Figure 5.4. Thus, the final state refers
to the distribution starting at 16:32:12 UT and the initial state refers to the distribution
starting at 16:32:03 UT. The top half of the table shows the values for ∆Tes, ∆T⊥s,
161
V
 p
e
rp
e
n
d
ic
u
la
r 
(1
0
0
0
 k
m
/s
)
-20
-10
0
10
20
V parallel (1000 km/s)
-20 -10 0 10 20
V parallel (1000 km/s)
-20 -10 0 10 20
V parallel (1000 km/s)
-20 -10 0 10 20
V parallel (1000 km/s)
-20 -10 0 10 20
-2000
-1000
0
1000
2000
Vp
e
rp
 
[km
/s
]
-2000 -1000 0 1000 2000
Vpara [km/s] -2000 -1000 0 1000 2000Vpara [km/s] -2000 -1000 0 1000 2000Vpara [km/s] -2000 -1000 0 1000 2000Vpara [km/s]
16:32:03 - 16:32:06 16:32:06 - 16:32:09 16:32:09 - 16:32:12 16:32:12 - 16:32:15
Pesa High
Eesa Low Eesa Low Eesa Low Eesa Low
Pesa HighPesa HighPesa High
Vsw
nsh
Vsw
nsh
Vsw
nsh
Vsw
nsh
16:32:03 - 16:32:06 16:32:06 - 16:32:09 16:32:09 - 16:32:12 16:32:12 - 16:32:15
Vsw
nsh
Vsw
nsh
Vsw
nsh
Vsw
nsh
Figure 5.4: Four particle distributions from both the PHB (top row) and ELB (bottom
row) detectors on the Wind 3DP instrument. The horizontal axis of each plot is parallel
to the magnetic field and the vertical axis is perpendicular in the plane created by the
solar wind velocity and magnetic field direction. Each pair of samples, ELB and PHB,
are outlined by a colored box which corresponds to the shaded regions in Figure 5.3.
The black solid line represents the projection of the solar wind velocity and the red
dotted line represents the shock normal vector projection. Note the sample times are
the same for each instrument.
∆T‖s, and ∆(T⊥/T‖)s for the core, halo, and entire distribution (i.e. core and halo
combined). The bottom of Table 5.2 shows the global changes (downstream, subscript
2, over upstream, subscript 1) across the 04/06/2000 event for the core, halo, and entire
distribution. Here, the subscripts 1 and 2 refer to asymptotic values for the upstream
and downstream estimated by averaging the quantities over 10 minute intervals in each
region (i.e. outside of the time period shown in Figure 5.3 and calculated for more than
just the four distributions shown herein). The quantities for global changes shown in
Table 5.2 are the average, perpendicular, and parallel temperatures.
The core electrons show the strongest heating in bulk, parallel, and perpendicular,
components while the halo dominates in the change in temperature anisotropy. Note
also that there appears to be a preference toward perpendicular heating as indicated by
162
Table 5.2: Wind 3DP ELB stats across the 04/06/2000 IP shock
Species ∆Tes ∆T⊥s ∆T‖s ∆(T⊥/T‖)s
Eesa Low Burst (Core/Halo)
Core 150% 174% 110% 30%
Halo 42% 67% 5% 58%
Eesa Low Burst (Entire Distribution)
Entire 137% 164% 96% 35%
Temperature Anisotropies (T⊥/T‖)s
16:32:03 16:32:06 16:32:09 16:32:12
Core 0.81 0.75 0.73 1.06
Halo 0.77 0.87 0.97 1.23
Entire 0.79 0.75 0.73 1.06
Global Changes Across Shock
(T2/T1)s (T⊥2/T⊥1)s (T‖2/T‖1)s
Core 3.3 3.8 2.5
Halo 1.4 1.8 0.8
Entire 3.0 3.7 2.3
∆T⊥s for all three electron components. The middle part of Table 5.2 shows the specific
values of (T⊥/T‖)s for the core, halo, and entire energy range for each distribution
shown in Figure 5.4. Notice that all three temperature anisotropies increase across the
shock ramp, with the core increasing by & 30% and the halo by & 58%. The global
changes across the shock show a preference for perpendicular heating as well, increasing
by almost a factor of four (see bottom of Table 5.2).
To be more quantitative, let us define the following parameter, α, used by
Omidi and Winske [1990] as a test of whether or not the shock is heating the particles
adiabatically:
α ≡ (T es,2
T es,1
)− (N 2
N 1
)γ−1 (5.1a)
α⊥ ≡ (T es,2⊥
T es,1⊥
)− (N 2
N 1
)γ−1 (5.1b)
where Tes,j is the asymptotic estimate of the electron temperature in for the regions
defined by the subscript, j, which represents the upstream (j = 1) and downstream (j
= 2), Tes,j⊥ are the perpendicular asymptotic estimates of the electron temperature, Nj
163
are the asymptotic estimates of the particle density, and γ is the ratio of specific heats
(here we used 5/3). If the factor α is equal to zero, then the electrons were heated adia-
batically. If the α >(<) 0, then the electrons were over(under) heated across the shock.
All the asymptotic values of interest are given in Table 5.3 for comparison with the same
definition for the ∆Q quantities as in Table 5.2. As one can see, the only shock with
positive values of α for the entire electron distribution is the 2000-04-06 event. Oddly
enough, the two weakest events, 1996-04-03 and 1996-04-08, have α > 0 for both the
core and halo electrons but α < 0 when the entire distribution is considered. Although
the values of ∆B and ∆T⊥,e for the 2000-04-06 event are roughly the same, this result
is unique to this event.
The differences in electron heating between the 04/06/2000 event and the four more-
Table 5.3: Wind 3DP Global ELB stats across the five IP shocks
Species ∆Tes ∆T⊥s ∆T‖s ∆B α α⊥
1996-04-03 Event
Core 45% 46% 45% 0.15 0.15
Halo 134% 125% 153% 1.03 0.95
Entire -14% -15% -13% 57% -0.45 -0.45
1996-04-08 Event
Core 54% 48% 65% 0.12 0.07
Halo 119% 108% 144% 0.78 0.66
Entire -15% -19% -6% 62% -0.56 -0.60
1997-10-24 Event
Core 54% 57% 49% -0.28 -0.26
Halo 87% 103% 60% 0.04 0.20
Entire 4% 10% -8% 92% -0.79 -0.72
1997-12-10 Event
Core 12% 13% 9% -0.72 -0.71
Halo 44% 38% 55% -0.40 -0.46
Entire 5% 6% 2% 121% -0.79 -0.78
2000-04-06 Event
Core 228% 281% 153% 0.83 1.36
Halo 37% 84% -23% -1.08 -0.61
Entire 209% 267% 128% 266% 0.63 1.22
typical events are: 1) a global decrease in Th,‖ across the shock, 2) an increase by over a
164
factor of three in Tce and Te across the shock, 3) a global increase in both Tc,⊥/Tc,‖ and
Th,⊥/Th,‖ across the shock, 4) a global decrease in both Th,‖/Tc,‖ and Th,⊥/Tc,⊥ across
the shock, and 5) sustained flattop electron distributions downstream for over an hour.
We suggest that these differences are the result of wave-particle heating.
Figure 5.5 shows four TDS samples plotted in FACs at the times shown by the
vertical lines in Figure 5.3. For each TDS sample, the magnetic field estimates were
determined by averaging the HTR MFI data over the time range of the TDS sample.
The angle of the magnetic field unit vector out of the XY-GSE plane, θBxy, for each
TDS sample is: +10◦ for A, +10◦ for B, -10◦ for C, and -31◦ for D. The relative scales
for both E‖ (shown in red) and E⊥ (shown in blue) are shown with the vertical arrows
in each plot (e.g. wave D is ∼160 mV/m pk-pk for E⊥). The peak values of |Exy | for
the four TDS samples are ∼39, ∼35, ∼20, and ∼166 mV/m. Note that these values
of |Exy | are a lower bound to the actual maximum amplitudes of the waves since, as
discussed in Section 5.2.1, we only measure two components of the electric field. Below
the four waveforms are their associated hodograms. The time ranges of the hodograms
are defined by the magenta boxes overlaying the waveforms. The solid green line corre-
sponds to the XY-GSE projection of the shock normal vector, nxy/|n|. The projection
of the shock normal vector was scaled to the maximum value of |Exy | for each TDS
sample for ease of comparison to the polarization of the electric fields.
Each panel in Figure 5.6 represents a Morlet wavelet transform power spectral den-
sity plot of a single component of a waveform from Figure 5.5, E‖ on the left and E⊥ on
the right. In each panel, two horizontal lines, a bowl shaped line, and multiple contours
are plotted. The two horizontal lines correspond to the fundamental and first harmonic
of the electron cyclotron frequency. The cyclotron frequency and first harmonic were
calculated by interpolating the magnitude of the magnetic field over the duration of each
TDS sample. The bowl-like line corresponds to the cone of influence and the contours
correspond to the 95% confidence level. The top three rows share the same power range
of 0.01 to 1000 (mV/m)2/Hz for the wavelets, while waveform D ranged from 0.05 to
45000 (mV/m)2/Hz because of its much larger amplitude.
Further examination of the wavelet transforms of waveforms A through D show
enhanced power near the fundamental and first harmonic of the electron cyclotron fre-
quency (fce) for E⊥ and E‖. Half-integer harmonic power intensifications are most easily
165
2 4 6 8 10 12 14 16
Time (ms)
2 4 6 8 10 12 14 16
Time (ms)
Electric Field vs. Time on:  04-06-2000 at:  16:32:09.380 UT
A E
E
~30 mV/m
-20  0 20
-20
 0
20A
E
E
A
-10 0 10
-10
 0
10
B
E
E
Electric Field vs. Time on:  04-06-2000 at:  16:32:09.447 UT
B ~15 mV/m
E
E
~32 mV/mB
C
-10
 0
10
-10  0 10
E
E
Electric Field vs. Time on:  04-06-2000 at:  16:32:09.886 UT
C
~10 mV/m E
E~8 mV/m
D
-50  0 50
-50
 0
50
E
E
Electric Field vs. Time on:  04-06-2000 at:  16:32:12.498 UT
D
~120 mV/m
E
E
~50 mV/m
~17 mV/m
~11 mV/m
~160 mV/m
~35 mV/m
~30 mV/m
Figure 5.5: Four waveform captures in and around the ramp region of the 04/06/2000
IP shock. The waves in Panels A (16:32:09.380 UT) and B (16:32:09.447 UT) are trains
of steepened waves while the waves in Panels C (16:32:09.886 UT) and D (16:32:12.498
UT) are solitary-like waves. The top panels of each waveform (red) corresponds to E‖
and the bottom (blue) corresponds to E⊥. Their respective peak-to-peak amplitudes
are marked by the vertical black arrows. Below the four waveforms are their respective
hodograms labeled with the same capital letters corresponding to the time range defined
by the magenta boxes in each TDS sample. The solid green line in each hodograms
represents the XY-projection of the shock normal vector in FACs.
166
1.0
10.0
Fr
e
q
u
e
n
cy
 (k
H
z)
0.01
1000
0.46
21.5
W
av
e
le
t 
P
o
w
e
r
Parallel Electric Field Perpendicular Electric Field
0.01
1000
0.46
21.5
W
av
e
le
t 
P
o
w
e
r
1.0
10.0
Fr
e
q
u
e
n
cy
 (k
H
z)
0.01
1000
0.46
21.5
W
av
e
le
t 
P
o
w
e
r
1.0
10.0
Fr
e
q
u
e
n
cy
 (k
H
z)
0.05
45000
4.83
466.
W
av
e
le
t 
P
o
w
e
r
1.0
10.0
Fr
e
q
u
e
n
cy
 (k
H
z)
Morlet Wavelet Power Spectra
AA
B B
CC
D D
Figure 5.6: The Morlet wavelet power spectrum for the four waveforms in Figure 5.5,
labeled A through D accordingly. The left column is for E‖ and the right for E⊥. The
top three rows share the same spectral range of 0.01 to 1000 for the wavelets while
waveform D ranged from 0.05 to 45000, while all four have been normalized to conserve
energy in the wavelet transformations. In each panel, two horizontal lines mark the
fundamental and first harmonic of the electron cyclotron frequency. The bowl-like line
plotted in each spectra marks the cone of influence while the contours mark the 95%
confidence levels [Torrence and Compo, 1998a].
167
seen in the E‖ component of waveform B, outlined by the black box in Figure 5.6. How-
ever, the large amplitude of the higher frequency components makes a detailed analysis
difficult for these events. Figures 5.7 through 5.10 present examples of snapshots of
windowed FFTs for the four waveforms from Figure 5.5 to provide a complementary
way to examine the power spectra. The left-hand column of panels shows E‖, the power
spectra (mV/m2/Hz) versus frequency (kHz) for the orange box, and the power spec-
tra versus frequency for the black box, respectively. The right-hand column shows the
same for E⊥. The vertical lines in the power spectra correspond to integer (green) and
half-integer (purple) multiples of the electron cyclotron frequency. All power spectra
plots in Figures 5.7 and 5.8 range from 10−5 to 4×10−1 while in Figures 5.9 and 5.10
they range from 10−6 to 2×100.
Figures 5.7 through 5.10 show that waveforms A throughD have mixtures of integer
half-integer multiples of the electron cyclotron frequency, fce, that change throughout
the TDS samples. Note that the maximum Doppler shifted IAW frequency for waveform
A(B) was estimated to be roughly 7.5 kHz(8.5 kHz), assuming wavelengths consistent
with the measurements of Fuselier and Gurnett [1984], and the background noise level
for this event is roughly 10−6 mV/m2/Hz. The images clearly illustrates how the wave-
forms shift power between integer and half-integer harmonics of fce during the duration
of the TDS samples. If we calculate the power spectrum using the entire time range for
each TDS sample, the shifting peaks smear together into one peak. Thus, the combina-
tion of wavelet and windowed FFT analysis shows that waveforms A through D exhibit
mixtures of integer and half-integer multiples of fce throughout the samples. Waveforms
C and D differ from waveforms A and B in that they do not show broad power en-
hancements above 4 kHz. The difference is likely due to the fact that waveforms A and
B are composed of IAWs and cyclotron harmonic waves whereas the power spectra for
waveforms C and D are dominated primarily by the solitary waves. Also, waveforms A
and B often have E⊥ > E‖, consistent with cyclotron harmonic or Bernstein-like waves
[Usui et al., 1999]. Note that 1/fce ∼ 1.95 ms(1.62 ms) for waveform C(D). Also, the
solitary wave in waveform C near 4 ms(8 ms) has a period of ∼1.33 ms(1.44 ms), slightly
smaller than the cyclotron period. The solitary wave in waveform D near 5 ms(8 ms)
has a period of ∼1.56 ms(1.67 ms), almost identical to the cyclotron period. Thus,
the Bernstein-like emissions are still simultaneously observed with the solitary waves
168
2 4 6 8 10 12 14 16
Time (ms)
W
av
e
 P
o
w
e
r
2 4 6 8 10 12 14 16
Time (ms)
fcen Peaks
fce(n + 1/2) Peaks
A
Electric Field vs. Time on:  04-06-2000 at:  16:32:09.380 UT
E
A~30 mV/mFrame: 230
E
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Frame: 230
7 8 11
3/2 5/2 9/2 19/2
9/2
13/2
19/2 21/2
821
Frame: 663 Frame: 663
10
-4
10
-3
10
-2
10
-1
fcen Peaks
fce(n + 1/2) Peaks
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
82 4
13/2 19/2 21/23/2
5/2
13/2
19/2 21/2
7/2
84 5
W
av
e
 P
o
w
e
r
10
-4
10
-3
10
-2
10
-1
Frame: 663
E
Frame: 663
E
Frame: 230
E
Frame: 230
E
~30 mV/m
Figure 5.7: Example snapshots of windowed FFTs of waveform A from Figure 5.5. The
left-hand column of panels shows E‖, the power spectra (mV/m
2/Hz) versus frequency
(kHz) for the orange box, and the power spectra versus frequency for the black box,
respectively. The right-hand column shows the same for E⊥. The vertical lines in the
power spectra correspond to integer (green) and half-integer (purple) multiples of the
electron cyclotron frequency. All power spectra plots in this figure range from 10−5 to
4×10−1.
169
up to ∼14 c/ωpi away from the shock ramp. These integer and/or half-integer har-
monic intensifications are consistent with previous observations of cyclotron harmonic
or Bernstein-like waves [Usui et al., 1999].
Figure 5.11 shows the 2D Hammer-Aitoff projection of 27 and 40 keV electrons from
fcen Peaks
fce(n + 1/2) Peaks
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
2 4 6 8 10 12 14 16
Time (ms)
2 4 6 8 10 12 14 16
Time (ms)
B
Electric Field vs. Time on:  04-06-2000 at:  16:32:09.447 UT
E
Frame: 002
E BFrame: 002
751 1110
5/2
7/2
17/2
84 7 11
3/2
5/2 13/2
11/2 19/2
Frame: 764 Frame: 764
W
av
e
 P
o
w
e
r
10
-4
10
-3
10
-2
10
-1
fcen Peaks
fce(n + 1/2) Peaks
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
951 11
3/2
5/2 9/2 13/2
11/2 15/2
2 4 951 3
13/2
11/2
21/2
15/2
W
av
e
 P
o
w
e
r
10
-4
10
-3
10
-2
10
-1
Frame: 002
E
Frame: 002
E
Frame: 764
E
Frame: 764
E
~30 mV/m~30 mV/m
Figure 5.8: Example snapshots of windowed FFTs of waveform B from Figure 5.5. The
format is the same as Figure 5.7. All power spectra plots in this figure range from 10−5
to 4×10−1.
the SST Foil instrument on Wind between 16:31:01 UT and 16:32:15 UT. The Hammer-
Aitoff projections display a full 4π steradian of the measured particles. The data are
plotted in units of phase-space density (s3cm−3km−3) with ranges of 5×10−22 (purple)
to 2.2×10−21 (red) for the 27 keV electrons and 5×10−22 (purple) to 1.1×10−21 (red) for
the 40 keV electrons. Different scales were used for the two energies to highlight the in-
tensification near the center of each plot. The SST Foil distributions are sampled every
12-13 seconds (∼4 spin periods) for this event. Four symbols are projected onto the 3D
170
maps corresponding to relevant vectors including: +, the magnetic field direction, ♦,
the anti-parallel magnetic field direction, ∗, the solar wind velocity direction (roughly
negative X-GSE for this event), and △, the shock normal vector. The dotted lines
denote 30◦ increments in both latitude and longitude. The beam-like feature, observed
in every SST distribution of Figure 5.11, parallel to the magnetic field is also intermit-
tently observed for nearly 20 minutes upstream of the shock ramp (not shown). It is
not clear whether the beam-like feature is a contributing source of free energy for the
solitary waves, but electron beams have been associated with solitary wave observations
in many studies (e.g. Ergun et al. [1998a]). Also, a beam-like feature roughly 30◦ to
the right of the shock normal and solar wind direction is observed in both energies with
strong enhancements in the last two panels. Correspondingly, there is another beam-like
enhancement roughly 180◦ to the right of the beam-like feature near the shock normal
and solar wind directions, suggesting they are contaminated data bins.
5.2.3 Discussion
Because the full three dimensional electric field is not measured, we can infer some
properties of the waves but cannot definitively determine the polarization. Recall that
θBxy is not zero for any of the four waveforms. If we assume the solitary waves prop-
agate along the magnetic field, the relative magnitude of E⊥ to E‖ should be smaller
for waveform D than C. The reason is that we should be measuring a unipolar E⊥ and
bipolar E‖, but the observation of a bipolar E⊥ we believe is the result of not measuring
Ez. Thus, the smaller bipolar E⊥ to E‖ in waveform D compared to C is likely due to
the larger θBxy.
Typical TDS samples in most IP shock ramps are consistent with Doppler shifted
IAWs [Wilson III et al., 2007]. For comparison to waveforms A and B, Figure 1.14
shows an example of a typical IAW observed in the 04/08/1996 IP shock ramp, one of
the four lower Mach number shocks examined by Wilson III et al. [2009]. The paral-
lel(perpendicular) components are plotted in red(blue) with the associated hodogram
to the right. The hodogram has the XY-GSE projection of the shock normal vector
plotted as a solid green line, as in the hodograms of Figure 5.5. The wavelet transforms
for the parallel and perpendicular components are plotted below the waveform captures.
The IAW wavelet shows a strong isolated band of power between ∼2-5 kHz for both
171
fcen Peaks
fce(n + 1/2) Peaks
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
2 4 6 8 10 12 14 16
Time (ms)
2 4 6 8 10 12 14 16
Time (ms)
C
Electric Field vs. Time on:  04-06-2000 at:  16:32:09.886 UT
E
Frame: 238
C ~15 mV/m
Frame: 238E
82 4 961 11
13/2 21/2
19/2
2 51 3
fcen Peaks
fce(n + 1/2) Peaks
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Frame: 665 Frame: 665
5/2 13/2
82 41 10
3/2
5/2 13/2
7/2 15/2
21/2
19/2
5
W
av
e
 P
o
w
e
r
10
-4
10
-3
10
-2
10
-1
10
-5
W
av
e
 P
o
w
e
r
10
-4
10
-3
10
-2
10
-1
10
-5
Frame: 238
E
Frame: 665
E
Frame: 238
E
Frame: 665
E
~15 mV/m
Figure 5.9: Example snapshots of windowed FFTs of waveform C from Figure 5.5. The
format is the same as Figure 5.7. All power spectra plots in this figure range from 10−6
to 2×100.
172
fcen Peaks
fce(n + 1/2) Peaks
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
2 4 6 8 10 12 14 16
Time (ms)
EFrame: 052
Electric Field vs. Time on:  04-06-2000 at:  16:32:12.498 UT
2 4 6 8 10 12 14 16
Time (ms)
~150 mV/m
ED D
Frame: 052
4 9 106
3/2
5/2
15/2
4 91 106
5/2
15/2
fcen Peaks
fce(n + 1/2) Peaks
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Frequency (kHz)
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
Frame: 598 Frame: 598
82 71 6
7/2
21/2
19/2
821
13/2
7/2
17/2
19/2
21/2
W
av
e
 P
o
w
e
r
10
-4
10
-3
10
-2
10
-1
10
-5
W
av
e
 P
o
w
e
r
10
-4
10
-3
10
-2
10
-1
10
-5
Frame: 052
E
Frame: 598
E
Frame: 052
E
Frame: 598
E
~150 mV/m
Figure 5.10: Example snapshots of windowed FFTs of waveform D from Figure 5.5.
The format is the same as Figure 5.7. All power spectra plots in this figure range from
10−6 to 2×100.
173
components. This isolated, relatively narrow, band of emission is typical of the IAWs
observed at the four lower Mach number IP shocks. The wavelets for waveforms A
through D are clearly different, supporting our argument that these waveforms are not
simple IAWs, rather mixtures of multiple modes. We also examined snapshot FFTs
of the IAWs in the four more-typical IP shock ramps and found that the waves show
slight enhancements below ∼1 kHz, but no noticeable enhancements near integer and
half-integer harmonics were observed. Waveforms A and B are quite different than the
typical IAWs observed at IP shocks [Wilson III et al., 2007] but similar to the waves
observed by Hull et al. [2006]. However, we argue that these waves are not simple IAWs
for the following reasons: 1) neither waveform electric field component oscillates sym-
metrically about zero; 2) both waveforms have mixtures of frequencies consistent with
IAWs and electron cyclotron harmonics in their power spectra; and 3) the polarizations
are correlated with the shock normal vector, not the magnetic field.
The 04/06/2000 event was the only shock of the five studied in detail to show strong
sustained core ion heating across the shock. Although the PL detector has a narrow field
of view (∼180◦ × 14◦) compared to PH (∼360◦ × 14◦), which could limit the accurate
measurement of the solar wind in the immediate downstream region of strong shocks.
We determined the relative accuracy of the PL measurements by comparing the down-
stream density measured by the PL detector to the downstream density estimated from
the plasma line (indicative of local density) seen by the WAVES thermal noise receiver
[Bougeret et al., 1995]. We only use the temperature increase determined from the PL
detector as a qualitative proxy for the bulk ion temperature increase. The sustained
ion temperature increase, measured with PL, across the 04/06/2000 shock was roughly
a factor of 7 for Ti,⊥ and 5 for Ti,‖ with spikes in downstream temperatures exceeding
factors of 8 and 6 respectively, consistent with previous observations [Thomsen et al.,
1985a]. However, the the four more-typical events showed ion temperature increases
that were less than a factor of three. The PH distributions showed qualitatively similar
changes.
A distinct difference in both electron and ion properties is observed across the
04/06/2000 shock compared to the other four events ofWilson III et al. [2009]. The ions
show strong heating in both the bulk of the ion distribution and high energy tail (ob-
served as diffuse ions for the 04/06/2000 event), while all four more-typical events only
174
2000-04-06/16:31:50-16:32:03 UT 2000-04-06/16:31:50-16:32:03 UT
2000-04-06/16:32:03-16:32:15 UT 2000-04-06/16:32:03-16:32:15 UT
df  (s  km  cm  )
3 -3-35.0 x 10
-22
2.2 x 10
-21
df  (s  km  cm  )
3 -3-35.0 x 10
-22
1.1 x 10
-21
2000-04-06/16:31:26-16:31:38 UT 2000-04-06/16:31:26-16:31:38 UT
2000-04-06/16:31:38-16:31:50 UT 2000-04-06/16:31:38-16:31:50 UT
Wind/3DP SST Foil (40.2 keV)
2000-04-06/16:31:01-16:31:13 UT
Wind/3DP SST Foil (27.0 keV)
2000-04-06/16:31:01-16:31:13 UT
2000-04-06/16:31:13-16:31:26 UT 2000-04-06/16:31:13-16:31:26 UT
Figure 5.11: Hammer-Aitoff projections of SST Foil observations from 16:31:01 UT to
16:32:15 UT of field-aligned enhancements of 27-40 keV electrons. The + symbol rep-
resents the magnetic field-aligned direction, diamond the anti-parallel field direction,
asterisks is the solar wind direction, and the triangle represents the shock normal di-
rection. The left column plots the 27 keV energy bin while the right plots the 40 keV
energy bin. All images in each column have the same color scales, shown at the bottom
of each column.
175
showed evidence for slight heating in the high energy ion tails. The 04/06/2000 event is
also the only event to show sustained (over an hour) electron heating downstream of the
shock ramp observed as flattop distributions, while the four more-typical events showed
weak heating consistent with adiabatic compression. The core and halo electron heating
is strongly anisotropic (T⊥ > T‖) in the 04/06/2000 event, while the electron heating in
the four more-typical events showed no particular preference towards parallel or perpen-
dicular, consistent with previous studies of marginally critical shocks [Thomsen et al.,
1985a]. Typical supercritical quasi-perpendicular shocks exhibit perpendicular heating
due to adiabatic compression and parallel heating due to a two-step process whereby the
cross-shock potential accelerates the electrons parallel to the magnetic field and then
the free energy associated with this accelerated beam excites microinstabilities, which
redistribute the electrons in phase space to form flattop distributions [Thomsen et al.,
1987]. This two-step process leads to a roughly isotropic electron distribution in the
downstream region of typical supercritical quasi-perpendicular shocks. The isotropy
increases as one moves further into the downstream region, consistent with relaxation
of the distributions due redistribution in phase space [Thomsen et al., 1985a, 1987].
The core electrons in the 04/06/2000 event, however, are observed to become more
anisotropic (Tc,⊥/Tc,‖ increases) as one progresses further downstream for up to an hour
after the shock encounter (not shown). Adiabatic compression due to the conservation
of the first adiabatic invariant cannot explain this observation since the magnetic field
magnitude does not correlate with Tc,⊥ in the downstream, as shown quantitatively in
Table 5.3. These observations suggest that wave heating is important, supported by the
observation of electron cyclotron harmonic waves (see Figure 1.17) and solitary waves
(not shown) similar to those of waveforms C and D, which would act to increase this
anisotropy.
5.3 Summary and Conclusions
The first part of this chapter focused on results from a statistical study of high
frequency (&1 kHz) waves at IP shocks. The results of the statistical study showed
that the amplitude of large amplitude (≥ 5 mV/m peak-to-peak) IAWs increased with
increasing Mach number and the shock strength, consistent with larger shock strengths
176
causing larger cross-field currents which may provide free energy for wave generation.
The study also found that large amplitude (≥ 5 mV/m peak-to-peak) IAWs had the
highest probability of occurrence in the ramp region, consistent with theory [Gary ,
1981].
The second part of this chapter focused on the conclusions from the study of a
specific IP shock event, namely the atypcial IP shock of 04/06/2000 Wilson III et al.
[2010]. The waveforms were the first observations of large amplitude (>100 mV/m
pk-pk) solitary waves and large amplitude (∼30 mV/m pk-pk) waves exhibiting charac-
teristics consistent with electron Bernstein waves at an interplanetary shock. Waveforms
A through D in Figures 5.7 through 5.10 all clearly show enhanced power near integer
and half-integer harmonics of the cyclotron frequency. Waveforms A and B show sig-
nificant power along the shock normal and are obliquely polarized with respect to the
magnetic field, consistent with the ECDI. If we use Equations 1.44 and 1.45 to estimate
the contribution of anomalous resistivity due to waveforms A and B we find ηIA ∼
920 and 740 Ωm, respectively. These values are over 108 times the classical Spitzer
collisional estimates, consistent with our arguments for their relative importance to
the observed heating. Strong particle heating in both the halo and core of the elec-
trons and ions is observed near these waves, consistent with the simulation results of
Matsukiyo and Scholer [2006a].
Waves with power spectra exhibiting characteristics of both IAWs and Bernstein-
like emissions are consistent with the predicted spectrum of the ECDI. The IAWs are
Doppler shifted and resonantly interact with the Bernstein-like emissions, coupling to
form a time-dependent diffuse frequency and wave vector spectrum
[Matsukiyo and Scholer , 2006a]. The ECDI is an attractive candidate for the event
herein because it can explain both the particle heating and the atypical waveforms.
The current produced by the relative drift between incident electrons and reflected ions
is unstable to the ECDI. Our observations suggest that this current is the source of free
energy for the waveforms observed in the 2000-04-06 event.
Further evidence to suggest that our observations are consistent with the ECDI is
shown using a simple test of linear instability. Using the ECDI instability criterion, at
the Debye length cutoff or (k λDe) > 1, determined from Equation 15 in Forslund et al.
177
[1972] under the observed conditions, we found that the ECDI instability criterion es-
timates are a factor of 20 or more above the threshold. Therefore, we argue that the
instability criterion determined by Forslund et al. [1972] is easily satisfied.
To further examine the consistence of the observed waves with the ECDI, we looked
at ion measurements from 10’s of eV to a few MeV, using the PH and SST Open
detectors, for reflected ion signatures, the source of free energy for the ECDI. No beam-
like or gyrophase-restricted features could be definitively discerned in high energy ion
SST Open measurements near the shock ramp. The SST measurements did show en-
hancements in the energy flux of 1-6.7 MeV ions and 100-500 keV electrons (not shown
herein) upstream of the 04/06/2000 event concurrent with the 12 shocklets observed
by Wilson III et al. [2009]. However, simultaneous increases in high energy particle
fluxes in and around low frequency waves is not unusual [Sanderson et al., 1985]. We
also examined distributions from the PH instrument (shown in Figure 5.4). We did
not detect reflected ion beams using the PH detector due to the combination of two
factors: 1) the 2000-04-06 event had very atypical ion distributions upstream called
diffuse ions, thought to be remnants of reflected ion beams scattered by wave-particle
interactions, [Paschmann et al., 1981] and 2) UV-light contamination. Note 12 shock-
lets of the type observed upstream of this event have been shown to have a one-to-one
correlation with reflected ions observed as diffuse ion distributions [Hoppe et al., 1981,
1982; Hoppe and Russell , 1983], as observed in this case as well. Thus, the simulta-
neous observation of shocklets with diffuse ions is evidence that reflected ions exist
upstream of the 04/06/2000 IP shock. A cursory comparison of the shock parameters
for this event with the critical Mach number estimates of Edmiston and Kennel [1984]
suggest that it is almost certainly supercritical, which would also entail ion reflection
[Greenstadt and Mellott , 1987]. Since the percentage of reflected ions relative to inci-
dent increases with increasing Mach number [Kennel et al., 1985], the 2000-04-06 event
likely reflects more ions than the four more-typical events. It is possible, therefore, that
the 2000-04-06 event surpassed a threshold for the minimum percentage of reflected ions
relative to incident ions necessary for the ECDI to become unstable. We also observe
a beam-like feature seen in the high energy electrons; although it likely has a very low
density compared to the background density, it may contribute to the free energy needed
to drive the observed waveforms. Note that we observe this beam-like feature for over
178
20 minutes upstream modulated in intensity concurrently with shocklets.
Some of the ion heating may be due to the large amplitude solitary waves (electron
phase space holes), which act like clumps of positive charge scattering and heating ions.
Observations have shown that the change in perpendicular ion temperature across a
train of electron holes can be comparable to the initial ion thermal energy [Ergun et al.,
1998b]. The solitary waves may also be contributing to the observed changes in the
electron distributions across the shock since their positive potentials can trap incident
electrons [Dyrud and Oppenheim, 2006; Lu et al., 2008]. Solitary waves can either cou-
ple to or directly cause the growth of IAWs [Dyrud and Oppenheim, 2006], whistler mode
waves [Lu et al., 2008], and/or electron acoustic waves [Matsukiyo and Scholer , 2006a].
Thus, solitary waves can directly heat/scatter particles or indirectly heat/scatter par-
ticles through the generation of or coupling to secondary waves.
This is the first study to report on the simultaneous observation of electron beams
with large amplitude Bernstein-like waves and solitary waves at an IP shock. The pref-
erence for perpendicular ion heating is consistent with previous bow shock observations
[Thomsen et al., 1985a], but the perpendicular electron heating is not. The polarizations
and frequencies of the wave modes observed are inconsistent with previous observations
of waves at IP shocks [Wilson III et al., 2007, 2009]. These results suggest a need for
further investigation into the detailed microphysics of collisionless shock dissipation,
particularly for higher Mach number events.
Chapter 6
Conclusion and Discussion
Collisionless shock waves have been studied for over 40 years, and still to this day sci-
entists debate about the possible mechanisms which limit wave steepening. Shock wave
formation requires a nonlinearly steepened wave to reach a point where the wave steepen-
ing is balanced by some form of irreversible energy dissipation. We observe shock waves
in the IP medium, but we still do not have a complete understanding of the microphysics
of these shocks. The ultimate goal of IP collisionless shock research is to fully under-
stand the mechanisms behind energy dissipation (e.g. particle reflection/acceleration)
that affect shock wave evolution/propagation for predictive capabilities applied to less
accessible shocks. This thesis has increased the understanding of energy dissipation in
low Mach number (Mf ∼1-4) IP shocks using electric and magnetic field and particle
distribution data from the Wind spacecraft. The increased understanding was gained
through the study of low frequency (∼0.2-10 Hz) magnetosonic whistler waves upstream
of the ramp regions, high frequency (&1 kHz) electric field measurements in the ramp
regions, and high frequency instabilities driven by reflected ions in the foot, ramp, and
immediate downstream regions of IP shocks.
Large scale (λ ∼ RE) low frequency (∼0.2-10 Hz) upstream waves are important
in the total energy dissipation buget of collisionless shocks. They are produced by
particles reflected by the shock, thus the particles carry energy away from the shock
front and radiate the waves. The waves can then scatter and diffuse particles result-
ing irreversible energy dissipation upstream or downstream of the shock. These effects
can also alter the Rankine-Hugoniot conservation relations. Part of this thesis focused
179
180
on Wind spacecraft observations of five IP shocks with low frequency upstream waves
[Wilson III et al., 2009]. Two types of low frequency waves were observed, both of which
lie on the magnetosonic whistler branch: precursor whistlers and shocklets. The precur-
sor whistler waves propagated at angles with respect to the magnetic field of 20◦ to 50◦
and large propagation angles with respect to the shock normal, thus they did not ap-
pear to be phase standing, consistent with previous studies [Russell et al., 1983]. In this
study, we presented the first observation of shocklets upstream of a quasi-perpendicular
IP shock. Almost 80% of the shocklets had θkB ≤ 45◦, consistent with the cometary
bow shock study by Le et al. [1989] and terrestrial bow shock studies [Hoppe et al.,
1981, 1982; Hoppe and Russell , 1983; Russell et al., 1971]. The shocklets appeared to
have higher frequencies leading the lower frequency waves, consistent with the theory
that ULF waves driven by reflected ions steepen into the shocklets [Omidi and Winske,
1990; Scholer et al., 2003; Tsubouchi and Lembe`ge, 2004]. Also, shocklets are only sup-
posed to exist upstream of quasi-parallel shocks with small radii of curvatures. Since
the spacecraft was over 50 RE in the X-GSE direction and over 35 RE in the Y-GSE
direction away from Earth, the shocklet source must be the IP shock and not the terres-
trial bow shock. Thus, our observation of 12 shocklets observed over an hour upstream
of a quasi-perpendicular IP shock has implications for shock structure and dynamics. A
possible explanation is that the shock front may not be of planar geometry. Meaning,
the IP shock could have quasi-parallel geometries adjacent to the quasi-perpendicular
crossing location. It is also possible, due to the supercritical nature of this IP shock, that
the shock itself was not stable and reforming. If the shock was undergoing reformation,
then the local geometry could oscillate between quasi-parallel and quasi-perpendicular.
Regardless of how the shocklets were generated, the observation was the first of its kind.
In addition to the observation of shocklets simultaneous with diffuse ion distributions
upstream of a quasi-perpendicular IP shock, we also examined low energy (≤1 keV) elec-
tron PADs for evidence of wave-particle interactions. The electrons showed evidence of
cyclotron interactions through pitch-angle diffusion and/or scattering. The study was
the first to report on electron distributions unstable to the whistler anisotropy and/or
heat flux criterion of Gary et al. [1994, 1999] observed simultaneously with whistler
waves at an IP shock. The associated electron heat flux calculated for each distribution
changed both in angle with respect to magnetic field and wave vector and in magnitude
181
across each shocklet. The maximum heat flux magnitude was observed just downstream
of the leading whistler wave packets on the upstream edge of the shocklets, consistent
with an electron heat flux free energy source. The angle between the heat flux vec-
tor and magnetic field direction also peaks just downstream of the steepened edge of
the shocklets, consistent with the shocklets pitch-angle scattering the heat flux vectors.
This was the first study to observe shocklets and precursor whistlers in association with
electron distributions unstable to whistler heat flux and/or whistler anisotropy insta-
bilities simultaneous with finite electron heat fluxes at quasi-perpendicular IP shocks.
Closer to the shock, we examined high frequency (&200 Hz) electric field measure-
ments in the ramp regions of IP shocks [Wilson III et al., 2007, 2010]. The first of two
studies was a statistical study of high frequency (&1 kHz) electric field measurements in
the ramp regions of IP shocks [Wilson III et al., 2007]. We established the first statisti-
cal evidence to suggest a dependence of large amplitude (>5 mV/m peak-to-peak) IAWs
in the ramp regions on the shock compression ratio and fast mode Mach number. The
dependence on shock compression ratio is consistent with larger shock strengths caus-
ing larger cross-field currents, which may provide free energy for wave generation. The
study also showed the first statistical evidence that large amplitude (≥ 5 mV/m peak-
to-peak) IAWs had the highest probability of occurrence in the ramp region, consistent
with theory [Gary , 1981]. In addition to their relatively high frequency of occurrence,
the waves were found to produce large anomalous resistivities. Quasi-linear estimates
of the anomalous resistivities produced by these large amplitude IAWs ranged from
∼1–856 Ω ·m (∼107 times greater than classical estimates). Recent Vlasov simulations
using realistic mass ratios have found resistivities that are 2-3 orders of magnitude larger
than their equivalent quasi-linear estimates [Petkaki et al., 2006; Petkaki and Freeman ,
2008; Yoon and Lui , 2006, 2007]. The argument is then that the effects due to these
high frequency waves in the ramp regions of collisionless shocks may be more important
than previously thought.
The second study of high frequency waves at IP shocks focused on the microphysics
of an atypical supercritical IP shock [Wilson III et al., 2010]. It provided the first ob-
servations of large amplitude (>100 mV/m pk-pk) solitary waves and large amplitude
(∼30 mV/m pk-pk) waves exhibiting characteristics consistent with electron Bernstein
waves at an IP shock. The observed waveforms all clearly show enhanced power near
182
integer and half-integer harmonics of the cyclotron frequency. Two of the waveforms
showed significant power along the shock normal and are obliquely polarized with re-
spect to the magnetic field, consistent with the ECDI. Strong particle heating in both
the halo and core of the electrons and ions is observed near these waves, consistent with
the simulation results of Matsukiyo and Scholer [2006a]. The discussions in Section 5.2
indicate that the waves are consistent with the ECDI because: 1) the polarizations are
primarily aligned with the shock normal direction, not the magnetic field direction; 2)
the frequency spectrum shows integer and half-integer cyclotron harmonics; 3) the broad
power intensifications at frequencies above 1 kHz of waveforms A and B are inconsistent
with typical IAWs; and 4) we observed strong and preferentially perpendicular electron
heating that is consistent with cyclotron heating as described in Forslund et al. [1972]
and Matsukiyo and Scholer [2006a]. The study showed the first observational evidence
of the ECDI at IP shocks and we presented the first observations of electron cyclotron
harmonic waves downstream of an IP shock in this thesis (Figure 1.17).
An examination of electron distributions at other supercritical shocks (not shown)
showed particle heating resulting in parallel flattop and perpendicular heated Maxwellians.
The heating was predominantly in the parallel direction immediately downstream of
the shock but isotropized as one progressed further into the downstream, in agreement
with previous observations [Feldman et al., 1983a; Thomsen et al., 1987]. Strong quasi-
static electric fields have been observed at collisionless shocks [Wygant et al., 1987;
Walker et al., 2004; Bale and Mozer , 2007] and used to help explain particle heating
[Scudder et al., 1986a; Thomsen et al., 1987]. Particle heating due to the cross-shock
potential through a two step process of acceleration and relaxation is expected to create
stronger heating of the core electrons parallel to the magnetic field than perpendicular
[Thomsen et al., 1987]. In contrast, the 04/06/2000 event showed perpendicular heating
dominating in the downstream and Tc,⊥/Tc,‖ increasing as one progressed farther down-
stream, suggesting wave heating is important. This study provided the first observation
evidence to definitively suggest that wave heating may dominate over the heating due
to the cross-shock potential.
183
6.1 Summary
The work presented in this thesis has helped increase the understanding of the micro-
physics of IP shocks in addition to raising new questions regarding the energy dissipation
mechanisms dominating in the ramp regions. This was performed by providing the first
detailed analysis of high time resolution waveform captures and particle distributions
observed simultaneously in IP shocks. The initial work focused on a statistical study
of high frequency waveforms in IP shock ramps. The study results suggested a re-
evaluation of the importance of anomalous resistivity due to wave-particle interactions
relative to dispersion, particle reflection, and macroscopic DC field effects. Further evi-
dence to support the need for a re-evaluation of wave-particle significance was reported
through the observational evidence of pitch-angle scattering and diffusion of electrons
by low frequency magnetosonic whistler waves upstream of IP shocks. The importance
of anomalous resistivity due to wave-particle interactions was examined in further de-
tail in the study of the atypical electron heating observed at a supercritical IP shock,
which we claimed clearly showed a dependence on the observed waveforms. The nearly
ubiquitous observations of large amplitude IAWs in the ramp regions of IP shocks raise
doubts about ignoring these high frequency fluctuations when examining the heating of
particle distributions and particle energization in collisionless shocks.
Thus, we conclude that in the analysis of IP shocks the microphysics can no longer
be disregarded. The results of the work presented in this thesis suggest a need for
further investigation into the detailed microphysics of collisionless shock dissipation,
particularly for higher Mach number events.
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Appendix A
Wind Measurements and Data
Analysis Issues
A.1 The Pesa High Detector: The Glitch
Table A.1: Pesa High Bad Data Bins
Mapcode Mapcode # of Bins Bin Elements
(Hex) (Long)
D4A4 54436 121 [0,1,8,9,16,17,24,25,35,36,43,44,51,52,59,60]
D4FE 54526 97 [0,1,5,6,10,11,15,16,23,24,28,29,33,34,38,39]
D5EC 54764 56 [0,1,2,8,9,10,19,20,21,27,28,29]
D6BB 54971 65 Unknown
D65D 54877 88 Unknown
On occasion, the PH detector has a glitch in the data which always occurs in the
same data bins, regardless of date/time. The data bins which are affected are specific
to the mode that PH is in, as seen in Table A.1. As a consequence, the set of data bins
are essentially useless because they contain incorrect energy levels, among other things.
The glitch appears to connected to a saturation due to the solar wind and solar UV
contamination1 .
1 This connection was subjectively made upon plotting the PH distribution functions. I found
216
217
The glitches occur consistently in the same data bins, regardless of day or year. The
Wind 3DP Pesa High Burst
2000-04-06/16:31:32-16:31:35
-2000
-1000
0
1000
2000
Vp
e
rp
 
[km
/s
]
Wind 3DP Pesa High Burst
2000-04-06/16:31:41-16:31:44
UV contamination 
(noise)
Induces Bifurcation
of Thermal Core
Pesa High “Glitch”
and
Solar Wind “Noise”
Pesa High “Glitch”
Not “Fixed”
-2000 -1000 0 1000 2000
Vpara [km/s]-2000 -1000 0 1000 2000Vpara [km/s]
-2000
-1000
0
1000
2000
Vp
e
rp
 
[km
/s
]
Pesa High “Glitch”
“Fixed”
UV contamination 
reduced and
Thermal Core
“Fixed”
Pesa High “Glitch”
and
Solar Wind “Noise”
greatly reduced
A B
C D
Figure A.1: Here is an example of an ion distribution seen upstream of an IP shock by
the Wind Pesa High instrument. The figure illustrates the differences between before
and after the glitch and solar wind noise were removed.
bins, however, depend upon the mode that the Pesa High detector is in. I have found
the following bins to be an issue for the sample modes seen in Table A.1. The only
two modes I have yet to find a glitch in are the two where the number of bins equals
65 or 88. This is largely due to the fact that I almost never find Pesa High data with
this number of data bins. Also, the glitch appears always at the same GSE azimuthal
angles, φ (degrees), in the Pesa High data structures. In each case a repeating pattern
is seen, as with the data bins, consistent with the spacecraft rotation rate remaining
that the excess observed in the contour plots along the projection of the solar wind velocity decreased
dramatically when these bins were removed from the data. The UV contamination seen in the thermal
core also changed as a result of these alterations.
218
approximately the same, within the angular resolution of the Pesa High detector.
Figure A.1 shows an example of two distributions from PHB where I have attempted
to correct/remove the glitch and reduce the solar wind noise. The top two distributions,
panels A and B, have not been corrected and show the characteristics of the UV and
solar wind contamination while the bottom two distributions, panels C and D, show
reduced levels of both. Though there still exists contamination, the reduced effects help
one to recognize more important and real phenomena in the distributions. The contour
plots illustrate typical diffuse ion distributions, as seen in Figure 1.11. The large ge-
ometry factor and response of the PH instrument causes the thermal core measurement
to saturate the detectors, thus the isolated peak in the cuts of Figures 1.9 and 1.10.
One can also see remnants of UV contamination that have not been removed. Notice
that there appears to be a beam-like feature at an oblique angle to the magnetic field in
both distributions, which I have outlined in panels A and B. This is not a real feature,
rather an artifact of the glitch. It can be misleading, but the beam-like feature follows
the sun direction in every distribution with the glitch. The feature is seen far enough
upstream that one could argue that the particles are not magnetically connected with
the shock, thus it is not a signature of any sort of particle reflection/energization. The
intensity of this effect varies from event to event depending on location and orientation
of the satellite.
Another aspect of particle distributions which can cause misinterpretations is con-
tour closing algorithms used in IDL. Though the contour levels are determined through
a systematic and automated algorithm, they can alter interpretations of the data. As
one examines panels A and C, one can see that after correcting for the noise a beam-like
feature appears parallel to the magnetic field at roughly 450 km/s, and a weaker and
more diffuse signal peaked at roughly 850 km/s. The feature is likely a real signal, but
it is not long lived. Regardless, if one examines panel A, the feature is not noticeable
and would otherwise be missed.
A.2 SCET Corrections
The TDS samples are waveform captures of electric and magnetic field data. The
data is triggered by the largest amplitude waves which exceed a specific threshold and
219
are then stored in a memory buffer. The TDS datation time is sampled after the event
is acquired which requires a delay buffer. The datation time requires two corrections.
The first correction arises from tagging the TDS datation with an associated spacecraft
major frame in house keeping (HK) data. The second correction removes the delay
buffer duration. Both inaccuracies are essentially artifacts of on ground derived values
in the archives created by the WINDlib software (K. Goetz, Personal Communication,
2008).
The WAVES instrument’s HK mode sends relevant low rate science back to ground
once every spacecraft major frame. If multiple TDS events occur in the same major
frame, it is possible for the WINDlib software to assign them the same SCETs. One
can correct these issues to within +10 ms. This is often not necessary since the highest
sampling rate of the HTR MFI data is only 22 Hz, however there are occasions where
the 0.33 Hz data differs from the HTR data by > 20◦. As a consequence, the polariza-
tion analysis (see hodogram in Figure 2.3) can be misleading.
The TDS receiver onboard calculations force the peak amplitude of the data to occur
roughly in the center of the waveform capture time window. Thus the data is sampled
from both before and after the peak. The time stamps associated with the data sampled
after the peak is easily obtained but the times before the peak require the delay buffer
memory. The delay buffer duration depends on sampling rate and waveform duration.
WINDlib attempts to fix the delay buffer uncertainty but it cannot correct the built-in
timing error of ∼300 ms when sampling at 120 kHz. One should note that for lower
sample rates, the timing error is larger (K. Goetz, Personal Communication, 2008).
The TDSF receiver (sampled at 120 kHz) time stamps retrieved from WINDlib,
before corrections, are accurate to +300 ms. The 300 ms uncertainty, due to the HK
corrections mentioned above, results from WINDlib trying to recreate the time stamp
after it has been telemetered back to ground. If an event stays in the TDS buffer
for extended periods of time (i.e. >2 days), the interpolation done by WINDlib can
make mistakes in the 11th significant digit. The positive definite nature of this uncer-
tainty is due to rounding errors associated with the onboard DPU clock rollover. The
DPU clock is a 24 bit integer clock sampling at ∼50,018.8 Hz. The clock rolls over at
220
∼5366.691244092221 seconds2 (K. Goetz, Personal Communication, 2008). The sam-
ple rate is a temperature sensitive issue and thus subject to change over time. From a
sample of 384 different points on 14 different days, a statistical estimate of the rollover
time is 5366.691124061162 ± 0.000478370049 seconds.
The method by which to correct the SCETs is as follows:
1. Retrieve the DPU clock times, SCETs, UR8 times, and DPU Major Frame Num-
bers from the Windlib libraries on the VAX/ALPHA systems for the TDSS(F)
data of interest.
2. Retrieve the same quantities from the HK data.
3. Match the HK event with the same DPU Major Frame Number as the TDSS(F)
event of interest.
4. Find the difference in DPU clock times between the TDSS(F) event of interest
and the HK event with matching major frame number (Note: The TDSS(F) DPU
clock time will always be greater than the HK DPU clock if they are the same
DPU Major Frame Number and the DPU clock has not rolled over).
5. Convert the difference to a UR8 time and add this to the HK UR8 time. The new
UR8 time is the corrected UR8 time to within +10 ms.
6. Find the difference between the new UR8 time and the UR8 time WINDlib asso-
ciates with the TDSS(F) event. Add the difference to the DPU clock time assigned
by WINDlib to get the corrected DPU clock time (Note: watch for the DPU clock
rollover).
7. Convert the new UR8 time to a SCET using either the IDL Windlib libraries or
TMLib libraries of available functions. This new SCET is accurate to within +10
ms.
2 The calculation is done by (16*224)/(50,018.8 Hz)
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A.3 Explicit Explanation of Actual Data Calculations
A.3.1 Wind 3DP Software Calculations
Two tests were performed to determine the stability of our original temperature
anisotropy estimates and to observe the effects on our comparisons to the instability
estimates of Gary et al. [1994, 1999]. The first test was to lower and raise our cutoff
energy bin for the minimum energy of the halo electrons to raise and lower our esti-
mate of nhe/ne. Occasionally our estimates of nhe/ne were a factor of 5-10 less (for the
12/10/1997 and 04/06/2000 events) than the estimate used by Gary et al. [1994], nhe/ne
= 0.05. The second test was a rigorous examination of our fit estimates in comparison
to the anisotropy estimates calculated directly from the 3DP particle data. Note, since
Gary et al. [1999] explicitly removed the strahl electrons from their calculations, we
ignored the component of the strahl electrons and only fit to the more isotropic halo
components when calculating temperature anisotropies for comparison to the threshold
estimates estimated by Gary et al. [1994, 1999].
The results of the first test showed the largest relative change in the relevant param-
eters of Gary et al. [1994, 1999] occurred in β‖c and T‖h/T‖c. When we lowered(raised)
the cutoff energy, both β‖c and T‖h/T‖c decreased(increased). However, for most distri-
butions T⊥h/T‖h increased(decreased). Since T⊥h/T‖h changed inversely with respect
to T‖h/T‖c, we concluded that changing the energy bins did not dramatically affect
the instability estimates. Almost all of the PADs which were unstable in our original
estimates were still unstable after lowering the lowest energy of the halo electrons. In
fact, many of the PADs for the four IP shocks without shocklets were more unstable
when we lowered the energy cutoff. For both estimates of break energies, almost all the
distributions within 30 seconds of all five IP shock ramps were unstable to the whistler
heat flux instability estimated by Gary et al. [1994].
The results of the second test almost always showed a stronger anisotropy in the core
and halo electrons than the direct calculations from the data. We found that the strahl
electrons, found almost entirely in the parallel component of the electron distribution,
lowered estimates of T⊥/T‖ for both the core and halo. The effect is seen more strongly
in the halo electrons, and Gary et al. [1994, 1999] suggested the halo is more important
for the whistler heat flux and anisotropy instabilities than the core. This implies that
222
the electron distributions we present may be more unstable than our estimates. There-
fore, we used our moment calculations as a lower bound on the halo/core temperature
anisotropies.
A.3.2 Specific Software Implementations
This section will illustrate how typical particle software calculates the particle mo-
ments in detail, specifically for the Wind/3DP software (consequently, this same soft-
ware is used for every Berkeley particle instrument, e.g. Cluster, STEREO, THEMIS
etc.).
To calculate the distribution function, one must first define a weighting factor, wt
(unitless). wt depends upon the energy, E, spacecraft potential, φsc, particle charge,
q (in units of fundamental charge), and differential energy, δE. When calculating the
weighting factor for real data, for mathematical and practical purposes, wt is constrained
by defining it as:
wt = 0.0 <
(
E + φscq
δE
+ 0.5
)
< 1.0 (A.1)
where Equation A.1 defines the mathematical result of wt. The next step is to shift the
energy of the particles by φsc to avoid using photo-electrons from the satellite in the
calculations. This is done by defining the differential velocity, δv (
√
eV ), as:
δv ≡
√(
E +
φsc
q
)
(A.2)
where δv is constrained to be > 0 always. Before going any further, one must convert
the data units to energy flux (Energy cm−2 sr−1 s−1 eV−1) so that our distribution
function is a function of energy and solid angle. At this point, one can calculate the
differential distribution function in the following manner:
δf ≡ δv
E
∗ d(E,Ω)
105
∗ δE
E
∗ dΩ ∗ wt (A.3)
where dΩ is the differential volume and the factor 105 is a conversion factor to multiply
by cm/km to get the resulting units eV−1/2km/s cm−3sr−1. One should note, that at
this point, δf is a 2-dimensional NE×Nb-array, where NE corresponds to the number
of energy bins and Nb corresponds to the number of data bins. Integrating over dΩ
223
(2nd dimension, or Nb) gives an estimate of the number density per root mass (units of
eV/c2 and c is in km/s) per energy bin. Summing over the energy bins and dividing by
a normalization factor given by:
no ≡
√∣∣∣∣2 ∗ qM s
∣∣∣∣ = 593.09544 (for electrons) (A.4)
yields the number per centimeter cubed, or the number density. Generally one would
refer to this as the zeroth order moment calculation. The mass is given by:
mass =
510, 990.6eV/c2
(2.99792458 × 105km/s)2 (A.5a)
= 5.6967578 × 10−6 eV/(km/s)2 . (A.5b)
and a particle charge normalization is given by:
0.010438871
eV
(km/s)2
=
938.27231 × 106eV/c2
(2.99792458 × 105km/s)2 (A.6)
which is used for determining the fractional charge (units of proton charge as eV/(km/s)2)
of the input particle type. For electrons, the rounded value of the particle mass over
this charge conversion constant is zero, thus the program assigns a value of -1 to the
variable charge.
The first moment comes from integrating (summing in IDL) δf δv rˆ, where rˆ is defined
as:
rˆ ≡ (cos θ cosφ) xˆ+ (cos θ sinφ) yˆ+ (sin θ) zˆ (A.7)
where θ and φ are the latitude and azimuthal angles in which ever coordinate system
you happen to be in. After some calculations, the first moment gives a flux density
(km/s cm−3) as a vector in the IDL routine.
The second moment is calculated in a similar manner but now we must define a new
variable, δf 1, as δf δv. So now we see that the second moment can be calculated as:
~Vl,m =
∫
d3v (δf 1 ∗ δv ∗ (rˆrˆ)l,m) (A.8)
where ~Vl,m is the (l,m)-component of the velocity flux (
√
eVkm/s cm−3).
The third moment is calculated in a similar manner but now we must define a new
224
variable, δf 2, as δf δv2. Thus one can show the third moment goes as:
~Ql,m,n =
∫
d3v (δf 2 ∗ δv ∗ (rˆrˆrˆ)l,m,n) (A.9)
where ~Ql,m,n is the general form of the heat flux. However, one assumes that the order
of the indices does not matter (due to symmetry assumption) which reduces the possible
combinations from 27 down to 10. The 10 variations of (l,m,n) are: Qx,x,x, Qx,y,y, Qx,z,z,
Qx,x,y, Qx,x,z, Qx,y,z, Qy,y,z, Qy,z,z, Qy,y,y, and Qz,z,z. Generally in any particle code, one
reduces this even further to a simple rank-2 tensor or 3×3 matrix where the sum of the
ith row results in the ith component of the resultant heat flux vector. The above method
is the same as the typical definition of the heat flux given by Equation A.10:
~q =
me
2
∫
d3v~vv2f(~x, ~v, t) (A.10)
where me is the electron mass (in the program this is done by multiplying the result of
Equation A.5b by the square of the result of Equation A.4), ~v the velocities, and f(~v,~x,t)
represents a general form of the distribution function. Note, the heat flux is only relevant
in the plasma rest frame so that f(~v,~x,t) should be appropriately transformed before
calculating Equation A.10.
A.3.3 Pressure and Temperature Calculation
This section will illustrate how typical particle software calculates the particle pres-
sure and temperature in detail. In theory, the particle pressure need not explicitly
depend upon a parameter called the temperature. In practice, however, the assumption
is made that the temperatures come directly from the pressure tensor. It is important
to note that the particle temperature, or average kinetic energy in the bulk flow frame,
is derived from the second moment of the distribution function. It should be derived
from carefully calculated numerical fits of the ‖ or ⊥ cut of the reduced distribution
function for every data structure (see Equations 2.2 through 2.11). However, this is
rarely done and often the temperature is calculated from the moments derived in the
particle software. There is a fundamental difference in the two calculations which should
be carefully regarded because it has some important implications on the interpretation
of the data.
225
The pressure tensor is calculated from the second moment of the distribution func-
tion, Vl,m (Equation A.8), where l and m correspond to the rows and columns of the
second moment tensor. To calculate the pressure tensor, one multiples Vl,m by two
normalization constants (corresponding to a charge and mass normalization) in the
software. In the software, the only indices (l,m) used for Vl,m are the following: Vx,x,
Vy,y, Vz,z, Vx,y, Vx,z, and Vy,z (thus a symmetric tensor is assumed). In typical soft-
ware calculations, the total pressure is calculated by taking one third of the trace of the
pressure tensor. Similarly, the average temperature is calculated by that same quantity
divided by the number density. Thus, <Te> = Tr[Pi,j]/ne.
Appendix B
Glossary and Acronyms
• Coronal Mass Ejection (CME) – A large eruption of plasma from the surface
of the sun releasing upwards of 50 billion tons of mass and propagating through
the interplanetary medium at up to 3000 km/s.
• Cyclotron(Gyro) Frequency (Ωcs for angular in radians per second or fcs for
circular in Hz) – The characteristic frequency associated with a particle in circular
motion in a uniform static magnetic field.
• Cyclotron(Gyro) Radius (ρgs) – The characteristic radius of a charged particle
orbit with a velocity perpendicular to the magnetic field.
• Electron Cyclotron Drift Instability (ECDI) – Instability driven by a rel-
ative drift between reflected ions and incident electrons at a quasi-perpendicular
collisionless shock. This results in a resonant interaction between Doppler-shifted
ion-acoustic waves and electron Bernstein (or cyclotron) waves.
• Electron Cyclotron Harmonic Waves (ECHWs) – A group of wave modes
that exhibit power enhancements at integer and half-integer harmonics of fce and
can include (n + 1/2), electron Bernstein-like, or ”totem pole” emissions which
are typically banded electrostatic waves propagating nearly perpendicular to the
magnetic field.
• Electrostatic[Electromagnetic] (ES[EM]) – A wave that has its wave vector
226
227
parallel(perpendicular) to the electric field fluctuations without(with) correspond-
ing magnetic field fluctuations, which is also a longitudinal(transverse) fluctuation.
• Electrostatic Solitary Waves (ESWs) – A nonlinear wave with a bipolar elec-
tric field signature parallel to the background magnetic field that is consistent
with a BGK phase space hole.
• Interplanetary Coronal Mass Ejection (ICME) – (See CME) A CME that
leaves the sun’s surface.
• Interplanetary Shock (IP Shock) – A shock wave typically propagating away
from the sun due to an ICME or a corotating interaction region acting as the
driver.
• Ion-Acoustic Wave (IAW) – A plasma oscillation which is composed of the
inertia carrying ions fluctuating as the wave propagates with the electrons provid-
ing the charge balance. In the electrostatic limit, the waves are linearly polarized
parallel to the magnetic (or at slightly oblique angles) field and propagate parallel
to the electric field oscillations.
• Lower Hybrid Wave (LHW) – A plasma oscillation that is typically ES with
phase fronts propagating nearly perpendicular to the ambient magnetic field car-
ried primarily by demagnetized ion motions with a frequency near the lower hybrid
resonance, flh2 = fce fci.
• Macro(Micro)instability – A fluid(kinetic) instability which generally has the
largest growth rates when kρgi ≪ 1(kρgi & 1).
• Modified Two-Stream Instability (MTSI) – Instability driven by relative
drift between electrons and ions across a magnetic field.
• Plasma Frequency (ωps for angular in radians per second or fps for circular in
Hz) – The frequency associated with a small net displacement of a particle species,
s, from equilbrium due to an electric field restoring force.
• Right[Left]-Hand Polarized Wave (RH[LH]) – A transverse electromagnetic
wave which has kˆ ‖ Bo but δB and δE rotate in a clock(counterclock)wise sense
228
when looking along Bo.
• Shocklet – A nonlinearly steepened magnetosonic wave which often exhibits a
high frequency wave packet on its leading edge and has a magnetic compression
ratio ≤2. They are thought to result from ultra low frequency foreshock waves
that interact with intermediate (or gyrating or gyrophase bunched) and diffuse
ion distributions which alter the index of refraction, thus steepening the waves.
• Short Large Amplitude Magnetic Structures (SLAMS) – Similar to shock-
lets except they have steepened to much larger amplitudes with magnetic com-
pression ratios often exceeding a factor of 4.
• Soliton – A nonlinear wave whose different Fourier components propagate at
different speeds for small amplitudes and have a constant shape as they propagate
because the nonlinear steepening is balanced by dispersive effects.