
Citation 
 Permanent Link:
 http://digital.auraria.edu/AA00006549/00001
Material Information
 Title:
 Magnetospheric whistler mode raytracing with the inclusion of finite electron and ion temperature
 Creator:
 Maxworth, Ashanthi S. ( author )
 Place of Publication:
 Denver, Colo.
 Publisher:
 University of Colorado Denver
 Publication Date:
 2017
 Language:
 English
 Physical Description:
 1 electronic file (102 pages) : ;
Thesis/Dissertation Information
 Degree:
 Doctorate ( Doctor of philosophy)
 Degree Grantor:
 University of Colorado Denver
 Degree Divisions:
 Department of Electrical Engineering, CU Denver
 Degree Disciplines:
 Electrical engineering
Subjects
 Subjects / Keywords:
 Electromagnetic waves ( lcsh )
Magnetosphere ( lcsh ) Electromagnetic waves ( fast ) Magnetosphere ( fast )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Review:
 Whistler mode waves are a type of a low frequency (100 Hz  30 kHz) wave, which exists only in a magnetized plasma. These waves play a major role in Earth's magnetosphere. Due to the impact of whistler mode waves in many fields such as space weather, satellite communications and lifetime of space electronics, it is important to accurately predict the propagation path of these waves. The method used to determine the propagation path of whistler waves is called numerical raytracing.
 Review:
 Numerical raytracing determines the power flow path of the whistler mode waves by solving a set of equations known as the Haselgrove's equations. In the majority of the previous work, raytracing was implemented assuming a cold background plasma (0 K), but the actual magnetosphere is at a temperature of about 1 eV (11600 K). In this work we have modified the numerical raytracing algorithm to work at finite electron and ion temperatures. The finite temperature effects have also been introduced into the formulations for linear cyclotron resonance wave growth and Landau damping, which are the primary mechanisms for whistler mode growth and attenuation in the magnetosphere. Including temperature increases the complexity of numerical raytracing, but the overall effects are mostly limited to increasing the group velocity of the waves at highly oblique wave normal angles.
 Bibliography:
 Includes bibliographical references.
 System Details:
 System requirements: Adobe Reader.
 Statement of Responsibility:
 by Ashanthi S. Maxworth.
Record Information
 Source Institution:
 University of Colorado Denver
 Holding Location:
 Auraria Library
 Rights Management:
 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 on10153 ( NOTIS )
1015317538 ( OCLC ) on1015317538
 Classification:
 LD1193.E54 2017d M39 ( lcc )

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MAGNETOSPHERIC WHISTLER MODE RAYTRACING WITH THE INCLUSION OF
FINITE ELECTRON AND ION TEMPERATURE
by
ASHANTHI S MAXWORTH
B.Sc (Hons), University of Moratuwa, 2011, M.S, University of Colorado Denver, 2014
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Electrical Engineering Program
2017
This thesis for the Doctor of Philosophy degree by Ashanthi S Maxworth has been approved for the Electrical Engineering Program by
Mark Golkowski, Advisor and Chair Martin Huber Tim Lei Yiming Deng Dana Carpenter Date: May 13, 2017
Maxworth, Ashanthi S (Ph.D, Electrical Engineering)
Magnetospheric Whistler Mode Raytracing with the Inclusion of Finite Electron and Ion Temperature
Thesis directed by Associate Professor Mark Golkowski
ABSTRACT
Whistler mode waves are a type of a low frequency (100 Hz 30 kHz) wave, which exists only in a magnetized plasma. These waves play a major role in Earths magnetosphere. Due to the impact of whistler mode waves in many fields such as space weather, satellite communications and lifetime of space electronics, it is important to accurately predict the propagation path of these waves. The method used to determine the propagation path of whistler waves is called numerical raytracing. Numerical raytracing determines the power flow path of the whistler mode waves by solving a set of equations known as the Haselgroves equations. In the majority of the previous work, ray tracing was implemented assuming a cold background plasma (0 K), but the actual magnetosphere is at a temperature of about 1 eV (11600 K). In this work we have modified the numerical raytracing algorithm to work at finite electron and ion temperatures. The finite temperature effects have also been introduced into the formulations for linear cyclotron resonance wave growth and Landau damping, which are the primary mechanisms for whistler mode growth and attenuation in the magnetosphere. Including temperature increases the complexity of numerical raytracing, but the overall effects are mostly limited to increasing the group velocity of the waves at highly oblique wave normal angles.
The form and content of this abstract are approved. I recommend its publication.
Approved: Mark Golkowski
This work is dedicated to my family, my teachers and my students.
ACKNOWLEDGMENTS
First I would like to thank my advisor Prof. Mark Golkowski. Prof.Golkowski hired a student coming from a small island named Sri Lanka located in the exact opposite side of the world. Over these four and half years the guidance and support provided by Prof. Golkowski for my academic development as well as career development is tremendous. I cannot thank him enough for all the support he has provided me over the years and I am forever grateful for that. As a student coming from a developing country I would not be able to come this far without the financial support provided by Prof. Golkowski.
I also would like to thank Prof. Stephen Gedney, chair of the Electrical Engineering department for all the trust he has in me, and giving me the opportunity to teach Electromagnetic Fields course, which is a huge step forward in my teaching career.
Electrical Engineering program assistant Mrs. Annie Bennet deserves a big thank you for all the help and motivation she gave me over these years as a coworker and as a friend. Same thanks goes to Miss. Karla Flores.
I also would like to thank my thesis committee members, Prof. Dana Carpenter, Prof. Martin Pluber, Prof. Tim Lei and Prof. Yiming Deng for their support and valuable suggestions provided on improving my thesis work. Sincere thanks go to all my research group members especially to my dear friends Naomi Watanabe and Ryan Jacobs for their friendship and motivation.
I dearly acknowledge late Prof. Titsa Papantoni for being a role model for all womenkind. I clearly remember the five word advice I received from her, that she gave only to her daughter. I was honored to receive such a motivational advice from her.
My other employers over the years, Campus Village Apartments, Metropolitan State University of Denver and Learning Resources Center, staff members at all these places deserves a thank you for all the support they have provided me over these five years in Denver.
My teachers and my students, that taught me and taught by me over the years deserves a huge thank you for all their support. Last but never the least this Ph.D would not be
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possible without the unconditional love and support from my family; my mother Nilanthie, my father Frank and my sister Menoji. Over these years we realized how strong we are as a family and completion of this PhD is an achievement of all four of us.
vi
TABLE OF CONTENTS
I. INTRODUCTION ......................................................... 2
1.1 NearEarth Space................................................... 2
1.2 Whistlers and Whistler Mode Waves.................................. 3
1.3 Fundamentals of Whistler Mode Waves................................ 6
1.4 Role of Whistler Mode Waves in Near Earth Space................... 10
1.4.1 Cold and Warm Plasma......................................... 14
1.5 Scientific Contributions.......................................... 17
1.6 Thesis Organization............................................... 18
II. THEORETICAL BACKGROUND .............................................. 19
2.1 Introduction to Numerical Raytracing.............................. 19
2.2 Refractive Index Surface ......................................... 20
2.3 Dispersion Relation in Cold Plasma................................ 22
2.4 Dispersion Relation with the Inclusion of Temperature Effects..... 26
2.5 Introducing Temperature Effects to Multiple Species............... 30
2.6 Landau Damping ................................................... 31
2.7 Numerical Raytracing.............................................. 34
III. MODIFICATION TO THE REFRACTIVE INDEX SURFACE........................ 39
3.1 Effect of Ions.................................................... 39
3.2 Inclusion of Finite Electron or Ion Temperature................... 39
3.3 Inclusion of Temperature for Both Electrons and Ions.............. 41
IV. MODIFICATION TO WHISTLER MODE RAY TRAJECTORIES FROM FINITE ELECTRON AND ION TEMPERATURE........................................ 45
4.1 Effect of Temperature on Landau Damping........................... 45
4.2 Launching Waves in Different Directions........................... 45
vii
4.3 Frequency Dependence of Wave Trajectories ........................ 47
4.4 Dependence on Initial Wave Normal Angle........................... 48
4.5 Observations of Waves Launched at L = 4........................... 51
V. COMPARISON WITH VAN ALLEN PROBE SPACECRAFT OBSERVATIONS . 65
5.1 Chorus and Hiss................................................... 65
5.2 Dependence on Magnetic Local Time ................................ 65
5.3 Van Allen Probe Spacecraft Observations........................... 66
5.4 Simulated Wave Power with Landau Damping.......................... 67
5.5 Wave Growth Analysis ............................................. 69
VI. SUMMARY AND CONCLUSIONS.............................................. 74
6.1 Future Work....................................................... 75
REFERENCES............................................................... 76
APPENDIX
A. Dispersion Relation for Free Space................................. 81
B. Dispersion Relation in a NonMagnetized Plasma.................... 84
C. Dispersion Relation in a Cold Magnetized Plasma ................... 86
D. Noncausal Filtering............................................... 88
E. Glossary .......................................................... 90
viii
FIGURES
1.1 Near Earth space environment, housing man made electronics for research, communication, navigation and military purposes. Image source: www.science.nasa.gov. 3
1.2 Observation of a sferic at a very low frequency antenna located in Chistochina, Alaska. Top panel shows the amplitude of the sferic captured by the antenna oriented in the geomagnetic NorthSouth, and bottom panel shows the amplitude
of the same sferic captured by the antenna oriented in the EastWest direction. 5
1.3 Image of the Earth Ionosphere Waveguide(EIW). Image source:vlf.stanford ... 6
1.4 Spectrogram of a whistler mode wave together with sferics and triggered emis
sions. Sferics are occurred with the lightening strikes. Image source: [Green and Inan( 2015)]................................................................ 7
1.5 Dispersion diagram of a whistler mode wave. The dispersion relation for free
space (us = ck) is also shown in the plot...................................... 9
1.6 Different modes of propagation for whistler mode waves; ducted and nonducted.
Here in the figure fw is the frequency of the wave and Jlhr is the lower hybrid resonance frequency. Ducted waves are guided by the geomagnetic held and the nonducted waves show multiple reflections. Image source: [Green and Inan( 2015)] 11
1.7 Illustration of the geomagnetic held. Image source: www.science.nasa.gov. ... 12
1.8 Illustration of the concept of L shells. Image source: en.wikipedia.org........... 12
1.9 Van Allen radiation belts and the satellites launched by NASA for Van Allen
belts observations. Image source: www.science.nasa.gov........................... 13
1.10 Trajectory of trapped higher energy particles and the magnetic conjugate point.
Image source: Handbook of Geophysics and the Space Environment (Air Force Research Laboratory ,1985)................................................. 14
IX
1.11 Precipitation of electrons trapped in Earths radiation belts with the interaction
of whistler mode waves from lightning. Image source: vlf.stanford.edu......... 15
1.12 MaxwellBoltzmann velocity distribution in one dimension...................... 15
2.1 Orientation of the wave normal vector and the group velocity vector in an
anisotropic medium............................................................ 21
2.2 (a) Isotropic, non homogeneous medium, (b) Non homogeneous and anisotropic 21
2.3 Orientation of the wave normal vector and the group velocity vector in an
anisotropic medium.(This figure is a recreation similar to Figure 3.13 in [HelUwell( 1965)])............................................................. 22
2.4 Progression of the refractive index surface, ray direction and the direction of the
wave normal for a 3 kHz wave.Image source: [Bortnik(2005)].................... 23
2.5 Change of the location of plasma pause with the kp index. Plasma pause is the
region where the electron density suddenly drops.............................. 38
3.1 Refractive Index surface for 2.3 kHz, at L = 2................................ 40
3.2 Refractive Index surface for 2.3 kHz, at L = 2, plotted with the same aspect ratio
for both axes................................................................... 40
3.3 Refractive Index surfaces for 2.3 kHz, at the equator of L = 2.................. 42
3.4 Refractive Index surfaces for 3 kHz, at the equator of L = 2.................... 43
3.5 Refractive Index surfaces for 10 kHz, at the equator of L = 2................... 43
3.6 Refractive Index surfaces for 3 kHz, at the equator of L = 2 with 1 eV temperature and with 4 eV temperature 44
4.1 (a) Trajectories of whistler mode waves with a frequency of 3.5 kHz, launched
at the equator of L = 2, with an initial wave normal angle of 88, with (Red) and without (Black) finite electron and ion temperature, (b) original power attenuation plots, (c) zoomed version of power attenuation plots.............. 46
x
4.2 Whistler mode waves with a frequency of 3 kHz, launched at the equator of L = 2, with (a).i initial wave normal angle of 85, (b).i initial wave normal angle of 85,(c).i initial wave normal angle of 95, (a).ii, (b).ii, (c).ii show the power
attenuation plots of the corresponding trajectories with and without temperature effects. In all the panels Black traces were produced with cold plasma conditions and Red traces were produced with warm plasma conditions............................. 54
4.3 Whistle mode wave trajectories with a frequency of 2.1 kHz launched at the equator of L = 2 with cold (Black) and warm (Red)plasma conditions, (a) shows
the original trajectories and (b) shows the zoomed in version........................ 55
4.4 (a) Power attenuation due to Landau damping for the trajectories shown in Figure
4.3. (b) Group velocity along the trajectories shown in Figure 4.3................... 55
4.5 Whistle mode wave trajectories with a frequency of 2.98 kHz launched at the equator of L = 2 with cold (Black) and warm (Red)plasma conditions, (a) shows
the original trajectories and (b) shows the zoomed in version........................ 56
4.6 (a) Power attenuation due to Landau damping for the trajectories shown in Figure
4.5. (b) Group velocity along the trajectories shown in Figure 4.5................... 56
4.7 Whistle mode wave trajectories with a frequency of 7 kHz launched at the equator
of L = 2 with cold (Black) and warm (Red)plasma conditions, (a) shows the original trajectories and (b) shows the zoomed in version.......................... 57
4.8 (a) Power attenuation due to Landau damping for the trajectories shown in Figure
4.7. (b) Group velocity along the trajectories shown in Figure 4.7.................... 57
4.9 Whistler mode wave trajectories of a 3 kHz wave launched with an initial wave
normal angle of 60, at the equator of L = 2 under cold (Black) and warm (Red) plasma conditions, (a)original wave and (b) shows the zoomed in version. ... 58
4.10 Group velocities of the wave trajectories shown in Figure 4.9......................... 58
xi
4.11 Whistler mode wave trajectories of a 3 kHz wave launched with an initial wave
normal angle of 89, at the equator of L = 2 under cold (Black) and warm (Red) plasma assumptions, (a)original wave and (b) shows the zoomed in version. . 59
4.12 Group velocities of the wave trajectories shown in Figure 4.11.................... 59
4.13 Whistler mode wave trajectories of a 3 kHz wave launched with an initial wave
normal angle of 89, at the equator of L = 2 under cold (Black) and warm (Red) plasma conditions, (a)original wave and (b) shows the zoomed in version. 60
4.14 Group velocities of the wave trajectories shown in Figure 4.13.................... 60
4.15 Magnetospheric reflection points for waves with a frequency of 3 kHz, launched at the equator of L = 2, with an initial wave normal angle of 89, with (Red)
and without (Black) temperature effects........................................... 61
4.16 Difference between wave frequency and local hybrid resonance as a 3 kHz wave propagates under warm plasma conditions with initial wave normal angle of 88.9
(top) Instantaneous wave normal angle along the trajectory (bottom)............. 62
4.17 Whistler mode waves launched with an initial wave normal angle of 89, at the equator of L = 4, with frequencies 250 Hz, 375 Hz and 1 kHz. Left hand panels show the original trajectories and panels on the Right hand side show the zoomed in versions. All trajectories in Black were produced with cold plasma assumptions
and all trajectories in Red were produced with warm plasma assumptions. ... 63
4.18 Whistler mode wave trajectories for a 380 Hz wave launched at the equator of
L = 4, with an initial wave normal angles 85, 89 and 89. All trajectories in Black were produced with cold plasma assumptions and all trajectories in Red were produced with warm plasma assumptions. Left hand panels show the original trajectories and panels on the Right hand side show the zoomed in versions.......................................................................... 64
5.1 Origination locations and spectrograms of chorus and hiss waves.Image source: [Bortnik et aZ. (2008)] ......................................................................... 66
xii
5.2 Regions of chorus and hiss energy confinements with respect to the Magnetic
Local Time.Image source: vlf.stanford.edu/Daniel Golden Thesis ............... 67
5.3 Artists illustraton of the Earths radiation belts and the trajectory of Van Allen
Probe spacecrafts. The twin spacecrafts are shown here. Image source:Andy Kale, University of Alberta ............................................... 67
5.4 (a) Wave energy measured by the Van Allen Probe spacecrafts, (b) Same wave
energy as shown in (a), normalized to the maximum at each frequency........... 68
5.5 Normalized wave power with the wave growth analysis under cold background plasma scenario. Wave power is normalized to the maximum at each frequency. 69
5.6 Normalized wave power with the wave growth analysis under warm background plasma scenario. Wave power is normalized to the maximum at each frequency. 70
5.7 Normalized wave power with the wave growth analysis under cold background plasma scenario. Wave power is normalized to the maximum at each frequency. 72
5.8 Normalized wave power with the wave growth analysis under warm background plasma scenario. Wave power is normalized to the maximum at each frequency. 73
D.l Graphical illustration of the smoothing process given in Equation (D.4), and the
truncation constant \n........................................................ 89
1
CHAPTER I
INTRODUCTION
1.1 NearEarth Space
NearEarth space is the region surrounding the Earth, starting from 70 km above the surface and extending to about 10 Earth radii. One Earth radii is 6371 km, hence nearEarth space extends to about 63,710 km from the surface of the Earth. The defining characteristic of nearEarth space is that matter is in the plasma state. Plasma is the fourth state of matter, where the particles are ionized. This ionized state of matter starts in the Earths ionosphere at altitude of approximately 70 km. Earths magnetosphere extends from an approximate altitude of 1000km, and is also in the plasma state. The magnetosphere gets its name from the fact that the overall density of particles is much lower than in the ionosphere, which makes the Earths magnetic held be the dominant force. In both regions, ionization occurs due to solar radiation.
The nearEarth space environment is of a significant interest for both physicists and engineers for several reasons. One main reason is the richness of energy dynamics in which the sun plays an important role. Bursts of additional plasma and energy from the sun as solar hares and coronal mass ejections directly impact nearEarth space and even distort the geomagnetic held. The energy interactions involve not only plasma particles but also different types of electromagnetic and electrostatic waves, both natural and manmade. Manmade whistler mode waves include monochromatic signals from ground based ELF/VLF transmitters, triggered ELF /VLF emissions from the large scale ionospheric heating facilities such as High frequency Active Auroral Research Program (HAARP). The large scale effects of such interactions are often denoted with the term space weather.
Furthermore, as shown in Figure 1.1, nearEarth space hosts manmade electronics launched for communication, navigation, monitoring and military purposes. Hence it contains equipment worth billions of dollars. The operational condition and lifetime of those manmade electronics are strongly affected by the surrounding plasma environment. There
2
fore understanding the physics of nearEarth Space is an important task of the physics and engineering communities. Various space missions were conducted in order to study whistler mode waves and their impact on different aspects of space weather, such as the CLUSTER space craft launched to observe the chorus (a naturally generated whistler mode wave type) waves and Van Allen Probe spacecrafts launched to study the Earths radiation belts.
RHESSI
Voyager (2)
TIMED
TWINS (2)
Geotail (JAXA)
STEREO (2)
SOHO (ESA) j a
I Implementation I Primary Ops I Extended Ops
Solar
Probe Plus
Van Allen Probes (2)
I
$ 1 Hinode (JAXA)
Solar
Orbiter (ESA)
THEMIS (3)
THEMIS MMS (4)
(ARTEMIS 2)
Figure 1.1: Near Earth space environment, housing man made electronics for research, communication, navigation and military purposes. Image source: www.science.nasa.gov.
1.2 Whistlers and Whistler Mode Waves
A dominant player in nearEarth space environment is a special type of electromagnetic wave called the whistler mode wave. The name of this class of waves derives from a specific occurrence of these waves that was first observed to make a whistling sound in audio equipment. During the trench warfare of World War I, long wires were laid on the ground for communication. When the German physicist Heinrich Barkhausen tried to eavesdrop on Allied communication lines, he reported intermittently hearing a very remarkable whistling note. During that time he was unable to explain the source of this unusual whistling sound heard
3
through the earth current detector communication system that he was using. After an unsuccessful attempt to recreate these whistling tones inside a lab, Barkhausen concluded that these waves are created due an unidentified natural phenomenon [Barkhausen( 1919)]. Due to the sound they made when passed through a speaker, these waves were called whistlers.
Years later, in 1935, English theoretical physicist and engineer Thomas Eckersley, observed that a whistler occurs about one second after a sferic. A sferic or an atmospheric (shown in Figure 1.2) is a broadband electromagnetic emission induced by a lightning strike. Whenever there is a lightning strike, the resulting ionized plasma channel acts as an antenna and radiates waves with many frequencies since a high amount of electric current (10 100 kA) flows in a very short (< 100/iSec) amount of time. Some of this wave energy is trapped in what is known as the EarthIonosphere waveguide (shown in Figure 1.3) and some of it is leaked into the upper layers of the ionosphere and magnetosphere. Sferics can be guided by the EarthIonosphere waveguide for thousands of kilometers before being attenuated. Based on this observation, Eckersley speculated that whistlers are related to lightning and are an echo of sferics [Eckersley (1935)], but he was not able to explain the main feature of whistlers, their whistling sound, which is caused by dispersion.
Whistlers are electromagnetic waves consisting of a continuous band of frequencies. Figure 1.4 shows a spectrogram of a whistler. One main observation which can be made with the spectrogram is that for whistlers, higher frequencies arrive at the receiver prior to the lower frequencies. In other words, different frequency components propagate at different velocities. This phenomenon is known as dispersion. It is due to the dependence of refractive index on frequency.
L.R.O Storey gave an explanation for the dispersive behavior of whistlers [Storey(1953)]. As mentioned earlier, when there is a lightning strike, it emits electromagnetic waves with different frequencies. Some of those waves are being guided between the Earth Ionosphere waveguide and some of those are leaked into the magnetosphere. Those waves which are leaked into the magnetosphere can be guided by the geomagnetic (Earths magnetic held)
4
Chistochina 21 Jul2011 UT N/S Antenna
Figure 1.2: Observation of a sferic at a very low frequency antenna located in Chistochina, Alaska. Top panel shows the amplitude of the sferic captured by the antenna oriented in the geomagnetic NorthSouth, and bottom panel shows the amplitude of the same sferic captured by the antenna oriented in the EastWest direction.
held lines and return to earth in the conjugate hemisphere. Conjugate point is the hypothetical location where a geomagnetic held line connects to the surface of the Earth at the opposite pole. Along this path, the waves propagate through the plasma in the magnetosphere, which is highly dispersive for wave frequencies below 30 kHz. Storey had thus succeeded in explaining Barkhausens observations made over 30 years earlier. Since the initially identified whistlers were created due to lightning, those were called lightning generated whistlers.
Storey and Barkhausen were dealing with waves induced by lightning; later on, it was identified that there are other types of waves created by natural and manmade sources, which can also propagate in the same mode. Nevertheless, the name from the original observations has stuck and the name of the propagation mode is the whistler mode, even when the source is not lightning related.
5
Magnetosphere
Ionosphere
Atmosphere
Earth
Figure 1.3: Image of the Earth Ionosphere Waveguide(EIW). Image source:vlf.Stanford
1.3 Fundamentals of Whistler Mode Waves
As mentioned above, whistler mode waves exist only in a magnetized plasma. Here we present the fundamentals of these waves and contrast their properties with those of simpler electromagnetic waves in free space. All electromagnetic phenomena derive from Maxwells equations:
VE
Pv
e
VB
V x E = V x B = peJ
= 0
<9B
<9eE
~df
(1.1)
Equation (1.1) shows Maxwells equations for any medium. For a macroscopically neutral medium, pv = 0. We start with assuming a time harmonic electric held with sinusoidal spatial variation which can be written in complex phasor notation as E = Eaek'r^ax. Here, E0 is the amplitude, t is the time, u is the wave frequency, k is the wave normal vector and ax is the unit vector in the x direction. Using the equations shown in Equation (1.1), we can
6
Palmer. Station^ March] 2,19921
Whistler
 Triggered j
. PmiceinniC 3
Emissions
Sferics
N 6.0
C 4.0
Li 2.0
2 3
Time (Seconds)
dB
to
20
30
Figure 1.4: Spectrogram of a whistler mode wave together with sferics and triggered emissions. Sferics are occurred with the lightening strikes. Image source: [Green and /nan(2015)]
arrive at the plane wave equation shown in Equation (1.2) for free space, p.o and eo are the permeability and permittivity in free space.
V2E p,0e0^2E (12)
For a non trivial solution the coefficients of the electric held should be equal to zero. This condition is called the dispersion relation. Hence for free space the dispersion relation can be written as:
a;2 A:V = 0 (1.3)
In Equation (1.3), c is the speed of light (A=), and k is the magnitude of the wave normal vector. Full derivation of this dispersion relation is given in Appendix A.
For a plasma medium in the absence of a static magnetic held the dispersion can be expressed as:
k2c2
ui
UJ,
'pe
Ur
(1.4)
7
where ujpe is the electron plasma frequency. The plasma frequency is the frequency at which the plasma particles oscillate at, if an external disturbance is introduced. The electron plasma frequency is given by where Ne is the electron density, me is the mass of an electron, qe is the charge of an electron. Derivation of this dispersion relation is given in Appendix B.
From the dispersion relation shown in Equation (1.4), it can be seen that for a wave to propagate through a plasma in the absence of a static magnetic held, the wave frequency should be higher than the plasma frequency. For wave frequencies lower than the plasma frequency, the wave normal vector (wave number) becomes imaginary. Waves with an imaginary wave number are being attenuated very quickly and those waves are called evanescent waves. Therefore, the cutoff frequency for propagation in a plasma in the absence of a static magnetic held is the plasma frequency.
Figure 1.3 illustrates how the frequency dependence of Equation (1.4) manifests itself for propagation of waves generated on Earth. For a wave to penetrate through the plasma, the wave frequency should be higher than the plasma frequency; otherwise, the waves are rehected back and forth between the Earth and the ionosphere and can propagate to long distances. Extremely Low Frequency (ELF) waves in the frequency range 300 Hz 3 kHz and Very Low Frequency (VLF) waves in the frequency range 3 kHz to 30 kHz are well below the plasma frequency. Therefore, those waves propagate within the Earth Ionosphere waveguide, whereas high frequency signals such as microwave signals propagate through the ionospheric plasma L
When there is a static magnetic held present, the dispersion relation becomes more complicated and is governed by the AppletonHartree equation that is presented in Chapter
2. Full derivation of the dispersion relation in a cold magnetized plasma is given in Appendix
C. At this point, we consider only the special case of propagation of waves parallel to the
1Here we have discussed the Earths ionosphere as unmagnetized plasma in effect ignoring the geomagnetic field. Such as assumption is accurate to first order in the lower ionosphere, since the collision frequency is sufficiently high to make magnetic field secondary. We will show later that the geomagnetic field does allow for a fraction of low frequency wave energy to propagate through the ionosphere via the whistler mode.
8
static magnetic field vector. In such a scenario, the dispersion relation simplifies to:
Ire2
ur
1 
UJ,
pe
(1.5)
OJ(OJ UJce) J
In Equation (1.5), ojce is the electron cyclotron frequency defined as ojce = The
electron cyclotron frequency is the precession frequency of electrons under the influence of a magnetic field. qe, me, B0 are charge of an electron, mass of an electron and the magnitude of the static magnetic field. The mode of propagation expressed in Equation (1.5) is called the whistler mode. Whistler mode propagation exists only in the presence of a static magnetic field. In order to propagate in the whistler mode the frequency of the waves should be lower than the plasma frequency and the electron cyclotron frequency. Frequencies higher than the electron cyclotron frequency results in an imaginary wave number. The dispersion relation diagram of a whistler mode wave is given in Figure 1.5.
t'ipe "> (,ice
Figure 1.5: Dispersion diagram of a whistler mode wave. The dispersion relation for free space (us = ck) is also shown in the plot.
Due to the fact that the plasma is a dispersive medium, waves with different frequencies propagate with different velocities. It is important to introduce two descriptions of wave velocity: group velocity and the phase velocity. Group velocity (vg) is the velocity at which the energy travels; phase velocity (vp) is the velocity in which the wave pattern travels. In
9
other words, for sinusoidal waves, phase velocity is the propagation velocity of a null or a crest. Group velocity vg is defined as duj/dk and the phase velocity is defined as uj/k. In free space, both group velocity and phase velocity converge to the speed of light for electromagnetic waves. Whistler mode waves are considered as a slow mode, since the group velocity of whistler mode waves is typically about a factor of a 100 smaller compared to the speed of light. From Figure 1.5, the slope of the free space mode and whistler mode can be compared. The slope of the free space line shows the speed of light is much larger than the slope of the whistler mode curve, which varies with frequency.
Although whistler mode waves are low frequency waves, they penetrate though the ionosphere and propagate into the magnetosphere through a leakage process. Once in the magnetosphere, whistler mode waves can propagate in two ways: ducted and nonducted. In ducted propagation, whistler mode waves are guided by the geomagnetic held lines and will return to Earth at the magnetic conjugate point. The magnetic conjugate point is where the magnetic held lines connect back to the surface of the Earth. Ducted propagation results from heldaligned plasma density irregularities and has many similarities to optical hber guiding. In the case of nonducted propagation, whistler mode waves leaked into the magnetosphere follow a much more complicated trajectory, and they are rehected multiple times before being completely attenuated. Nonducted whistler mode waves waves might encounter multiple rehections depending on the local refractive index. These type of whistler mode waves are called magnetospherically rehected whistlers or MR whistlers. Figure 1.6 shows an illustration of whistlers propagated along ducted and non ducted trajectories 2
1.4 Role of Whistler Mode Waves in Near Earth Space
Whistler mode waves play a dominant role in the Earths magnetosphere [Kulkarni et al. (2015)]. As mentioned above, Earths magnetic held is the dominant force in the magnetosphere Figure 1.7 shows an artists illustration of the geomagnetic held. Distance measurements in the magnetosphere are expressed in terms of L shells. L values are in the
2Storey and Barkhausen were investigating ducted whistler mode waves.
10
Figure 1.6: Different modes of propagation for whistler mode waves; ducted and nonducted. Here in the figure fw is the frequency of the wave and Jlhr is the lower hybrid resonance frequency. Ducted waves are guided by the geomagnetic field and the nonducted waves show multiple reflections. Image source:[Green and /nan(2015)]
units of Earth radii at the magnetic equator and describe the geomagnetic field line. For example, L = 1 is one Earth radii from the center of the Earth; hence, it is located on the surface of the Earth. Similarly, L = 2 is located two Earth radii from the center of the Earth and one Earth radii from the surface of the Earth. Figure 1.8 shows a graphical illustration of L shells.
Our Earth is surrounded by two regions of high energy plasma particles known as the Van Allen Belts, discovered by American space scientist James Van Allen based on the data collected by Explorer 1 and 4. High energy electrons created due to cosmic ray interactions are trapped in these radiation belts. Figure 1.9 shows an illustration of the Van Allen Belts. The inner belt extends from an altitude of about 1000 km up to L = 3. The inner belt consists of high energy electrons typically in the energy range between 0.04 MeV to 4.5 MeV [Green and /nan(2015)]. The outer radiation belt, which extends from L = 5 to L = 7, consists of high energy electrons up to 7 MeV. The region between the two radiation belts
11
Figure 1.7: Illustration of the geomagnetic field. Image source: www.science.nasa.gov.
Figure 1.8: Illustration of the concept of L shells. Image source: en.wikipedia.org.
(between L=3 and 5) is known as the slot region.
These trapped high energy electrons gyrate around the geomagnetic held lines as shown in Figure 1.10 and, in the absence of any perturbation, can remain trapped indefinitely. Whistler mode waves can change the momentum of these electrons via resonance interactions. Figure 1.11 shows the resonant interactions of whistler mode waves from lightning with trapped higher energy particles and subsequent precipitation of electrons. These resonance interactions can modify the trajectory of these trapped electrons with respect to the geomagnetic held. Hence the pitch angle of the electric held of the waves can be reduced. Lower pitch angles causes electron trajectories to come closer to the Earth. When the pitch angle
12
Figure 1.9: Van Allen radiation belts and the satellites launched by NASA for Van Allen belts observations. Image source: www.science.nasa.gov.
is 0, electrons are no longer trapped and will impact the atmosphere guided into it by the geomagnetic held lines. Such electrons which make contact with the atmosphere (specifically the ionosphere) deposit their energy in collisions and are said to precipitate. The Aurora Borealis (at the geomagnetic north pole) and Aurora Australis (at the geomagnetic south pole) are natural effects of electron precipitations. These high energy electrons interact with various atoms in the atmosphere and raise their electrons to reach higher energy (excited) states. When those excited electrons relax back to their ground state, they release energy as light observed as auroras.
As mentioned above, whistler mode waves are a key driver of the energy dynamics in nearEarth space [Bell et al.(2002), Bortnik et aZ.(2006a), Bortnik et aZ.(2006b), Bortnik et al. (2007a), Shprits et al. (2008), Li et al. (2011)]. Interaction of whistler mode waves with the trapped higher energy particles in the radiation belts is one of the main processes of particle loss from radiation belts, wave amplification and wave growth [Kennel and Petscheck(1966), Lyons et al.(1972), Huang and Goertz( 1983), Abel and Thome( 1998a), Abel and Thome( 1998b), Bortnik et a/.(2007b), Ornura et a/.(2008)]. Hence the propagation characteristics of whistler mode waves are of great interest.
13
FLUX TUBE
NORTH
MAGNETIC CONJUGATE POINT
Figure 1.10: Trajectory of trapped higher energy particles and the magnetic conjugate point. Image source: Handbook of Geophysics and the Space Environment (Air Force Research Laboratory ,1985).
As discussed later in Chapter 2, Earths magnetosphere is an anisotropic medium; hence, the propagation characteristics of a wave depend on the propagation direction. The propagation of magnetospherically reflection whistlers can be tracked by a numerical process called raytracing [Ha,selgrove( 1955), Inan and Bell( 1977)] and is the topic of this thesis.
1.4.1 Cold and Warm Plasma
Before concluding this chapter, we disucss one additional important parameter of plasmas that affects how waves propagate through them. This parameter is the plasma temperature and can be rigorously discussed in the same manner as for ideal gasses.
The kinetic energy of a particle with mass m and speed u can be written as \mu2. For
a number N of particles with masses m* and speeds tq, average kinetic energy is given by
1 2 2N Â£i=l
When a plasma is in thermal equilibrium at temperature T, particles have a range velocities. The velocity distribution of the particles can be expressed by a Maxwellian or MaxwellBoltzmann distribution. For a onedimensional distribution, MaxwellBoltzmann velocity distribution f(u) can be given by;
/() =
1 mu^
2 T
(1.6)
14
Precipitation
Figure 1.11: Precipitation of electrons trapped in Earths radiation belts with the interaction of whistler mode waves from lightning. Image source: vlf.stanford.edu.
In Equation (1.6), A is a multiplier which can be found by the total density N, ks is the Boltzmann constant which is equal to 1.38 x 10 23 JK~l and T is the absolute temperature in Kelvin. Figure 1.12 illustrates the Maxwellian velocity distribution.
Figure 1.12: MaxwellBoltzmann velocity distribution in one dimension.
The Maxwellian velocity distribution can be used to find the average kinetic energy of
the particles as shown Equation (1.7).
foo \mu2f{u)du
Ioo f(u)du
(1.7)
15
For a one dimensional system, the average energy of particles is \ksT. This result can be extended for a three dimensional system, in which the average kinetic energy is Â§ksT.
The energy of particles in plasmas, is expressed in units of electron volts (eV) instead of the standard unit of Joules. In order to express the energy in eV, the kinetic energy should be divided by the magnitude of the charge of an electron (1.602 x 1019). It is important to note here that the temperature T is an indicator of kinetic energy. And it is thus also referred to as a kinetic temperature.
As we will be discussing later, for the cold plasma we will be considering in Chapter 2, the absolute temperature is 0 K. And the kinetic energy of particles is zero. Hence the random thermal velocity of particles under the cold plasma assumption is also zero in other words, under cold plasma assumptions, the particles are motionless when the plasma is unperturbed by an external held [Stix( 1992)].
Hence the absolute temperature of the bulk particles in the Earths magnetosphere (not including the small population of high energy particles making up the radiation belts) is about 11600K. It is worth mentioning here that, although the absolute temperature is a very high value, it does not indicate that the Earths magnetosphere is hot or is a reservoir of large amounts of thermal energy. The particle density in the Earths magnetosphere is approximately 106 m~3, whereas the particle density in the atmosphere is in the orders of 1025 m~3. Due to the lower particle density in the magnetosphere compared to the atmosphere, the heat capacity of the magnetosphere is much lower than the heat capacity of the atmosphere.
In order to distinguish the two cases between 0 K absolute temperature and 11600 K temperature, we use the terms cold and warm, respectively. The term hot plasmais often used to describe the high energy particles making up the radiation belts. It is worth reemphasizing that the high energy radiation belts particles are a small minority of all the charged particles in the magnetosphere.
16
Later in the thesis we will be considering these high energy electrons trapped in the radiation belts, when introducing Landau damping. Landau damping is the attenuation of wave power when the wave encounters a high energy electron distribution. We will be introducing the mathematical formulation of Landau damping in Chapter 2. Chapter 4 discusses the the simulated ray trajectories in detail.
1.5 Scientific Contributions
In this work we study the effect of whistler mode ray trajectories with the inclusion of finite electron and ion temperature.
1. This is the first time analysis of whistler mode trajectories has been done with finite thermal effects.
2. This project presents a fully functioning raytracer which takes temperature into account. The raytracer is modified to work with or without finite electron and ion temperature. The user can define the electron temperature, ion temperature simultaneously.
3. Also in this work we introduce an updated equation for wave power attenuation with Landau damping; by modifying the existing Landau damping equation for cold plasma. Landau damping formulations are included into the main raytracer, and the power attenuation of the waves was studied under cold and warm background plasma assumptions.
4. In the presence of a significant temperature anisotropy, the waves experience a power growth. We have conducted the wave growth analysis, under cold and warm plasma assumptions, using a biMaxwellian velocity distribution. More details of wave growth analysis will be given in Chapter 5.
5. Comparison of our simulation results with VanAlien Probe spacecraft observations show that, when Landau damping and wave power growth are taken into consideration, there is a better agreement under warm plasma assumptions.
17
1.6 Thesis Organization
In Chapter 2 we present the process of numerical raytracing that we have used. Starting from cold plasma raytracing, we discuss the modifications we made to the existing 3D raytracer. We also discuss the modifications made to the power attenuation of whistler mode waves. Chapter 3 presents the modifications observed with the refractive index surface with the inclusion of finite electron and ion temperature. Since in raytracing medium properties enter the ray tracing equations in terms of the refractive index, calculation of the refractive index surface is a very important step.
In Chapter 4 we present the modifications observed in the ray trajectories specifically when finite temperature effects are included. We discuss the modifications observed with respect to wave frequency, initial wave normal angle and launched location.
Chapter 5 presents the comparison of wave simulations with Van Allen Probe observations. Here in this chapter we consider wave damping and wave growth due to resonance interactions. Chapter 6 presents conclusions from our results and proposed future work. Additional work we have performed in terms of statistical modelling and identifying the outliers is presented in Appendix D.
18
CHAPTER II
THEORETICAL BACKGROUND
2.1 Introduction to Numerical Raytracing
Numerical raytracing is the process of determining the powerflow path of a whistler wave by solving the Haselgrove equations [Haselgrove(1955)].Haselgroves equations are given in Equation(2.31), and will be explained in detail later in this chapter. Graphical ray tracing solutions to wave propagation in the Earths magnetosphere were carried out as early as 1956 by [Maeda and Kimura( 1956)]. The effect of ions on whistler mode raytracing was first studied by [Kimura(1966)]. Since the 1970s a very large body of magnetospheric whistler mode raytracing work has been carried out [Inan and Bell( 1977), Huang and Goerte(1983), Cairo and Lefeuvre{ 1970), Bortnik et aZ. (2006a), Bortnik et aZ.(2006b), Churn and Santolik(2005), Gotkowski and Inan(2008), Chen et al.(2009), Golden(2011), Kulkarni et a/.(2006)].
However, with a few exceptions, the majority of previous works have relied on the ideal cold plasma assumption (both electrons and ions are at 0 K) in calculation of the wave refractive index surface even though the temperature of the background electrons and ions in the magnetosphere is known to be in the range of a few eV [Decreau et a/.(1982)]. This background plasma should not be confused with the radiation belts which contain hot plasma particles in the keV MeV range but with overall densities too small to affect the propagation trajectories directly. [Aubry et a/.(1970)] used warm plasma adiabatic theory for the frequencies around the plasma frequency and the upper hybrid resonance frequency to derive the equations related to the dispersion relation as well as for the group velocity. In that work all the equations were derived by considering only isotropic temperature distribution of electrons. [Bitoun et a/.(1970)] demonstrated the procedure for determining the ray trajectory in a warm plasma using the equations derived by [Aubry et al. (1970)]. In that work the authors used the raytracing observations to interpret the top side resonance at upper hybrid frequency as oblique echoes. Top side resonances are the plasma resonances observed
19
around the plasma frequency and the upper hybrid frequency in the top side ionosphere (600 km 1000 km).
2.2 Refractive Index Surface
For whistler mode waves the Earths magnetosphere is a nonhomogeneous and anisotropic medium. In a homogeneous medium, the medium properties are the same in every location. In isotropic media, the properties are the same in all directions. The magnetosphere is nonhomogeneous since the plasma density and the strength and direction of the geomagnetic held vary from location to location. It is anisotropic because the geomagnetic held imposes a directional dependence on the wave properties.
Before proceeding further, here we introduce some of the key terms used in this work. As mentioned above, due to geomagnetic held Earths magnetosphere becomes an anisotropic medium. The vector drawn perpendicular to the wave fronts is called the wave normal vector. The angle between the wave normal and the geomagnetic held is known as the wave normal angle. Throughout this work the wave normal angle will be indicated by the Greek letter 0. The surface plot obtained once the refractive index is plotted at each wave normal angle, is known as the refractive index surface. The vector drawn perpendicular to the refractive index surface at a location is the direction of the group velocity or the ray direction at that point. When a wave is going through a magnetospheric rehection, the ray direction changes from positive to negative or vice versa. In order to satisfy that condition, the refractive index needs to be closed. In other words closure of the refractive index surface is required for the occurrence of magnetospheric reflections. In Figure 2.1, the angle measured from the geomagnetic held direction to a reddashed line, that angle is know as the resonance cone. If a wave is launched with an initial wave normal angle greater than the resonance cone, it would not propagate.
Figure 2.2 (a) shows the refractive index surface for an isotropic medium. The arrows in black show the direction of wave normal and the arrows in purple show the direction of group velocity. The direction of group velocity is the direction of wave energy propagation.
20
Figure 2.1: Orientation of the wave normal vector and the group velocity vector in an anisotropic medium
Figure 2.2: (a) Isotropic, non homogeneous medium, (b) Non homogeneous and anisotropic
21
Direction of wave normal.
~ Direction of
Mr .
group velocity.
B
Figure 2.3: Orientation of the wave normal vector and the group velocity vector in an anisotropic medium.(This figure is a recreation similar to Figure 3.13 in [Helliwell( 1965)])
In an isotropic medium the wave normal and the group velocity both are oriented in the same direction. In contrast to the case in Figure 2.2 (a), in an anisotropic medium (b), the refractive index is not the same in all directions, and the wave normal and group velocity are not in the same direction. Figure 2.3 further shows the propagation of a wave packet in anisotropic medium from point A to point B.
Figure 2.4, shows the progression of the refractive index surface, direction of the ray and the direction of the wave normal, at the first magnetospheric reflection point for a 3 kHz wave launched at latitude 30, at an altitude of 1000 km. As the wave goes through a magnetospheric reflection, wave normal angle becomes 90, and the ray direction alters.
2.3 Dispersion Relation in Cold Plasma
If the medium contains only mobile electrons (ions are stationary) the refractive index can be calculated by the well known AppletonHartree equation given in Equation (2.1).
pe
1 1/2
(2.1)
22
Figure 2.4: Progression of the refractive index surface, ray direction and the direction of the wave normal for a 3 kHz wave.Image source: [Bortnik(2005)].
23
In Equation (2.1), ojpe and uce represent plasma frequency and cyclotron frequency of electrons calculated as a function of electron density Ne, charge of an electron qe, mass of an electron me, magnitude of the geomagnetic held B0 and the permittivity of free space eo as in Equations (2.2), and 9 is the wave normal angle.
LOpe
^ce
Neql
qeB0
me
(2.2)
In this work, the medium is composed of multiple species such as electrons H+, He+ and 0+ ions which will yield a modified version of Equation (2.1). To derive the dispersion relation including these ions, we use the approach shown in [Bortnik(2005)].
Using the Maxwells equations and the Langevin equation of particle motion, the linearized time harmonic plane wave equations can be obtained as:
<9B
VxE = sF
V x B /x ( J + f
Du ^
ms~Dt = ^ (E + u x B)
(2.3)
In Equation (2.3) E is the intensity of the electric held perturbation, B it the magnetic perturbation, k is the wave normal vector and t is time. ms and qs are the mass and charge of a particle from species s. u is the particle velocity. For a constant ambient magnetic held, B is a constant; hence, the time derivative of the magnetic held equals to zero. D/Dt indicates the convective derivative, given by D/Dt = d/dt + u V. Solving above set of equations given in (2.3) leads to the following relationship in Equation (2.4).
^ V) ]L)[uj) fi sin tf cos o rjx
0 E = 0 (2.4)
(S(uj) fj2 sin2 9) jD{uj) jD{uj) S{uj) n2
V
fj, sm 9 cos i
0 B(uj) /i2 sin2 9 y
Eo,
E< /
0
24
In Equation (2.4), n is the real part of refractive index, Ex,Ey and Ez represent the x,y and z components of the electric held, and S, P, R, L, D represent Stix [Stix( 1992)] parameters dehned as;
R(oj)
L{uj)
ph = 1 E
S
, ,2
= iE
S
= iE
S{uj) =
>2
ps
oj (cu E cacs)
, ,2
^ps
UJ (oj UJCS)
R(ca) E L{uS)
D{uj)
R(uj) L{uj)
(2.5)
In the Stix parameters. ujps and ujcs represent the plasma frequency and cyclotron frequency of the species s:
Nsq.2
ps 
t0ms
_ qsB0
Mcs 
ms
(2.6)
In the above set of equations in (2.6), Ns is the particle density, Although the Stix parameters are functions of angular frequency oj, from here onwards we will be using the notations S,D,P,R and L to address those parameters.
By setting the determinant of Equation (2.4), to zero we can arrive at the dispersion relation of a cold, collisionless, unbounded plasma given in Equation (2.7).
E Bq/i2 E Cq = 0 (27)
25
ji is the refractive index. The coefficients Aq, B0 and Co are functions of the Stix parameters and the wave normal angle 9. Those coefficients are given in the set of equations:
A0 = S sin2 9 + P cos2 9 B0 = RL sin2 9SP{ 1 + cos2 9)
Co = PRL
(2.8)
As expressed in Equation (2.7) the dispersion relation of a wave propagating under cold plasma conditions is a fourth order equation. Once we consider fj2 as a single quantity we can solve the above Equation (2.7) for fj2, using the quadratic formula, and we arrive at the following two solutions given in Equation (2.9);
2 ~Bo ~ \/Bo ~ 4A0Co
h =,do >9
2A0
2C0
Bo + \J Bq AAqCq
Bo < 0
(2.9)
2.4 Dispersion Relation with the Inclusion of Temperature Effects
Inclusion of temperature effects increases the complexity of the dispersion relation.
[.Aubry et a/.(1970)] derived the modification to the dielectric tensor K, with the inclusion of temperature effects. Authors of [Aubry et al. (1970)] started with the plane wave solution of a wave propagating in a homogeneous plasma given in Equation (2.10) as expressed in [Stix( 1992)]. In Equation (2.10) k\\ is the parallel component of the wave normal vector the geomagnetic held. In Equation (2.11), 5^ represents the elements of a unity matrix and are components of the dielectric tensor.
(kkkl{2+ ^KjE = 0 (2.10)
For the above Equation (2.10) to have a solution it should satisfy the dispersion relation, hence the following equation should be satisfied.
26
det \kikj k25ij + uj2/c2Ky] = 0 (2.11)
Equation (2.11) is then solved for the dielectric tensor using the classical equation of conductivity given in Equation (2.12) where / is the unity matrix and er(k,u;) is the plasma
conductivity tensor.
K (k, u) = / + j
.a (k,Â£n)
UJto
(2.12)
Introduction of temperature effects the modifies the overall dielectric tensor as given in Equation (2.13).
K = K(0) + tK(1) (2.13)
In Equation (2.13), denotes the dielectric tensor under cold plasma conditions and denotes the dielectric tensor with temperature effects taken into account. The small parameter r is defined as k2ksT/msuj2, where k is the magnitude of the wave normal vector. [Aubry et a/.(1970)] defined the dielectric tensor under cold plasma conditions as in Equation (2.14).
1 t4t9 0
o
K()
x
1V2 jXY
1 
X
(2.14)
1V2 ^ lV2
0 0 1 X
In Equation (2.14), X and Y are functions of plasma frequency ups and cyclotron fre quency ojcs of species s as in Equation (2.15).
2
X
cq
'ps
OJ*
(2.15)
Y =
UJ
The dielectric tensor under cold plasma conditions given in Equation (2.14) can also be expressed using the Stix parameters as in Equation (2.16).
27
5 jD 0 jD S 0
(2.16)
K()
OOP
It is worth noting that both equations (2.14) and (2.16) represent the same dielectric tensors but in two different notations. For easiness of expression, in our work to represent the cold plasma dielectric tensor components we will be using the tensor as in Equation
(2.16).
Assuming that the temperature effects were given to particle species s, the dispersion relation of under warm plasma conditions can be expressed as in Equation (2.17);
qsTAisfi6 + (A0 + q&TB\sj fi4 + (Po + qsTCis) + Co 0 (217)
with qsT for particle species s defined as in Equation (2.18), where kB is the Boltzmanns constant, Ts is the temperature in Kelvin of species s. A0,B0 and C0 are the coefficients of the cold plasma dispersion relation calculated as in Equation (2.8), also:
T
kBTs
m.c2
(2.18)
In the dispersion relation of warm plasma (Equation (2.17)), the coefficients Ais,Bis and C\s are defined as in Equation (2.19) for plasma species s.
Ais = Kii.^ sin2 0 + K^A cos2 0 + 2KiSs(A sin 0 cos 0 (2.19) Bls = (iFlls(1)A + K22sA)S + 2jDK12sA^ sin2 0 K338W (S + S cos2 0) P(S + S cos2 0) + 2 sin 0 cos 0 (]DK2is^ SK13gA Cu = KsJA ((JjD)2 + S2) + P (2jDK12sA) + SK22sA) + SKns{1))
28
The parameters S, P and D are the Stix parameters (Equation (2.5)). The additional KijJ'1'1 parameters are the components of the warm plasma dielectric tensor defined as in the set of equations given in Equation (2.20) for species s [Aubry et al.(1970), Kulkarni et al.( 2015)];
Klls{l)
X
3 sin2 9
1 + SY2
K22sw
X
_ 1 T2 \ 1 4Y2 (1 Y2)
2 a
cos2 9
1 + 8Y2 1 + 3T2
1 Y2 V 1 4Y2
sin
(i Y2y
cos
k33sw =
X 3 cos2 9 +
i+i,s(1) = K2UW
jx
6 sin2 9
3 + Y2
sin U
1 Y2
cos2 9
K23s(1) = =
32 s
_1Y2 [14Y2 (1Y2)2
j (3 T2) sin 9 cos
(1 Y2y
X13s(1) = =
31s
2X
_(1 Y2Y
sm U cos
(2.20)
The dispersion relation for warm plasma given in Equation (2.17) is a sixth order equation. Once we consider n2 as a single variable we need to solve a cubic equation. In order to solve the cubic equation we use the formulation developed by the Italian mathematician Cardano in [Cardano( 1545)].
According to Cardanos s formulation the roots of an equation in the form of an3 + bn2 + cn + d = 0 is can be expressed as
n2
n3
p + w 2
p + w 2
b
ni = p + w 6a
b_ jy/3(pw) 3 a 2
b jV?>{p w)
3 a 2
(2.21)
29
where the parameters q,r,p,w are given as:
3ac b2
(2.22)
Q 9 a2
r
9 abc 27 a2d 2 b3 54^
P
\J r + \fq3 + r
.2.
w
\Jr y/q3 + r
2
Out of the three roots of the cubic equations, we need to select the solution which is always real. Because we consider the plasma is collisionless, our result of the refractive index should be a real quantify signifying wave propagation and not attenuation of an evanescent mode. Therefore out of the three solutions we select the n\ solution.
2.5 Introducing Temperature Effects to Multiple Species
In the published work of [Aubry et al. (1970)], the formulation were derived considering only electrons as the plasma species. [Kulkarni et al. (2015)], considered introducing temperature to electrons and H+ ions both but one type of species at a time. In this work we extend the work of [Kulkarni et al. (2015)] in order to introduce temperature to both electrons and ions simultaneously. Without the lack of generality we introduce the following parameters for electrons and ions as in Equation (2.23) and we calculate the KijJ^ for electrons and ions separately using the plasma frequency and cyclotron frequency for electrons and ions while at thermal equilibrium.
In the above Equation (2.23) me and to* are the masses of electrons and ions. Te and Ti are the temperatures given for electrons and ions. This gives the flexibility of setting
(2.23)
30
two temperatures for different species. Hence we calculated the qeT and qiT parameters for electrons and ions separately. Then we introduce those to the dispersion relation equation given by
qJAisfi6 + ^40 + Qs Bisj fi4 + QjCisj fj2 + Co 0. (2.24)
In the Equation (2.24) subscript s refers to different species in the magnetospheric plasma. In this work we have introduced temperature effects to electrons and H+ species.
2.6 Landau Damping
As the whistler wave propagates through the magnetospheric plasma it can interact with highly energetic electron populations quantified in so called suprathermal electron distributions. Typically these suprathermal electron distributions are in the energy range of 100 eV to 1 Mev. Depending on the specifics of the suprathermal electron flux distribution, the whistler mode wave propagating might either grow or attenuate. The interaction between the suprathermal electron distribution and the wave introduces an imaginary component to the index of refraction given by x
X
ch
(2.25)
UJ
The process of Landau damping can be explained with an analogy of surfers on water waves. Surfers are analogous to the hot electron particles, and the water waves are analogous to the whistler waves interacting with them. Assume that each water wave front propagates with the phase velocity vp. When the surfers move along with the water wave front with the same velocity vp, neither the surfers nor the water wave loose energy. Hence there is no acceleration or deceleration. Similarly when the hot energy particles propagate in the same velocity as the whistler wave, there are is no acceleration or deceleration nor wave growth or damping.
Whereas lets consider the situation where the surfers move slower than the water wave, they paddle into the wave. Hence the surfers gain energy from the wave. This results
31
a reduction in waves gravitational potential energy. Similarly when the whistler waves encounter a hot electron distribution moving slower than the wave, the hot energy particles gain energy from the wave, causing a wave energy reduction. This is known as damping. This is the most common situation encountered by whistler waves.
When the surfers move faster than the water wave, they decelerate themselves by bending the surfing trajectory. Hence they loose energy to the wave. In this situation the water wave gets energy from the surfers. Similarly when the whistler waves encounter hot electron particles moving faster than the phase velocity of the wave, particles loose energy to the wave. Increment of wave energy in this process is called a wave growth. For a large number of electrons quantified by a distribution function, damping or growth will be determined by the relative number of electrons moving faster or slower than the wave. Hence the sign of the derivative of the distribution function in velocity space, evaluated at the resonance velocity determines the growth vs damping.
From Tfenne/(1966)] and [5nnca(1972)] x can be expressed as follows;
X
5
4//(2Ao/i2 B0) [(i? 
[(p2 sin2 d P) Ti
 (L ^2)Jm+i]2Gl 2[(S /j,2 cos2 6){S n2) D2}Ai JmG2
 2/i2 sin 9 cos 6Ti [(R fi2)Jmi + (L fJ2)Jm+i]G2\
In the above equation A0 and B0 are the coefficients of fi4 and fj2 in the cold plasma dispersion relation and S, P, R, L, D are the Stix parameters as defined in Equation (2.5). Jm represent the first kind Bessels function of order m of argument f3 defined below together with other parameters used to calculate x in Equation (2.26). The high energy electron velocity distribution is considered to be / (v) = 2 x 10nu4m6s3 as in [Bortnik(2005)].
32
1
(2.26)
G\
1 
v/bq?;
p = me jv B = 
Tfie 7 UJce
kzvz\ df kzv_l 5/
a;
<9w i
oj dv7
Go
Jr *
1 m^ceX 5/ mwcevz df kz J dvz uv dv
In the above equations k and kz denote the perpendicular and parallel components of the wave normal, v and vz denote the perpendicular and parallel electron velocity components, v represents group velocity. Based on the warm plasma corrections to the dielectric tensor shown in [Kulkarni et aZ. (2015)], we introduce new variables
Gnew> k)m u' Pnew, Pnew> k n, tr dn( !r and Bnew in Equation (2.27),
Pnew = S + TeKe n(1') + TiKni^k Dnew = D + TeKe 12(1) + Tjidjn Pnew = P + TeKe 33d) + TiKm^
Pnew Pnew I Dnew
(2.27)
Pnew Pnew Dnew
In the above equations Kj'1'1 and id/1) are the dielectric tensors with warm plasma corrections for electrons and ions. The variable r is defined for electrons and ions in Equation (2.28);
re = qeTp2 (2.28)
T 2 Ti = Qi p
33
We also introduce two more new variables Anew and Brnew based on the warm plasma
dispersion relation equation as in Equation (2.29):
A
B
(^A0 + Y QTBi (b0 + Y qTC1
(2.29)
With all the warm plasma corrections above the equation for y can be expressed as follows;
X
8
Afi(2A newRnew ) [i^Rnew ) Jrn 1 [Lnew
[{fj2 sin2 0 Pnew) Vi
 /J>2)Jm\i]2Gi 2[(Snew fj2 cos2 9)(Snew fj2) D^iew]AiJmG2
2fJj sill 0 COS 0Ti \iyRnew ) Jm 1 H (Lnew ^2]
( r\ ^ V Al )
( 2 \ (11
27T2Wpe2 V^oo roo u)kz Z^m=00 JO tluL El r dvz J 00 ^
{ V ) lVz)
(2.30)
2.7 Numerical Raytracing
Raytracing is the process of determining the power flow path of a wave. Numerical raytracing is implemented by solving the Haselgroves equations [Haselgrove( 1955)] listed below, in spherical coordinates.
34
dr
dt
dp
tr\'J'r 'V
dtp
dt
dp
rp2 ^ dpv
d(f)
dt
1
r/iz sm ip
jJjfj) fJj
dp
0
dpr
d/jitf,
dt
dt
dpv 1/15/x dt r \p dp 1 f 1 dp dr
1 dp, drtp p dr ^ dt dr
Pipr + P4.J7 sint/?
d(f)
dt
(2.31)
d(f)
^Tt+r^coslf
dp
. o, sin^r^cos^
r sm
In the above equations r is the geocentric distance, ip is the zenith angle and is the longitude, p is the real part of the complex refractive index. The variables pr, pv and p^ are the components of the refractive index vector in the corresponding directions of r, ip and (f). And t is the independent variable of integration which is also the time of phase travel along the ray trajectory scaled by the velocity of light.
Numerical raytracing works well in mediums which are smoothly varying(no sudden density drops)and no mode coupling.
In the above set of equations properties of the medium enters via the refractive index p. Therefore in this work it is extremely important to find the magnitude and direction of refractive index vector.
The starting point is a cold plasma raytracing code developed at Stanford University [Inan and Bell(1977)]. In the actual raytracing process it is important to make sure that the dispersion relation is always equal to zero as this is the condition of propagating wave solutions. Once the wave is propagating through the magnetosphere this condition should be always enforced.
In order to make sure that the dispersion relation always equal to zero the following differential equation approach is used in the currently available cold plasma models.
35
Step 1 = the dispersion relation F is defined as a conserved quantity, and it is a function of the wave number k (related to the refractive index according to the AppletonHartree equation), frequency of the wave oj and the position(r) of the wave. Along the trajectory of the ray F(k,u,r) = 0, due to F being a conserved quantity. In order to derive a relationship between F and the time t, we start by assuming all k, oj and r, are functions of a dummy variable r. Hence
dr
dr
dk
dr
du
dr
VfcF
VrF
dF = ~dt
Step 2 = Using the conservation of F, the total derivative of it with respect to r becomes:
dF dk OF duj dr OF dt
Ih = VFW + IkoTr + VrFW + 1ft Tt = '
Therefore the following relationships can be derived from the above equation.
dt dF
dr du dr VfcF
dt dF/dur dk VrF dt dF/dur
These set of equations are analogous to the Snells Law for a wave packet propagating in a slow varying medium. These compact derivatives are implemented in the raytracer in discrete steps using the partial derivative of F, with respect to frequency, position
36
and the wave normal.
(2.32)
dF F(k + Ak,uj,r) F(k Ak,uj,r)
dk 2A k
dF F(k,u + Au,r) F(k,u Au,r)
duo 2A uj
dF F(k,u,r + Ar) F(k,u,r Ar)
dr 2A r
The relationship given by the group velocity of the wave.
Step 3 = Integrate the dispersion relation, using the fourth order RungeKutta formula with respect to time and find a new space location. Force the dispersion relation to be zero at the new location by solving the dispersion relation. If the new location is outside the resonance cone, reduce the step size and recalculate the location.
Step 4 = Repeat this process until the maximum time steps are reached.
Before proceeding on to simulations of complete whistler mode trajectories we make a few important comments regarding refractive index surfaces in the context of the results of [Kulkarni et al. (2015)]. [Kulkarni et al. (2015)] used the Carpenter and Anderson model [Carpenter and Anders on (1992)] for background density and examined refractive index surfaces with nonzero temperatures for either hydrogen ions or electrons. All the work presented in here was done with the Global Core Plasmasphere Model (GCPM)[Gallagher et al.(2002)]. In the GCPM model there are four types of magnetospheric species, namely the electrons, Fd+ ions, Fte+ ions, and 0+. Taking into account the lower mass and higher densities of electrons and Fd+ ions compared to the other species, in this work we consider only the effects of electrons and H+ ions.
With GCMP model, we are capable of changing the location of plasma pause (Lpp), according to Equation (2.33).
37
(2.33)
Lpp = 5.6 0A6.KP
In the above equation Lpp is the starting point of the plasma pause in earth radii and Kp is an index which is used to change the location of the plasmapause. Figure 2.5 shows location of the plasma pause with two different Kp values.
Figure 2.5: Change of the location of plasma pause with the kP index. Plasma pause is the region where the electron density suddenly drops.
38
CHAPTER III
MODIFICATION TO THE REFRACTIVE INDEX SURFACE
3.1 Effect of Ions
[Hines( 1957), Smith and Angerami( 1968)], identified without thermal effects inclusion of ions closes an otherwise open refractive index surface for frequencies below the lower hybrid resonance. [Kulkarni et a/.(2015)], have done the analysis with respect to the lower hybrid resonance frequency. Lower hybrid resonance at a given location can be calculated using Equation (3.1).
Ill , N
2 22 1 (31)
^LH ci + Upi ^ci^ce
In Equation (3.1), is the lower hybrid resonance frequency, uCi and uce are the ion and electron gyrofrequencies respectively and uPi is the ion plasma frequency. Figure 3.1 shows the refractive index surface plots for a 2.3 kHz wave launched at the equator of L = 2 with and without the inclusion of ions. The lower hybrid resonance at the launching location is 2.57 kHz. Hence the selected wave frequency is 270 Hz below the lower hybrid resonance frequency of the launching location. As shown in Figure 3.1 when only electrons were considered in the refractive index calculation process, the refractive index surface remains open for highly oblique angles. Whereas when both electrons and ions are considered in solving for the refractive index, the surface is being closed for highly oblique angles. In all the raytracing work we have performed in this work, we have considered electrons and also the effect of multiple types of ions, namely H+, He+ and 0+.
3.2 Inclusion of Finite Electron or Ion Temperature
As [Kulkarni et al. (2015)] observed, when wave frequency increases above the lower hybrid resonance frequency, inclusion of ions does not change an otherwise open refractive index surface. As shown in Figure 3.4 for a 3 kHz wave at the equator at L = 2, although ions are considered for the calculations, refractive index surface remains open for highly oblique
39
Figure 3.1: Refractive Index surface for 2.3 kHz, at L = 2
1 1000 800 600 400 200 0 200 400 600 800 1000
H Perpendicular
Figure 3.2: Refractive Index surface for 2.3 kHz, at L = 2, plotted with the same aspect ratio for both axes.
angles. Hence if a wave was launched at a highly oblique angle it will resonate within the resonance cone without showing any magnetospheric reflections. This observation is also true for frequencies which are a couple of kHz above the lower hybrid resonance frequency at a location as shown in Figure 3.5.
As discussed above, closure of the refractive index surface is necessary in order to observe magnetopsheric reflections. Without the closure of the refractive index surface, the wave would not have propagating solutions in for all wave normal angles. Therefore it is necessary to include temperature effects in the process of refractive index calculations.
In this work we recreated the refractive index surface observations done by [Kulkarni et al. (2015)]. As [Kulkarni et al. (2015)] observed, if the wave frequency is less than the lower hybrid resonance frequency, inclusion of temperature does not modify the refractive index
40
surface. As shown in Figure 3.3, for a 2.3 kHz wave the refractive index surface generated with and without temperature effects are overlapping on each other. Hence we should not see a modified ray trajectory with the inclusion of finite temperature effects.
Figure 3.4, shows the refractive index surfaces with and without temperature effects for a 3 kHz wave at the equator of L = 2. As mentioned above, inclusion of temperature closes an otherwise open refractive index surface. The selected frequency is 430 Hz above the lower hybrid resonance at the equator of L = 2. Inclusion of 1 eV ion temperature closes the refractive index surface tighter than the case with only electron temperature. Therefore as the authors of [Kulkarni et al. (2015)] observed, ion temperature plays a dominant role than the electron temperature at frequencies slightly above the lower hybrid resonance frequency.
Once the frequency of the wave is much higher than the lower hybrid resonance frequency, electron temperature becomes dominant compared to the ion temperature. This scenario is shown in Figure 3.5, for a wave with 10 kHz, launched at the equator of L = 2. Inclusion of a 1 eV temperature to electrons produces a tighter refractive index surface than with the case with only ion temperature.
3.3 Inclusion of Temperature for Both Electrons and Ions
In this work we included temperature effects to both electrons and H+ ions simultaneously. Once the temperature effects are given to both electrons and ions, that produces the tightest refractive index surface for frequencies higher than the lower hybrid resonance. Hence the magnetospheric reflections should be hastened with that inclusion.
As shown in Figure 3.3 if the frequency of the wave is lower than the lower hybrid resonance, inclusion of temperature does not modify the refractive index surface. Hence we expect no changes in the ray trajectories, produced with cold and warm plasma assumptions for every wave normal angle.
But as we increase the frequency of the wave above the lower hybrid resonance, inclusion of temperature to both species tightens the refractive index surface. As shown in Figure 3.4, the refractive index surface produced with both electron and ion temperature
41
is approximately 19% reduced than the refractive index surface produced only with the ion temperature.
Similarly at 10 kHz frequency also the tightest refractive index surface is produced when both species were given temperature effects, but that reduction from the refractive index surface with only electron temperature is about 0.77% as shown in Figure 3.5.
Hence the maximum deviation between cold and warm plasma ray trajectories is expected to be observed at frequencies slightly above the lower hybrid resonance and at highly oblique wave normal angles.
For the completeness we have tried increasing the electron and ion temperature to 4 eV. The refractive index surface plot is shown in Figure 3.6 in comparison with the refractive index surface plot obtained with 1 eV temperature given to both species for a 3 kHz wave at the equator at L = 2.
Figure 3.3: Refractive Index surfaces for 2.3 kHz, at the equator of L = 2.
42
Figure 3.4: Refractive Index surfaces for 3 kHz, at the equator of L = 2.
Figure 3.5: Refractive Index surfaces for 10 kHz, at the equator of L = 2.
43
Figure 3.6: Refractive Index surfaces for 3 kHz, at the equator of L = 2 with I eV temperature and with 4 eV temperature
44
CHAPTER IV
MODIFICATION TO WHISTLER MODE RAY TRAJECTORIES FROM FINITE ELECTRON AND ION TEMPERATURE
4.1 Effect of Temperature on Landau Damping
Before proceeding into modification of the ray trajectories from thermal effects, it is important to first quantify the modifications introduced to Landau damping with the inclusion of finite electron and ion temperature. This is because Landau damping is the main source of attenuation of whistler mode waves and determines how long a whistler mode wave packet will have a nonnegligible amount of energy. In magnetospheric ray tracing it is typical to define the lifetime of a ray as the time it takes the wave amplitude to decrease by a factor of 10 dB [Bortnik et al. (2006c)]. Figure 4.1 shows whistler mode wave trajectories with and without thermal effects for a frequency of 3.5 kHz launched at the equator at L = 2, with an initial wave normal angle of 88. Both trajectories seem to have similar lifetimes. An expanded view of the power attenuation plot is shown in panel (c), where it can be seen that the attenuation slightly increases with the inclusion of finite temperature effects. This observation can be made with most of the frequencies at all wave normal angles. In all the work presented here we take power attenuation due to Landau damping into consideration. And all the trajectories presented here were traced for the duration of the signal lifetime. For all the results presented in this chapter dipole model was used as the geomagnetic field model.
4.2 Launching Waves in Different Directions
The raytracer is capable of launching waves in different directions. The initial direction of a wave depends on the initial wave normal angle. Figure 4.2 shows 3 kHz waves launched at the equator of L = 2, with three different initial wave normal angles and their power attenuation plots with and without temperature effects. In Figure 4.2 (a).i the waves were launched with an initial wave normal angle of 85. This initial wave normal angle is well below the resonance cone wave normal angle of 89.1. The wave is launched away from the
45
x (Earth Radii)
(b)
Figure 4.1: (a) Trajectories of whistler mode waves with a frequency of 3.5 kHz, launched at the equator of L = 2, with an initial wave normal angle of 88, with (Red) and without (Black) finite electron and ion temperature, (b) original power attenuation plots, (c) zoomed version of power attenuation plots.
Earth towards the North pole. When the initial wave normal angle is 85, as shown in Figure 4.2 (b).i, the wave is launched towards the Earth and towards the North pole. If the wave needs to be launched towards the South pole outward the Earth, the initial wave normal angle should be set to 95, as shown in Figure 4.2 (c).i. Since all wave normal angles are below the resonance cone, power attenuation with and without temperature effects are the same.
From the refractive index surfaces presented in Chapter 3, we can make the following predictions about the ray trajectories.
1. For frequencies below the lower hybrid resonance frequency, inclusion of finite electron and ion temperature should not modify the whistler mode ray trajectories.
2. When the frequency of the wave is slightly above the lower hybrid resonance frequency, inclusion of temperature effects to both electrons and ions should modify the ray trajectories at highly oblique angles.
3. For frequencies much higher than the lower hybrid resonance frequency, inclusion of
46
finite electron and ion temperature is still expected to modify the ray trajectories at highly oblique angles.
4.3 Frequency Dependence of Wave Trajectories
Figure 4.3 shows the ray trajectories for 2.1 kHz waves launched at the equator of L = 2. Here we have selected a highly oblique angle, since if there is a trajectory modification it should be observable at a highly oblique angle. From the Landau damping plots shown in Figure 4.4, the lifetimes of the waves under cold and warm assumptions are the same. Both ray trajectories show the same number of reflections. As it was expected based on the refractive index surfaces, the trajectories obtained under cold and warm background plasma are the same for frequencies below the lower hybrid resonance frequency even at highly oblique angles.
Figure 4.5 shows the ray trajectories for 2.98 kHz waves launched at the equator of L = 2. The launching angle is again 89 as in the previous case. Landau damping curves suggest that the damping under warm plasma assumptions is slightly higher than the damping observed with a cold background plasma. The lifetime of the whistler mode wave with a cold background plasma is about 0.2 seconds higher than the lifetime of the wave with thermal effects. In this case the wave propagates a longer distance with temperature effects compared to the case with no thermal effects. Hence we considered that it will be interesting to see the properties of the group velocity of those waves as shown in Figure 4.6.
In the previous case where the frequency of the waves was 2.1 kHz the group velocities of the waves under cold and warm plasma conditions were the same as expected (Figure 4.4). But as the frequency increases above the lower hybrid resonance, the group velocities also increases with the inclusion of temperature effects. As shown in Figure 4.6, when the frequency of the wave is above the lower hybrid resonance frequency, inclusion of temperature increases the group velocity compared to the case with no temperature effects. Hence the wave propagates faster with temperature effects compared to the case without the temperature effects.
47
When the frequency of the wave is much higher than the lower hybrid resonance frequency, inclusion of temperature introduces propagation solutions at highly oblique angles as shown in Figure 4.7. For a wave at 7 kFlz at an initial wave normal angle of 89, the refractive index surface is open, without temperature effects. Flence with no temperature effects there will be no propagation solutions at this initial wave normal angle. But once the temperature effects are taken into consideration, it introduces new propagation solutions, due to the closure of the refractive index surface. From the Landau damping curve it is observed that at such higher frequencies and at highly oblique angles, wave power attenuates within a fraction of a second. Flence those waves might play a nonimportant role in actual spacecraft observations. As shown in Figure 4.8, although the wave propagates under warm plasma conditions the group velocity is very small compared to the previously considered two cases.
4.4 Dependence on Initial Wave Normal Angle
In the previous section we discussed the effect of temperature at different frequencies. We noticed that when the frequency is below the lower hybrid resonance frequency, inclusion of temperature does not modify the ray trajectories even at highly oblique angles. When the frequencies are much higher than the lower hybrid resonance frequency, inclusion of temperature introduces propagation solutions at highly oblique angles, but those waves are being attenuated within a fraction of a second. When the frequency is slightly higher than the lower hybrid resonance, inclusion of temperature causes a longer trajectory and increases the group velocity of the wave.
In this section we study the effect of temperature at different wave normal angles. All the whistler mode waves we are considering in this section are launched with a frequency of 3 kHz, at the equator of L = 2. From the zoomed in trajectories shown in Figure 4.9, there is a negligible difference between the two trajectories. The selected initial wave normal angle of 60, is well below the resonance cone of the refractive index surface. In other words, the refractive index values obtained with cold and warm plasma conditions are the same.
48
Hence although the wave frequency is higher than the lower hybrid resonance frequency, if the wave normal angle is well below the resonance cone, the trajectories would be the same with and without temperature effects. This is further supported by the group velocity plot which contain overlapping group velocity plots. The lifetimes of both the waves were observed to be 7 seconds. Figure 4.10 shows the group velocities of the two trajectories. At this particular wave normal angle, group velocities obtained under both cold and warm plasma conditions are the same.
Figure 4.11 shows the original and zoomed in trajectories of a 3 kHz wave launched at the equator for L = 2, but now with an initial wave normal angle of 89. For a 3 kHz frequency at the equator of L = 2, the resonance cone of the refractive index surface is 89.1. Hence this selected initial wave normal angle is almost on the resonance cone. In this case the lifetime of both waves was approximately 1 second and under warm plasma conditions the lifetime was about 0.02 seconds lower compared to the case with no temperature effects. The zoomed in version of the trajectories shows an additional reflection when the thermal effects are taken into consideration. Which means although the lifetime was lower, distance travelled by the wave under warm plasma assumptions was longer than the distance travelled with no temperature effects. This observation can be well explained with the group velocity plot shown in Figure 4.12. The group velocity with the inclusion of temperature effects was considerably higher than the group velocity obtained with no thermal effects. Hence the waves propagate faster with the inclusion of finite electron and ion temperature.
Until this point all the initial wave normal angles we have considered were positive. As mentioned at the beginning of this chapter, a positive wave normal angle indicates that the wave was launched away from the Earth. For completeness, here we are considering an initial wave normal angle of 89 as shown in Figure 4.13. Negative initial wave normal angles indicate that the waves were launched toward the Earth. Based on the symmetry of the refractive index surface we should expect similar wave characteristics as above. In this case also the selected initial wave normal angle is about 1, below the resonance cone. Here
49
also we can make the same observations we made with the initial wave normal angle of 89. The trajectory with temperature effects was going through an additional magnetospheric reflection. Also the group velocity under warm plasma conditions is considerably, higher than the group velocity under cold plasma conditions. Lifetime of the waves was the same as observed with the initial wave normal angle of 89. Here also although the damping was higher under warm plasma conditions, the waves travel a longer distance compared to the cold plasma scenario, due to the higher group velocity as shown in Figure 4.14.
We need to bring the attention here to an interesting observation about the magnetospheric reflection point. [Kulkarni et a/.(2015)] predicted that the inclusion of thermal effects should hasten the magnetospheric reflections. Here we have shown the zoomed in trajectories for waves with a frequency of 3 kHz launched at the equator of L = 2 with initial wave normal angles of 89 and 89. In both cases the lifetime of the waves is approximately 1 second. As shown above during the lifetime trajectories produced under cold plasma conditions do not go though a magnetospheric reflection. But here in this case we ran the simulation for 2 seconds in order to observe the magnetospheric reflection point under cold plasma assumptions. From Figure 4.15 shown below, the prediction made in [Kulkarni et a/.(2015)] can be confirmed. Although both waves were launched at the same location, trajectories under warm plasma conditions encounter magnetospheric reflections quicker than the trajectories under cold plasma conditions. This observation can be made at both initial wave normal angles.
For the L = 2 launch location only highly oblique waves in the frequency range of 2.93.0 kHz have their trajectory modified by the inclusion of finite temperature. We take a closer look at the conditions along this trajectory. The top panel of Figure 4.16 shows the difference between wave frequency and the local lower hybrid resonance frequency as the wave propagates away from the launching location under warm plasma conditions. The wave is launched at L = 2 with a frequency of 3.0 kHz and with an initial wave normal angle of 88.9. The bottom panel of Figure 4.16 shows the wave normal angle along this trajectory.
50
Although the actual wave attenuates 10 dB within 1 second, here we ran the simulation for 5 seconds to show the propagation characteristics. It can be seen that both the wave normal angle and frequency difference have an oscillatory pattern of the same period but are inversely phased. As the difference between the wave frequency and the local lower hybrid resonance reduces and crosses zero, the wave normal angle is increasing to achieve it maximum oblique value (almost 90). As was discussed in Chapter 3, the finite temperature effects change the refractive index surface only when the wave frequency is above the lower hybrid resonance frequency and only for highly oblique waves. The joint dynamics of the frequency difference and the wave normal angle shown in Figure 4.16 suggests that the conditions for maximum effect of finite temperatures are largely avoided by the wave thus explaining why the trajectory changes are very small even though parts of the refractive index are changed dramatically in the warm plasma model.
4.5 Observations of Waves Launched at L = 4
Here we present the observations made with the waves launched at the equator of L =
4. As mentioned in a previous chapter for all the raytracing done in this work the kp index was set to 4. Hence the location of the plasmapause is at L = 3.76. Therefore the launching location we selected here is beyond the plasma pause. Hence plasma densities at the launching location are much lower compared to the densities at the equator of L = 2. The lower hybrid resonance frequency at the equator of L = 4 is 300 Hz. Another interesting observation at this launching location is that the lifetime of the whistler mode waves is very low compared to the waves launched inside the plasmasphere.
Here also we studied the behavior of trajectories at different frequencies at a highly oblique wave normal angle and at a fixed frequency with different initial wave normal angles at different positions of the refractive index surface.
The top panel of Figure 4.17, shows the wave trajectories (original and zoomed) for a frequency of 250 Hz, launched at the equator of L = 4 with an initial wave normal angle of 89. The selected frequency is 50 Hz lower than the lower hybrid resonance frequency at the
51
launching location. Similar trajectories were obtained, with and without temperature effects, for frequencies lower than the lower hybrid resonance frequency. The lifetime observed for both trajectories was 0.25 seconds.
As shown in the middle panel of Figure 4.17, once the frequency of the wave is higher than the lower hybrid resonance frequency at the initial location, inclusion of temperature makes the wave propagates faster. The selected frequency of 375 Hz is 75 Hz higher than the lower hybrid resonance frequency of the equator of L = 4. Hence inclusion of finite electron and ion temperature has increased the group velocity of the wave. In this case lifetime of both waves was observed to be around 0.1 seconds.
For 1 kHz waves as shown in the bottom panel of Figure 4.17, inclusion of temperature has introduced propagation solutions which did not exist under cold plasma conditions, at highly oblique angles. The frequency of 1 kHz is 700 Hz higher than the lower hybrid resonance frequency at the launching location. But the waves with much higher frequencies compared to the lower hybrid resonance frequency, are being attenuated within about 0.001 seconds.
As we observed in the previous section with the waves launched at the equator of L = 2, when the frequency of the wave is below the the lower hybrid resonance frequency, inclusion of temperature does not modify the trajectory. Here we made similar observations with waves launched at the equator of L = 4. The trajectory modifications were observed for frequencies above the lower hybrid resonance frequency.
Now we keep the frequency of the wave fixed at 380 Hz and change the initial wave normal angle. As shown in the top panel of Figure 4.18, when the initial wave normal angle of the wave is 85, which is well below the resonance cone at this location, inclusion of temperature does not modify the trajectory. In other words, although the frequency of the waves were higher than the lower hybrid resonance frequency, if the initial wave normal angle is well below the resonance cone, inclusion of finite electron and ion temperature does not modify the trajectory. Once the initial wave normal angle becomes highly oblique 89, or much
52
closer to the resonance cone angle, inclusion of temperature increases the group velocity hence the wave propagates a longer distance under warm plasma conditions as shown in the middle panel of Figure 4.18.
Similarly when the waves were launched towards the Earth with an initial wave normal angle of 89, inclusion of thermal effects makes the propagation path of the wave longer compared to the case with no thermal effects as shown in the bottom panel of Figure 4.18.
53
Figure 4.2: Whistler mode waves with a frequency of 3 kHz, launched at the equator of L = 2, with (a).i initial wave normal angle of 85, (b).i initial wave normal angle of 85,(c).i initial wave normal angle of 95, (a).ii, (b).ii, (c).ii show the power attenuation plots of the corresponding trajectories with and without temperature effects. In all the panels Black traces were produced with cold plasma conditions and Red traces were produced with warm plasma conditions.
54
Figure 4.3: Whistle mode wave trajectories with a frequency of 2.1 kHz launched at the equator of L = 2 with cold (Black) and warm (Red)plasma conditions, (a) shows the original trajectories and (b) shows the zoomed in version.
(b)
Figure 4.4: (a) Power attenuation due to Landau damping for the trajectories shown in
Figure 4.3. (b) Group velocity along the trajectories shown in Figure 4.3
55
x (km) X104
(b)
Figure 4.5: Whistle mode wave trajectories with a frequency of 2.98 kHz launched at the equator of L = 2 with cold (Black) and warm (Red)plasma conditions, (a) shows the original trajectories and (b) shows the zoomed in version.
Figure 4.6: (a) Power attenuation due to Landau damping for the trajectories shown in
Figure 4.5. (b) Group velocity along the trajectories shown in Figure 4.5
56
Figure 4.7: Whistle mode wave trajectories with a frequency of 7 kHz launched at the equator of L = 2 with cold (Black) and warm (Red)plasma conditions, (a) shows the original trajectories and (b) shows the zoomed in version.
i_______________________________________________i ^
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Time (sec)
(b)
Figure 4.8: (a) Power attenuation due to Landau damping for the trajectories shown in
Figure 4.7. (b) Group velocity along the trajectories shown in Figure 4.7
57
Figure 4.9: Whistler mode wave trajectories of a 3 kHz wave launched with an initial wave normal angle of 60, at the equator of L = 2 under cold (Black) and warm (Red) plasma conditions, (a)original wave and (b) shows the zoomed in version.
Time (sec)
Figure 4.10: Group velocities of the wave trajectories shown in Figure 4.9.
58
x (Earth Radii)
x (km) X104
(b)
Figure 4.11: Whistler mode wave trajectories of a 3 kHz wave launched with an initial wave normal angle of 89, at the equator of L = 2 under cold (Black) and warm (Red) plasma assumptions, (a)original wave and (b) shows the zoomed in version.
Time (sec)
Figure 4.12: Group velocities of the wave trajectories shown in Figure 4.11.
59
x (Earth Radii)
(a)
(b)
Figure 4.13: Whistler mode wave trajectories of a 3 kHz wave launched with an initial wave normal angle of 89, at the equator of L = 2 under cold (Black) and warm (Red) plasma conditions, (a)original wave and (b) shows the zoomed in version.
Time (sec)
Figure 4.14: Group velocities of the wave trajectories shown in Figure 4.13.
60
x106
x (m) x 107
Figure 4.15: Magnetospheric reflection points for waves with a frequency of 3 kHz, launched at the equator of L = 2, with an initial wave normal angle of 89, with (Red) and without (Black) temperature effects.
61
ff. UD for 3kHz, 0 = 88.9
LH K
Wave Normal Angle for 3kHz, 0 = 88.9
Time (sec)
Figure 4.16: Difference between wave frequency and local hybrid resonance as a 3 kHz wave propagates under warm plasma conditions with initial wave normal angle of 88.9 (top) Instantaneous wave normal angle along the trajectory (bottom).
62
f= 250Hz 0= 89 at L = 4
1 2 3 4 5
x (Earth Radii)
(a,).i
f = 375Hz Q = 89 at L = 4
1 2 3 4 5
x (Earth Radii)
(b).i
f= 1kHz Q = 89 atL = 4
1 2 3 4 5
x (Earth Radii)
(e).i
f= 250Hz Q=89 at L = 4
4000 H
H
3500 n
*
**
3000 * *
i *
2500 * *
V *
^ 2000 a *
N
1500 a
a a
1000 a
500 a a
0 1
2.4 2.5 2.6 2.7
x (km) x1,
(a.).ii
f= 375Hz 0 = 89 at L = 4
E
4630
4620
4610
4600
4590
4580
4570
4560
4550
2,41 2,412
x (km)
(b).ii
f= 1kHz 0 = 89 at L = 4
Figure 4.17: Whistler mode waves launched with an initial wave normal angle of 89, at the equator of L = 4, with frequencies 250 Hz, 375 Hz and 1 kHz. Left hand panels show the original trajectories and panels on the Right hand side show the zoomed in versions. All trajectories in Black were produced with cold plasma assumptions and all trajectories in Red were produced with warm plasma assumptions.
63
f = 380Hz Q = 85 at L = 4
1 2 3 4 5
x (Earth Radii)
(a,).i
f= 380Hz 6 =89 atL = 4
1 2 3 4 5
x (Earth Radii)
(b).i
f = 380Hz 0 = 89 at L = 4
1 2 3 4 5
x (Earth Radii)
(e).i
J*= 380Hz 0= 85 at L = 4
x 10
(a).ii
f= 380Hz Q = 89 at L = 4
2.435 2.44 2.445 2.45 2.455 2.46
x (km) xio4
(e).ii
Figure 4.18: Whistler mode wave trajectories for a 380 Hz wave launched at the equator of L = 4, with an initial wave normal angles 85, 89 and 89. All trajectories in Black were produced with cold plasma assumptions and all trajectories in Red were produced with warm plasma assumptions. Left hand panels show the original trajectories and panels on the Right hand side show the zoomed in versions.
64
CHAPTER V
COMPARISON WITH VAN ALLEN PROBE SPACECRAFT
OBSERVATIONS
5.1 Chorus and Hiss
Until this point we presented the characteristics of whistler mode waves in general. In this chapter it is worth introducing two specific types of whistler mode waves; chorus and hiss. Chorus and hiss both are whistler mode waves and the names are due to the sounds they make when passed through a speaker. Chorus waves exists in two frequency bands. Given the equatorial electron cyclotron frequency as fce, lower chorus band is around 0.34 x fce and the upper chorus frequency band is around 0.53 x fce. The characteristic which distinguishes chorus from hiss is the frequency and time coherence observed in chorus that cannot be observed with hiss. For hiss, the wave energy is distributed within all frequency components in the band (1 kHz 3 kHz). Both chorus and hiss waves are assumed to be generated from cyclotron resonant wave particle interactions in the magnetosphere.
The highest chorus energy concentration is observed around L = 5 and the highest hiss energy concentration is observed around L = 2. In other words the high wave power observed outside the plasmapause is considered to the chorus energy and the wave power observed inside the plasmasphere is mainly due to the hiss power. Figure 5.1 shows the spectrograms of chorus and hiss.
5.2 Dependence on Magnetic Local Time
Since chorus and hiss waves are created due to space dynamics, origination regions of hiss and chorus changes with the Magnetic Local Time or MLT. At MLT = 12, the magnetic meridian is facing the sun. Hence it is a day time observation (noon). Whereas at MLT = 00, the sun is at the opposite side of the magnetic meridian. Therefore it is midnight in magnetic local time. Figure 5.2 shows the origination locations of hiss and chorus with respect to the magnetic local time.
65
I
b Chorus
ut: 13:01:12 13:01:17 13:01:22
Re: 4.43 4.44 4.44
MLAT: 29.41 29.45 29.50
MLT: 7.25 7.25 7.25
L: 5.81 5.82 5.83
19 November 2001
ut: 13:49:24 13:49:29 13:49:34
flE: 4.02 4.02 4.02
MLAT: 11.59 11.64 11.69
MLT: 1.50 1.50 1.50
L: 4.20 4.20 4 February 2001 4.20
Figure 5.1: Origination locations and spectrograms of chorus and hiss waves.Image source: [Bortnik et al. (2008)]
5.3 Van Allen Probe Spacecraft Observations
The Van Allen Probes are the spacecrafts launched by National Atmospheric and Space Agency (NASA) in order to monitor the characteristics of Earths radiation belts or the Van Allen Belts [iVTS'T(2012)]. Figure 5.3 shows the trajectory of Van Allen Probe spacecrafts. Figure 5.4 shows the normalized power plot observed by the NASA Van Allen Probe spacecraft at Magnetic Local Time (MLT) 06. The power was normalized to the maximum of each frequency. During the observation period the plasmapause was observed to be in between L = 3 and 4. In Figure 5.4, all measurements were indicated with respect to the plasma pause. Negative distances are the measurements taken towards the Earth and positive measurements are measured outward the Earth. All the measurements were taken in the absence of sudden magnetic disturbances such as solar flares, hence during magnetic quite conditions [.Malaspina et e/(2016)].
66
18
Dusk
Dawn
Figure 5.2: Regions of chorus and hiss energy confinements with respect to the Magnetic Local Time.Image source: vlf.stanford.edu/Daniel Golden Thesis
Figure 5.3: Artists illustraton of the Earths radiation belts and the trajectory of Van Allen Probe spacecrafts. The twin spacecrafts are shown here. Image source: Andy Kale, University of Alberta
5.4 Simulated Wave Power with Landau Damping
In order to compare simulation results with the Van Allen probe observations, here we are launching multiple rays. Since the Van Allen Probe observations were done at the dawn time, we have also set the simulation time to MLT = 6. In order to be realistic with the actual magnetospheric conditions, we have used the actual measured electron and ion temperatures at MLT = 6 from [Decreau et aZ.(1982)]. [Decreau et aZ.(1982)] observed the electron and ion temperatures at different magnetic local times and at different altitudes using satellites. Based on [Decreau et a/.(1982)] measurements we have used the electron temperature as 26,500 K (2.3 eV) and the ion temperature as 11,600 K (1 eV). All the waves were launched
67
Figure 5.4: (a) Wave energy measured by the Van Allen Probe spacecrafts, (b) Same wave energy as shown in (a), normalized to the maximum at each frequency.
at the equator of L = 5 since L = 5 is considered as the origination location of chorus waves. The plasma pause location was set at L = 3.76, which is equivalent to the actual observed conditions. The waves were launched within a frequency range of 100 Hz to 3.5 kHz. Waves with each frequency were launched with a range of initial wave normal angles from 70 to 20, with increments of 5. It is worth recalling here that negative initial wave normal angle indicate that the wave was launched towards the Earth. All waves were launched with equal power. Dipole model was used as the geomagnetic held. The originating location and the wave normal angle distribution of the waves were selected according to [Bortnik et al. (2008)].
If the temperature contributing the parallel propagation velocity and the perpendicular propagation velocity are different it is called a temperature anisotropy. Even in the presence of a significant anisotropy, the main source of power damping is Landau damping which was explained in Chapter 2. Landau damping is not affected by the temperature anisotropy. Hence in this case also in order to calculate the power attenuation due to Landau damping, we are using the isotropic high energy electron distribution that we used in Chapter 2, which is / (v) = 2 x 10ni4'm_6s3 [Bortnik(2005)]. All the waves were traced until the power of the wave was attenuated 10 dB from the initial power. Figure 5.5 shows the whistler mode
68
wave power normalized to the maximum at each frequency in a cold background plasma. And Figure 5.6 shows the whistler mode wave power with the warm plasma conditions, normalized to the maximum at each frequency. Similar observations can be made with both figures, but since the warm plasma conditions introduce additional propagating solutions, which are otherwise not possible with cold plasma assumptions, there are more data points observed under warm plasma conditions.
Normalized Power Cold Plasma
3500
3000
2500
c 2000 CD =5 CT
CD
ll 1500 1000 500
2 1 0 1 2 3 4 "
Distance from the Plasma Pause AL
Figure 5.5: Normalized wave power with the wave growth analysis under cold background plasma scenario. Wave power is normalized to the maximum at each frequency.
5.5 Wave Growth Analysis
Whistler waves encounter wave growth when there is a temperature anisotropy in the parallel and perpendicular directions. In this simulation we consider, an isotropic background plasma distribution and a hot energy electron distribution with a low anisotropy of the dawn side on the Magnetosphere. The model is described below [Hikishima et al. (2008), Harid( 2015)].
f(u h wl)
______Uh______
[;2^Z!2)U,Kpl/xv
1
exp
exp
~u y
69
Normalized Power Warm Plasma
3500 3000 2500
>,
c 2000 0 3 cr 0
ll 1500 1000 500
2 1 0 1 2 3 4 "
Distance from the Plasma Pause AL
Figure 5.6: Normalized wave power with the wave growth analysis under warm background plasma scenario. Wave power is normalized to the maximum at each frequency.
The velocity distribution function of the isotropic background plasma is considered Maxwellian. And the velocity distribution of the hot anisotropic plasma distribution function is considered modifiedMaxwellian as given by above. It is defined in the parallel and perpendicular momentum space (u, u). The parameters of the modified Maxwellian distribution, are defined as follows and c is the speed of light.
7 = [1 + 4)/cr1/2 (5.1)
u\\ = yen (5.2)
u = jv (5.3)
In modified Maxwellian distribution, /?,/, is the hot energetic electron density which is taken as 2 x 104 per cubic meter. Uth\\ and Uth are parallel and perpendicular components of thermal momentum per unit mass, which are considered to be 0.2c and 0.33c respectively (c is the speed of light). At the equator Loss cone is the term used for the minimum pitch angle for which electrons can be trapped in the Earths radiation belts. The term /3 in the above modified Maxwellian distribution is the parameter which determines the range of the loss cone. Increasing (3, makes the loss cone larger. For this simulation the /5 value was set
70
to 0.3, based on the measurements taken at the equator.
Vr = cSZ (l  (5.4)
Equation (5.4), expresses the resonant condition between the whistlers propagating parallel to the Earths magnetic fields and the hot energy electrons. The parameters Â£ and 5 are given as follows:
UJ (uce cu) (5.5)
^pe
;2 1 (5.6)
k2 ~ 1 + Â£2
UJi = 7TUJce (1 ) 7J (Vro) [A (Vro) Ac] (5.7)
V UceJ
In the above equations upe is the plasma frequency and uce is the electron cyclotron frequency. Linear growth rate of a parallel propagating whistler mode wave is given in Equation (5.7) where:
V (VR0)
2tt
(jJrp
vFdv\v =Vro
(5.8)
A (Vrq)
2 J0 vFdv
^= Vro
(5.9)
Ac = ] (5.10)
cuce/u 1
the term rj represents the total number of electrons at resonance with the propagating whistler wave, which is a positive number. Also A (Vro) is the temperature anisotropy of hot energy electrons at resonance with the propagating whistler wave, and Ac is the critical temperature anisotropy.
71
Figure 5.7, shows the growth of wave power assuming a cold isotropic background plasma. Figure 5.8 shows the wave power growth assuming a warm isotropic background plasma. All the waves were launched at the equator of L = 5, and the initial wave normal angles were varied from 70 to 20, with increments of 5. Both electrons and ions were given temperatures based on the observations made by [Decreau et a/.(1982)]. All waves were launched with equal power. Dipole model was used as the geomagnetic held. The location of the plasma pause was set at L = 3.76. Figures 5.7 and 5.8, show similar power how propagation pattern. In Figure 5.4 (b), the highest power concentration moves closer to the Earth as the frequency increases. A similar pattern is observed in both hgures 5.7 and 5.8. But the power how path is more confined to the region similar to observation made in Figure 5.4, when the background plasma is considered as warm.
Normalized Power Cold Plasma
2101234 Distance from the Plasma Pause AL
Figure 5.7: Normalized wave power with the wave growth analysis under cold background plasma scenario. Wave power is normalized to the maximum at each frequency.
This suggests the importance of warm plasma corrections in being able to match raytracing simulations to data. In particular even though Unite temperature had a small effect on individual ray trajectories, the the cumulative effect on many rays is significant.
72
Normalized Power Warm Plasma
3500
3000
2500
c 2000
0
3
cr
0
ll 1500
1000
500
21 01 234
Distance from the Plasma Pause AL
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 5.8: Normalized wave power with the wave growth analysis under warm background plasma scenario. Wave power is normalized to the maximum at each frequency.
73
CHAPTER VI
SUMMARY AND CONCLUSIONS
At the beginning of this thesis we presented the warm plasma corrections that need to be introduced to the raytracing platform. Incorporating warm plasma corrections increases the complexity of the existing ray tracing equations. We have successfully introduced these effects to the cold plasma raytracer and performed the predictions of ray trajectories. In this thesis we introduced a modified version of the Landau damping calculations, which takes warm plasma corrections in to account. Calculation of the refractive index is of utmost importance in raytracing, since the medium properties enters the raytracing equations via the refractive index at each point. Previous researchers have shown that even without temperature effects, ions are necessary to cause magnetic reflections. For frequencies below the lower hybrid resonance inclusion of ions closes an otherwise open refractive index surface created by only taking electrons into account. For frequencies above the lower hybrid resonance frequency, inclusion of ions does not close the refractive index surface unless thermal corrections are taken into account. According to the published literature for frequencies in the close vicinity of the lower hybrid resonance frequency, inclusion of ion temperature produces a tighter refractive index surface compared to the refractive index surface created considering only electron temperature. But for frequencies much higher than the lower hybrid resonance frequency, inclusion of electron temperature produces a tighter refractive index surface compared to the ion temperature. For frequencies below the lower hybrid resonance, inclusion of temperature does not modify the refractive index surface.
In this thesis, we extended the refractive index surface analysis such that we can introduce thermal corrections to both electrons and ions simultaneously. This approach increases the accuracy of the model, since for a natural plasma in thermal equilibrium electrons and ions can posses two different temperature distributions. The main observation were, for frequencies below the lower hybrid resonance frequency, inclusion of temperature to both species did not modify the refractive index surface. But for frequencies above the lower
74
hybrid resonance frequency, inclusion of temperature to both electrons and ions produced the tightest refractive index surface.
In Chapter 4, we presented the modifications to the whistler mode ray trajectories introduced by the thermal corrections. We have observed the waves launched at two different L shells, 2 and 4. Similar observations were made in both cases. Propagation trajectory of whistler mode waves were modified in response to thermal effects, only at frequencies above the lower hybrid resonance and at highly oblique initial wave normal angles. No modifications were observed when the frequency of the waves were lower than the lower hybrid resonance and the launching wave normal angles are well below the resonance cone. Another observation was that the Landau damping increases with the inclusion of temperature effects, but that increment of damping is negligible.
Chapter 5 presented the comparison of simulation results with Van Allen probe spacecraft observations. We presented the simulated wave power by considering only Landau damping into account and by considering linear wave growth. In each case we performed the analysis by assuming a cold background plasma and a warm background plasma. The normalized power distribution obtained only considering Landau damping was the same for both cases with cold background plasma and warm background plasma. But a clear match was observable between simulations and observations when wave power growth was considered, in a warm background plasma.
6.1 Future Work
For the simulations in Chapter 5, we have considered a single source location (L = 5) and all waves were launched with equal power. The accuracy of the simulation results can be improved if multiple sources were to be considered, with different initial power. Also Appendix D presents a noncausal filtering method that can be applied for whistler mode raytracing to detect outliers that can lead to simulation inaccuracies. Implementation of this noncausal filtering method on the mainraytracer with an increased robustness will improve the accuracy of the ray tracer in general.
75
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APPENDIX A. Dispersion Relation for Free Space
Maxwellss equations for any region can be expressed as follows;
V.E
Pv
e
V.B
VxE = V x B = pj + (it
= 0
<9B
~dt
<9E
~dt
(A.l)
In the above equations E, B, J represent the electric held intensity, magnetic held intensity and the current density. pv is the volume charge density and p, e respectively represent the permeability and permittivity of the medium.
For free space volume charge density pv is zero and the current density also goes to zero. The permittivity and permeability can be expressed in terms of free space permittivity and permeability e0 and po Hence the above Maxwells equations can be rewritten in as follows for a charge free region such as free space as follows;
V.E = 0
V.B = 0
___ <9B
VxE=aF
dE
V x B po^o^rr o t
(A.2)
Using the following vector identity for any vector A
V x (V x A) = V(V.A) V2A (A.3)
Lets apply the above vector identity on the electric held intensity vector;
81
V x (V x E) = V (V.E) V2E
For a charge free region V.E is zero. Hence
V x (V x E) = V2E
From Maxwells third equation VxE=^. Therefore
V x (V x E) = 
V2E = 
<9V x B dt
<9V x B
dt
d ( <9E
VESU,Â£
V2E jlQtQ
d2E
~dÂ¥
Given that the electric held intensity in the form of E = ejGt_kz); where uj is the angular frequency, k is the wave number and z is the direction of propagation.
Hence;
V2E = jjLQtQUj2 E
82
V2E = = k2 E
oz2
(k2 izotou2^) E = 0
For a non trivial solution electric held intensity E cannot be zero. Therefore;
k2 izotou2 = 0 (A.4)
The relationship given in Equation (A.6) is called the dispersion relationship for free space.
82
Given c is the speed of light 3 x 108ms 1, the following relationship holds;
1
yjli> 0^0
Hence the dispersion relation can be rewritten as follows;
(A.5)
c2k2 uj2
(A.6)
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APPENDIX B. Dispersion Relation in a NonMagnetized Plasma
In this section we present the dispersion relation of nonmagnetized plasma. The term nonmagnetized indicates that the background magnetic held is zero. Before proceeding it is worth mentioning the plasma frequency up, which is the frequency that the particles rearrange themselves when disturbed with an external disturbance. Lets assume that the plasma is cold, hence the temperature of the plasma is at 0 K. If we consider only the motion of electrons the plasma frequency can be defined as follows;
_ Neqe2
COpe
mee0
In the above equation Ne is the electron density, qe is the charge of an electron, me is the electron mass and eo is the permittivity of free space.
J = Neqeue (B.l)
Equation (B.l) is the current density J, which is the rate of change of charge Neqe times the velocity ue. When the plasma is disturbed with an external electric held with intensity E the current density created due to the external electric held can be written as;
Neq2 E
(B.2)
Due to this external current density present, the effective permittivity of the medium ee// changes to the following;
Â£e//
Nege2
tomeuj2
UJ,
pe
UJ*
(B.3)
84
V.E = 0
V.B = 0
V x E VxBi
5Bx
~dt~
<9E
jit
(B4)
In Equations (B.4) represent the Maxwells equations for a non magnetized plasma. Here we want to emphasize that that magnetic held indicated by Bx is the magnetic held created due to the electric held E. For an electric held intensity E in the form of e(wtkz) Maxwells equations can be re written as follows;
V.E = 0 (B.5)
V.B = 0
V x E = jtnBi VxBx = ja;/ioeeffE
Applying the same vector identity that we applied in Appendix I, we can derive the following relationship
V2E = jÂ£n2E (B.6)
Hence the dispersion relation becomes;
k2 HoteffUJ2 0
(B.7)
or
i _
uj'2
(B.8)
85
APPENDIX C. Dispersion Relation in a Cold Magnetized Plasma
We start with the Maxwells equations for timeharmonic uniform plane waves;
k x B = jji0J utoiioEi
k x E = o;B
kE
JP
Â£o
kB = 0
along with the momentum equation
mnojuju = qN0(E + u x B).
Assuming an infinite cold, collisionless and homogeneous plasma, by dehnitiion we have:
E = 1 E
In the above equation ep is the dielectric tensor in the plasma.
Using the above relations the following relationships can be derived;
^ = w_^\ (E + uxB)
jujeo jujeo \jumeJ
UJ.
pe
E+
0JZ
J X z
jueo \ uj j \uj* / \ uj
The above equation was derived assuming that the ambient magnetic held is oriented in the z direction. By separating the components in the x, y and z directions,
f
V
e_L
jtx
0
e_L
0
0
0
\ / F \
ell /
E,,
Ez /
tp E
(C.l)
Using the dehnition for the refractive index, the above dielectric tensor can be written as follows.
86
^ (e jJ2 sin2 9) jtx jj2 sin 6 cos 6 ^
Jtx
e_L /r
0
2 ,
y ir sin 6 cos 6 0 e\\ fi2 sin 9 y
^ Ex ^
E
V E* )
(C.2)
(C.3)
= 1  ('cpe
UJ2 UJce2
(Uce\ , i2 ^pe
IcJ UJ2 UJce2
Â£ll , i2 i ^Pe UJ2
For a non trivial solution the determinant of the above Equation (C.2) should be equal to zero. Hence we get the dispersion relation of a cold magnetized plasma as:
2
k2c2
UJ
1 
W2 sin2 Q
ipVy
u>2e sin2 9 2(w2w2e)
sin
1/2
(C.4)
87
APPENDIX D. Noncausal Filtering
Here we present a noncausal filtering method that can be applied to eliminate outliers occurring in raytracing. This method was first introduce in [Kazakos and Papantoni( 1989)]. In the raytracer the maximum error is defined as the difference between two consecutive refractive index values. This parameter can be set by the user. If the user sets a high error value, the raytracer results outliers. Therefore it is important to find the nominal error bound to avoid the occurrence of outliers.
Method
For Gaussian density function f0s and /on, we select some probability of outlier occurrence Â£n(0, 1) and some finite non negative integer m. Let {....X_i, X0, Xi...} and
{...IFi, Wo, W\...}, denote the sequences of random variables that are generated by the
above Gaussian density functions f0s and /0n respectively. Given some integer k and some nonnegative integer n, let A^n+i.fc and M2n+i,k respectively denote the autocovariance matrices E{Wp: (W&) l/o} and E{Xp: (*Â£;) /o.>. Let +u denote the (+l) row of the matrix M2n+i,k
^2n+l,k M2n+l,k + ^2ra+l,fc (D.l)
And let (a^Iln, a^+r); n > l denote the optimal mean squared interpolation operation
at the Gaussian density function f0s for the datums Xk, given x^zh and xk^. Let us then define the sets {dk>n,j\ k n < j < k l, k + l < j < k + n} and {bk,n,j\ k n < j < k + n} of the coefficients as follows, where A2ra+i;fc is assumed non singular.
kl k\n
! 9kl ) ^ ^ dktn,jXj T ^ ^ dktn,jXj (D.2)
j=kn j=k\l
[bk,n,kn i ? b>k,n,k+n\ = a2n+l,k^2n+l,k (D.3)

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MAGNETOSPHERICWHISTLERMODERAYTRACINGWITHTHEINCLUSIONOF FINITEELECTRONANDIONTEMPERATURE by ASHANTHISMAXWORTH B.ScHons,UniversityofMoratuwa,2011,M.S,UniversityofColoradoDenver,2014 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy ElectricalEngineeringProgram 2017
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ThisthesisfortheDoctorofPhilosophydegreeby AshanthiSMaxworth hasbeenapprovedforthe ElectricalEngineeringProgram by MarkGolkowski,AdvisorandChair MartinHuber TimLei YimingDeng DanaCarpenter Date:May13,2017 ii
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Maxworth,AshanthiSPh.D,ElectricalEngineering MagnetosphericWhistlerModeRaytracingwiththeInclusionofFiniteElectronandIon Temperature ThesisdirectedbyAssociateProfessorMarkGolkowski ABSTRACT WhistlermodewavesareatypeofalowfrequencyHz30kHzwave,whichexists onlyinamagnetizedplasma.ThesewavesplayamajorroleinEarth'smagnetosphere. Duetotheimpactofwhistlermodewavesinmanyeldssuchasspaceweather,satellite communicationsandlifetimeofspaceelectronics,itisimportanttoaccuratelypredictthe propagationpathofthesewaves.Themethodusedtodeterminethepropagationpathof whistlerwavesiscallednumericalraytracing.Numericalraytracingdeterminesthepower owpathofthewhistlermodewavesbysolvingasetofequationsknownastheHaselgrove's equations.Inthemajorityofthepreviouswork,raytracingwasimplementedassuming acoldbackgroundplasmaK,buttheactualmagnetosphereisatatemperatureof about1eVK.Inthisworkwehavemodiedthenumericalraytracingalgorithmto workatniteelectronandiontemperatures.Thenitetemperatureeectshavealsobeen introducedintotheformulationsforlinearcyclotronresonancewavegrowthandLandau damping,whicharetheprimarymechanismsforwhistlermodegrowthandattenuationin themagnetosphere.Includingtemperatureincreasesthecomplexityofnumericalraytracing, buttheoveralleectsaremostlylimitedtoincreasingthegroupvelocityofthewavesat highlyobliquewavenormalangles. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:MarkGolkowski iii
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Thisworkisdedicatedtomyfamily,myteachersandmystudents. iv
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ACKNOWLEDGMENTS FirstIwouldliketothankmyadvisorProf.MarkGolkowski.Prof.Golkowskihiredastudent comingfromasmallislandnamedSriLankalocatedintheexactoppositesideoftheworld. OverthesefourandhalfyearstheguidanceandsupportprovidedbyProf.Golkowskifor myacademicdevelopmentaswellascareerdevelopmentistremendous.Icannotthankhim enoughforallthesupporthehasprovidedmeovertheyearsandIamforevergratefulfor that.AsastudentcomingfromadevelopingcountryIwouldnotbeabletocomethisfar withoutthenancialsupportprovidedbyProf.Golkowski. IalsowouldliketothankProf.StephenGedney,chairoftheElectricalEngineering departmentforallthetrusthehasinme,andgivingmetheopportunitytoteachElectromagneticFieldscourse,whichisahugestepforwardinmyteachingcareer. ElectricalEngineeringprogramassistantMrs.AnnieBennetdeservesabigthankyou forallthehelpandmotivationshegavemeovertheseyearsasacoworkerandasafriend. SamethanksgoestoMiss.KarlaFlores. Ialsowouldliketothankmythesiscommitteemembers,Prof.DanaCarpenter,Prof. MartinHuber,Prof.TimLeiandProf.YimingDengfortheirsupportandvaluablesuggestionsprovidedonimprovingmythesiswork.Sincerethanksgotoallmyresearchgroup membersespeciallytomydearfriendsNaomiWatanabeandRyanJacobsfortheirfriendship andmotivation. IdearlyacknowledgelateProf.TitsaPapantoniforbeingarolemodelforallwomenkind.IclearlyrememberthevewordadviceIreceivedfromher,thatshegaveonlytoher daughter.Iwashonoredtoreceivesuchamotivationaladvicefromher. Myotheremployersovertheyears,CampusVillageApartments,MetropolitanState UniversityofDenverandLearningResourcesCenter,stamembersatalltheseplacesdeservesathankyouforallthesupporttheyhaveprovidedmeovertheseveyearsinDenver. Myteachersandmystudents,thattaughtmeandtaughtbymeovertheyearsdeserves ahugethankyouforalltheirsupport.LastbutnevertheleastthisPh.Dwouldnotbe v
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possiblewithouttheunconditionalloveandsupportfrommyfamily;mymotherNilanthie, myfatherFrankandmysisterMenoji.Overtheseyearswerealizedhowstrongweareasa familyandcompletionofthisPhDisanachievementofallfourofus. vi
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TABLEOFCONTENTS I.INTRODUCTION...................................2 1.1NearEarthSpace................................2 1.2WhistlersandWhistlerModeWaves......................3 1.3FundamentalsofWhistlerModeWaves....................6 1.4RoleofWhistlerModeWavesinNearEarthSpace..............10 1.4.1ColdandWarmPlasma.........................14 1.5ScienticContributions.............................17 1.6ThesisOrganization...............................18 II.THEORETICALBACKGROUND..........................19 2.1IntroductiontoNumericalRaytracing.....................19 2.2RefractiveIndexSurface............................20 2.3DispersionRelationinColdPlasma......................22 2.4DispersionRelationwiththeInclusionofTemperatureEects........26 2.5IntroducingTemperatureEectstoMultipleSpecies.............30 2.6LandauDamping................................31 2.7NumericalRaytracing..............................34 III.MODIFICATIONTOTHEREFRACTIVEINDEXSURFACE..........39 3.1EectofIons...................................39 3.2InclusionofFiniteElectronorIonTemperature................39 3.3InclusionofTemperatureforBothElectronsandIons............41 IV.MODIFICATIONTOWHISTLERMODERAYTRAJECTORIESFROMFINITEELECTRONANDIONTEMPERATURE..................45 4.1EectofTemperatureonLandauDamping..................45 4.2LaunchingWavesinDierentDirections....................45 vii
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4.3FrequencyDependenceofWaveTrajectories.................47 4.4DependenceonInitialWaveNormalAngle..................48 4.5ObservationsofWavesLaunchedat L =4...................51 V.COMPARISONWITHVANALLENPROBESPACECRAFTOBSERVATIONS.65 5.1ChorusandHiss.................................65 5.2DependenceonMagneticLocalTime.....................65 5.3VanAllenProbeSpacecraftObservations...................66 5.4SimulatedWavePowerwithLandauDamping................67 5.5WaveGrowthAnalysis.............................69 VI.SUMMARYANDCONCLUSIONS..........................74 6.1FutureWork...................................75 REFERENCES.......................................76 APPENDIX A.DispersionRelationforFreeSpace........................81 B.DispersionRelationinaNonMagnetizedPlasma................84 C.DispersionRelationinaColdMagnetizedPlasma...............86 D.NoncausalFiltering................................88 E.Glossary......................................90 viii
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FIGURES 1.1NearEarthspaceenvironment,housingmanmadeelectronicsforresearch,communication,navigationandmilitarypurposes.Imagesource:www.science.nasa.gov.3 1.2ObservationofasfericataverylowfrequencyantennalocatedinChistochina, Alaska.Toppanelshowstheamplitudeofthesfericcapturedbytheantenna orientedinthegeomagneticNorthSouth,andbottompanelshowstheamplitude ofthesamesfericcapturedbytheantennaorientedintheEastWestdirection.5 1.3ImageoftheEarthIonosphereWaveguideEIW.Imagesource:vlf.stanford...6 1.4Spectrogramofawhistlermodewavetogetherwithsfericsandtriggeredemissions.Sfericsareoccurredwiththelighteningstrikes.Imagesource:[ Greenand Inan ]......................................7 1.5Dispersiondiagramofawhistlermodewave.Thedispersionrelationforfree space = ck isalsoshownintheplot.......................9 1.6Dierentmodesofpropagationforwhistlermodewaves;ductedandnonducted. Hereinthegure f w isthefrequencyofthewaveand f LHR isthelowerhybridresonancefrequency.Ductedwavesareguidedbythegeomagneticeld andthenonductedwavesshowmultiplereections.Imagesource:[ Greenand Inan ].....................................11 1.7Illustrationofthegeomagneticeld.Imagesource:www.science.nasa.gov....12 1.8IllustrationoftheconceptofLshells.Imagesource:en.wikipedia.org.......12 1.9VanAllenradiationbeltsandthesatelliteslaunchedbyNASAforVanAllen beltsobservations.Imagesource:www.science.nasa.gov..............13 1.10Trajectoryoftrappedhigherenergyparticlesandthemagneticconjugatepoint. Imagesource:HandbookofGeophysicsandtheSpaceEnvironmentAirForce ResearchLaboratory,1985..............................14 ix
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1.11PrecipitationofelectronstrappedinEarth'sradiationbeltswiththeinteraction ofwhistlermodewavesfromlightning.Imagesource:vlf.stanford.edu......15 1.12MaxwellBoltzmannvelocitydistributioninonedimension.............15 2.1Orientationofthewavenormalvectorandthegroupvelocityvectorinan anisotropicmedium..................................21 2.2aIsotropic,nonhomogeneousmedium.bNonhomogeneousandanisotropic21 2.3Orientationofthewavenormalvectorandthegroupvelocityvectorinan anisotropicmedium.ThisgureisarecreationsimilartoFigure3.13in[ Helliwell ].....................................22 2.4Progressionoftherefractiveindexsurface,raydirectionandthedirectionofthe wavenormalfora3kHzwave.Imagesource:[ Bortnik ]...........23 2.5Changeofthelocationofplasmapausewiththe k p index.Plasmapauseisthe regionwheretheelectrondensitysuddenlydrops..................38 3.1RefractiveIndexsurfacefor2.3kHz,at L =2...................40 3.2RefractiveIndexsurfacefor2.3kHz,at L =2,plottedwiththesameaspectratio forbothaxes.....................................40 3.3RefractiveIndexsurfacesfor2.3kHz,attheequatorof L =2...........42 3.4RefractiveIndexsurfacesfor3kHz,attheequatorof L =2............43 3.5RefractiveIndexsurfacesfor10kHz,attheequatorof L =2...........43 3.6RefractiveIndexsurfacesfor3kHz,attheequatorof L =2with1eVtemperatureandwith4eVtemperature...........................44 4.1aTrajectoriesofwhistlermodewaveswithafrequencyof3.5kHz,launched attheequatorof L =2,withaninitialwavenormalangleof88 ,withRed andwithoutBlackniteelectronandiontemperature.boriginalpower attenuationplots,czoomedversionofpowerattenuationplots.........46 x
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4.2Whistlermodewaveswithafrequencyof3kHz,launchedattheequatorof L =2,witha.iinitialwavenormalangleof85 ,b.iinitialwavenormalangle of )]TJ/F15 11.9552 Tf 9.299 0 Td [(85 ,c.iinitialwavenormalangleof95 ,a.ii,b.ii,c.iishowthepower attenuationplotsofthecorrespondingtrajectorieswithandwithouttemperature eects.InallthepanelsBlacktraceswereproducedwithcoldplasmaconditions andRedtraceswereproducedwithwarmplasmaconditions...........54 4.3Whistlemodewavetrajectorieswithafrequencyof2.1kHzlaunchedatthe equatorof L =2withcoldBlackandwarmRedplasmaconditions.ashows theoriginaltrajectoriesandbshowsthezoomedinversion...........55 4.4aPowerattenuationduetoLandaudampingforthetrajectoriesshowninFigure 4.3.bGroupvelocityalongthetrajectoriesshowninFigure4.3........55 4.5Whistlemodewavetrajectorieswithafrequencyof2.98kHzlaunchedatthe equatorof L =2withcoldBlackandwarmRedplasmaconditions.ashows theoriginaltrajectoriesandbshowsthezoomedinversion...........56 4.6aPowerattenuationduetoLandaudampingforthetrajectoriesshowninFigure 4.5.bGroupvelocityalongthetrajectoriesshowninFigure4.5........56 4.7Whistlemodewavetrajectorieswithafrequencyof7kHzlaunchedattheequator of L =2withcoldBlackandwarmRedplasmaconditions.ashowsthe originaltrajectoriesandbshowsthezoomedinversion.............57 4.8aPowerattenuationduetoLandaudampingforthetrajectoriesshowninFigure 4.7.bGroupvelocityalongthetrajectoriesshowninFigure4.7........57 4.9Whistlermodewavetrajectoriesofa3kHzwavelaunchedwithaninitialwave normalangleof60 ,attheequatorof L =2undercoldBlackandwarmRed plasmaconditions.aoriginalwaveandbshowsthezoomedinversion....58 4.10GroupvelocitiesofthewavetrajectoriesshowninFigure4.9...........58 xi
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4.11Whistlermodewavetrajectoriesofa3kHzwavelaunchedwithaninitialwave normalangleof89 ,attheequatorof L =2undercoldBlackandwarmRed plasmaassumptions.aoriginalwaveandbshowsthezoomedinversion...59 4.12GroupvelocitiesofthewavetrajectoriesshowninFigure4.11...........59 4.13Whistlermodewavetrajectoriesofa3kHzwavelaunchedwithaninitialwave normalangleof )]TJ/F15 11.9552 Tf 9.299 0 Td [(89 ,attheequatorof L =2undercoldBlackandwarm Redplasmaconditions.aoriginalwaveandbshowsthezoomedinversion.60 4.14GroupvelocitiesofthewavetrajectoriesshowninFigure4.13...........60 4.15Magnetosphericreectionpointsforwaveswithafrequencyof3kHz,launched attheequatorof L =2,withaninitialwavenormalangleof89 ,withRed andwithoutBlacktemperatureeects.......................61 4.16Dierencebetweenwavefrequencyandlocalhybridresonanceasa3kHzwave propagatesunderwarmplasmaconditionswithinitialwavenormalangleof88 : 9 top.Instantaneouswavenormalanglealongthetrajectorybottom.....62 4.17Whistlermodewaveslaunchedwithaninitialwavenormalangleof89 ,atthe equatorof L =4,withfrequencies250Hz,375Hzand1kHz.Lefthandpanels showtheoriginaltrajectoriesandpanelsontheRighthandsideshowthezoomed inversions.AlltrajectoriesinBlackwereproducedwithcoldplasmaassumptions andalltrajectoriesinRedwereproducedwithwarmplasmaassumptions....63 4.18Whistlermodewavetrajectoriesfora380Hzwavelaunchedattheequatorof L =4,withaninitialwavenormalangles85 ; 89 and )]TJ/F15 11.9552 Tf 9.299 0 Td [(89 .Alltrajectories inBlackwereproducedwithcoldplasmaassumptionsandalltrajectoriesin Redwereproducedwithwarmplasmaassumptions.Lefthandpanelsshow theoriginaltrajectoriesandpanelsontheRighthandsideshowthezoomedin versions........................................64 5.1Originationlocationsandspectrogramsofchorusandhisswaves.Imagesource:[ Bortnik etal. ].....................................66 xii
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5.2RegionsofchorusandhissenergyconnementswithrespecttotheMagnetic LocalTime.Imagesource:vlf.stanford.edu/DanielGoldenThesis........67 5.3Artist'sillustratonoftheEarth'sradiationbeltsandthetrajectoryofVanAllen Probespacecrafts.Thetwinspacecraftsareshownhere.Imagesource:Andy Kale,UniversityofAlberta.............................67 5.4aWaveenergymeasuredbytheVanAllenProbespacecrafts.bSamewave energyasshownina,normalizedtothemaximumateachfrequency......68 5.5Normalizedwavepowerwiththewavegrowthanalysisundercoldbackground plasmascenario.Wavepowerisnormalizedtothemaximumateachfrequency.69 5.6Normalizedwavepowerwiththewavegrowthanalysisunderwarmbackground plasmascenario.Wavepowerisnormalizedtothemaximumateachfrequency.70 5.7Normalizedwavepowerwiththewavegrowthanalysisundercoldbackground plasmascenario.Wavepowerisnormalizedtothemaximumateachfrequency.72 5.8Normalizedwavepowerwiththewavegrowthanalysisunderwarmbackground plasmascenario.Wavepowerisnormalizedtothemaximumateachfrequency.73 D.1GraphicalillustrationofthesmoothingprocessgiveninEquationD.4,andthe truncationconstant n ................................89 1
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CHAPTERI INTRODUCTION 1.1NearEarthSpace NearEarthspaceistheregionsurroundingtheEarth,startingfrom70kmabovethe surfaceandextendingtoabout10Earthradii.OneEarthradiiis6371km,hencenearEarth spaceextendstoabout63,710kmfromthesurfaceoftheEarth.Thedeningcharacteristic ofnearEarthspaceisthatmatterisintheplasmastate.Plasmaisthefourthstateofmatter, wheretheparticlesareionized.ThisionizedstateofmatterstartsintheEarth'sionosphere ataltitudeofapproximately70km.Earthsmagnetosphereextendsfromanapproximate altitudeof1000km,andisalsointheplasmastate.Themagnetospheregetsitsnamefrom thefactthattheoveralldensityofparticlesismuchlowerthanintheionosphere,which makestheEarth'smagneticeldbethedominantforce.Inbothregions,ionizationoccurs duetosolarradiation. ThenearEarthspaceenvironmentisofasignicantinterestforbothphysicistsand engineersforseveralreasons.Onemainreasonistherichnessofenergydynamicsinwhich thesunplaysanimportantrole.Burstsofadditionalplasmaandenergyfromthesunas solararesandcoronalmassejectionsdirectlyimpactnearEarthspaceandevendistort thegeomagneticeld.Theenergyinteractionsinvolvenotonlyplasmaparticlesbutalso dierenttypesofelectromagneticandelectrostaticwaves,bothnaturalandmanmade.ManmadewhistlermodewavesincludemonochromaticsignalsfromgroundbasedELF/VLF transmitters,triggeredELF/VLFemissionsfromthelargescaleionosphericheatingfacilities suchasHighfrequencyActiveAuroralResearchProgramHAARP.Thelargescaleeects ofsuchinteractionsareoftendenotedwiththetermspaceweather". Furthermore,asshowninFigure1.1,nearEarthspacehostsmanmadeelectronics launchedforcommunication,navigation,monitoringandmilitarypurposes.Henceitcontainsequipmentworthbillionsofdollars.Theoperationalconditionandlifetimeofthose manmadeelectronicsarestronglyaectedbythesurroundingplasmaenvironment.There2
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foreunderstandingthephysicsofnearEarthSpaceisanimportanttaskofthephysicsand engineeringcommunities.Variousspacemissionswereconductedinordertostudywhistler modewavesandtheirimpactondierentaspectsofspaceweather,suchastheCLUSTER spacecraftlaunchedtoobservethechorusanaturallygeneratedwhistlermodewavetype wavesandVanAllenProbespacecraftslaunchedtostudytheEarth'sradiationbelts. Figure1.1:NearEarthspaceenvironment,housingmanmadeelectronicsforresearch,communication,navigationandmilitarypurposes.Imagesource:www.science.nasa.gov. 1.2WhistlersandWhistlerModeWaves AdominantplayerinnearEarthspaceenvironmentisaspecialtypeofelectromagnetic wavecalledthewhistlermodewave.Thenameofthisclassofwavesderivesfromaspecic occurrenceofthesewavesthatwasrstobservedtomakeawhistlingsoundinaudioequipment.DuringthetrenchwarfareofWorldWarI,longwireswerelaidonthegroundforcommunication.WhentheGermanphysicistHeinrichBarkhausentriedtoeavesdroponAllied communicationlines,hereportedintermittentlyhearingaveryremarkablewhistlingnote. Duringthattimehewasunabletoexplainthesourceofthisunusualwhistlingsoundheard 3
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throughtheearthcurrentdetectorcommunicationsystemthathewasusing.Afteranunsuccessfulattempttorecreatethesewhistlingtonesinsidealab,Barkhausenconcludedthat thesewavesarecreateddueanunidentiednaturalphenomenon[ Barkhausen ].Due tothesoundtheymadewhenpassedthroughaspeaker,thesewaveswerecalledwhistlers. Yearslater,in1935,EnglishtheoreticalphysicistandengineerThomasEckersley,observedthatawhistleroccursaboutonesecondafterasferic.Asfericoranatmospheric showninFigure1.2isabroadbandelectromagneticemissioninducedbyalightningstrike. Wheneverthereisalightningstrike,theresultingionizedplasmachannelactsasanantenna andradiateswaveswithmanyfrequenciessinceahighamountofelectriccurrent )]TJ/F15 11.9552 Tf 11.898 0 Td [(100 kAowsinaveryshort < 100 Secamountoftime.Someofthiswaveenergyistrapped inwhatisknownastheEarthIonospherewaveguideshowninFigure1.3andsomeofitis leakedintotheupperlayersoftheionosphereandmagnetosphere.Sfericscanbeguidedby theEarthIonospherewaveguideforthousandsofkilometersbeforebeingattenuated.Based onthisobservation,Eckersleyspeculatedthatwhistlersarerelatedtolightningandarean echoofsferics[ Eckersley ],buthewasnotabletoexplainthemainfeatureofwhistlers, theirwhistlingsound,whichiscausedbydispersion. Whistlersareelectromagneticwavesconsistingofacontinuousbandoffrequencies.Figure1.4showsaspectrogramofawhistler.Onemainobservationwhichcanbemadewith thespectrogramisthatforwhistlers,higherfrequenciesarriveatthereceiverpriortothe lowerfrequencies.Inotherwords,dierentfrequencycomponentspropagateatdierentvelocities.Thisphenomenonisknownasdispersion.Itisduetothedependenceofrefractive indexonfrequency. L.R.OStoreygaveanexplanationforthedispersivebehaviorofwhistlers[ Storey ]. Asmentionedearlier,whenthereisalightningstrike,itemitselectromagneticwaveswith dierentfrequencies.SomeofthosewavesarebeingguidedbetweentheEarthIonosphere waveguideandsomeofthoseareleakedintothemagnetosphere.Thosewaveswhichare leakedintothemagnetospherecanbeguidedbythegeomagneticEarth'smagneticeld 4
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Figure1.2:ObservationofasfericataverylowfrequencyantennalocatedinChistochina, Alaska.Toppanelshowstheamplitudeofthesfericcapturedbytheantennaorientedin thegeomagneticNorthSouth,andbottompanelshowstheamplitudeofthesamesferic capturedbytheantennaorientedintheEastWestdirection. eldlinesandreturntoearthintheconjugatehemisphere.ConjugatepointisthehypotheticallocationwhereageomagneticeldlineconnectstothesurfaceoftheEarthatthe oppositepole.Alongthispath,thewavespropagatethroughtheplasmainthemagnetosphere,whichishighlydispersiveforwavefrequenciesbelow30kHz.Storeyhadthus succeededinexplainingBarkhausen'sobservationsmadeover30yearsearlier.Sincetheinitiallyidentiedwhistlerswerecreatedduetolightning,thosewerecalledlightninggenerated whistlers. StoreyandBarkhausenweredealingwithwavesinducedbylightning;lateron,itwas identiedthatthereareothertypesofwavescreatedbynaturalandmanmadesources, whichcanalsopropagateinthesamemode.Nevertheless,thenamefromtheoriginal observationshasstuckandthenameofthepropagationmodeisthewhistlermode,even whenthesourceisnotlightningrelated. 5
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Figure1.3:ImageoftheEarthIonosphereWaveguideEIW.Imagesource:vlf.stanford 1.3FundamentalsofWhistlerModeWaves Asmentionedabove,whistlermodewavesexistonlyinamagnetizedplasma.Herewe presentthefundamentalsofthesewavesandcontrasttheirpropertieswiththoseofsimpler electromagneticwavesinfreespace.AllelectromagneticphenomenaderivefromMaxwell's equations: r E = v .1 r B =0 r E = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(@ B @t r B = J + @ E @t Equation.1showsMaxwell'sequationsforanymedium.Foramacroscopicallyneutral medium, v =0.Westartwithassumingatimeharmonicelectriceldwithsinusoidalspatial variationwhichcanbewrittenincomplexphasornotationas E = E o e j !t )]TJ/F37 7.9701 Tf 6.586 0 Td [(k r a x .Here, E 0 istheamplitude, t isthetime, isthewavefrequency, k isthewavenormalvectorand a x istheunitvectorinthexdirection.UsingtheequationsshowninEquation.1,wecan 6
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Figure1.4:Spectrogramofawhistlermodewavetogetherwithsfericsandtriggeredemissions.Sfericsareoccurredwiththelighteningstrikes.Imagesource:[ GreenandInan ] arriveattheplanewaveequationshowninEquation.2forfreespace. 0 and 0 arethe permeabilityandpermittivityinfreespace. r 2 E = )]TJ/F21 11.9552 Tf 9.298 0 Td [( 0 0 2 E .2 Foranontrivialsolutionthecoecientsoftheelectriceldshouldbeequaltozero. Thisconditioniscalledthedispersionrelation.Henceforfreespacethedispersionrelation canbewrittenas: 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 c 2 =0.3 InEquation.3, c isthespeedoflight 1 p 0 0 ,and k isthemagnitudeofthewave normalvector.FullderivationofthisdispersionrelationisgiveninAppendixA. Foraplasmamediumintheabsenceofastaticmagneticeldthedispersioncanbe expressedas: k 2 c 2 2 = 1 )]TJ/F21 11.9552 Tf 13.151 8.087 Td [(! pe 2 2 ; .4 7
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where pe istheelectronplasmafrequency.Theplasmafrequencyisthefrequencyat whichtheplasmaparticlesoscillateat,ifanexternaldisturbanceisintroduced.Theelectron plasmafrequencyisgivenby N e q 2 e o m e ,where N e istheelectrondensity, m e isthemassofan electron, q e isthechargeofanelectron.Derivationofthisdispersionrelationisgivenin AppendixB. FromthedispersionrelationshowninEquation.4,itcanbeseenthatforawaveto propagatethroughaplasmaintheabsenceofastaticmagneticeld,thewavefrequency shouldbehigherthantheplasmafrequency.Forwavefrequencieslowerthantheplasma frequency,thewavenormalvectorwavenumberbecomesimaginary.Waveswithanimaginarywavenumberarebeingattenuatedveryquicklyandthosewavesarecalledevanescent waves.Therefore,thecutofrequencyforpropagationinaplasmaintheabsenceofastatic magneticeldistheplasmafrequency. Figure1.3illustrateshowthefrequencydependenceofEquation.4manifestsitself forpropagationofwavesgeneratedonEarth.Forawavetopenetratethroughtheplasma, thewavefrequencyshouldbehigherthantheplasmafrequency;otherwise,thewavesare reectedbackandforthbetweentheEarthandtheionosphereandcanpropagatetolong distances.ExtremelyLowFrequencyELFwavesinthefrequencyrange300Hz3kHz andVeryLowFrequencyVLFwavesinthefrequencyrange3kHzto30kHzarewell belowtheplasmafrequency.Therefore,thosewavespropagatewithintheEarthIonosphere waveguide,whereashighfrequencysignalssuchasmicrowavesignalspropagatethroughthe ionosphericplasma 1 Whenthereisastaticmagneticeldpresent,thedispersionrelationbecomesmore complicatedandisgovernedbytheAppletonHartreeequationthatispresentedinChapter 2.FullderivationofthedispersionrelationinacoldmagnetizedplasmaisgiveninAppendix C.Atthispoint,weconsideronlythespecialcaseofpropagationofwavesparalleltothe 1 HerewehavediscussedtheEarth'sionosphereasunmagnetizedplasmaineectignoringthegeomagnetic eld.Suchasassumptionisaccuratetorstorderinthelowerionosphere,sincethecollisionfrequencyis sucientlyhightomakemagneticeldsecondary.Wewillshowlaterthatthegeomagneticelddoesallow forafractionoflowfrequencywaveenergytopropagatethroughtheionosphereviathewhistlermode. 8
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staticmagneticeldvector.Insuchascenario,thedispersionrelationsimpliesto: k 2 c 2 2 = 1 )]TJ/F21 11.9552 Tf 30.064 8.087 Td [(! pe 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(! ce : .5 InEquation.5, ce istheelectroncyclotronfrequencydenedas ce = q e B o m e .The electroncyclotronfrequencyistheprecessionfrequencyofelectronsundertheinuenceofa magneticeld. q e ;m e ;B o arechargeofanelectron,massofanelectronandthemagnitudeof thestaticmagneticeld.ThemodeofpropagationexpressedinEquation.5iscalledthe whistlermode.Whistlermodepropagationexistsonlyinthepresenceofastaticmagnetic eld.Inordertopropagateinthewhistlermodethefrequencyofthewavesshouldbelower thantheplasmafrequencyandtheelectroncyclotronfrequency.Frequencieshigherthanthe electroncyclotronfrequencyresultsinanimaginarywavenumber.Thedispersionrelation diagramofawhistlermodewaveisgiveninFigure1.5. Figure1.5:Dispersiondiagramofawhistlermodewave.Thedispersionrelationforfree space = ck isalsoshownintheplot. Duetothefactthattheplasmaisadispersivemedium,waveswithdierentfrequencies propagatewithdierentvelocities.Itisimportanttointroducetwodescriptionsofwave velocity:groupvelocityandthephasevelocity.Groupvelocity v g isthevelocityatwhich theenergytravels;phasevelocity v p isthevelocityinwhichthewavepatterntravels.In 9
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otherwords,forsinusoidalwaves,phasevelocityisthepropagationvelocityofanullor acrest.Groupvelocity v g isdenedas d!=dk andthephasevelocityisdenedas !=k Infreespace,bothgroupvelocityandphasevelocityconvergetothespeedoflightfor electromagneticwaves.Whistlermodewavesareconsideredasaslowmode,sincethegroup velocityofwhistlermodewavesistypicallyaboutafactorofa100smallercomparedtothe speedoflight.FromFigure1.5,theslopeofthefreespacemodeandwhistlermodecanbe compared.Theslopeofthefreespacelineshowsthespeedoflightismuchlargerthanthe slopeofthewhistlermodecurve,whichvarieswithfrequency. Althoughwhistlermodewavesarelowfrequencywaves,theypenetratethoughthe ionosphereandpropagateintothemagnetospherethroughaleakageprocess.Onceinthe magnetosphere,whistlermodewavescanpropagateintwoways:ductedandnonducted. Inductedpropagation,whistlermodewavesareguidedbythegeomagneticeldlinesand willreturntoEarthatthemagneticconjugatepoint.Themagneticconjugatepointis wherethemagneticeldlinesconnectbacktothesurfaceoftheEarth.Ductedpropagation resultsfromeldalignedplasmadensityirregularitiesandhasmanysimilaritiestooptical berguiding.Inthecaseofnonductedpropagation,whistlermodewavesleakedintothe magnetospherefollowamuchmorecomplicatedtrajectory,andtheyarereectedmultiple timesbeforebeingcompletelyattenuated.Nonductedwhistlermodewaveswavesmight encountermultiplereectionsdependingonthelocalrefractiveindex.Thesetypeofwhistler modewavesarecalledmagnetosphericallyreectedwhistlersorMRwhistlers.Figure1.6 showsanillustrationofwhistlerspropagatedalongductedandnonductedtrajectories 2 1.4RoleofWhistlerModeWavesinNearEarthSpace WhistlermodewavesplayadominantroleintheEarth'smagnetosphere[ Kulkarni etal. ].Asmentionedabove,Earth'smagneticeldisthedominantforceinthe magnetosphere.Figure1.7showsanartist'sillustrationofthegeomagneticeld.Distance measurementsinthemagnetosphereareexpressedintermsof L shells. L valuesareinthe 2 StoreyandBarkhausenwereinvestigatingductedwhistlermodewaves. 10
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Figure1.6:Dierentmodesofpropagationforwhistlermodewaves;ductedandnonducted. Hereinthegure f w isthefrequencyofthewaveand f LHR isthelowerhybridresonance frequency.Ductedwavesareguidedbythegeomagneticeldandthenonductedwaves showmultiplereections.Imagesource:[ GreenandInan ] unitsofEarthradiiatthemagneticequatoranddescribethegeomagneticeldline.For example, L =1isoneEarthradiifromthecenteroftheEarth;hence,itislocatedonthe surfaceoftheEarth.Similarly, L =2islocatedtwoEarthradiifromthecenteroftheEarth andoneEarthradiifromthesurfaceoftheEarth.Figure1.8showsagraphicalillustration ofLshells. OurEarthissurroundedbytworegionsofhighenergyplasmaparticlesknownasthe VanAllenBelts,discoveredbyAmericanspacescientistJamesVanAllenbasedonthedata collectedbyExplorer1and4.Highenergyelectronscreatedduetocosmicrayinteractions aretrappedintheseradiationbelts.Figure1.9showsanillustrationoftheVanAllenBelts. Theinnerbeltextendsfromanaltitudeofabout1000kmupto L =3.Theinnerbelt consistsofhighenergyelectronstypicallyintheenergyrangebetween0.04MeVto4.5MeV [ GreenandInan ].Theouterradiationbelt,whichextendsfrom L =5to L =7, consistsofhighenergyelectronsupto7MeV.Theregionbetweenthetworadiationbelts 11
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Figure1.7:Illustrationofthegeomagneticeld.Imagesource:www.science.nasa.gov. Figure1.8:IllustrationoftheconceptofLshells.Imagesource:en.wikipedia.org. betweenL=3and5isknownastheslotregion. Thesetrappedhighenergyelectronsgyratearoundthegeomagneticeldlinesasshown inFigure1.10and,intheabsenceofanyperturbation,canremaintrappedindenitely. Whistlermodewavescanchangethemomentumoftheseelectronsviaresonanceinteractions.Figure1.11showstheresonantinteractionsofwhistlermodewavesfromlightningwith trappedhigherenergyparticlesandsubsequentprecipitationofelectrons.Theseresonance interactionscanmodifythetrajectoryofthesetrappedelectronswithrespecttothegeomagneticeld.Hencethepitchangleoftheelectriceldofthewavescanbereduced.Lower pitchanglescauseselectrontrajectoriestocomeclosertotheEarth.Whenthepitchangle 12
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Figure1.9:VanAllenradiationbeltsandthesatelliteslaunchedbyNASAforVanAllen beltsobservations.Imagesource:www.science.nasa.gov. is0 ,electronsarenolongertrappedandwillimpacttheatmosphereguidedintoitbythe geomagneticeldlines.Suchelectronswhichmakecontactwiththeatmospherespecically theionospheredeposittheirenergyincollisionsandaresaidtoprecipitate".TheAurora BorealisatthegeomagneticnorthpoleandAuroraAustralisatthegeomagneticsouth polearenaturaleectsofelectronprecipitations.Thesehighenergyelectronsinteractwith variousatomsintheatmosphereandraisetheirelectronstoreachhigherenergyexcited states.Whenthoseexcitedelectronsrelaxbacktotheirgroundstate,theyreleaseenergyas lightobservedasauroras. Asmentionedabove,whistlermodewavesareakeydriveroftheenergydynamics innearEarthspace[ Belletal. Bortniketal. a, Bortniketal. b, Bortniketal. a, Shpritsetal. Lietal. ].Interactionofwhistlermodewaves withthetrappedhigherenergyparticlesintheradiationbeltsisoneofthemainprocessesofparticlelossfromradiationbelts,waveamplicationandwavegrowth[ Kenneland Petscheck Lyonsetal. HuangandGoertz AbelandThorne a, Abel andThorne b, Bortniketal. b, Omuraetal. ].Hencethepropagationcharacteristicsofwhistlermodewavesareofgreatinterest. 13
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Figure1.10:Trajectoryoftrappedhigherenergyparticlesandthemagneticconjugatepoint. Imagesource:HandbookofGeophysicsandtheSpaceEnvironmentAirForceResearch Laboratory,1985. AsdiscussedlaterinChapter2,Earth'smagnetosphereisananisotropicmedium;hence, thepropagationcharacteristicsofawavedependonthepropagationdirection.Thepropagationofmagnetosphericallyreectionwhistlerscanbetrackedbyanumericalprocesscalled raytracing[ Haselgrove InanandBell ]andisthetopicofthisthesis. 1.4.1ColdandWarmPlasma Beforeconcludingthischapter,wedisucssoneadditionalimportantparameterofplasmas thataectshowwavespropagatethroughthem.Thisparameteristheplasmatemperature andcanberigorouslydiscussedinthesamemannerasforidealgasses. Thekineticenergyofaparticlewithmass m andspeed u canbewrittenas 1 2 mu 2 .For anumber N ofparticleswithmasses m i andspeeds u i ,averagekineticenergyisgivenby 1 2 N P N i =1 m i u i 2 Whenaplasmaisinthermalequilibriumattemperature T ,particleshavearange velocities.ThevelocitydistributionoftheparticlescanbeexpressedbyaMaxwellianor MaxwellBoltzmanndistribution.Foraonedimensionaldistribution,MaxwellBoltzmann velocitydistribution f u canbegivenby; f u = Ae )]TJ/F20 5.9776 Tf 7.782 3.258 Td [(1 2 mu 2 k B T .6 14
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Figure1.11:PrecipitationofelectronstrappedinEarth'sradiationbeltswiththeinteraction ofwhistlermodewavesfromlightning.Imagesource:vlf.stanford.edu. InEquation.6, A isamultiplierwhichcanbefoundbythetotaldensity N k B isthe Boltzmannconstantwhichisequalto1 : 38 10 )]TJ/F19 7.9701 Tf 6.586 0 Td [(23 JK )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 and T istheabsolutetemperature inKelvin.Figure1.12illustratestheMaxwellianvelocitydistribution. Figure1.12:MaxwellBoltzmannvelocitydistributioninonedimension. TheMaxwellianvelocitydistributioncanbeusedtondtheaveragekineticenergyof theparticlesasshownEquation.7. E av = R 1 1 2 mu 2 f u du R 1 f u du .7 15
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Foraonedimensionalsystem,theaverageenergyofparticlesis 1 2 k B T .Thisresultcan beextendedforathreedimensionalsystem,inwhichtheaveragekineticenergyis 3 2 k B T Theenergyofparticlesinplasmas,isexpressedinunitsofelectronvoltseVinsteadof thestandardunitofJoules.InordertoexpresstheenergyineV,thekineticenergyshould bedividedbythemagnitudeofthechargeofanelectron : 602 10 )]TJ/F19 7.9701 Tf 6.587 0 Td [(19 .Itisimportant tonoteherethatthetemperature T isanindicatorofkineticenergy.Anditisthusalso referredtoasakinetictemperature. Aswewillbediscussinglater,forthecoldplasmawewillbeconsideringinChapter 2,theabsolutetemperatureis0K.Andthekineticenergyofparticlesiszero.Hencethe randomthermalvelocityofparticlesunderthecoldplasmaassumptionisalsozero{in otherwords,undercoldplasmaassumptions,theparticlesaremotionlesswhentheplasma isunperturbedbyanexternaleld[ Stix ]. HencetheabsolutetemperatureofthebulkparticlesintheEarth'smagnetospherenot includingthesmallpopulationofhighenergyparticlesmakinguptheradiationbeltsis about11600K.Itisworthmentioningherethat,althoughtheabsolutetemperatureisa veryhighvalue,itdoesnotindicatethattheEarth'smagnetosphereishotorisareservoir oflargeamountsofthermalenergy.TheparticledensityintheEarth'smagnetosphereis approximately10 6 m )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 ,whereastheparticledensityintheatmosphereisintheorders of10 25 m )]TJ/F19 7.9701 Tf 6.587 0 Td [(3 .Duetothelowerparticledensityinthemagnetospherecomparedtothe atmosphere,theheatcapacityofthemagnetosphereismuchlowerthantheheatcapacity oftheatmosphere. Inordertodistinguishthetwocasesbetween0Kabsolutetemperatureand11600K temperature,weusethetermscoldandwarm,respectively.Thetermhotplasma"isoften usedtodescribethehighenergyparticlesmakinguptheradiationbelts.Itisworthreemphasizingthatthehighenergyradiationbeltsparticlesareasmallminorityofallthe chargedparticlesinthemagnetosphere. 16
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Laterinthethesiswewillbeconsideringthesehighenergyelectronstrappedinthe radiationbelts,whenintroducingLandaudamping.Landaudampingistheattenuation ofwavepowerwhenthewaveencountersahighenergyelectrondistribution.Wewillbe introducingthemathematicalformulationofLandaudampinginChapter2.Chapter4 discussesthethesimulatedraytrajectoriesindetail. 1.5ScienticContributions Inthisworkwestudytheeectofwhistlermoderaytrajectorieswiththeinclusionof niteelectronandiontemperature. 1.Thisisthersttimeanalysisofwhistlermodetrajectorieshasbeendonewithnite thermaleects. 2.Thisprojectpresentsafullyfunctioningraytracerwhichtakestemperatureintoaccount.Theraytracerismodiedtoworkwithorwithoutniteelectronandion temperature.Theusercandenetheelectrontemperature,iontemperaturesimultaneously. 3.Alsointhisworkweintroduceanupdatedequationforwavepowerattenuationwith Landaudamping;bymodifyingtheexistingLandaudampingequationforcoldplasma. Landaudampingformulationsareincludedintothemainraytracer,andthepowerattenuationofthewaveswasstudiedundercoldandwarmbackgroundplasmaassumptions. 4.Inthepresenceofasignicanttemperatureanisotropy,thewavesexperienceapower growth.Wehaveconductedthewavegrowthanalysis,undercoldandwarmplasma assumptions,usingabiMaxwellianvelocitydistribution.Moredetailsofwavegrowth analysiswillbegiveninChapter5. 5.ComparisonofoursimulationresultswithVanAllenProbespacecraftobservations showthat,whenLandaudampingandwavepowergrowtharetakenintoconsideration, thereisabetteragreementunderwarmplasmaassumptions. 17
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1.6ThesisOrganization InChapter2wepresenttheprocessofnumericalraytracingthatwehaveused.Starting fromcoldplasmaraytracing,wediscussthemodicationswemadetotheexisting3Draytracer.Wealsodiscussthemodicationsmadetothepowerattenuationofwhistlermode waves.Chapter3presentsthemodicationsobservedwiththerefractiveindexsurfacewith theinclusionofniteelectronandiontemperature.Sinceinraytracingmediumproperties entertheraytracingequationsintermsoftherefractiveindex,calculationoftherefractive indexsurfaceisaveryimportantstep. InChapter4wepresentthemodicationsobservedintheraytrajectoriesspecically whennitetemperatureeectsareincluded.Wediscussthemodicationsobservedwith respecttowavefrequency,initialwavenormalangleandlaunchedlocation. Chapter5presentsthecomparisonofwavesimulationswithVanAllenProbeobservations.Hereinthischapterweconsiderwavedampingandwavegrowthduetoresonance interactions.Chapter6presentsconclusionsfromourresultsandproposedfuturework.Additionalworkwehaveperformedintermsofstatisticalmodellingandidentifyingtheoutliers ispresentedinAppendixD. 18
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CHAPTERII THEORETICALBACKGROUND 2.1IntroductiontoNumericalRaytracing Numericalraytracingistheprocessofdeterminingthepowerowpathofawhistler wavebysolvingtheHaselgroveequations[ Haselgrove ].Haselgrove'sequationsaregiven inEquation.31,andwillbeexplainedindetaillaterinthischapter.GraphicalraytracingsolutionstowavepropagationintheEarth'smagnetospherewerecarriedoutasearly as1956by[ MaedaandKimura ].Theeectofionsonwhistlermoderaytracing wasrststudiedby[ Kimura ].Sincethe1970'saverylargebodyofmagnetospheric whistlermoderaytracingworkhasbeencarriedout[ InanandBell HuangandGoertz CairoandLefeuvre Bortniketal. a, Bortniketal. b, Chum andSantolik GolkowskiandInan Chenetal. Golden Kulkarni etal. ]. However,withafewexceptions,themajorityofpreviousworkshavereliedontheideal coldplasmaassumptionbothelectronsandionsareat0Kincalculationofthewaverefractiveindexsurfaceeventhoughthetemperatureofthebackgroundelectronsandionsin themagnetosphereisknowntobeintherangeofafeweV[ Decreauetal. ].Thisbackgroundplasmashouldnotbeconfusedwiththeradiationbeltswhichcontainhotplasma particlesinthekeVMeVrangebutwithoveralldensitiestoosmalltoaectthepropagationtrajectoriesdirectly.[ Aubryetal. ]usedwarmplasmaadiabatictheoryforthe frequenciesaroundtheplasmafrequencyandtheupperhybridresonancefrequencytoderivetheequationsrelatedtothedispersionrelationaswellasforthegroupvelocity.Inthat workalltheequationswerederivedbyconsideringonlyisotropictemperaturedistribution ofelectrons.[ Bitounetal. ]demonstratedtheprocedurefordeterminingtheraytrajectoryinawarmplasmausingtheequationsderivedby[ Aubryetal. ].Inthatwork theauthorsusedtheraytracingobservationstointerpretthetopsideresonanceatupper hybridfrequencyasobliqueechoes.Topsideresonancesaretheplasmaresonancesobserved 19
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aroundtheplasmafrequencyandtheupperhybridfrequencyinthetopsideionosphere km1000km. 2.2RefractiveIndexSurface ForwhistlermodewavestheEarth'smagnetosphereisanonhomogeneousandanisotropic medium.Inahomogeneousmedium,themediumpropertiesarethesameineverylocation. Inisotropicmedia,thepropertiesarethesameinalldirections.Themagnetosphereisnonhomogeneoussincetheplasmadensityandthestrengthanddirectionofthegeomagnetic eldvaryfromlocationtolocation.Itisanisotropicbecausethegeomagneticeldimposes adirectionaldependenceonthewaveproperties. Beforeproceedingfurther,hereweintroducesomeofthekeytermsusedinthiswork.As mentionedabove,duetogeomagneticeldEarth'smagnetospherebecomesananisotropic medium.Thevectordrawnperpendiculartothewavefrontsiscalledthewavenormalvector. Theanglebetweenthewavenormalandthegeomagneticeldisknownasthewavenormal angle.ThroughoutthisworkthewavenormalanglewillbeindicatedbytheGreekletter .Thesurfaceplotobtainedoncetherefractiveindexisplottedateachwavenormalangle, isknownastherefractiveindexsurface.Thevectordrawnperpendiculartotherefractive indexsurfaceatalocationisthedirectionofthegroupvelocityortheraydirectionatthat point.Whenawaveisgoingthroughamagnetosphericreection,theraydirectionchanges frompositivetonegativeorviceversa.Inordertosatisfythatcondition,therefractive indexneedstobeclosed.Inotherwordsclosureoftherefractiveindexsurfaceisrequired fortheoccurrenceofmagnetosphericreections.InFigure2.1,theanglemeasuredfromthe geomagneticelddirectiontoareddashedline,thatangleisknowastheresonancecone. Ifawaveislaunchedwithaninitialwavenormalanglegreaterthantheresonancecone,it wouldnotpropagate. Figure2.2ashowstherefractiveindexsurfaceforanisotropicmedium.Thearrows inblackshowthedirectionofwavenormalandthearrowsinpurpleshowthedirectionof groupvelocity.Thedirectionofgroupvelocityisthedirectionofwaveenergypropagation. 20
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Figure2.1:Orientationofthewavenormalvectorandthegroupvelocityvectorinan anisotropicmedium Figure2.2:aIsotropic,nonhomogeneousmedium.bNonhomogeneousandanisotropic 21
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Figure2.3:Orientationofthewavenormalvectorandthegroupvelocityvectorinan anisotropicmedium.ThisgureisarecreationsimilartoFigure3.13in[ Helliwell ] Inanisotropicmediumthewavenormalandthegroupvelocitybothareorientedinthe samedirection.IncontrasttothecaseinFigure2.2a,inananisotropicmediumb,the refractiveindexisnotthesameinalldirections,andthewavenormalandgroupvelocity arenotinthesamedirection.Figure2.3furthershowsthepropagationofawavepacketin anisotropicmediumfrompointAtopointB. Figure2.4,showstheprogressionoftherefractiveindexsurface,directionoftheray andthedirectionofthewavenormal,attherstmagnetosphericreectionpointfora3 kHzwavelaunchedatlatitude30 ,atanaltitudeof1000km.Asthewavegoesthrougha magnetosphericreection,wavenormalanglebecomes90 ,andtheraydirectionalters. 2.3DispersionRelationinColdPlasma Ifthemediumcontainsonlymobileelectronsionsarestationarytherefractiveindex canbecalculatedbythewellknownAppletonHartreeequationgiveninEquation.1. 2 = k 2 c 2 2 =1 )]TJ/F22 7.9701 Tf 123.164 15.646 Td [(! 2 pe 2 1 )]TJ/F22 7.9701 Tf 17.435 5.699 Td [(! 2 ce sin 2 2 2 )]TJ/F22 7.9701 Tf 6.586 0 Td [(! 2 pe 2 ce sin 2 2 2 )]TJ/F22 7.9701 Tf 6.587 0 Td [(! 2 pe 2 + 2 ce 2 sin 2 # 1 = 2 .1 22
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Figure2.4:Progressionoftherefractiveindexsurface,raydirectionandthedirectionofthe wavenormalfora3kHzwave.Imagesource:[ Bortnik ]. 23
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InEquation.1, pe and ce representplasmafrequencyandcyclotronfrequencyof electronscalculatedasafunctionofelectrondensity N e ,chargeofanelectron q e ,massof anelectron m e ,magnitudeofthegeomagneticeld B 0 andthepermittivityoffreespace 0 asinEquations.2,and isthewavenormalangle. pe = N e q 2 e o m e .2 ce = q e B o m e Inthiswork,themediumiscomposedofmultiplespeciessuchaselectrons H + He + and O + ionswhichwillyieldamodiedversionofEquation.1.Toderivethedispersion relationincludingtheseions,weusetheapproachshownin[ Bortnik ]. UsingtheMaxwell'sequationsandtheLangevinequationofparticlemotion,thelinearizedtimeharmonicplanewaveequationscanbeobtainedas: r E = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(@ B @t .3 r B = J + 0 @ E @t m s D u Dt = q s E + u B InEquation2.3 E istheintensityoftheelectriceldperturbation, B itthemagnetic perturbation, k isthewavenormalvectorand t istime. m s and q s arethemassandcharge ofaparticlefromspecies s u istheparticlevelocity.Foraconstantambientmagnetic eld, B isaconstant;hence,thetimederivativeofthemagneticeldequalstozero. D=Dt indicatestheconvectivederivative,givenby D=Dt = @=@t + u r .Solvingabovesetof equationsgivenin.3leadstothefollowingrelationshipinEquation.4. 0 B B B B @ S )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 sin 2 )]TJ/F21 11.9552 Tf 9.298 0 Td [(jD 2 sin cos jD S )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 0 2 sin cos 0 P )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 sin 2 1 C C C C A 0 B B B B @ E x E y E z 1 C C C C A =0.4 24
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InEquation.4, istherealpartofrefractiveindex, E x ;E y and E z representthe x;y and z componentsoftheelectriceld,and S;P;R;L;D representStix[ Stix ] parametersdenedas; P =1 )]TJ/F27 11.9552 Tf 11.955 11.358 Td [(X s 2 ps 2 .5 R =1 )]TJ/F27 11.9552 Tf 11.955 11.357 Td [(X s 2 ps + c s L =1 )]TJ/F27 11.9552 Tf 11.955 11.358 Td [(X s 2 ps )]TJ/F21 11.9552 Tf 11.955 0 Td [(! c s S = R + L 2 D = R )]TJ/F21 11.9552 Tf 11.955 0 Td [(L 2 IntheStixparameters. ps and cs representtheplasmafrequencyandcyclotronfrequencyofthespecies s : ps = N s q 2 s o m s .6 cs = q s B o m s Intheabovesetofequationsin.6, N s istheparticledensity,AlthoughtheStixparameters arefunctionsofangularfrequency ,fromhereonwardswewillbeusingthenotations S;D;P;R and L toaddressthoseparameters. BysettingthedeterminantofEquation.4,tozerowecanarriveatthedispersion relationofacold,collisionless,unboundedplasmagiveninEquation.7. A 0 4 + B 0 2 + C 0 =0.7 25
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istherefractiveindex.Thecoecients A 0 B 0 and C 0 arefunctionsoftheStix parametersandthewavenormalangle .Thosecoecientsaregiveninthesetofequations: A 0 = S sin 2 + P cos 2 .8 B 0 = )]TJ/F21 11.9552 Tf 9.299 0 Td [(RL sin 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(SP +cos 2 C 0 = PRL AsexpressedinEquation.7thedispersionrelationofawavepropagatingundercold plasmaconditionsisafourthorderequation.Onceweconsider 2 asasinglequantitywe cansolvetheaboveEquation.7for 2 ,usingthequadraticformula,andwearriveatthe followingtwosolutionsgiveninEquation.9; 2 = )]TJ/F21 11.9552 Tf 9.299 0 Td [(B 0 )]TJ/F27 11.9552 Tf 11.955 10.395 Td [(p B 2 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 A 0 C 0 2 A 0 ; B 0 > 0.9 2 = 2 C 0 )]TJ/F21 11.9552 Tf 9.299 0 Td [(B 0 + p B 2 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 A 0 C 0 ; B 0 < 0 2.4DispersionRelationwiththeInclusionofTemperatureEects Inclusionoftemperatureeectsincreasesthecomplexityofthedispersionrelation. [ Aubryetal. ]derivedthemodicationtothedielectrictensor K ,withtheinclusion oftemperatureeects.Authorsof[ Aubryetal. ]startedwiththeplanewavesolution ofawavepropagatinginahomogeneousplasmagiveninEquation.10asexpressedin [ Stix ].InEquation.10 k jj istheparallelcomponentofthewavenormalvectorthe geomagneticeld.InEquation.11, ij representstheelementsofaunitymatrixand K ij arecomponentsofthedielectrictensor. k k )]TJ/F21 11.9552 Tf 11.955 0 Td [(k jj 2 + 2 c 2 K E =0.10 FortheaboveEquation.10tohaveasolutionitshouldsatisfythedispersionrelation, hencethefollowingequationshouldbesatised. 26
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det k i k j )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 ij + 2 =c 2 K ij =0.11 Equation.11isthensolvedforthedielectrictensorusingtheclassicalequationofconductivitygiveninEquation.12where I istheunitymatrixand k ;! istheplasma conductivitytensor. K k ;! = I + j k ;! 0 .12 Introductionoftemperatureeectsthemodiestheoveralldielectrictensorasgivenin Equation.13. K = K 0 + K 1 .13 InEquation.13, K 0 denotesthedielectrictensorundercoldplasmaconditionsand K 1 denotesthedielectrictensorwithtemperatureeectstakenintoaccount.Thesmall parameter isdenedas k 2 k B T=m s 2 ,where k isthemagnitudeofthewavenormalvector. [ Aubryetal. ]denedthedielectrictensorundercoldplasmaconditionsasinEquation .14. K 0 = 2 6 6 6 6 4 1 )]TJ/F22 7.9701 Tf 20.238 4.707 Td [(X 1 )]TJ/F22 7.9701 Tf 6.586 0 Td [(Y 2 jXY 1 )]TJ/F22 7.9701 Tf 6.586 0 Td [(Y 2 0 )]TJ/F22 7.9701 Tf 6.587 0 Td [(jXY 1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(Y 2 1 )]TJ/F22 7.9701 Tf 20.238 4.707 Td [(X 1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(Y 2 0 001 )]TJ/F21 11.9552 Tf 11.955 0 Td [(X 3 7 7 7 7 5 .14 InEquation.14, X and Y arefunctionsofplasmafrequency ps andcyclotronfrequency cs ofspecies s asinEquation.15. X = ps 2 2 .15 Y = cs ThedielectrictensorundercoldplasmaconditionsgiveninEquation2.14canalsobe expressedusingtheStixparametersasinEquation.16. 27
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K 0 = 2 6 6 6 6 4 S )]TJ/F21 11.9552 Tf 9.298 0 Td [(jD 0 jDS 0 00 P 3 7 7 7 7 5 .16 Itisworthnotingthatbothequations.14and.16representthesamedielectric tensorsbutintwodierentnotations.Foreasinessofexpression,inourworktorepresent thecoldplasmadielectrictensorcomponentswewillbeusingthetensorasinEquation .16. Assumingthatthetemperatureeectsweregiventoparticlespecies s ,thedispersion relationofunderwarmplasmaconditionscanbeexpressedasinEquation.17; q s T A 1 s 6 + )]TJ/F21 11.9552 Tf 5.48 9.684 Td [(A 0 + q s T B 1 s 4 + )]TJ/F21 11.9552 Tf 5.479 9.684 Td [(B 0 + q s T C 1 s 2 + C 0 =0.17 with q s T forparticlespecies s denedasinEquation.18,where k B istheBoltzmann's constant, T s isthetemperatureinKelvinofspecies s A 0 ;B 0 and C 0 arethecoecientsof thecoldplasmadispersionrelationcalculatedasinEquation.8,also: q s T = k B T s m s c 2 .18 InthedispersionrelationofwarmplasmaEquation.17,thecoecients A 1 s ;B 1 s and C 1 s aredenedasinEquation.19forplasmaspecies s A 1 s = K 11 s sin 2 + K 33 s cos 2 +2 K 13 s sin cos .19 B 1 s = )]TJ/F27 11.9552 Tf 11.291 13.271 Td [( K 11 s S + K 22 s S +2 jDK 12 s sin 2 )]TJ/F21 11.9552 Tf 387.149 26.969 Td [(K 33 s )]TJ/F21 11.9552 Tf 5.48 9.684 Td [(S + S cos 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(P )]TJ/F21 11.9552 Tf 5.479 9.684 Td [(S + S cos 2 +2sin cos jDK 23 s )]TJ/F21 11.9552 Tf 11.955 0 Td [(SK 13 s C 1 s = K 33 s )]TJ/F15 11.9552 Tf 5.479 9.684 Td [( jD 2 + S 2 + P 2 jDK 12 s + SK 22 s + SK 11 s 28
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Theparameters S P and D aretheStixparametersEquation.5.Theadditional K ijs parametersarethecomponentsofthewarmplasmadielectrictensordenedasin thesetofequationsgiveninEquation.20forspecies s [ Aubryetal. Kulkarniet al. ]; K 11 s = )]TJ/F21 11.9552 Tf 9.298 0 Td [(X 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y 2 3sin 2 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(4 Y 2 + 1+3 Y 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y 2 2 cos 2 .20 K 22 s = )]TJ/F21 11.9552 Tf 9.298 0 Td [(X 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y 2 1+8 Y 2 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(4 Y 2 sin 2 + 1+3 Y 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y 2 2 cos 2 K 33 s = )]TJ/F21 11.9552 Tf 9.299 0 Td [(X 3cos 2 + sin 2 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(Y 2 K 12 s = )]TJ/F21 11.9552 Tf 9.299 0 Td [(K 21 s = jX 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y 2 6sin 2 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(4 Y 2 + 3+ Y 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y 2 2 cos 2 K 23 s = )]TJ/F21 11.9552 Tf 9.298 0 Td [(K 32 s = )]TJ/F21 11.9552 Tf 9.298 0 Td [(j XY )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y 2 2 )]TJ/F15 11.9552 Tf 5.479 9.683 Td [(3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y 2 sin cos K 13 s = K 31 s = 2 X )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y 2 2 sin cos ThedispersionrelationforwarmplasmagiveninEquation.17isasixthorderequation.Onceweconsider 2 asasinglevariableweneedtosolveacubicequation.Inorder tosolvethecubicequationweusetheformulationdevelopedbytheItalianmathematician Cardanoin[ Cardano ]. AccordingtoCardanos'sformulationtherootsofanequationintheformof an 3 + bn 2 + cn + d =0iscanbeexpressedas n 1 = p + w )]TJ/F21 11.9552 Tf 16.661 8.088 Td [(b 3 a .21 n 2 = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(p + w 2 )]TJ/F21 11.9552 Tf 16.661 8.088 Td [(b 3 a + j p 3 p )]TJ/F21 11.9552 Tf 11.955 0 Td [(w 2 n 3 = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(p + w 2 )]TJ/F21 11.9552 Tf 16.661 8.088 Td [(b 3 a )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(j p 3 p )]TJ/F21 11.9552 Tf 11.955 0 Td [(w 2 : 29
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wheretheparameters q;r;p;w aregivenas: q = 3 ac )]TJ/F21 11.9552 Tf 11.955 0 Td [(b 2 9 a 2 .22 r = 9 abc )]TJ/F15 11.9552 Tf 11.955 0 Td [(27 a 2 d )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 b 3 54 a 3 p = 3 q r + p q 3 + r 2 ; w = 3 q r )]TJ/F27 11.9552 Tf 11.956 10.949 Td [(p q 3 + r 2 : Outofthethreerootsofthecubicequations,weneedtoselectthesolutionwhichis alwaysreal.Becauseweconsidertheplasmaiscollisionless,ourresultoftherefractiveindex shouldbearealquantifysignifyingwavepropagationandnotattenuationofanevanescent mode.Thereforeoutofthethreesolutionsweselectthe n 1 solution. 2.5IntroducingTemperatureEectstoMultipleSpecies Inthepublishedworkof[ Aubryetal. ],theformulationwerederivedconsidering onlyelectronsastheplasmaspecies.[ Kulkarnietal. ],consideredintroducingtemperaturetoelectronsand H + ionsbothbutonetypeofspeciesatatime.Inthisworkweextend theworkof[ Kulkarnietal. ]inordertointroducetemperaturetobothelectronsand ionssimultaneously.Withoutthelackofgeneralityweintroducethefollowingparameters forelectronsandionsasinEquation.23andwecalculatethe K ijs forelectronsand ionsseparatelyusingtheplasmafrequencyandcyclotronfrequencyforelectronsandions whileatthermalequilibrium. q e T = k B T e m e c 2 .23 q i T = k B T i m i c 2 IntheaboveEquation.23 m e and m i arethemassesofelectronsandions. T e and T i arethetemperaturesgivenforelectronsandions.Thisgivestheexibilityofsetting 30
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twotemperaturesfordierentspecies.Hencewecalculatedthe q e T and q i T parametersfor electronsandionsseparately.Thenweintroducethosetothedispersionrelationequation givenby X s q T s A 1 s 6 + A 0 + X s q T s B 1 s 4 + B 0 + X s q T s C 1 s 2 + C 0 =0 : .24 IntheEquation.24subscript s referstodierentspeciesinthemagnetosphericplasma. Inthisworkwehaveintroducedtemperatureeectstoelectronsand H + species. 2.6LandauDamping Asthewhistlerwavepropagatesthroughthemagnetosphericplasmaitcaninteractwith highlyenergeticelectronpopulationsquantiedinsocalledsuprathermalelectrondistributions.Typicallythesesuprathermalelectrondistributionsareintheenergyrangeof100 eVto1Mev.Dependingonthespecicsofthesuprathermalelectronuxdistribution,the whistlermodewavepropagatingmighteithergroworattenuate.Theinteractionbetween thesuprathermalelectrondistributionandthewaveintroducesanimaginarycomponentto theindexofrefractiongivenby = ck i .25 TheprocessofLandaudampingcanbeexplainedwithananalogyofsurfersonwater waves.Surfersareanalogoustothehotelectronparticles,andthewaterwavesareanalogous tothewhistlerwavesinteractingwiththem.Assumethateachwaterwavefrontpropagates withthephasevelocity v p .Whenthesurfersmovealongwiththewaterwavefrontwith thesamevelocity v p ,neitherthesurfersnorthewaterwavelooseenergy.Hencethereisno accelerationordeceleration.Similarlywhenthehotenergyparticlespropagateinthesame velocityasthewhistlerwave,thereareisnoaccelerationordecelerationnorwavegrowth ordamping. Whereaslet'sconsiderthesituationwherethesurfersmoveslowerthanthewaterwave, theypaddleintothewave.Hencethesurfersgainenergyfromthewave.Thisresults 31
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areductioninwave'sgravitationalpotentialenergy.Similarlywhenthewhistlerwaves encounterahotelectrondistributionmovingslowerthanthewave,thehotenergyparticles gainenergyfromthewave,causingawaveenergyreduction.Thisisknownasdamping. Thisisthemostcommonsituationencounteredbywhistlerwaves. Whenthesurfersmovefasterthanthewaterwave,theydeceleratethemselvesbybending thesurngtrajectory.Hencetheylooseenergytothewave.Inthissituationthewaterwave getsenergyfromthesurfers.Similarlywhenthewhistlerwavesencounterhotelectron particlesmovingfasterthanthephasevelocityofthewave,particleslooseenergytothe wave.Incrementofwaveenergyinthisprocessiscalledawavegrowth.Foralargenumber ofelectronsquantiedbyadistributionfunction,dampingorgrowthwillbedeterminedby therelativenumberofelectronsmovingfasterorslowerthanthewave.Hencethesignofthe derivativeofthedistributionfunctioninvelocityspace,evaluatedattheresonancevelocity determinesthegrowthvsdamping. From[ Kennel ]and[ Brinca ], canbeexpressedasfollows; = 4 A 0 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(B 0 )]TJ/F21 11.9552 Tf 10.46 9.684 Td [( 2 sin 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(P )]TJ/F19 7.9701 Tf 7.314 1.793 Td [(1 [ R )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 J m )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 + L )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 J m +1 ] 2 G 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2[ S )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 cos 2 S )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(D 2 ] 1 J m G 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 sin cos )]TJ/F19 7.9701 Tf 7.314 1.793 Td [(1 [ R )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 J m )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 + L )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 J m +1 ] G 2 Intheaboveequation A 0 and B 0 arethecoecientsof 4 and 2 inthecoldplasma dispersionrelationand S;P;R;L;D aretheStixparametersasdenedinEquation.5. J m representtherstkindBessel'sfunctionoforder m ofargument denedbelowtogether withotherparametersusedtocalculate inEquation.26.Thehighenergyelectron velocitydistributionisconsideredtobe f v =2 10 11 v )]TJ/F19 7.9701 Tf 6.587 0 Td [(4 m )]TJ/F19 7.9701 Tf 6.586 0 Td [(6 s 3 asin[ Bortnik ]. 32
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= 1 q 1 )]TJ/F27 11.9552 Tf 11.955 9.684 Td [()]TJ/F22 7.9701 Tf 6.675 4.977 Td [(v c 2 .26 p ? = m e v ? = k ? p ? m e ce G 1 = 1 )]TJ/F21 11.9552 Tf 13.15 8.088 Td [(k z v z @f @v ? + k z v ? @f @v z G 2 = J m 1+ m! ce k z @f @v z )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(m! ce v z !v ? @f @v ? Intheaboveequations k ? and k z denotetheperpendicularandparallelcomponentsofthewavenormal, v ? and v z denotetheperpendicularandparallelelectronvelocitycomponents. v representsgroupvelocity.Basedonthewarmplasmacorrectionstothedielectrictensorshownin[ Kulkarnietal. ],weintroducenewvariables S new ;D new ;P new ;R new ;L new ;A new and B new asinEquation.27; S new = S + e K e 11 + i K i 11 .27 D new = D + e K e 12 + i K i 11 P new = P + e K e 33 + i K i 11 R new = S new + D new L new = S new )]TJ/F21 11.9552 Tf 11.955 0 Td [(D new Intheaboveequations K e and K i arethedielectrictensorswithwarmplasma correctionsforelectronsandions.Thevariable isdenedforelectronsandionsinEquation .28; e = q e T 2 .28 i = q i T 2 33
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Wealsointroducetwomorenewvariables A new and B new basedonthewarmplasma dispersionrelationequationasinEquation.29: A new = A 0 + X s q T B 1 .29 B new = B 0 + X s q T C 1 Withallthewarmplasmacorrectionsabovetheequationfor canbeexpressedas follows; = 4 A new 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(B new )]TJ/F21 11.9552 Tf 10.461 9.684 Td [( 2 sin 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(P new )]TJ/F19 7.9701 Tf 7.314 1.793 Td [(1 [ R new )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 J m )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 + L new )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 J m +1 ] 2 G 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2[ S new )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 cos 2 S new )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(D 2 new ] 1 J m G 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 sin cos )]TJ/F19 7.9701 Tf 7.315 1.793 Td [(1 [ R new )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 J m )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 + L new )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 J m +1 ] G 2 0 B @ )]TJ/F19 7.9701 Tf 7.314 1.794 Td [(1 1 1 C A = 2 2 pe 2 !k z P 1 m = R 1 0 dv ? 0 B @ v ? 2 v ? 1 C A R 1 dv z 0 B @ 1 v z 1 C A v z )]TJ/F21 11.9552 Tf 11.955 0 Td [(v z res .30 2.7NumericalRaytracing Raytracingistheprocessofdeterminingthepowerowpathofawave.Numerical raytracingisimplementedbysolvingtheHaselgrove'sequations[ Haselgrove ]listed below,insphericalcoordinates. 34
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dr dt = 1 2 r )]TJ/F21 11.9552 Tf 11.956 0 Td [( @ @ r d' dt = 1 r 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( @ @ d dt = 1 r 2 sin )]TJ/F21 11.9552 Tf 11.955 0 Td [( @ @ d r dt = 1 @ @r + d dt + d dt sin d dt = 1 r 1 @ @' )]TJ/F21 11.9552 Tf 11.955 0 Td [( dr dt + r d dt cos d dt = 1 r sin 1 @ @ )]TJ/F21 11.9552 Tf 11.955 0 Td [( dr dt sin )]TJ/F21 11.9552 Tf 11.955 0 Td [(r d' dt cos .31 Intheaboveequations r isthegeocentricdistance, isthezenithangleand isthe longitude. istherealpartofthecomplexrefractiveindex.Thevariables r and are thecomponentsoftherefractiveindexvectorinthecorrespondingdirectionsof r and And t istheindependentvariableofintegrationwhichisalsothetimeofphasetravelalong theraytrajectoryscaledbythevelocityoflight. Numericalraytracingworkswellinmediumswhicharesmoothlyvaryingnosudden densitydropsandnomodecoupling. Intheabovesetofequationspropertiesofthemediumentersviatherefractiveindex .Thereforeinthisworkitisextremelyimportanttondthemagnitudeanddirectionof refractiveindexvector. ThestartingpointisacoldplasmaraytracingcodedevelopedatStanfordUniversity [ InanandBell ].Intheactualraytracingprocessitisimportanttomakesurethat thedispersionrelationisalwaysequaltozeroasthisistheconditionofpropagatingwave solutions.Oncethewaveispropagatingthroughthemagnetospherethisconditionshould bealwaysenforced. Inordertomakesurethatthedispersionrelationalwaysequaltozerothefollowing dierentialequationapproachisusedinthecurrentlyavailablecoldplasmamodels. 35
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Step1=thedispersionrelation F isdenedasaconservedquantity,anditisafunction ofthewavenumber k relatedtotherefractiveindexaccordingtotheAppletonHartreeequation,frequencyofthewave andthepositionrofthewave.Alongthe trajectoryoftheray F k;!;r =0,dueto F beingaconservedquantity.Inorderto derivearelationshipbetween F andthetime t ,westartbyassumingall k and r arefunctionsofadummyvariable .Hence dr d = r k F dk d = r r F d! d = @F @t Step2=Usingtheconservationof F ,thetotalderivativeofitwithrespectto becomes: dF d = r k F: dk d + @F @! d! d + r r F: dr d + @F @t dt d =0 : Thereforethefollowingrelationshipscanbederivedfromtheaboveequation. dt d = )]TJ/F21 11.9552 Tf 10.494 8.087 Td [(@F @! dr dt = )]TJ 16.243 8.088 Td [(r k F @F=@! dk dt = r r F @F=@! ThesesetofequationsareanalogoustotheSnell'sLawforawavepacketpropagating inaslowvaryingmedium.Thesecompactderivativesareimplementedintheraytracer indiscretestepsusingthepartialderivativeof F ,withrespecttofrequency,position 36
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andthewavenormal. .32 @F @k = F k + k;!;r )]TJ/F21 11.9552 Tf 11.955 0 Td [(F k )]TJ/F15 11.9552 Tf 11.955 0 Td [( k;!;r 2 k @F @! = F k;! + !;r )]TJ/F21 11.9552 Tf 11.955 0 Td [(F k;! )]TJ/F15 11.9552 Tf 11.955 0 Td [( !;r 2 @F @r = F k;!;r + r )]TJ/F21 11.9552 Tf 11.955 0 Td [(F k;!;r )]TJ/F15 11.9552 Tf 11.955 0 Td [( r 2 r Therelationshipgivenby dr dt ,thegroupvelocityofthewave. Step3=Integratethedispersionrelation,usingthefourthorderRungeKuttaformula withrespecttotimeandndanewspacelocation.Forcethedispersionrelationto bezeroatthenewlocationbysolvingthedispersionrelation.Ifthenewlocationis outsidetheresonancecone,reducethestepsizeandrecalculatethelocation. Step4=Repeatthisprocessuntilthemaximumtimestepsarereached. Beforeproceedingontosimulationsofcompletewhistlermodetrajectorieswemake afewimportantcommentsregardingrefractiveindexsurfacesinthecontextoftheresultsof[ Kulkarnietal. ].[ Kulkarnietal. ]usedtheCarpenterandAnderson model[ CarpenterandAnderson ]forbackgrounddensityandexaminedrefractiveindexsurfaceswithnonzerotemperaturesforeitherhydrogenionsorelectrons.Allthework presentedinherewasdonewiththeGlobalCorePlasmasphereModelGCPM[ Gallagher etal. ].IntheGCPMmodeltherearefourtypesofmagnetosphericspecies,namely theelectrons, H + ions, He + ions,and O + .Takingintoaccountthelowermassandhigher densitiesofelectronsand H + ionscomparedtotheotherspecies,inthisworkweconsider onlytheeectsofelectronsandH + ions. WithGCMPmodel,wearecapableofchangingthelocationofplasmapause L pp accordingtoEquation.33. 37
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L pp =5 : 6 )]TJ/F15 11.9552 Tf 11.955 0 Td [(0 : 46 :K p .33 Intheaboveequation L pp isthestartingpointoftheplasmapauseinearthradiiand K p isanindexwhichisusedtochangethelocationoftheplasmapause.Figure2.5shows locationoftheplasmapausewithtwodierent K p values. Figure2.5:Changeofthelocationofplasmapausewiththe k p index.Plasmapauseisthe regionwheretheelectrondensitysuddenlydrops. 38
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CHAPTERIII MODIFICATIONTOTHEREFRACTIVEINDEXSURFACE 3.1EectofIons [ Hines SmithandAngerami ],identiedwithoutthermaleectsinclusionof ionsclosesanotherwiseopenrefractiveindexsurfaceforfrequenciesbelowthelowerhybrid resonance.[ Kulkarnietal. ],havedonetheanalysiswithrespecttothelowerhybrid resonancefrequency.Lowerhybridresonanceatagivenlocationcanbecalculatedusing Equation.1. 1 2 LH = 1 2 ci + 2 pi + 1 ci ce .1 InEquation.1, 2 LH isthelowerhybridresonancefrequency, ci and ce aretheion andelectrongyrofrequenciesrespectivelyand pi istheionplasmafrequency.Figure3.1 showstherefractiveindexsurfaceplotsfora2.3kHzwavelaunchedattheequatorof L =2 withandwithouttheinclusionofions.Thelowerhybridresonanceatthelaunchinglocation is2.57kHz.Hencetheselectedwavefrequencyis270Hzbelowthelowerhybridresonance frequencyofthelaunchinglocation.AsshowninFigure3.1whenonlyelectronswere consideredintherefractiveindexcalculationprocess,therefractiveindexsurfaceremains openforhighlyobliqueangles.Whereaswhenbothelectronsandionsareconsideredin solvingfortherefractiveindex,thesurfaceisbeingclosedforhighlyobliqueangles.Inall theraytracingworkwehaveperformedinthiswork,wehaveconsideredelectronsandalso theeectofmultipletypesofions,namely H + ;He + and O + 3.2InclusionofFiniteElectronorIonTemperature As[ Kulkarnietal. ]observed,whenwavefrequencyincreasesabovethelower hybridresonancefrequency,inclusionofionsdoesnotchangeanotherwiseopenrefractive indexsurface.AsshowninFigure3.4fora3kHzwaveattheequatorat L =2,althoughions areconsideredforthecalculations,refractiveindexsurfaceremainsopenforhighlyoblique 39
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Figure3.1:RefractiveIndexsurfacefor2.3kHz,at L =2 Figure3.2:RefractiveIndexsurfacefor2.3kHz,at L =2,plottedwiththesameaspect ratioforbothaxes. angles.Henceifawavewaslaunchedatahighlyobliqueangleitwillresonatewithinthe resonanceconewithoutshowinganymagnetosphericreections.Thisobservationisalso trueforfrequencieswhichareacoupleofkHzabovethelowerhybridresonancefrequency atalocationasshowninFigure3.5. Asdiscussedabove,closureoftherefractiveindexsurfaceisnecessaryinordertoobserve magnetopshericreections.Withouttheclosureoftherefractiveindexsurface,thewave wouldnothavepropagatingsolutionsinforallwavenormalangles.Thereforeitisnecessary toincludetemperatureeectsintheprocessofrefractiveindexcalculations. Inthisworkwerecreatedtherefractiveindexsurfaceobservationsdoneby[ Kulkarniet al. ].As[ Kulkarnietal. ]observed,ifthewavefrequencyislessthanthelower hybridresonancefrequency,inclusionoftemperaturedoesnotmodifytherefractiveindex 40
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surface.AsshowninFigure3.3,fora2.3kHzwavetherefractiveindexsurfacegenerated withandwithouttemperatureeectsareoverlappingoneachother.Henceweshouldnot seeamodiedraytrajectorywiththeinclusionofnitetemperatureeects. Figure3.4,showstherefractiveindexsurfaceswithandwithouttemperatureeectsfor a3kHzwaveattheequatorof L =2.Asmentionedabove,inclusionoftemperaturecloses anotherwiseopenrefractiveindexsurface.Theselectedfrequencyis430Hzabovethelower hybridresonanceattheequatorof L =2.Inclusionof1eViontemperatureclosesthe refractiveindexsurfacetighterthanthecasewithonlyelectrontemperature.Thereforeas theauthorsof[ Kulkarnietal. ]observed,iontemperatureplaysadominantrolethan theelectrontemperatureatfrequenciesslightlyabovethelowerhybridresonancefrequency. Oncethefrequencyofthewaveismuchhigherthanthelowerhybridresonancefrequency, electrontemperaturebecomesdominantcomparedtotheiontemperature.Thisscenariois showninFigure3.5,forawavewith10kHz,launchedattheequatorof L =2.Inclusion ofa1eVtemperaturetoelectronsproducesatighterrefractiveindexsurfacethanwiththe casewithonlyiontemperature. 3.3InclusionofTemperatureforBothElectronsandIons Inthisworkweincludedtemperatureeectstobothelectronsand H + ionssimultaneously.Oncethetemperatureeectsaregiventobothelectronsandions,thatproduces thetightestrefractiveindexsurfaceforfrequencieshigherthanthelowerhybridresonance. Hencethemagnetosphericreectionsshouldbehastenedwiththatinclusion. AsshowninFigure3.3ifthefrequencyofthewaveislowerthanthelowerhybrid resonance,inclusionoftemperaturedoesnotmodifytherefractiveindexsurface.Hencewe expectnochangesintheraytrajectories,producedwithcoldandwarmplasmaassumptions foreverywavenormalangle. Butasweincreasethefrequencyofthewaveabovethelowerhybridresonance,inclusionoftemperaturetobothspeciestightenstherefractiveindexsurface.Asshownin Figure3.4,therefractiveindexsurfaceproducedwithbothelectronandiontemperature 41
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isapproximately19%reducedthantherefractiveindexsurfaceproducedonlywiththeion temperature. Similarlyat10kHzfrequencyalsothetightestrefractiveindexsurfaceisproducedwhen bothspeciesweregiventemperatureeects,butthatreductionfromtherefractiveindex surfacewithonlyelectrontemperatureisabout0 : 77%asshowninFigure3.5. Hencethemaximumdeviationbetweencoldandwarmplasmaraytrajectoriesisexpectedtobeobservedatfrequenciesslightlyabovethelowerhybridresonanceandathighly obliquewavenormalangles. Forthecompletenesswehavetriedincreasingtheelectronandiontemperatureto4eV. TherefractiveindexsurfaceplotisshowninFigure3.6incomparisonwiththerefractive indexsurfaceplotobtainedwith1eVtemperaturegiventobothspeciesfora3kHzwave attheequatorat L =2. Figure3.3:RefractiveIndexsurfacesfor2.3kHz,attheequatorof L =2. 42
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Figure3.4:RefractiveIndexsurfacesfor3kHz,attheequatorof L =2. Figure3.5:RefractiveIndexsurfacesfor10kHz,attheequatorof L =2. 43
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Figure3.6:RefractiveIndexsurfacesfor3kHz,attheequatorof L =2with1eVtemperatureandwith4eVtemperature 44
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CHAPTERIV MODIFICATIONTOWHISTLERMODERAYTRAJECTORIESFROM FINITEELECTRONANDIONTEMPERATURE 4.1EectofTemperatureonLandauDamping Beforeproceedingintomodicationoftheraytrajectoriesfromthermaleects,itisimportanttorstquantifythemodicationsintroducedtoLandaudampingwiththeinclusion ofniteelectronandiontemperature.ThisisbecauseLandaudampingisthemainsource ofattenuationofwhistlermodewavesanddetermineshowlongawhistlermodewavepacket willhaveanonnegligibleamountofenergy.Inmagnetosphericraytracingitistypicalto denethelifetime"ofarayasthetimeittakesthewaveamplitudetodecreasebyafactor of10dB[ Bortniketal. c].Figure4.1showswhistlermodewavetrajectorieswithand withoutthermaleectsforafrequencyof3.5kHzlaunchedattheequatorat L =2,with aninitialwavenormalangleof88 .Bothtrajectoriesseemtohavesimilarlifetimes.An expandedviewofthepowerattenuationplotisshowninpanelc,whereitcanbeseen thattheattenuationslightlyincreaseswiththeinclusionofnitetemperatureeects.This observationcanbemadewithmostofthefrequenciesatallwavenormalangles.Inallthe workpresentedherewetakepowerattenuationduetoLandaudampingintoconsideration. Andallthetrajectoriespresentedhereweretracedforthedurationofthesignallifetime. Foralltheresultspresentedinthischapterdipolemodelwasusedasthegeomagneticeld model. 4.2LaunchingWavesinDierentDirections Theraytraceriscapableoflaunchingwavesindierentdirections.Theinitialdirection ofawavedependsontheinitialwavenormalangle.Figure4.2shows3kHzwaveslaunched attheequatorof L =2,withthreedierentinitialwavenormalanglesandtheirpower attenuationplotswithandwithouttemperatureeects.InFigure4.2a.i,thewaveswere launchedwithaninitialwavenormalangleof85 .Thisinitialwavenormalangleiswell belowtheresonanceconewavenormalangleof89 : 1 .Thewaveislaunchedawayfromthe 45
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Figure4.1:aTrajectoriesofwhistlermodewaveswithafrequencyof3.5kHz,launched attheequatorof L =2,withaninitialwavenormalangleof88 ,withRedandwithout Blackniteelectronandiontemperature.boriginalpowerattenuationplots,czoomed versionofpowerattenuationplots. EarthtowardstheNorthpole.Whentheinitialwavenormalangleis )]TJ/F15 11.9552 Tf 9.298 0 Td [(85 ,asshownin Figure4.2b.i,thewaveislaunchedtowardstheEarthandtowardstheNorthpole.If thewaveneedstobelaunchedtowardstheSouthpoleoutwardtheEarth,theinitialwave normalangleshouldbesetto95 ,asshowninFigure4.2c.i.Sinceallwavenormalangles arebelowtheresonancecone,powerattenuationwithandwithouttemperatureeectsare thesame. FromtherefractiveindexsurfacespresentedinChapter3,wecanmakethefollowing predictionsabouttheraytrajectories. 1.Forfrequenciesbelowthelowerhybridresonancefrequency,inclusionofniteelectron andiontemperatureshouldnotmodifythewhistlermoderaytrajectories. 2.Whenthefrequencyofthewaveisslightlyabovethelowerhybridresonancefrequency, inclusionoftemperatureeectstobothelectronsandionsshouldmodifytheraytrajectoriesathighlyobliqueangles. 3.Forfrequenciesmuchhigherthanthelowerhybridresonancefrequency,inclusionof 46
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niteelectronandiontemperatureisstillexpectedtomodifytheraytrajectoriesat highlyobliqueangles. 4.3FrequencyDependenceofWaveTrajectories Figure4.3showstheraytrajectoriesfor2.1kHzwaveslaunchedattheequatorof L =2. Herewehaveselectedahighlyobliqueangle,sinceifthereisatrajectorymodicationit shouldbeobservableatahighlyobliqueangle.FromtheLandaudampingplotsshown inFigure4.4,thelifetimesofthewavesundercoldandwarmassumptionsarethesame. Bothraytrajectoriesshowthesamenumberofreections.Asitwasexpectedbasedon therefractiveindexsurfaces,thetrajectoriesobtainedundercoldandwarmbackground plasmaarethesameforfrequenciesbelowthelowerhybridresonancefrequencyevenat highlyobliqueangles. Figure4.5showstheraytrajectoriesfor2.98kHzwaveslaunchedattheequatorof L =2. Thelaunchingangleisagain89 asinthepreviouscase.Landaudampingcurvessuggest thatthedampingunderwarmplasmaassumptionsisslightlyhigherthanthedamping observedwithacoldbackgroundplasma.Thelifetimeofthewhistlermodewavewith acoldbackgroundplasmaisabout0.2secondshigherthanthelifetimeofthewavewith thermaleects.Inthiscasethewavepropagatesalongerdistancewithtemperatureeects comparedtothecasewithnothermaleects.Henceweconsideredthatitwillbeinteresting toseethepropertiesofthegroupvelocityofthosewavesasshowninFigure4.6. Inthepreviouscasewherethefrequencyofthewaveswas2.1kHzthegroupvelocities ofthewavesundercoldandwarmplasmaconditionswerethesameasexpectedFigure 4.4.Butasthefrequencyincreasesabovethelowerhybridresonance,thegroupvelocitiesalsoincreaseswiththeinclusionoftemperatureeects.AsshowninFigure4.6,when thefrequencyofthewaveisabovethelowerhybridresonancefrequency,inclusionoftemperatureincreasesthegroupvelocitycomparedtothecasewithnotemperatureeects. Hencethewavepropagatesfasterwithtemperatureeectscomparedtothecasewithoutthe temperatureeects. 47
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Whenthefrequencyofthewaveismuchhigherthanthelowerhybridresonancefrequency,inclusionoftemperatureintroducespropagationsolutionsathighlyobliqueangles asshowninFigure4.7.Forawaveat7kHzataninitialwavenormalangleof89 ,the refractiveindexsurfaceisopen,withouttemperatureeects.Hencewithnotemperature eectstherewillbenopropagationsolutionsatthisinitialwavenormalangle.Butoncethe temperatureeectsaretakenintoconsideration,itintroducesnewpropagationsolutions, duetotheclosureoftherefractiveindexsurface.FromtheLandaudampingcurveitisobservedthatatsuchhigherfrequenciesandathighlyobliqueangles,wavepowerattenuates withinafractionofasecond.Hencethosewavesmightplayanonimportantroleinactual spacecraftobservations.AsshowninFigure4.8,althoughthewavepropagatesunderwarm plasmaconditionsthegroupvelocityisverysmallcomparedtothepreviouslyconsidered twocases. 4.4DependenceonInitialWaveNormalAngle Intheprevioussectionwediscussedtheeectoftemperatureatdierentfrequencies. Wenoticedthatwhenthefrequencyisbelowthelowerhybridresonancefrequency,inclusion oftemperaturedoesnotmodifytheraytrajectoriesevenathighlyobliqueangles.When thefrequenciesaremuchhigherthanthelowerhybridresonancefrequency,inclusionof temperatureintroducespropagationsolutionsathighlyobliqueangles,butthosewavesare beingattenuatedwithinafractionofasecond.Whenthefrequencyisslightlyhigherthan thelowerhybridresonance,inclusionoftemperaturecausesalongertrajectoryandincreases thegroupvelocityofthewave. Inthissectionwestudytheeectoftemperatureatdierentwavenormalangles.All thewhistlermodewavesweareconsideringinthissectionarelaunchedwithafrequency of3kHz,attheequatorof L =2.FromthezoomedintrajectoriesshowninFigure4.9, thereisanegligibledierencebetweenthetwotrajectories.Theselectedinitialwavenormal angleof60 ,iswellbelowtheresonanceconeoftherefractiveindexsurface.Inotherwords, therefractiveindexvaluesobtainedwithcoldandwarmplasmaconditionsarethesame. 48
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Hencealthoughthewavefrequencyishigherthanthelowerhybridresonancefrequency, ifthewavenormalangleiswellbelowtheresonancecone,thetrajectorieswouldbethe samewithandwithouttemperatureeects.Thisisfurthersupportedbythegroupvelocity plotwhichcontainoverlappinggroupvelocityplots.Thelifetimesofboththewaveswere observedtobe7seconds.Figure4.10showsthegroupvelocitiesofthetwotrajectories. Atthisparticularwavenormalangle,groupvelocitiesobtainedunderbothcoldandwarm plasmaconditionsarethesame. Figure4.11showstheoriginalandzoomedintrajectoriesofa3kHzwavelaunchedat theequatorfor L =2,butnowwithaninitialwavenormalangleof89 .Fora3kHz frequencyattheequatorof L =2,theresonanceconeoftherefractiveindexsurfaceis89 : 1 Hencethisselectedinitialwavenormalangleisalmostontheresonancecone.Inthiscase thelifetimeofbothwaveswasapproximately1secondandunderwarmplasmaconditions thelifetimewasabout0.02secondslowercomparedtothecasewithnotemperatureeects. Thezoomedinversionofthetrajectoriesshowsanadditionalreectionwhenthethermal eectsaretakenintoconsideration.Whichmeansalthoughthelifetimewaslower,distance travelledbythewaveunderwarmplasmaassumptionswaslongerthanthedistancetravelled withnotemperatureeects.Thisobservationcanbewellexplainedwiththegroupvelocity plotshowninFigure4.12.Thegroupvelocitywiththeinclusionoftemperatureeectswas considerablyhigherthanthegroupvelocityobtainedwithnothermaleects.Hencethe wavespropagatefasterwiththeinclusionofniteelectronandiontemperature. Untilthispointalltheinitialwavenormalangleswehaveconsideredwerepositive.As mentionedatthebeginningofthischapter,apositivewavenormalangleindicatesthat thewavewaslaunchedawayfromtheEarth.Forcompleteness,hereweareconsideringan initialwavenormalangleof )]TJ/F15 11.9552 Tf 9.299 0 Td [(89 asshowninFigure4.13.Negativeinitialwavenormal anglesindicatethatthewaveswerelaunchedtowardtheEarth.Basedonthesymmetryof therefractiveindexsurfaceweshouldexpectsimilarwavecharacteristicsasabove.Inthis casealsotheselectedinitialwavenormalangleisabout1 ,belowtheresonancecone.Here 49
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alsowecanmakethesameobservationswemadewiththeinitialwavenormalangleof89 Thetrajectorywithtemperatureeectswasgoingthroughanadditionalmagnetospheric reection.Alsothegroupvelocityunderwarmplasmaconditionsisconsiderably,higher thanthegroupvelocityundercoldplasmaconditions.Lifetimeofthewaveswasthesame asobservedwiththeinitialwavenormalangleof89 .Herealsoalthoughthedampingwas higherunderwarmplasmaconditions,thewavestravelalongerdistancecomparedtothe coldplasmascenario,duetothehighergroupvelocityasshowninFigure4.14. Weneedtobringtheattentionheretoaninterestingobservationaboutthemagnetosphericreectionpoint.[ Kulkarnietal. ]predictedthattheinclusionofthermaleects shouldhastenthemagnetosphericreections.Herewehaveshownthezoomedintrajectories forwaveswithafrequencyof3kHzlaunchedattheequatorof L =2withinitialwavenormal anglesof89 and )]TJ/F15 11.9552 Tf 9.299 0 Td [(89 .Inbothcasesthelifetimeofthewavesisapproximately1second. Asshownaboveduringthelifetimetrajectoriesproducedundercoldplasmaconditionsdo notgothoughamagnetosphericreection.Buthereinthiscaseweranthesimulationfor2 secondsinordertoobservethemagnetosphericreectionpointundercoldplasmaassumptions.FromFigure4.15shownbelow,thepredictionmadein[ Kulkarnietal. ]can beconrmed.Althoughbothwaveswerelaunchedatthesamelocation,trajectoriesunder warmplasmaconditionsencountermagnetosphericreectionsquickerthanthetrajectories undercoldplasmaconditions.Thisobservationcanbemadeatbothinitialwavenormal angles. Forthe L =2launchlocationonlyhighlyobliquewavesinthefrequencyrangeof 2.9 )]TJ/F15 11.9552 Tf 9.298 0 Td [(3.0kHzhavetheirtrajectorymodiedbytheinclusionofnitetemperature.Wetake acloserlookattheconditionsalongthistrajectory.ThetoppanelofFigure4.16shows thedierencebetweenwavefrequencyandthelocallowerhybridresonancefrequencyasthe wavepropagatesawayfromthelaunchinglocationunderwarmplasmaconditions.Thewave islaunchedat L =2withafrequencyof3.0kHzandwithaninitialwavenormalangleof 88 : 9 .ThebottompanelofFigure4.16showsthewavenormalanglealongthistrajectory. 50
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Althoughtheactualwaveattenuates10dBwithin1second,hereweranthesimulation for5secondstoshowthepropagationcharacteristics.Itcanbeseenthatboththewave normalangleandfrequencydierencehaveanoscillatorypatternofthesameperiodbut areinverselyphased.Asthedierencebetweenthewavefrequencyandthelocallower hybridresonancereducesandcrosseszero,thewavenormalangleisincreasingtoachieveit maximumobliquevaluealmost90 .AswasdiscussedinChapter3,thenitetemperature eectschangetherefractiveindexsurfaceonlywhenthewavefrequencyisabovethelower hybridresonancefrequencyandonlyforhighlyobliquewaves.Thejointdynamicsofthe frequencydierenceandthewavenormalangleshowninFigure4.16suggeststhatthe conditionsformaximumeectofnitetemperaturesarelargelyavoidedbythewavethus explainingwhythetrajectorychangesareverysmalleventhoughpartsoftherefractive indexarechangeddramaticallyinthewarmplasmamodel. 4.5ObservationsofWavesLaunchedat L =4 Herewepresenttheobservationsmadewiththewaveslaunchedattheequatorof L = 4.Asmentionedinapreviouschapterforalltheraytracingdoneinthisworkthe k p indexwassetto4.Hencethelocationoftheplasmapauseisat L =3 : 76.Thereforethe launchinglocationweselectedhereisbeyondtheplasmapause.Henceplasmadensitiesat thelaunchinglocationaremuchlowercomparedtothedensitiesattheequatorof L =2. Thelowerhybridresonancefrequencyattheequatorof L =4is300Hz.Anotherinteresting observationatthislaunchinglocationisthatthelifetimeofthewhistlermodewavesisvery lowcomparedtothewaveslaunchedinsidetheplasmasphere. Herealsowestudiedthebehavioroftrajectoriesatdierentfrequenciesatahighly obliquewavenormalangleandataxedfrequencywithdierentinitialwavenormalangles atdierentpositionsoftherefractiveindexsurface. ThetoppanelofFigure4.17,showsthewavetrajectoriesoriginalandzoomedfora frequencyof250Hz,launchedattheequatorof L =4withaninitialwavenormalangleof 89 .Theselectedfrequencyis50Hzlowerthanthelowerhybridresonancefrequencyatthe 51
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launchinglocation.Similartrajectorieswereobtained,withandwithouttemperatureeects, forfrequencieslowerthanthelowerhybridresonancefrequency.Thelifetimeobservedfor bothtrajectorieswas0.25seconds. AsshowninthemiddlepanelofFigure4.17,oncethefrequencyofthewaveishigher thanthelowerhybridresonancefrequencyattheinitiallocation,inclusionoftemperature makesthewavepropagatesfaster.Theselectedfrequencyof375Hzis75Hzhigherthanthe lowerhybridresonancefrequencyoftheequatorof L =4.Henceinclusionofniteelectron andiontemperaturehasincreasedthegroupvelocityofthewave.Inthiscaselifetimeof bothwaveswasobservedtobearound0.1seconds. For1kHzwavesasshowninthebottompanelofFigure4.17,inclusionoftemperature hasintroducedpropagationsolutionswhichdidnotexistundercoldplasmaconditions,at highlyobliqueangles.Thefrequencyof1kHzis700Hzhigherthanthelowerhybrid resonancefrequencyatthelaunchinglocation.Butthewaveswithmuchhigherfrequencies comparedtothelowerhybridresonancefrequency,arebeingattenuatedwithinabout0.001 seconds. Asweobservedintheprevioussectionwiththewaveslaunchedattheequatorof L =2, whenthefrequencyofthewaveisbelowthethelowerhybridresonancefrequency,inclusion oftemperaturedoesnotmodifythetrajectory.Herewemadesimilarobservationswith waveslaunchedattheequatorof L =4.Thetrajectorymodicationswereobservedfor frequenciesabovethelowerhybridresonancefrequency. Nowwekeepthefrequencyofthewavexedat380Hzandchangetheinitialwavenormal angle.AsshowninthetoppanelofFigure4.18,whentheinitialwavenormalangleofthe waveis85 ,whichiswellbelowtheresonanceconeatthislocation,inclusionoftemperature doesnotmodifythetrajectory.Inotherwords,althoughthefrequencyofthewaveswere higherthanthelowerhybridresonancefrequency,iftheinitialwavenormalangleiswell belowtheresonancecone,inclusionofniteelectronandiontemperaturedoesnotmodify thetrajectory.Oncetheinitialwavenormalanglebecomeshighlyoblique89 ,ormuch 52
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closertotheresonanceconeangle,inclusionoftemperatureincreasesthegroupvelocity hencethewavepropagatesalongerdistanceunderwarmplasmaconditionsasshowninthe middlepanelofFigure4.18. SimilarlywhenthewaveswerelaunchedtowardstheEarthwithaninitialwavenormal angleof )]TJ/F15 11.9552 Tf 9.299 0 Td [(89 ,inclusionofthermaleectsmakesthepropagationpathofthewavelonger comparedtothecasewithnothermaleectsasshowninthebottompanelofFigure4.18. 53
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Figure4.2:Whistlermodewaveswithafrequencyof3kHz,launchedattheequatorof L =2,witha.iinitialwavenormalangleof85 ,b.iinitialwavenormalangleof )]TJ/F15 11.9552 Tf 9.298 0 Td [(85 ,c.i initialwavenormalangleof95 ,a.ii,b.ii,c.iishowthepowerattenuationplotsofthe correspondingtrajectorieswithandwithouttemperatureeects.InallthepanelsBlack traceswereproducedwithcoldplasmaconditionsandRedtraceswereproducedwithwarm plasmaconditions. 54
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Figure4.3:Whistlemodewavetrajectorieswithafrequencyof2.1kHzlaunchedatthe equatorof L =2withcoldBlackandwarmRedplasmaconditions.ashowsthe originaltrajectoriesandbshowsthezoomedinversion. Figure4.4:aPowerattenuationduetoLandaudampingforthetrajectoriesshownin Figure4.3.bGroupvelocityalongthetrajectoriesshowninFigure4.3 55
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Figure4.5:Whistlemodewavetrajectorieswithafrequencyof2.98kHzlaunchedatthe equatorof L =2withcoldBlackandwarmRedplasmaconditions.ashowstheoriginal trajectoriesandbshowsthezoomedinversion. Figure4.6:aPowerattenuationduetoLandaudampingforthetrajectoriesshownin Figure4.5.bGroupvelocityalongthetrajectoriesshowninFigure4.5 56
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Figure4.7:Whistlemodewavetrajectorieswithafrequencyof7kHzlaunchedatthe equatorof L =2withcoldBlackandwarmRedplasmaconditions.ashowsthe originaltrajectoriesandbshowsthezoomedinversion. Figure4.8:aPowerattenuationduetoLandaudampingforthetrajectoriesshownin Figure4.7.bGroupvelocityalongthetrajectoriesshowninFigure4.7 57
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Figure4.9:Whistlermodewavetrajectoriesofa3kHzwavelaunchedwithaninitialwave normalangleof60 ,attheequatorof L =2undercoldBlackandwarmRedplasma conditions.aoriginalwaveandbshowsthezoomedinversion. Figure4.10:GroupvelocitiesofthewavetrajectoriesshowninFigure4.9. 58
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Figure4.11:Whistlermodewavetrajectoriesofa3kHzwavelaunchedwithaninitialwave normalangleof89 ,attheequatorof L =2undercoldBlackandwarmRedplasma assumptions.aoriginalwaveandbshowsthezoomedinversion. Figure4.12:GroupvelocitiesofthewavetrajectoriesshowninFigure4.11. 59
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Figure4.13:Whistlermodewavetrajectoriesofa3kHzwavelaunchedwithaninitialwave normalangleof )]TJ/F15 11.9552 Tf 9.298 0 Td [(89 ,attheequatorof L =2undercoldBlackandwarmRedplasma conditions.aoriginalwaveandbshowsthezoomedinversion. Figure4.14:GroupvelocitiesofthewavetrajectoriesshowninFigure4.13. 60
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Figure4.15:Magnetosphericreectionpointsforwaveswithafrequencyof3kHz,launched attheequatorof L =2,withaninitialwavenormalangleof89 ,withRedandwithout Blacktemperatureeects. 61
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Figure4.16:Dierencebetweenwavefrequencyandlocalhybridresonanceasa3kHzwave propagatesunderwarmplasmaconditionswithinitialwavenormalangleof88 : 9 top. Instantaneouswavenormalanglealongthetrajectorybottom. 62
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Figure4.17:Whistlermodewaveslaunchedwithaninitialwavenormalangleof89 ,atthe equatorof L =4,withfrequencies250Hz,375Hzand1kHz.Lefthandpanelsshowthe originaltrajectoriesandpanelsontheRighthandsideshowthezoomedinversions.All trajectoriesinBlackwereproducedwithcoldplasmaassumptionsandalltrajectoriesinRed wereproducedwithwarmplasmaassumptions. 63
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Figure4.18:Whistlermodewavetrajectoriesfora380Hzwavelaunchedattheequator of L =4,withaninitialwavenormalangles85 ; 89 and )]TJ/F15 11.9552 Tf 9.298 0 Td [(89 .AlltrajectoriesinBlack wereproducedwithcoldplasmaassumptionsandalltrajectoriesinRedwereproducedwith warmplasmaassumptions.Lefthandpanelsshowtheoriginaltrajectoriesandpanelson theRighthandsideshowthezoomedinversions. 64
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CHAPTERV COMPARISONWITHVANALLENPROBESPACECRAFT OBSERVATIONS 5.1ChorusandHiss Untilthispointwepresentedthecharacteristicsofwhistlermodewavesingeneral.In thischapteritisworthintroducingtwospecictypesofwhistlermodewaves;chorusand hiss.Chorusandhissbotharewhistlermodewavesandthenamesareduetothesoundsthey makewhenpassedthroughaspeaker.Choruswavesexistsintwofrequencybands.Given theequatorialelectroncyclotronfrequencyas f ce ,lowerchorusbandisaround0 : 34 f ce and theupperchorusfrequencybandisaround0 : 53 f ce .Thecharacteristicwhichdistinguishes chorusfromhissisthefrequencyandtimecoherenceobservedinchorusthatcannotbe observedwithhiss.Forhiss,thewaveenergyisdistributedwithinallfrequencycomponents inthebandkHz3kHz.Bothchorusandhisswavesareassumedtobegeneratedfrom cyclotronresonantwaveparticleinteractionsinthemagnetosphere. Thehighestchorusenergyconcentrationisobservedaround L =5andthehighesthiss energyconcentrationisobservedaround L =2.Inotherwordsthehighwavepowerobserved outsidetheplasmapauseisconsideredtothechorusenergyandthewavepowerobserved insidetheplasmasphereismainlyduetothehisspower.Figure5.1showsthespectrograms ofchorusandhiss. 5.2DependenceonMagneticLocalTime Sincechorusandhisswavesarecreatedduetospacedynamics,originationregionsof hissandchoruschangeswiththeMagneticLocalTimeorMLT.At MLT =12,themagnetic meridianisfacingthesun.Henceitisadaytimeobservationnoon.Whereasat MLT =00, thesunisattheoppositesideofthemagneticmeridian.Thereforeitismidnightinmagnetic localtime.Figure5.2showstheoriginationlocationsofhissandchoruswithrespecttothe magneticlocaltime. 65
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Figure5.1:Originationlocationsandspectrogramsofchorusandhisswaves.Image source:[ Bortniketal. ] 5.3VanAllenProbeSpacecraftObservations TheVanAllenProbesarethespacecraftslaunchedbyNationalAtmosphericandSpace AgencyNASAinordertomonitorthecharacteristicsofEarth'sradiationbeltsortheVan AllenBelts[ NASA ].Figure5.3showsthetrajectoryofVanAllenProbespacecrafts. Figure5.4showsthenormalizedpowerplotobservedbytheNASAVanAllenProbespacecraftatMagneticLocalTimeMLT06.Thepowerwasnormalizedtothemaximumofeach frequency.Duringtheobservationperiodtheplasmapausewasobservedtobeinbetween L =3and4.InFigure5.4,allmeasurementswereindicatedwithrespecttotheplasma pause.NegativedistancesarethemeasurementstakentowardstheEarthandpositivemeasurementsaremeasuredoutwardtheEarth.Allthemeasurementsweretakenintheabsence ofsuddenmagneticdisturbancessuchassolarares,henceduringmagneticquiteconditions [ Malaspinaetel ]. 66
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Figure5.2:RegionsofchorusandhissenergyconnementswithrespecttotheMagnetic LocalTime.Imagesource:vlf.stanford.edu/DanielGoldenThesis Figure5.3:Artist'sillustratonoftheEarth'sradiationbeltsandthetrajectoryofVanAllen Probespacecrafts.Thetwinspacecraftsareshownhere.Imagesource:AndyKale,University ofAlberta 5.4SimulatedWavePowerwithLandauDamping InordertocomparesimulationresultswiththeVanAllenprobeobservations,hereweare launchingmultiplerays.SincetheVanAllenProbeobservationsweredoneatthedawntime, wehavealsosetthesimulationtimeto MLT =6.Inordertoberealisticwiththeactual magnetosphericconditions,wehaveusedtheactualmeasuredelectronandiontemperatures at MLT =6from[ Decreauetal. ].[ Decreauetal. ]observedtheelectronand iontemperaturesatdierentmagneticlocaltimesandatdierentaltitudesusingsatellites. Basedon[ Decreauetal. ]measurementswehaveusedtheelectrontemperatureas 26,500K.3eVandtheiontemperatureas11,600KeV.Allthewaveswerelaunched 67
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Figure5.4:aWaveenergymeasuredbytheVanAllenProbespacecrafts.bSamewave energyasshownina,normalizedtothemaximumateachfrequency. attheequatorof L =5since L =5isconsideredastheoriginationlocationofchoruswaves. Theplasmapauselocationwassetat L =3 : 76,whichisequivalenttotheactualobserved conditions.Thewaveswerelaunchedwithinafrequencyrangeof100Hzto3.5kHz.Waves witheachfrequencywerelaunchedwitharangeofinitialwavenormalanglesfrom )]TJ/F15 11.9552 Tf 9.299 0 Td [(70 to 20 ,withincrementsof5 .Itisworthrecallingherethatnegativeinitialwavenormalangle indicatethatthewavewaslaunchedtowardstheEarth.Allwaveswerelaunchedwithequal power.Dipolemodelwasusedasthegeomagneticeld.Theoriginatinglocationandthe wavenormalangledistributionofthewaveswereselectedaccordingto[ Bortniketal. ]. Ifthetemperaturecontributingtheparallelpropagationvelocityandtheperpendicular propagationvelocityaredierentitiscalledatemperatureanisotropy.Eveninthepresence ofasignicantanisotropy,themainsourceofpowerdampingisLandaudampingwhichwas explainedinChapter2.Landaudampingisnotaectedbythetemperatureanisotropy. HenceinthiscasealsoinordertocalculatethepowerattenuationduetoLandaudamping, weareusingtheisotropichighenergyelectrondistributionthatweusedinChapter2,which is f v =2 10 11 v )]TJ/F19 7.9701 Tf 6.587 0 Td [(4 m )]TJ/F19 7.9701 Tf 6.586 0 Td [(6 s 3 [ Bortnik ].Allthewavesweretraceduntilthepowerof thewavewasattenuated10dBfromtheinitialpower.Figure5.5showsthewhistlermode 68
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wavepowernormalizedtothemaximumateachfrequencyinacoldbackgroundplasma. AndFigure5.6showsthewhistlermodewavepowerwiththewarmplasmaconditions, normalizedtothemaximumateachfrequency.Similarobservationscanbemadewithboth gures,butsincethewarmplasmaconditionsintroduceadditionalpropagatingsolutions, whichareotherwisenotpossiblewithcoldplasmaassumptions,therearemoredatapoints observedunderwarmplasmaconditions. Figure5.5:Normalizedwavepowerwiththewavegrowthanalysisundercoldbackground plasmascenario.Wavepowerisnormalizedtothemaximumateachfrequency. 5.5WaveGrowthAnalysis Whistlerwavesencounterwavegrowthwhenthereisatemperatureanisotropyinthe parallelandperpendiculardirections.Inthissimulationweconsider,anisotropicbackground plasmadistributionandahotenergyelectrondistributionwithalowanisotropyofthe dawnsideontheMagnetosphere.Themodelisdescribedbelow[ Hikishimaetal. Harid ]. f u jj ;u ? = n h 3 = 2 U th jj U 2 th ? exp )]TJ/F21 11.9552 Tf 9.298 0 Td [(u 2 jj 2 U 2 th jj : 1 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( exp )]TJ/F21 11.9552 Tf 9.298 0 Td [(u 2 ? 2 U 2 th ? )]TJ/F21 11.9552 Tf 11.955 0 Td [(exp )]TJ/F21 11.9552 Tf 9.298 0 Td [(u 2 ? 2 U 2 th ? 69
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Figure5.6:Normalizedwavepowerwiththewavegrowthanalysisunderwarmbackground plasmascenario.Wavepowerisnormalizedtothemaximumateachfrequency. Thevelocitydistributionfunctionoftheisotropicbackgroundplasmaisconsidered Maxwellian.Andthevelocitydistributionofthehotanisotropicplasmadistributionfunction isconsideredmodiedMaxwellianasgivenbyabove.Itisdenedintheparallelandperpendicularmomentumspace u jj ;u ? .TheparametersofthemodiedMaxwelliandistribution, aredenedasfollowsand c isthespeedoflight. = 1 )]TJ/F27 11.9552 Tf 11.955 9.683 Td [()]TJ/F21 11.9552 Tf 5.479 9.683 Td [(v 2 jj + v 2 ? =c 2 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 = 2 .1 u jj = v jj .2 u ? = v ? .3 InmodiedMaxwelliandistribution, n h isthehotenergeticelectrondensitywhichis takenas2 10 4 percubicmeter. U th jj and U th ? areparallelandperpendicularcomponents ofthermalmomentumperunitmass,whichareconsideredtobe0 : 2 c and0 : 33 c respectively c isthespeedoflight.AttheequatorLossconeisthetermusedfortheminimumpitch angleforwhichelectronscanbetrappedintheEarth'sradiationbelts.Theterm inthe abovemodiedMaxwelliandistributionistheparameterwhichdeterminestherangeofthe losscone.Increasing ,makesthelossconelarger.Forthissimulationthe valuewasset 70
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to0.3,basedonthemeasurementstakenattheequator. V R = c 1 )]TJ/F21 11.9552 Tf 13.151 8.087 Td [(! ce .4 Equation.4,expressestheresonantconditionbetweenthewhistlerspropagatingparalleltotheEarth'smagneticeldsandthehotenergyelectrons.Theparameters and aregivenasfollows: 2 = ce )]TJ/F21 11.9552 Tf 11.956 0 Td [(! 2 pe .5 2 =1 )]TJ/F21 11.9552 Tf 17.422 8.087 Td [(! 2 c 2 k 2 = 1 1+ 2 .6 i = ce 1 )]TJ/F21 11.9552 Tf 16.983 8.088 Td [(! ce 2 V R 0 [ A V R 0 )]TJ/F21 11.9552 Tf 11.955 0 Td [(A c ].7 Intheaboveequations pe istheplasmafrequencyand ce istheelectroncyclotron frequency.Lineargrowthrateofaparallelpropagatingwhistlermodewaveisgivenin Equation.7where: V R 0 =2 ce )]TJ/F21 11.9552 Tf 11.955 0 Td [(! k Z 1 0 v ? Fdv ? j v jj = V R 0 .8 A V R 0 = R 1 0 v jj @F @v ? )]TJ/F21 11.9552 Tf 11.955 0 Td [(v ? @F @v jj v 2 ? v jj dv ? 2 R 1 0 v ? Fdv ? v jj = V R 0 .9 A c = 1 ce =! )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 .10 theterm representsthetotalnumberofelectronsatresonancewiththepropagating whistlerwave,whichisapositivenumber.Also A V R 0 isthetemperatureanisotropyof hotenergyelectronsatresonancewiththepropagatingwhistlerwave,and A c isthecritical temperatureanisotropy. 71
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Figure5.7,showsthegrowthofwavepowerassumingacoldisotropicbackgroundplasma. Figure5.8showsthewavepowergrowthassumingawarmisotropicbackgroundplasma. Allthewaveswerelaunchedattheequatorof L =5,andtheinitialwavenormalangles werevariedfrom )]TJ/F15 11.9552 Tf 9.298 0 Td [(70 to20 ,withincrementsof5 .Bothelectronsandionsweregiven temperaturesbasedontheobservationsmadeby[ Decreauetal. ].Allwaveswere launchedwithequalpower.Dipolemodelwasusedasthegeomagneticeld.Thelocation oftheplasmapausewassetat L =3 : 76.Figures5.7and5.8,showsimilarpowerow propagationpattern.InFigure5.4b,thehighestpowerconcentrationmovesclosertothe Earthasthefrequencyincreases.Asimilarpatternisobservedinbothgures5.7and5.8. ButthepowerowpathismoreconnedtotheregionsimilartoobservationmadeinFigure 5.4,whenthebackgroundplasmaisconsideredaswarm. Figure5.7:Normalizedwavepowerwiththewavegrowthanalysisundercoldbackground plasmascenario.Wavepowerisnormalizedtothemaximumateachfrequency. Thissuggeststheimportanceofwarmplasmacorrectionsinbeingabletomatchraytracingsimulationstodata.Inparticulareventhoughnitetemperaturehadasmalleect onindividualraytrajectories,thethecumulativeeectonmanyraysissignicant. 72
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Figure5.8:Normalizedwavepowerwiththewavegrowthanalysisunderwarmbackground plasmascenario.Wavepowerisnormalizedtothemaximumateachfrequency. 73
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CHAPTERVI SUMMARYANDCONCLUSIONS Atthebeginningofthisthesiswepresentedthewarmplasmacorrectionsthatneedto beintroducedtotheraytracingplatform.Incorporatingwarmplasmacorrectionsincreases thecomplexityoftheexistingraytracingequations.Wehavesuccessfullyintroducedthese eectstothecoldplasmaraytracerandperformedthepredictionsofraytrajectories.Inthis thesisweintroducedamodiedversionoftheLandaudampingcalculations,whichtakes warmplasmacorrectionsintoaccount.Calculationoftherefractiveindexisofutmostimportanceinraytracing,sincethemediumpropertiesenterstheraytracingequationsviathe refractiveindexateachpoint.Previousresearchershaveshownthatevenwithouttemperatureeects,ionsarenecessarytocausemagneticreections.Forfrequenciesbelowthelower hybridresonance,inclusionofionsclosesanotherwiseopenrefractiveindexsurfacecreated byonlytakingelectronsintoaccount.Forfrequenciesabovethelowerhybridresonance frequency,inclusionofionsdoesnotclosetherefractiveindexsurfaceunlessthermalcorrectionsaretakenintoaccount.Accordingtothepublishedliteratureforfrequenciesinthe closevicinityofthelowerhybridresonancefrequency,inclusionofiontemperatureproduces atighterrefractiveindexsurfacecomparedtotherefractiveindexsurfacecreatedconsideringonlyelectrontemperature.Butforfrequenciesmuchhigherthanthelowerhybrid resonancefrequency,inclusionofelectrontemperatureproducesatighterrefractiveindex surfacecomparedtotheiontemperature.Forfrequenciesbelowthelowerhybridresonance, inclusionoftemperaturedoesnotmodifytherefractiveindexsurface. Inthisthesis,weextendedtherefractiveindexsurfaceanalysissuchthatwecanintroducethermalcorrectionstobothelectronsandionssimultaneously.Thisapproachincreases theaccuracyofthemodel,sinceforanaturalplasmainthermalequilibriumelectronsand ionscanpossestwodierenttemperaturedistributions.Themainobservationwere,for frequenciesbelowthelowerhybridresonancefrequency,inclusionoftemperaturetoboth speciesdidnotmodifytherefractiveindexsurface.Butforfrequenciesabovethelower 74
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hybridresonancefrequency,inclusionoftemperaturetobothelectronsandionsproduced thetightestrefractiveindexsurface. InChapter4,wepresentedthemodicationstothewhistlermoderaytrajectoriesintroducedbythethermalcorrections.Wehaveobservedthewaveslaunchedattwodierent L shells,2and4.Similarobservationsweremadeinbothcases.Propagationtrajectoryof whistlermodewavesweremodiedinresponsetothermaleects,onlyatfrequenciesabove thelowerhybridresonanceandathighlyobliqueinitialwavenormalangles.Nomodicationswereobservedwhenthefrequencyofthewaveswerelowerthanthelowerhybrid resonanceandthelaunchingwavenormalanglesarewellbelowtheresonancecone.AnotherobservationwasthattheLandaudampingincreaseswiththeinclusionoftemperature eects,butthatincrementofdampingisnegligible. Chapter5presentedthecomparisonofsimulationresultswithVanAllenprobespacecraftobservations.WepresentedthesimulatedwavepowerbyconsideringonlyLandau dampingintoaccountandbyconsideringlinearwavegrowth.Ineachcaseweperformedthe analysisbyassumingacoldbackgroundplasmaandawarmbackgroundplasma.ThenormalizedpowerdistributionobtainedonlyconsideringLandaudampingwasthesameforboth caseswithcoldbackgroundplasmaandwarmbackgroundplasma.Butaclearmatchwas observablebetweensimulationsandobservationswhenwavepowergrowthwasconsidered, inawarmbackgroundplasma. 6.1FutureWork ForthesimulationsinChapter5,wehaveconsideredasinglesourcelocation L =5 andallwaveswerelaunchedwithequalpower.Theaccuracyofthesimulationresultscan beimprovedifmultiplesourcesweretobeconsidered,withdierentinitialpower.Also AppendixDpresentsanoncausallteringmethodthatcanbeappliedforwhistlermode raytracingtodetectoutliersthatcanleadtosimulationinaccuracies.Implementationofthis noncausallteringmethodonthemainraytracerwithanincreasedrobustnesswillimprove theaccuracyoftheraytraceringeneral. 75
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REFERENCES [ AbelandThorne a]Abel,B.,andR.M.Thornea,Electronscatteringlossin theEarthsinnermagnetosphere:1.Dominantphysicalprocesses, J.Geophys.Res.,104, 23852396 ,doi:10.1029/97JA02919.,Correction, J.Geophys.Res.,104,4627 ,1999. [ AbelandThorne b]Abel,B.,andR.M.Thorneb,Electronscatteringlossin theEarthsinnermagnetosphere:2.Sensitivitytomodelparameters, J.Geophys.Res., 103,23972408 ,Correction, J.Geophys.Res.,104,4627 ,1999. [ Aubryetal. ]Aubry,M.P,J.Bitoun,andPh.Gra,PropagationandGroup VelocityinaWarmMagnetoplasma, RadioSci,5,635645 1970 [ Bakeretal. ]Baker,D.N.,A.N.Jaynes,V.C.Hoxie,R.M.Thorne,J.C.Foster,X.Li, J.F.Fennell,J.R.Wygant,S.G.Kanekal,P.J.Erickson,W.Kurth,W.Li,Q.Ma,Q. Schiller,L.Blum,D.M.Malaspina,A.Gerrard,L.J.Lanzerotti,AnimpenetrablebarriertoultrarelativisticelectronsintheVanAllenradiationbelt,Nature,515, 531534,doi:10.1038/nature13956.2014 [ Barkhausen ]Barkhausen,1919.PhysikZeit.,20,401. [ Belletal. ]BellT.F.,U.S.Inan,J.Bortnik,andJ.D.Scudder,TheLandaudamping ofmagnetosphericallyreectedwhistlerswithintheplasmasphere, Geophys.Res.Lett., 29,15 ,2002. [ Bortnik ]Bortnik.J,PrecipitationofRadiationBeltElectronsbyLightninggeneratedManetosphericallyReectingWhistlerWaves, DoctoralThesis,StanfordUniversity ,January2005 [ Bortniketal. a]BortnikJ.,U.S.Inan,andT.F.Bell,TemporalsignaturesofradiationbeltelectronprecipitationinducedbylightninggeneratedMRwhistlerwaves.Part I:Methodology, J.Geophys.Res.,111,A02204 ,doi:10.1029/2005JA011182,2006. [ Bortniketal. b]BortnikJ.,U.S.Inan,andT.F.Bell,TemporalsignaturesofradiationbeltelectronprecipitationinducedbylightninggeneratedMRwhistlerwaves.Part II:Globalsignatures, J.Geophys.Res.,111,A02205 ,doi:10.1029/2005JA011398,2006. [ Bortniketal. c]BortnikJ.,U.S.Inan,andT.F.Bell,Landaudampingandresultantunidirectionalpropagationofchoruswaves Geophys.Res.letters,33,L03102 doi:10.1029/2005GL024553,2006. [ Bortniketal. a]Bortnik,J.,R.M.Thorne,N.P.Meredth,andO.Santolik,Ray tracingofpenetratingchorusanditsimplicationsfortheradiationbelts, Geophys.Res. Lett.,34,15 ,CiteIDL15109,doi:10.1029/2007GL030040,2007. [ Bortniketal. b]Bortnik,J.,andR.M.Thorne,ThedualroleofELF/VLF choruswavesintheaccelerationandprecipitationofradiationbeltelectrons, J.Atmos. Sol.Terr.Phys.,69,378386 ,doi:10.1016/j.jastp.2006.05.030.2007 76
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[ Bortniketal. ]Bortnik,J.,R.M.ThorneandN.P.Meredith8,Theunexpectedoriginofplasmaspherichissfromdiscretechorusemissions Nature,452,6266 doi:10.1038/nature06741.2008 [ Bitounetal. ]Bitoun,J,Ph.GraandM.PAubry,,Raytracinginwarmmagnetoplasmaandapplicationstotopsideresonances, RadioSci,5,13411349 1970 [ Brinca ]Kennel,A.L.,OntheStabilityofObliquelyPropagatingWhistlers, J.Geophys.Res.,77,19 1972 [ CairoandLefeuvre ]Cairo,L.andLefeuvre,F..Localizationofsourcesof ELF/VLFhissobservedinthemagnetosphere:Threedimensionalraytracing. J.Geophys.Res.:SpacePhysics ,91A4,43524364. [ Cardano ]Cardano.J,ArsMagna:GreatArt, Book,publisherunknown ,1545 [ CarpenterandAnderson ]Carpenter,D.L.,andR.R.Anderson,An ISEE/Whistlermodelofequatorialelectrondensityinthemagnetosphere, Geophys. Res.,97,1097 1992. [ CarpenterandLemaire ]Carpenter,D.L.,andJ.Lemaire,ThePlasmasphere BoundaryLayer, AnnalesGeophysicae22:42914298 2004. [ Chenetal. ]Chen,L.,J.Bortnik,R.M.Thorne,R.B.Horne,andV.K.Jordanova .ThreedimensionalraytracingofVLFwavesinamagnetosphericenvironment containingaplasmasphericplume. Geophys.Res.Lett. ,36. [ ChumandSantolik ]Chum,J.,andO.Santolk,Propagationofwhistlermode chorustolowaltitides:Divergentraytrajectoriesandgroundacessibility, Ann.Geophys. ,23,37273738. [ Decreauetal. ]Decreau,P.M.E.,C.BeghinandM.Parret,Globalcharacteristicsofthecoldplasmaintheequatorialplasmasphereregionasdeducedfromthe GEOS1mutialimpedanceprobe. J.Geophys.ResA87,695 1982 [ Eckersley ]Eckersley,T.L.,.Musicalatmospherics, Nature,13 1041051935. [ Gallagheretal. ]Gallagher,D.L.,P.D.Craven,andR.H.Comfort,GlobalCore PlasmaModel, J.Geophys.Res.,105,1881918833 ,2000. [ Goldenetal. ]Golden,D.I,M.Spasojevic,F.R.Foust,N.G.Lehtinen,N.P.Meredith,andU.S.Inan,Roleoftheplasmapauseindictatingthegroundaccessibilityof ELF/VLFchorus, J.Geophys.Res.,115,A11211 ,doi:10.1029/2010JA015955,2010 [ Golden ]GoldenD.I,SourceVariationandGroundAccessibilityofMegnetosphericMidlatitudeELF/VLFChorusandHiss, DoctoralThesis,StanfordUniversity ,January2005 77
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[ GolkowskiandInan ]Golkowski,M.,andU.S.Inan,MultistationobservationsofELF/VLFwhistlermodechorus, J.Geophys.Res.,113 ,A08210, doi:10.1029/2007JA012977. [ GreenandInan ]GreenJ.L,U.S.Inan,LightningEects onSpacePlasmasandApplications, PlasmaPhysicsApplied URL: https://www.researchgate.net/publication/255611268 [ Harid ]Harid.V,,CoherentInteractionsBetweenWhistlerModeWavesand EnergeticElectronsintheEarth'sRadiationBelts, DoctoralThesis,StanfordUniversity March2015 [ Haselgrove ]Haselgrove,J.,RayTheoryandaNewMethodforRayTracing, inPhysicsoftheIonosphere, PhysicsoftheIonosphereConference,Cambridge,London, 355364 ,1954 [ HelliwellandGehrels ]Helliwell,R.A.,andE.Gehrels,Observationsof magnetoionicductpropagationusingmanmadesignalsofverylowfrequency, Proc. IRE.,46 7857871958. [ Helliwell ]Helliwell,R.A.,Whistlersandrelatedionosphericphenomena, StanfordUniversityPress. [ Hikishimaetal. ]Hikishima.M,S.Yagitani,Y.OmuraandI.Nagano,Full particlesimulationofwhistlermoderisingchorusemissionsinthemagnetosphere, J. Geophys.Res.,114,A01203 ,doi:10.1029/2008JA013625,2009 [ Hines ]Hines,C.O.,Heavyioneectsinaudiofrequencyradiopropagation, J. Atmos.andTerrest.Phys.,11 ,3642. [ HuangandGoertz ]Huang,C.Y.andGoertz,C.K..Raytracingstudiesand pathintegratedgainsofELFunductedwhistlermodewavesintheEarth'smagnetosphere. JournalofGeophysicalResearch:SpacePhysics ,88A8,61816187. [ InanandBell ]Inan,U.S.,andT.F.Bell,TheplasmapauseasaVLFwaveguide, J.Geophys.Res.,82,28192827 ,doi:10.1029/JA082i019p02819 [ Inanetal. ]Inan,U.S.,T.F.Bell,J.Bortnik,andJ.M.Albert,Controlled precipitationofradiationbeltelectrons, Geophys.Res.,108A5,1186 2003 [ InanandGolkowski ]Inan,U.S,M.Golkowski011,PrinciplesofPlasmaPhysics forEngineersandScientists,CambridgeUniversityPress [ KazakosandPapantoni ]Kazakos.D,PapantoniKazakos,P., D etectionand Estimation,ComputerSciencePress,1989 [ KennelandPetscheck ]Kennel,C.F.,andH.E.Petschek,Limitonstably trappedparticleuxes, J.Geophys.Res.,71,128 1966 78
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[ Kennel ]Kennel,C.F.,LowFrequencyWhistlerMode, ThePhysicsofFluids, 9,11 1966 [ Kimura ]Kimura,I.,EectsofIonsonWhistlerModeRayTracing, Radio Science,1,269283 1966. [ Kletzingetal ]Kletzing,C.A.,etal.,TheElectricandMagneticFieldInstrumentSuiteandIntegratedScienceEMFISISonRBSP, SpaceSci.Rev.,179,10 127181 ,doi:10.1007/s1121401399936. [ Kulkarnietal. ]Kulkarni.P,U.S.InanandT.F.Bell,Whistlermodeillumination oftheplasmasphericresonantcavityviainsituinjectionofELF/VLFwaves, Geophys. Res.,A10215 ,2015 [ Kulkarnietal. ]Kulkarni.P,M.Golkowski,U.S.InanandT.F.Bell,Theeectof electronandiontemperatureontherefractiveindexsurfaceof110kHzwhistlermode wavesintheinnermagnetosphere, Geophys.Res.SpacePhysics,120,581591 ,2015 [ Kumaretal. ]Kumar.A,A.KumarandD.P.Singh,PlasmapauseandIncrease inWhistlerModeWavesGrowth, AppliedChemistry,1,3,1519 ,2012 [ Lietal. ]Li,W.,R.M.Thorne,J.Bortnik,Y.Y.Shprits,Y.Nishimura,V.Angelopoulos,C.Chaston,O.LeContel,andJ.W.Bonnell,Typicalpropertiesofrisingandfallingtonechoruswaves, Geophys.Res.Lett.,38,14 ,CiteIDL14103.doi: 10.1029/2011GL047925,2011. [ Lyonsetal. ]Lyons,L.R.,R.M.Thorne,andC.F.Kennel,Pitchangle diusionofradiationbeltelectronswithintheplasmasphere, J.Geophys.Res.,77,3455 1972 [ MaedaandKimura ]Maeda,K.,andI.Kimura,ATheoreticalInvestigation onthePropagationPathoftheWhistlingAtmospherics, Rept.IonosphereRes.10, 105123 1956. [ Malaspinaetel ]Malaspina,DavidM.andJaynes,AllisonN.andBoul,Coryand Bortnik,JacobandThaller,ScottA.andErgun,RobertE.andKletzing,CraigA.and Wygant,JohnR.,Thedistributionofplasmaspherichisswavepowerwithrespect toplasmapauselocation, J.Geophys.Res.,43,7878 ,doi:10.1002/2016GL069982 [ Moldwinetal ]Moldwin,M.B.,L.Downward,H.K.Rassoul,R.Amin,andR.R. Anderson,Anewmodelofthelocationoftheplasmapause:CRRESresults, J. Geophys.Res., 107A11,1339,doi:10.1029/2001JA009211 [ NASA ]NationalAeronauticsandSpaceAdminstration,RBSPMission Overview. www.nasa.gov .Retrieved3March2012. [ OlsenandPtzer ]Olson,W.P.,andK.A.Ptzer,Aquantitative modelofthemagnetosphericmagneticeld, J.Geophys.Res.,79,37393748 doi:10.1029/JA079i025p03739 79
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[ OlsenandPtzer ]Olson,W.P.,andK.A.Ptzer,Magnetosphericmagnetic eldmodeling, Ann.Sci.Rep.F4462075C0033 ,AirForceO.ofSci.Res.,Arlington, Va [ Omuraetal. ]Omura,Y.,Y.Katoh,andD.Summers,Theoryandsimulationofthegenerationofwhistlermodechorus, J.Geophys.Res.,113,A04223 doi:10.1029/2007JA012622,2008 [ SerbuandMaier ]Serbu,G.P.,andE.J.R.Maier,ObservationsfromOGO 5ofthethermaliondensityandtemperaturewithinthemagnetosphere, J.Geophys. Res.,78,6102 ,1970. [ Shpritsetal. ]Shprits,Y.Y.,D.A.Subbotin,N.P.Meredith,andS.R.Elkington ,ReviewofmodelingoflossesandsourcesofrelativisticelectronsintheouterradiationbeltII:Localaccelerationandloss, JournalofAtmosphericandSolarTerrestrial Physics,70,1694,1713 doi:10.1016/j.jastp.2008.06.014. [ SmithandAngerami ]Smith,R.L.,andJ.A.Angerami,MagnetosphericpropertiesdeducedfromOGO1observationsofductedandnonductedwhistlers, J.Geophys. Res.,,73 [ Stix ]Stix,H.,WavesinPlasmas,SpingerVerlag,NewYork. [ Storey ]Storey,L.R.O.,Aninvestigationofwhistlingatmospherics, Phil. Trans.Roy.Soc.,246 113141,1953. [ T horneandHorne94]Thorne,R.M.,andR.B.Horne94,LandauDampingofMagnetosphericallyRefectedWhistlers, JournalofGeophysicalResearchSpacePhysics,99 A9,1724917258 ,1994. 80
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APPENDIXA.DispersionRelationforFreeSpace Maxwells'sequationsforanyregioncanbeexpressedasfollows; r : E = v A.1 r : B = 0 r E = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(@ B @ t r B = J + @ E @ t Intheaboveequations E ; B ; J representtheelectriceldintensity,magneticeldintensityandthecurrentdensity. v isthevolumechargedensityand ; respectivelyrepresent thepermeabilityandpermittivityofthemedium. Forfreespacevolumechargedensity v iszeroandthecurrentdensityalsogoestozero. Thepermittivityandpermeabilitycanbeexpressedintermsoffreespacepermittivityand permeability 0 and 0 .HencetheaboveMaxwell'sequationscanberewritteninasfollows forachargefreeregionsuchasfreespaceasfollows; r : E = 0 A.2 r : B = 0 r E = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(@ B @ t r B = 0 0 @ E @ t Usingthefollowingvectoridentityforanyvector A r r A = r r : A )222(r 2 A A.3 Let'sapplytheabovevectoridentityontheelectriceldintensityvector; 81
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r r E = r r : E )222(r 2 E Forachargefreeregion r : E iszero.Hence r r E = r 2 E FromMaxwell'sthirdequation r E = )]TJ/F22 7.9701 Tf 10.494 4.707 Td [(@ B @t .Therefore r r E = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(@ r B @t r 2 E = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(@ r B @t r 2 E = @ @t 0 0 @ E @t r 2 E = 0 0 @ 2 E @t 2 Giventhattheelectriceldintensityintheformof E = e j t )]TJ/F37 7.9701 Tf 6.586 0 Td [(kz ,where istheangular frequency,kisthewavenumberandzisthedirectionofpropagation. Hence; r 2 E = )]TJ/F21 11.9552 Tf 9.299 0 Td [( 0 0 2 E r 2 E = @ 2 @z 2 e j !t )]TJ/F37 7.9701 Tf 6.587 0 Td [(k z = )]TJ/F21 11.9552 Tf 9.299 0 Td [(k 2 E )]TJ/F21 11.9552 Tf 5.48 9.683 Td [(k 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 0 0 2 E =0 Foranontrivialsolutionelectriceldintensity E cannotbezero.Therefore; k 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 0 0 2 =0A.4 TherelationshipgiveninEquationA.6iscalledthedispersionrelationshipforfree space. 82
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Given c isthespeedoflight3 10 8 ms )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 ,thefollowingrelationshipholds; c = 1 p 0 0 A.5 Hencethedispersionrelationcanberewrittenasfollows; c 2 k 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(! 2 =0A.6 83
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APPENDIXB.DispersionRelationinaNonMagnetizedPlasma Inthissectionwepresentthedispersionrelationofnonmagnetizedplasma.Theterm nonmagnetizedindicatesthatthebackgroundmagneticeldiszero.Beforeproceeding itisworthmentioningtheplasmafrequency p ,whichisthefrequencythattheparticles rearrangethemselveswhendisturbedwithanexternaldisturbance.Let'sassumethatthe plasmaiscold,hencethetemperatureoftheplasmaisat0K.Ifweconsideronlythemotion ofelectronstheplasmafrequencycanbedenedasfollows; pe = N e q e 2 m e 0 Intheaboveequation N e istheelectrondensity, q e isthechargeofanelectron, m e is theelectronmassand 0 isthepermittivityoffreespace. J = N e q e u e B.1 EquationB.1isthecurrentdensity J ,whichistherateofchangeofcharge N e q e times thevelocity u e .Whentheplasmaisdisturbedwithanexternalelectriceldwithintensity E thecurrentdensitycreatedduetotheexternalelectriceldcanbewrittenas; J = N e q e 2 E m e j! B.2 Duetothisexternalcurrentdensitypresent,theeectivepermittivityofthemedium eff changestothefollowing; eff = 0 1 )]TJ/F21 11.9552 Tf 17.364 8.088 Td [(N e q e 2 0 m e 2 = 0 1 )]TJ/F21 11.9552 Tf 13.15 9.168 Td [(! 2 pe 2 B.3 84
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r : E = 0 B.4 r : B = 0 r E = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(@ B 1 @ t r B 1 = @ E @ t InEquationsB.4representtheMaxwell'sequationsforanonmagnetizedplasma.Here wewanttoemphasizethatthatmagneticeldindicatedby B 1 isthemagneticeldcreated duetotheelectriceld E .Foranelectriceldintensity E intheformof e !t )]TJ/F37 7.9701 Tf 6.586 0 Td [(k : z Maxwell's equationscanberewrittenasfollows; r : E = 0 B.5 r : B = 0 r E = )]TJ/F16 11.9552 Tf 9.299 0 Td [(j B 1 r B 1 = j 0 e E ApplyingthesamevectoridentitythatweappliedinAppendix1,wecanderivethe followingrelationship r 2 E = 0 eff 2 E B.6 Hencethedispersionrelationbecomes; k 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 0 eff 2 =0B.7 or k 2 c 2 = 1 )]TJ/F21 11.9552 Tf 13.151 9.168 Td [(! 2 pe 2 B.8 85
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APPENDIXC.DispersionRelationinaColdMagnetizedPlasma WestartwiththeMaxwell'sequationsfortimeharmonicuniformplanewaves; k B = j 0 J )]TJ/F21 11.9552 Tf 11.955 0 Td [(! 0 0 E k E = B k E = j 0 k B =0 alongwiththemomentumequation mn 0 j! u = qN 0 E + u B : Assuminganinnitecold,collisionlessandhomogeneousplasma,bydenitiionwehave: p E = J j! 0 + E Intheaboveequation p isthedielectrictensorintheplasma. Usingtheaboverelationsthefollowingrelationshipscanbederived; J j! 0 = N 0 q e j! 0 q e j!m e E + u B J j! 0 = )]TJ/F27 11.9552 Tf 11.291 16.857 Td [( pe 2 2 E + ce 2 J z 2 Theaboveequationwasderivedassumingthattheambientmagneticeldisorientedin thezdirection.Byseparatingthecomponentsinthex,yandzdirections, 0 B B B B @ ? )]TJ/F21 11.9552 Tf 9.298 0 Td [(j x 0 j x ? 0 00 jj 1 C C C C A 0 B B B B @ E x E y E z 1 C C C C A = p E C.1 Usingthedenitionfortherefractiveindex,theabovedielectrictensorcanbewritten asfollows. 86
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0 B B B B @ ? )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 sin 2 )]TJ/F21 11.9552 Tf 9.299 0 Td [(j x 2 sin cos j x ? )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 0 2 sin cos 0 jj )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 sin 2 1 C C C C A 0 B B B B @ E x E y E z 1 C C C C A =0C.2 C.3 ? =1 )]TJ/F21 11.9552 Tf 28.751 9.168 Td [(! 2 pe 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(! ce 2 x = ce 2 pe 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(! ce 2 jj =1 )]TJ/F21 11.9552 Tf 13.15 9.168 Td [(! 2 pe 2 ForanontrivialsolutionthedeterminantoftheaboveEquationC.2shouldbeequal tozero.Hencewegetthedispersionrelationofacoldmagnetizedplasmaas: 2 = k 2 c 2 2 =1 )]TJ/F22 7.9701 Tf 123.164 15.646 Td [(! 2 pe 2 1 )]TJ/F22 7.9701 Tf 17.435 5.699 Td [(! 2 ce sin 2 2 2 )]TJ/F22 7.9701 Tf 6.586 0 Td [(! 2 pe 2 ce sin 2 2 2 )]TJ/F22 7.9701 Tf 6.587 0 Td [(! 2 pe 2 + 2 ce 2 sin 2 # 1 = 2 C.4 87
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APPENDIXD.NoncausalFiltering Herewepresentanoncausallteringmethodthatcanbeappliedtoeliminateoutliers occurringinraytracing.Thismethodwasrstintroducein[ KazakosandPapantoni ]. Intheraytracerthemaximumerrorisdenedasthedierencebetweentwoconsecutive refractiveindexvalues.Thisparametercanbesetbytheuser.Iftheusersetsahigherror value,theraytracerresultsoutliers.Thereforeitisimportanttondthenominalerror boundtoavoidtheoccurrenceofoutliers. Method ForGaussiandensityfunction f 0 s and f 0 N ,weselectsomeprobabilityofoutlieroccurrence N ; 1andsomenitenonnegativeintegerm.Let f ::::::X )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 ;X 0 ;X 1 ::::: g and f ::::::W )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 ;W 0 ;W 1 ::::: g ,denotethesequencesofrandomvariablesthataregeneratedbythe aboveGaussiandensityfunctions f 0 s and f 0 N respectively.Givensomeinteger k andsome nonnegativeinteger n ,let N 2 n +1 ;k and M 2 n +1 ;k respectivelydenotetheautocovariancematrices E f W k + n k )]TJ/F22 7.9701 Tf 6.587 0 Td [(n )]TJ/F21 11.9552 Tf 5.479 9.684 Td [(W k + n k )]TJ/F22 7.9701 Tf 6.587 0 Td [(n j f 0 N g and E f X k + n k )]TJ/F22 7.9701 Tf 6.587 0 Td [(n )]TJ/F21 11.9552 Tf 5.479 9.684 Td [(X k + n k )]TJ/F22 7.9701 Tf 6.586 0 Td [(n j f 0 s g .Let a T 2 n +1 ;k denotethe n +1 th row ofthematrix M 2 n +1 ;k 2 n +1 ;k = M 2 n +1 ;k + N 2 n +1 ;k D.1 Andlet g 0 kl )]TJ/F21 11.9552 Tf 5.479 9.683 Td [(x k )]TJ/F22 7.9701 Tf 6.586 0 Td [(l k )]TJ/F22 7.9701 Tf 6.586 0 Td [(n ;x k + n k + l ; n l denotetheoptimalmeansquaredinterpolationoperation attheGaussiandensityfunction f 0 s forthedatums x k ,given x k )]TJ/F22 7.9701 Tf 6.587 0 Td [(l k )]TJ/F22 7.9701 Tf 6.587 0 Td [(n and x k + n k + l .Letusthen denethesets f d k;n;j ; k )]TJ/F21 11.9552 Tf 12.004 0 Td [(n j k )]TJ/F21 11.9552 Tf 12.004 0 Td [(l;k + l j k + n g and f b k;n;j ; k )]TJ/F21 11.9552 Tf 12.004 0 Td [(n j k + n g ofthecoecientsasfollows,where 2 n +1 ;k isassumednonsingular. f d k;n;j : g 0 kl )]TJ/F21 11.9552 Tf 5.48 9.684 Td [(x k )]TJ/F22 7.9701 Tf 6.587 0 Td [(l k )]TJ/F22 7.9701 Tf 6.587 0 Td [(n ;x k + n k + l = k )]TJ/F22 7.9701 Tf 6.586 0 Td [(l X j = k )]TJ/F22 7.9701 Tf 6.586 0 Td [(n d k;n;j x j + k + n X j = k + l d k;n;j x j D.2 [ b k;n;k )]TJ/F22 7.9701 Tf 6.586 0 Td [(n ;:::::::::::;b k;n;k + n ]= a T 2 n +1 ;k )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 2 n +1 ;k D.3 88
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Letusnowdene g s n x = 8 > < > : x if j x j n n sgnxotherwise D.4 FigureD.1:GraphicalillustrationofthesmoothingprocessgiveninEquationD.4,and thetruncationconstant n Where c = n a T 2 n +1 ;k )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 2 n +1 ;k a 2 n +1 ;k )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 = 2 ,suchthat c + c )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 c =2 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 1+ )]TJ/F21 11.9552 Tf 11.956 0 Td [( N )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 D.5 Let x s k;n denotestheestimateofthesignaldatum x k ,fromtheobservationvector y k + n k )]TJ/F22 7.9701 Tf 6.586 0 Td [(n Thentheestimate x s k;n isdesignedas; x s k;n = 8 > < > : g s n )]TJ/F21 11.9552 Tf 5.479 9.684 Td [(a T 2 n +1 ;k )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 2 n +1 ;k y k + n k )]TJ/F22 7.9701 Tf 6.587 0 Td [(n if n m g 0 kl )]TJ/F21 11.9552 Tf 5.479 9.684 Td [(x k )]TJ/F22 7.9701 Tf 6.586 0 Td [(l k )]TJ/F22 7.9701 Tf 6.586 0 Td [(n ;x k + n k + l n>m D.6 where x i j = h x i j;m :::::::::::::;x s j;m i ; i>j Letusdene r s n n = a T 2 n +1 ;k )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 2 n +1 ;k a 2 n +1 ;k Then r s n n representsavariancegaininestimatingthesignaldatum x k fromtheobservationvector y k + n k )]TJ/F22 7.9701 Tf 6.587 0 Td [(n ,atthezeromeanGaussiannoisedensity. Therefore r s n n ismonotonicallynondecreasingwith n .Given N ,thesamemonotonicitycharacterizesthetruncationconstant n ,whosemaximumvalue 1 equalsto clim n !1 [ r s n n ] 1 = 2 ,wherecisthesolutiontotheEquationD.5. 89
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APPENDIXE.Glossary E =totalelectriceld B =totalmagneticeld v =volumechargedensity 0 =permeabilityoffreespace 0 =permittivityoffreespace =angularfrequency k ,k=wavenormalvector,wavenumber c=speedoflight pe =electronplasmafrequency ce =electroncyclotronfrequency pe =ionplasmafrequency ce =ioncyclotronfrequency LH =lowerhybridresonancefreqency N e =electrondensity q e =chargeofanelectron m e =massofanelectron B 0 =magnitudeofthegeomagneticeld k B =Boltzmann'sconstant T=absolutetemperature t=time N s =densityofspeciess q s =chargeofanelementfromspeciess =magnitudeoftherefractiveindex r=geocentricdistance =zenithangle =longitude r ; ; p hi =componentsoftherefractiveindexinr, and directions m s =massofanelementfromspeciess J =currentdensity u =particlevelocity =wavenormalangle S,D,P,R,L=Stixparameters m=modenumber J m =rstkindBessel'sfunctionoforderm v=groupvelotiy f=hotelectrondistribution v x ;v z =XandZcomponentsofgroupvelocity v jj ;v ? =parallelandperpendicularcomponentsofgroupvelocity k x ;k z =XandZcomponentsofthewavenormalvector 90

