VLF and ULF Waves Associated with Magnetospheric Substorms

VLF and ULF Waves Associated with Magnetospheric Substorms

Andrew B. Collier

PhD Thesis

Department of Space and Plasma Physics School of Electrical Engineering

Royal Institute of Technology Stockholm, Sweden

May 2006

Andrew B. Collier

VLF and ULF Waves Associated with Magnetospheric Substorms PhD Thesis.

Department of Space and Plasma Physics, School of Electrical Engineering, Royal Institute of Technology, Stockholm, Sweden.

May 2006.

Abstract

A magnetospheric substorm is manifested in a variety of phenomena observed both in space and on the ground. Two electromagnetic signatures are the Substorm Chorus Event (SCE) and Pi2 pulsations.

The SCE is a Very Low Frequency (VLF) radio phenomenon observed on the ground after the onset of the substorm expansion phase. It consists of a band of VLF chorus with rising upper and lower cutoff frequencies. These emissions are thought to result from Doppler-shifted cyclotron resonance between whistler mode waves and energetic electrons which drift into an observer’s field of view from an injection site around midnight. The ascending frequency of the emission envelope has been attributed to the combined effects of energy dispersion due to gradient and curvature drifts and the modification of the resonance conditions resulting from the radial component of the E × B drift. Two numerical models have been developed which simulate the production of a SCE. One accounts for both radial and azimuthal electron drifts but treats the wave-particle interaction in an approximate fashion, while the other retains only the azimuthal drift but rigorously calculates both the electron anisotropy and the wave growth rate. Results from the latter model indicate that the injected electron population should have an enhanced high-energy tail in order to produce a realistic SCE.

Pi2 are damped Ultra Low Frequency (ULF) pulsations with periods between 40 and 150 s.

The impulsive metamorphosis of the nightside inner magnetosphere during the onset of the substorm expansion phase is accompanied by a broad spectrum of magnetohydrodynamic (MHD) waves. Over a limited range of local times around midnight these waves excite field line resonances (FLRs) on field lines connected with the auroral zone. Compressional waves propagate into the inner magnetosphere, where they generate cavity mode resonances. The uniform frequency of Pi2 pulsations at middle and low latitudes is a consequence of these cavity modes. A number of Pi2 events were identified at times when the Cluster constellation was located in the nightside inner magnetosphere. Electric and magnetic field data from Cluster were used to establish the existence of both cavity and field line resonances during these events. The associated Poynting flux indicated negligible radial or field-aligned energy flow but an appreciable azimuthal flux directed away from midnight.

i

Acknowledgements

I am indebted to a few people who have make this work both possible and pleasurable: my su- pervisor, Professor Arthur R. W. Hughes, for stimulation, guidance, friendship and for sending me to some awesome places; my co-supervisor, Professor Lars G. Blomberg, for his enthusi- astic patronage, helpful advice and being an impeccable host; the various purveyors of data for their indispensable raw produce and generous practical support; Dr P. R. Sutcliffe for his constructive criticism and suggestions for further reading; Professor A. D. M. Walker for a number of highly enlightening discussions; the folk down the corridor, Drs Erhard Mravlag and Judy Stephenson, who have always been good company; Ann Nolte, who cheerfully and efficiently organised flights, hotels, purchase requisitions, and so many other perplexing things;

Struan Cockcroft for being great company and an outstanding cocoa mentor; Charles Forman, who must be credited with finally propelling me into the twenty-first century; Tommy Eriks- son, who directed me out of some tight corners and dead ends; the staff of the mechanical and electronic workshop for their support and assistance; my parents for their perennial en- couragement, patience and faith in me; Storm and Emma for their tolerance, stimulation and being the best damn caching partners; and, finally, my running mates: the miles we put in together have kept me sane.

A few of you may not have the patience or inclination to read all of this, so here is something for you:

1 9 6 4

8 5 3 2

2 7 5

1 5 9 3 2

5 2 8

5 8 2 4

8 1 5 6

iii

Contents

1 Introduction 1

1.1 VLF Phenomena . . . . 1

1.2 ULF Phenomena . . . . 1

1.3 Outline . . . . 3

2 The Magnetosphere 4 2.1 Structure . . . . 4

2.2 Magnetospheric Substorms . . . . 5

2.2.1 Particle Injections . . . . 7

2.2.2 Magnetic Variations . . . . 10

3 VLF Waves 14 3.1 The Whistler Mode . . . . 14

3.2 Whistlers . . . . 16

3.3 Doppler-Shifted Cyclotron Resonance . . . . 16

3.4 Chorus . . . . 18

3.5 Substorm Chorus Events . . . . 20

4 ULF Waves 21 4.1 Magnetohydrodynamics . . . . 21

4.1.1 Magnetosonic Waves . . . . 24

4.1.2 Shear Alfv´en Waves . . . . 25

4.1.3 Poynting Flux . . . . 25

4.2 ULF Pulsations . . . . 26

4.2.1 Classification . . . . 26

4.2.2 MHD Waves in a Box Magnetosphere . . . . 27

4.2.3 MHD Waves in Dipole Coordinates . . . . 27

4.2.4 Field Line Resonance . . . . 30

4.2.5 Cavity Resonance . . . . 37

4.2.6 Effects of the Ionosphere . . . . 44

4.3 Pi2 Pulsations . . . . 49

4.3.1 Observations . . . . 49

4.3.2 Theories . . . . 51

5 Instrumentation 53 5.1 Cluster . . . . 53

5.1.1 FGM . . . . 55

5.1.2 EFW . . . . 55

5.2 Polar . . . . 57

5.3 Terrestrial Magnetometers . . . . 57

v

6.2 Paper 2 . . . . 62

6.3 Paper 3 . . . . 64

6.4 Paper 4 . . . . 66

6.5 Conclusion . . . . 75

A Coordinate Systems 80 A.1 Geophysical Coordinate Systems . . . . 80

A.2 Dipole Coordinates . . . . 80

A.3 Mean Field-Aligned Coordinates . . . . 82

B Spectral Techniques 83 B.1 The Fourier Transform . . . . 83

B.2 The Discrete Fourier Transform . . . . 85

B.3 Filtering . . . . 87

B.4 Complex Demodulation . . . . 87

C Polarisation Parameters 92

Bibliography 96

vi

List of Symbols

f frequency

ω angular frequency

γ growth rate

k wave normal vector

θ wave normal angle

m azimuthal wave number n refractive index

R reflection coefficient

Π Poynting vector

Ω cyclotron frequency ω p plasma frequency Λ normalised frequency

W kinetic energy

α pitch angle

L McIlwain’s parameter λ magnetic latitude r geocentric distance

R E mean radius of the Earth (6371 km) ν, φ, µ dipole coordinates

ρ mass density

N number density

p pressure

v velocity

ξ displacement

σ conductivity

Σ conductance

B static magnetic flux density

b perturbation magnetic flux density E electric field strength

j current density

vii

List of Figures

1.1 SCEs recorded at SANAE–IV. . . . 2

1.2 Pi2 observations in day and night hemispheres. . . . . 3

2.1 The terrestrial magnetosphere. . . . 5

2.2 Locations of ACE and WIND with corresponding IMF B z data. . . . 6

2.3 Polar UVI data. . . . 8

2.4 SOPA geosynchronous electron flux data. . . . 9

2.5 Model explaining geosynchronous flux dropouts. . . . 10

2.6 Schematic of SCW with associated magnetic perturbations. . . . 11

2.7 Substorm magnetic bays at selected observatories. . . . 12

2.8 AU and AL indices from the IMAGE magnetometer network. . . . 13

3.1 Whistler ray direction as a function of wave normal angle. . . . 15

3.2 Whistler mode coupling between the atmosphere and the ionosphere. . . . 16

3.3 Spectrograms of whistlers. . . . 17

3.4 Spectrograms of chorus. . . . 19

4.1 Coupling between torsional and compressional modes. . . . 29

4.2 Phase relationships for a fundamental toroidal standing wave. . . . 32

4.3 Phase relationships for a second harmonic toroidal standing wave. . . . 33

4.4 Particle density, Alfv´en speed, turning point and FLR frequencies versus L. . . 35

4.5 Magnetic and electric field perturbations in a fundamental cavity mode. . . . . 38

4.6 Field line configuration in a fundamental cavity mode. . . . 39

4.7 Phase relationships for a fundamental radially transverse standing wave. . . . 40

4.8 Phase relationships for a second harmonic radially transverse standing wave. . 41

4.9 Dependence of the turning point location on azimuthal wavenumber. . . . . . 42

4.10 Boundaries of the cavities formed within the magnetosphere. . . . 44

4.11 Polarisation of obliquely incident shear Alfv´en and fast mode waves. . . . 45

4.12 Ionospheric components of a shear Alfv´en wave. . . . 47

5.1 Location of Cluster at perigee. . . . 54

5.2 Removal of background magnetic field variation. . . . 56

5.3 Stations in the IMAGE magnetometer network. . . . 58

6.1 Simulated evolution of a drifting electron population. . . . 60

6.2 Comparison of simulated SCEs with different injection regions. . . . 62

6.3 Lower frequency cutoff caused by upper cutoff in W _k . . . . 63

6.4 Annualised lightning flash rate density over southern Africa. . . . 65

6.5 Residuals of FGM data after removing the background trend. . . . . 68

6.6 Magnetic footprint of the Cluster reference spacecraft. . . . 70

6.7 Orbit of the Polar satellite in terms of L and MLT. . . . 72

6.8 Plasma density profiles from Polar EFI. . . . 72

6.9 Filtered EFW and FGM waveforms for 10 May 2001. . . . 73

ix

6.12 Phase of b µ relative to E φ on 10 May 2001. . . . 76

6.13 Phase of b _ν relative to E _φ on 10 May 2001. . . . 76

6.14 Phase of b φ relative to E ν on 10 May 2001. . . . 77

6.15 Polarisation parameters as a function of frequency on 10 May 2001. . . . 77

6.16 Polarisation parameters as a function of time on 10 May 2001. . . . 78

6.17 Poynting flux on 10 May 2001. . . . 78

A.1 Definition of mean field-aligned coordinates. . . . 82

B.1 Filter characteristics for the Pi2 band. . . . 87

B.2 Illustration of instantaneous PSDs and CPSD using complex demodulation. . . 89

B.3 Envelope and instantaneous frequency from the analytical signal. . . . . 91

C.1 Sense of ellipticity and azimuth. . . . 93

C.2 Relationship between phase and azimuth. . . . 94

x

List of Papers

The following papers form the basis of this thesis:

1. Collier, A. B. and Hughes, A. R. W.: Modelling and Analysis of Substorm-Related Chorus Events, Advances in Space Research, 34(8), 1819–1823, 2004a.

2. Collier, A. B. and Hughes, A. R. W.: Modelling substorm chorus events in terms of dispersive azimuthal drift, Annales Geophysicae, 22(12), 4311–4327, 2004b.

3. Collier, A. B., Hughes, A. R. W., Lichtenberger, J., and Steinbach, P.: Seasonal and diurnal variation of lightning activity over southern Africa and correlation with European whistler observations, Annales Geophysicae, 24(2), 529–542, 2006b.

4. Collier, A. B., Hughes, A. R. W., Blomberg, L. G., and Sutcliffe, P. R.: Evidence of standing waves during a Pi2 pulsation event observed on Cluster, Annales Geophysicae, Submitted for publication, 2006a.

These papers are referred to below as Paper 1, Paper 2, Paper 3 and Paper 4 respectively.

Contributions were also made to several other papers which do not conform to the topic of this thesis but were published during the course of its execution [Collier and Hughes, 2002;

Neubert et al., 2005; Marshall et al., 2005; Sundberg et al., 2005; Collier et al., 2006c].

xi

1 Introduction

There’s no sense in being precise when you don’t even know what you’re talking about.

John von Neumann

1.1 VLF Phenomena

Naturally occurring Very Low Frequency (VLF) radio waves, originating either in the magne- tosphere or from terrestrial sources, take on a variety of forms such as whistlers, hiss, chorus, discrete and periodic emissions [Helliwell, 1965]. These emerge as a consequence of the proper- ties of electromagnetic waves propagating through the magnetospheric and ionospheric plasma [Walker, 1993]. The analysis of these phenomena has led to a deeper understanding of the plasma environment persisting in near-Earth space.

VLF chorus emissions are routinely observed both on the ground and by satellites in the magnetosphere. The Substorm Chorus Event (SCE), which consists of a band of VLF chorus emissions with rising upper and lower cutoff frequencies, is correlated with the onset of the substorm expansion phase [Smith et al., 1996, 1999]. Examples of SCEs recorded at SANAE–

IV (71.7 ^◦ S 2.8 ^◦ W GEO, L = 4.36) are presented in Figure 1.1. SCEs are typically observed in the midnight-dawn quadrant and have duration ∼10 min to a few hours. These events are thought to arise from the amplification of whistler mode waves by cyclotron resonance with energetic electrons injected into the inner magnetosphere around midnight.

1.2 ULF Phenomena

Fluctuations of the Earth’s magnetic field occur at all levels of geomagnetic activity, and may be due to large scale currents or hydromagnetic waves. The perturbations associated with a geomagnetic substorm may be broadly divided in two categories: the substorm magnetic bay, a prolonged disturbance of the magnetic field with a time scale of tens of minutes, caused by currents flowing along field lines and through the auroral ionosphere, and Ultra Low Frequency (ULF) pulsations with periods of tens of seconds, which are initiated by the dipolarisation of the Earth’s magnetic field and propagate through the magnetosphere as hydromagnetic waves.

The latter class are known as Pi2 pulsations.

The characteristics of Pi2 pulsations observed at high latitudes, where they are confined to a limited range of local times around midnight, are quite distinct from those at middle and low latitudes, where they are found at all local times. Figure 1.2 illustrates Pi2 pulsations observed simultaneously at two low latitude stations, Hermanus (34.4 ^◦ S 19.2 ^◦ E GEO, L = 1.83) and Kakioka (36.2 ^◦ N 140.2 ^◦ E GEO, L = 1.32), located near midnight (02:46 LT) and noon (10:50 LT) respectively. The discovery of dayside Pi2 was only relatively recent, and has led to the understanding that they are caused by cavity mode resonances which are probably confined within the plasmasphere.

1

0 2 4 6 8 10

frequency [kHz]

08:00 09:00 10:00 11:00 12:00 UT

(a) 22 January 2002

0 1 2 3 4 5

frequency [kHz]

03:00 04:00 UT

(b) 21 February 2002

0 1 2 3 4 5

frequency [kHz]

05:00 06:00 UT

(c) 6 March 2002

0 2 4 6 8 10

frequency [kHz]

05:00 05:30 UT

(d) 23 June 2002

0 1 2 3 4 5

frequency [kHz]

05:00 06:00 07:00 08:00 UT

(e) 18 November 2002

0 2 4 6 8 10

frequency [kHz]

06:00 07:00 08:00 09:00 10:00 UT

(f) 26 February 2003

Figure 1.1: SCEs recorded at SANAE–IV. After 09:00 UTC the sampling interval changes

from 1-in-5 to 1-in-15. In (a) data for 08:35 UTC are unavailable.

3

00:30 00:45 01:00 01:15 01:30 01:45 02:00 02:15 02:30

UT -0.9

-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

nT

(a) Hermanus

00:30 00:45 01:00 01:15 01:30 01:45 02:00 02:15 02:30

UT -1.6

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

nT

(b) Kakioka

Figure 1.2: A Pi2 event on 27 March 2001 observed simultaneously in the H-component at two stations separated in local time by 8 h.

The Cluster constellation is an ideal platform for the investigation of ULF pulsations as the electric and magnetic field instruments provide data which enable the detailed characterisation of the waves. Furthermore, observations on multiple spacecraft facilitate the estimation of wavelength and phase velocity.

1.3 Outline

This thesis is concerned with substorm signatures in the VLF and ULF ranges, specifically

SCEs and Pi2 pulsations. In Chapter 2 a brief overview of the magnetosphere and a description

of the magnetospheric substorm are presented. In Chapter 3 various aspects of VLF waves in

the magnetosphere are discussed, a description of wave-particle interactions is given and the

mechanisms which result in a SCE are outlined. Chapter 4 surveys selected topics relating to

ULF waves in the magnetosphere. Chapter 5 describes some of the instruments from which

the data used in this thesis were derived. Finally, in Chapter 6, the papers which form the

basis of the thesis are summarised.

2.1 Structure

The configuration of the terrestrial magnetosphere is indicated schematically in Figure 2.1.

The Earth’s magnetic field presents an obstacle to the supersonic flow of the solar wind. A bow shock exists sunward of the Earth where the flow is slowed, heated and diverted around the planet. Since the solar wind plasma is essentially collisionless, the physical processes associated with the shock are mediated by electromagnetic forces. The region of space behind the bow shock is known as the magnetosheath. The plasma density is higher in the magnetosheath than in the solar wind because the flow rate is reduced.

The dynamic pressure of the solar wind confines the Earth’s magnetic field to form the magnetospheric cavity, which is bounded by the magnetopause. The magnetosphere is elon- gated in the anti-solar direction to form a tail consisting of two lobes containing tenuous plasma (N . 0.1 cm ⁻³ ) and oppositely directed magnetic fields. The lobes are separated by the plasma sheet, which is populated by relatively hot (∼1 keV) plasma. While the magnetic field in the lobes maps down to the polar caps, the plasma sheet magnetic field is connected to the auroral zones. A cross-tail current flows from dawn to dusk across the plasma sheet. The Chapman-Ferraro current flows along the magnetopause. Closer to the Earth, in a region of higher plasma density, the ring current flows in a westerly direction. Field-aligned Birkeland currents flow to and from the magnetosphere and are closed through the ionosphere.

The dynamics of charged particles in the magnetosphere are determined principally by the magnetic field configuration, which gives rise to gradient and curvature drifts. Electric fields, due to the rotation of the Earth and the interaction of the Interplanetary Magnetic Field (IMF) with the Earth’s magnetic field, also produce drift motions. The combined action of these drifts partitions the magnetospheric plasma into two distinct populations, one of which corotates with the Earth, while the other is convected sunward from the tail and returns tailward along the flanks of the magnetosphere. The plasmapause, a magnetic field-aligned density discontinuity, separates the plasmasphere, populated by cold (∼1 eV) dense (N ∼ 10 ³ cm ⁻³ ) plasma of predominantly ionospheric origin, from the plasmatrough, which contains a relatively sparse (N ∼ 1 cm ⁻³ ) plasma originating primarily in the solar wind [Carpenter, 1963; Gringauz, 1963].

The form of the magnetosphere and the processes occurring within it are substantially determined by the speed and density of the solar wind as well as the orientation of the IMF.

Magnetic reconnection on the dayside magnetopause allows interplanetary and terrestrial mag- netic field lines to merge. Solar wind and magnetospheric plasma can then mingle along open field lines. The reconnection rate is regulated by the north-south component of the IMF, being most intense when the IMF is directed antiparallel to the Earth’s magnetic field. Reconnected field lines are dragged across the Earth’s magnetic poles by the motion of the solar wind and draped over the magnetotail. The antisunward flow of open field lines is, on average, balanced by the sunward convection of closed field lines. Reconnection also takes place continuously within the magnetotail at a distant neutral line some 100 R E down the tail.

Periodically a spontaneous reconfiguration of the magnetosphere occurs. Such events are

4

5

Figure 2.1: Schematic representation of the terrestrial magnetosphere.

known as magnetospheric substorms.

2.2 Magnetospheric Substorms

A magnetospheric substorm is a process in which a substantial quantity of energy accumulated in the magnetotail is released and deposited in the inner magnetosphere [Rostoker et al., 1980;

Rostoker, 1996]. A substorm consists of three distinct phases: growth, expansion and recovery.

The growth phase is initiated by a southward turning of the IMF, which allows enhanced reconnection on the dayside of the magnetosphere and the consequent transfer of plasma and magnetic flux to the tail. Although the subsequent change of the IMF to a northward orientation often triggers a substorm, it is not necessary and substorms may occur during periods of sustained southward IMF. An illustrative event is presented in Figure 2.2. The release of the energy amassed in the tail occurs in the expansion phase. Following the activity of the expansion phase the magnetosphere gradually returns to its unperturbed state during the recovery phase, which has a duration of ∼1 h [Akasofu, 1964b].

The precise mechanism involved in triggering the substorm expansion phase is still unre- solved, although a variety of plausible theories exist [Akasofu, 1968; McPherron et al., 1973;

Rostoker et al., 1980; Shiokawa et al., 1998]. Most theories agree on the various processes

involved: the formation of a near-Earth neutral line, disruption of the cross-tail current, dipo-

larisation of the magnetic field and the formation of the substorm current wedge (SCW). The

sequence in which these processes occurs is still a contentious issue and has led to numerous

opposing theories, two of which espouse the near-Earth neutral line [Baker et al., 1996] or

current disruption [Lui et al., 1988; Voigt, 1995] as the initiating influence.

(a) Locations of ACE and WIND

0 3 6 9 12 15 18 21 24

UT -8

-6 -4 -2 0 2 4 6 8

nT

0 3 6 9 12 15 18 21 24

UT -8

-6 -4 -2 0 2 4 6 8

nT

(b) ACE

0 3 6 9 12 15 18 21 24

UT -6

-4 -2 0 2 4 6

nT

0 3 6 9 12 15 18 21 24

UT -6

-4 -2 0 2 4 6

nT

(c) WIND

Figure 2.2: (a) Locations of ACE and WIND during a substorm on 21 February 2003. ACE was close to the L1 point, while WIND was near the Sun-Earth line but earthward of L1.

Variation of the IMF B z at (b) ACE and (c) WIND on 21 February 2003. The onset of the

substorm is indicated by the vertical grey line. Around 14:30 UTC the IMF turned southward

and remained so for some time, then reverted to a northward orientation about an hour before

substorm onset. The solar wind takes roughly 1 h to travel from L1 to Earth, so that the

northward IMF was incident upon the magnetopause at a time coincident with the substorm

onset.

7

In the near-Earth neutral line model the expansion phase is activated by the formation of a magnetic neutral line around 30 R E down the tail. Reconnection at the neutral line accelerates plasma which is subsequently transported earthward. In the current disruption model an instability near the inner edge of the plasma sheet impedes the flow of the cross- tail current. The reduction in cross-tail current causes dipolarisation of the magnetic field.

Within the disruption region, which expands both in local time and radius, particles are locally energised.

In addition to those phenomena described in Chapter 1, the substorm expansion phase has several other manifestations [Rostoker et al., 1980; Yeoman et al., 1994; Rostoker, 2002]:

enhanced auroral luminosity, elevated ionospheric plasma density, particle injections at geo- stationary orbit and auroral kilometric radiation. None of these appear to provide consistent timing of the substorm onset [Liou et al., 1999, 2001].

The most conspicuous substorm signature is the aurora, which results from the interaction of precipitating charged particles with the upper atmosphere. Figure 2.3 illustrates the evo- lution of the aurora during a substorm with onset at 15:31 UTC on 3 April 1996. Dynamic auroral displays are generally accompanied by magnetic field fluctuations and the injection of particles into the nightside inner magnetosphere.

2.2.1 Particle Injections

The earthward motion of field lines during the substorm expansion phase drags fresh plasma into the inner magnetosphere, where it is detected as a dispersionless particle injection by geostationary satellites located around midnight [Reeves et al., 1992, 1996; Birn et al., 1997;

Reeves, 1997, 1998; Li et al., 1998, 2003]. Injections observed at geosynchronous orbit consist of abrupt flux enhancements (a few orders of magnitude above ambient levels) of particles with energies of tens to hundreds of keV [Parks et al., 1980; Sandholt and Farrugia, 2001]. The increase in particle flux is significantly less spectacular at lower energies . 25 keV [Birn et al., 1997]. The diamagnetic effect of the injected particles can account for a large fraction of the magnetic perturbation observed in orbit [Parks et al., 1980]. Figure 2.4 shows geosynchronous electron flux data for three separate substorms. The injection region typically extends over a few hours of local time [Reeves et al., 1992], and appears to have reasonably well defined azimuthal boundaries [Friedel et al., 1996] although some observations suggest that there is a central injection region flanked by injection peripheries [Reeves et al., 1991]. The injected plasma is subject to energy dependent azimuthal drifts, with the result that satellites located at later local times detect the injected electrons with progressively greater dispersion. This effect is apparent in Figure 2.4a.

Injections may be preceded by an appreciable reduction of the particle flux. These pre- injection flux dropouts have typical duration of around 40 min [Sauvaud and Winckler, 1980]

and are most dramatic for satellites located outside the geomagnetic equatorial plane. A flux

dropout is evident in Figure 2.4b but absent in Figure 2.4c. Since the Earth’s magnetic axis

is inclined with respect to its rotation axis, geosynchronous satellites at different longitudes

lie at various distances from the magnetic equatorial plane. As illustrated in Figure 2.5,

this determines whether or not a satellite is likely to experience a flux dropout. During

the substorm growth phase the field lines may become so distorted that a satellite located

some distance from the magnetic equatorial plane is left outside the trapping boundary and is

therefore exposed to reduced flux. If, however, the satellite is closer to the magnetic equatorial

15:27:04 15:27:23 15:28:17 15:28:36 15:30:45 15:31:03

15:31:58 15:32:17 15:33:12 15:33:31 15:35:39 15:35:58

15:36:53 15:37:11 15:38:06 15:38:25 15:40:33 15:40:52

15:41:47 15:42:06 15:43:01 15:43:19 15:45:28 15:45:47

15:46:41 15:47:00 15:47:55 15:48:14 15:50:22 15:50:41

15:51:36 15:51:55 15:52:49 15:53:08 15:55:17 15:55:35

15:56:30 15:56:49 15:57:44 15:58:03 16:00:11 16:00:30

16:01:07 16:01:43 16:02:38 16:02:57 16:03:52 16:04:11

Figure 2.3: Polar UVI data showing the auroral signature of a substorm with onset at 15:31 UTC on 3 April 1996. This event was also discussed by Liou et al. [1999] and Takahashi et al.

[2002].

9

20:00 20:30 21:00 21:30 22:00 22:30 23:00 23:30 24:00 UT

10

10

10

10

10

10

1991-080

21:33 UT

10

10

10

10

10

10

1994-084

21:33 UT

10

10

10

10

10

10

LANL-97A

21:33 UT

10

10

10

10

10

10

LANL-02A

21:33 UT

fl u x

 cm − 2 s − 1 sr − 1 k eV − 1 

(a) 17 February 2002

12 13 14 15 16 17 18 19 20 21 22 23 24

UT 10

10

10

10

10

10

LANL-97A

16:00 UT

(b) 3 January 2003

12 13 14 15 16 17 18 19 20 21 22 23 24

UT 10

10

10

10

10

10

LANL-97A

16:52 UT

(c) 23 May 2003

Figure 2.4: Electron flux data from SOPA instruments on LANL geosynchronous satellites.

The four traces in each panel represent different energy channels; from top to bottom they are

50–75, 75–105, 105–150 and 150–225 keV. In the upper right corner of each panel is an inset

indicating the local time of the satellite at the event epoch (the arrow points sunward).

11^◦

Figure 2.5: Model explaining flux dropouts. Geosynchronous satellites are located at various distances from the magnetic equatorial plane. Those at greater distances may fall outside the trapping boundary during the substorm growth phase.

plane, or the field contortion is not too severe, then it remains within the trapping region during the growth phase.

McIlwain [1974] hypothesised that electrons are deposited into a region outside a well defined boundary at the onset of the substorm expansion phase. The region of space in which a dispersionless injection is observed is known as the injection region, and its earthward limit is the injection boundary, described as a function of local time by [McIlwain, 1974; Mauk and McIlwain, 1974]

L = 122 − 10Kp

T (φ) − 7.3 (2.1)

where

T (φ) =

( 12φ/π for 3π/2 ≤ φ ≤ 2π,

12φ/π + 24 for 0 < φ ≤ π/2, (2.2)

and Kp enters as a parameter. An expression similar to (2.1) was derived independently by Kivelson et al. [1980]. These injection boundary models produce inward spirals from dusk to dawn. The boundaries are located at lower L with increasing Kp, consistent with observational evidence that the radial penetration of injected particles depends on Kp [Lopez et al., 1990], with events reaching lower L at more geomagnetically disturbed times.

2.2.2 Magnetic Variations

The magnetic activity accompanying a substorm assumes two widely disparate forms: pertur- bations of the field with duration of tens of minutes to hours, produced by modification of the large scale current systems in the magnetosphere, and ULF pulsations with periods of tens of seconds.

The collapse of the field in the near-Earth tail occurs via the establishment of the sub-

storm current wedge [McPherron et al., 1973; Sauvaud and Winckler, 1980]. The accompanying

field-aligned currents and the auroral electrojet produce magnetic signatures which are readily

detected in space and on the ground. The sense of the magnetic field perturbations observed

on the ground is indicated in Figure 2.6. The resulting variations involve both the magni-

11

NorthernHemisphereSouthernHemisphere

Figure 2.6: Schematic view of the field-aligned and ionospheric components of the substorm current wedge (red), with associated magnetic fields (blue) and predicted Pi2 polarisation azimuths (black). After Lester et al. [1983, Figure 3].

tude and orientation of the horizontal component of the magnetic field. Figure 2.7a displays the X component at selected stations in the International Monitor for Auroral Geomagnetic Effects (IMAGE) network. The lower latitude stations detect the start of the substorm bay simultaneously, but some delay is incurred before it arrives at the northernmost stations. This can be explained by the poleward expansion of the auroral electrojet during the course of the substorm. The initial enhancement of the field at lower latitudes results from a reduction in the tail current due to its diversion through the SCW [Clauer and McPherron, 1974]. The azimuthal position of a station with respect to the SCW determines the sense of the D mag- netic bay, which, in the northern hemisphere, is positive to the west of the bay and negative to the east [Nagai, 1982, Figure 11]. This is demonstrated in Figure 2.7b.

Figure 2.6 also reflects the predicted variation in Pi2 polarisation across the SCW if mid- latitude pulsations are assumed to arise directly from the field-aligned current system. Lester et al. [1983] found good agreement between observed Pi2 azimuths and those predicted from a current wedge model.

Under quiescent conditions an eastward auroral electrojet flows during the afternoon and

evening, while a westward electrojet flows in the late evening and early morning. An inten-

sification of the westward electrojet near midnight occurs during a substorm. The AE and

related indices provide a measure of global auroral electrojet activity [Mayaud, 1980] and are

commonly used to identify the substorm onset time. Local values of the AE indices are esti-

mated using data from the IMAGE magnetometer chain. Data for two events are presented

in Figure 2.8.

19 20 21 22 UT

-50 0 50 -100 0 100 -100 0 100 -200 0 200 -200 0 200 -500 0 500 -500 0 500 -200 0 200 -50 0 50

◦

PEL λ=63.6◦ φ=104.7◦

SOD λ=64.0◦ φ=107.1◦

MUO λ=64.8◦ φ=105.0◦

MAS λ=66.3◦ φ=106.2◦

KEV λ=66.4◦ φ=109.0◦

BJN λ=71.5◦ φ=107.7◦

HOR λ=74.2◦ φ=109.2◦

LYR λ=75.3◦ φ=111.6◦

NAL λ=76.3◦ φ=110.8◦

(a) X component on 14 January 2001

15 16 17 18 19 20 21

UT -500

0 500 -200 0 200 -500 0 500 -500 0 500 -20 0 20 -10 0 10 -50 0 50 -20 0 20 -20 0 20

MMB λ=37.3◦ φ=−144.3◦

BMT λ=34.8◦ φ=−171.1◦

IRT λ=47.6◦ φ=177.3◦

LZH λ=30.7◦ φ=176.2◦

QSB λ=27.7◦ φ=107.4◦

TRO λ=66.7◦ φ=102.7◦

ABK λ=65.4◦ φ=101.5◦

NUR λ=56.9◦ φ=102.1◦

SOD λ=64.0◦ φ=107.1◦

(b) D component on 21 January 2003

Figure 2.7: Substorm magnetic bays at selected observatories. The onset of the substorm is

indicated by the vertical grey line. The magnitude scale is in nT and the curves have been

shifted to zero mean. The site code, CGM latitude (λ) and CGM longitude (φ) are given to

the right of each panel.

13

0 2 4 6 8 10 12 14 16 18 20 22 24

UT -700

-650 -600 -550 -500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150

nT

(a) 22 February 2002

0 2 4 6 8 10 12 14 16 18 20 22 24

UT -700

-650 -600 -550 -500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200

nT

(b) 21 February 2003

Figure 2.8: Local AU (blue) and AL (red) indices from the IMAGE magnetometer network.

Substorm onset is indicated by the vertical grey line. The AL variation during (a) is more

significant since in this instance the stations are located below the westward electrojet, while

for (b) the stations are at an earlier local time and the indices are thus influenced more by

the eastward electrojet.

The nominal VLF range extends from 3 kHz to 30 kHz, although in practice lower frequencies are often considered. Whistler mode waves, which are the most significant form of electro- magnetic radiation within the terrestrial magnetosphere in this frequency range, may originate from a variety of sources. In space they can be generated by wave-particle interactions, while on Earth they are produced by either natural (lightning discharges) or anthropogenic (power line harmonic radiation (PLHR), navigation and military transmitters) sources. Two naturally occurring VLF phenomena are whistlers and chorus.

3.1 The Whistler Mode

At frequencies significantly higher than the ion gyrofrequencies the propagation of electro- magnetic waves through a cold homogeneous plasma is described by the Appleton-Hartree equation [Helliwell, 1965]

n ² = 1 − X

1 − Y ² sin ² θ ± pY ⁴ sin ⁴ θ + 4Y ² cos ² θ(1 − X) ² 2(1 − X)

(3.1)

where collisions have been neglected, θ is the angle between the wave normal vector and the static magnetic field, X = ω ² _p /ω ² and Y = Ω/ω = 1/Λ. If the direction of propagation is close to that of the static magnetic field then the terms involving sin θ can be neglected in (3.1), yielding

n ² = ω _p ²

ω(Ω cos θ − ω) , (3.2)

which is known as the quasi-longitudinal (QL) approximation [Helliwell, 1965, p. 27]. In the magnetosphere this approximation is valid for an appreciable range of θ. The refractive index depends on both the frequency and direction of propagation, and whistler mode waves are thus dispersive and anisotropic.

The denominator in (3.2) is zero when

cos θ = ω

Ω . (3.3)

This condition defines the oblique resonance cone. When (3.3) is satisfied the waves become electrostatic [Walker, 1993, p. 39]. At frequencies less than the lower hybrid resonance the effects of positive ions become significant and the resonance cone disappears [Walker, 1993, p. 209]. Since cos θ < 1 the whistler mode is restricted to frequencies less than the electron gyrofrequency.

As a result of the anisotropy in (3.2) the whistler mode is guided by the static magnetic field. This effect is illustrated in Figure 3.1, which is a plot of ray direction versus wave normal angle for a range of normalised frequencies. For moderate Λ the ray does not deviate significantly from the static magnetic field direction irrespective of the wave normal angle,

14

15

-90˚

-60˚

-30˚

0˚

30˚

0˚ 10˚ 20˚ 30˚ 40˚ 50˚ 60˚ 70˚ 80˚ 90˚

Λ = 0.00 Λ = 0.20

Λ = 0.40 Λ = 0.60

Λ = 0.80

Λ = 0.90

Λ = 0.95

θ

β

Figure 3.1: Whistler ray direction, β, as a function of wave normal angle, θ.

and the wave energy is thus directed along the magnetic field. As Λ → 1 the guiding effect of the magnetic field is lost. Field-aligned channels of enhanced or depleted plasma density (known respectively as crest or trough ducts) also play a role in determining the direction of propagation of whistler mode energy. The refractive index gradient within a duct causes the ray to progress along the magnetic field in a manner analogous to light travelling along an optical fibre. Only those waves with wave normal angles within the trapping cone become confined in a duct. The width of the trapping cone is determined by the normalised frequency and the degree of enhancement or depletion across the duct. The maximum frequency for trapping in a crest duct is half the local electron gyrofrequency.

The anisotropy of the whistler mode also influences the coupling of waves across the in- terface between the neutral atmosphere and the ionosphere. This situation is depicted in Figure 3.2. In the neutral atmosphere the refractive index surface is spherical. Within the ionised upper atmosphere it is a surface of revolution around the magnetic field direction.

Snell’s law determines the mapping of the wave normal vector from the atmosphere into the

ionosphere, defining a range of angles known as the transmission cone. The intersection of

the transmission and trapping cones determines a range of wave normal directions in the at-

mosphere which become ducted in the magnetosphere. The degree of overlap is determined

by the opening angles of the two cones and the inclination of the magnetic field. Coupling is

most efficient when the magnetic field is perpendicular to the ionosphere. For an oblique mag-

netic field waves which are incident from higher latitudes experience more effective coupling

[Helliwell, 1965, Figure 3-23].

Atmosphere Ionosphere

B

transmission cone

trapping cone atmospheric trapping cone

trapped waves

Figure 3.2: Coupling of whistler mode waves from the atmosphere to the ionosphere. After Helliwell [1965, Figure 3-21].

3.2 Whistlers

Whistlers are brief descending VLF tones generated by terrestrial lightning strikes [Storey, 1953; Helliwell, 1965]. Lightning is a fleeting source of electromagnetic radiation, producing a broad-band pulse which propagates through the atmosphere in the Earth-ionosphere waveg- uide. Since the atmosphere is a neutral medium it is non-dispersive: all frequencies travel at the same rate and are detected simultaneously as a spheric. Some fraction of the energy may penetrate through the ionosphere and enter the magnetosphere. These waves may couple to a duct and be guided to the conjugate hemisphere, where a portion of the incident radi- ation can penetrate through to the ground. The dispersive passage of the pulse through the magnetospheric plasma causes a frequency-dependent delay which results in the characteris- tic frequency-time structure of a whistler. Samples of whistler spectrograms are presented in Figure 3.3. Whistlers act as passive magnetospheric probes. Their analysis has led to an enhanced understanding of the near-Earth space environment [e.g., Carpenter, 1963].

3.3 Doppler-Shifted Cyclotron Resonance

Whistler mode waves are right-hand circularly polarised. The sense of rotation of the wave electric field is thus the same as that of the electron cyclotron motion. If the wave frequency and the electron gyrofrequency are matched then this can lead to the transfer of energy between the wave and the electron. Since electrons generally have finite velocity parallel to the magnetic field the Doppler-shift of the wave must be taken into account. Doppler-shifted cyclotron resonance occurs when the frequency of the wave matches the gyrofrequency of the counter-streaming electrons in their guiding centre frame. The condition for the fundamental resonance is

Ω = ω − k · v (3.4)

17

52 53 54 55 56 57 58 59

0 2 4 6 8

f [k H z] f [k H z]

t [s]

t [s]

(a) Sutherland [L = 1.76] at 23:16 UTC on 28 August 2003

5 6 7 8 9 10 11 12

0 2 4 6 8

f [k H z] f [k H z]

t [s]

t [s]

(b) SANAE–IV [L = 4.36] at 07:15 UTC on 22 June 2002

Figure 3.3: Spectrograms of whistlers.

where the relative motion of the particle and wave, k · v < 0, shifts the wave frequency up to the electron gyrofrequency. Using (3.2) one can derive the condition [Hargreaves, 1992, p.

365]

W k = B ² 2µ

N

Ω ω

 1 − ω

Ω

 3

(3.5) which relates the parallel energy of a resonant electron to the normalised wave frequency and magnetic energy per electron. A qualitative description of the resonant interaction is given by Dungey [1997].

The resonance condition (3.5) assumes that the wave vector lies along the magnetic field.

This is not necessarily the case. Lefeuvre et al. [1982], for example, find that signals originating from a terrestrial transmitter have wave normals with θ ≃ 130 ^◦ . A simple modification of (3.5) caters for oblique propagation, which results in larger W _k with increasing θ.

Plasma distributions which are not isotropic may be unstable to the amplification of whistler mode waves. Kennel and Petschek [1966] derived the growth rate for whistler mode waves interacting with a population of electrons:

γ = πΩ

 1 − ω

Ω

 2

η(A − A c ) (3.6a)

where the anisotropy,

A = Z π/2

0

tan α ∂f

∂α tan α sec ² α dα 2

Z π/2 0

f tan α sec ² α dα        

 W =W

sec

α

(3.6b)

characterises the distribution of electron pitch angles and

η = 2πv _k ³ N

Z π/2 0

f tan α sec ² α dα    

 W =W

sec

α

(3.6c)

represents the relative number of particles in resonance. To achieve amplification it is necessary that the anisotropy should exceed a critical value,

A c = 1

Ω/ω − 1 . (3.7)

3.4 Chorus

Chorus is a VLF emission which derives its name from an aural similarity to the twitter and

warble of a flock of birds. Sample spectrograms illustrating chorus emissions are displayed in

Figure 3.4. The spectral structure of chorus is composed of a multitude of discrete elements,

typically rising tones, falling tones, constant tones and hooks or inverted hooks. Although

dispersion certainly applies to these elements, it is not the source of their frequency-time char-

acteristics, which are more likely to be caused by a non-linear generation mechanism. Chorus

is thought to be created in close proximity to the geomagnetic equatorial plane [Tsurutani and

19

0 10 20 30 40 50 60

0 1 2 3 4

f[kHz]f[kHz]

(a) 06:20 UTC 26 April 2002

0 10 20 30 40 50 60

0 1 2 3 4

f[kHz]f[kHz]

(b) 07:30 UTC 3 May 2003

0 10 20 30 40 50 60

0 1 2 3 4

f[kHz]f[kHz]

(c) 10:30 UTC 1 February 2002

0 10 20 30 40 50 60

0 1 2 3 4

f[kHz]f[kHz]

(d) 12:00 UTC 14 October 2002

0 10 20 30 40 50 60

0 1 2 3 4

f[kHz]f[kHz]

(e) 14:30 UTC 20 November 2002

0 10 20 30 40 50 60

0 1 2 3 4

f[kHz]f[kHz]

t [s]

t [s]

(f) 15:00 UTC 20 January 2003

Figure 3.4: One minute samples of chorus recorded at SANAE–IV.

Smith, 1974, 1977; Santol´ık et al., 2005] by the transfer of energy from anisotropic hot electrons to whistler mode waves via the Doppler-shifted cyclotron resonance interaction [Sazhin and Hayakawa, 1992]. Chorus is commonly observed during magnetospheric storms and substorms [Tsurutani and Smith, 1974, 1977], and has become associated with particle injections [e.g., Isenberg et al., 1982].

Chorus emission frequencies in space extend up to Ω, with a notch at Ω/2 [Tsurutani and Smith, 1974]. The extinction at Ω/2 may be due to Landau damping. In the lower band chorus elements have relatively small wave normal angles, but in the upper band they lie close to the resonance cone [Hayakawa et al., 1984]. Observations of chorus on the ground are limited to frequencies below Ω/2, since this is the cutoff for ducted propagation.

3.5 Substorm Chorus Events

The Substorm Chorus Event (SCE) is an upward-drifting band of chorus observed primarily between midnight and dawn. Examples of SCEs were presented in Figure 1.1 on page 2. The frequency of the upper boundary of the emission envelope increases with time, typically at a rate in the range 20 to 1000 Hz/min [Smith et al., 1996].

The possible origins of SCEs were reviewed by Collier [2004]. In pr´ecis they arise from drift- ing electrons injected into the nightside inner magnetosphere during the substorm expansion phase [Smith et al., 1996]. The injected electrons are not isotropic [˚ Asnes et al., 2005a,b], and their subsequent drift is dispersive (both in terms of energy and pitch angle), which exacer- bates the anisotropy. The cyclotron resonant interaction of an anisotropic electron population with whistler mode waves leads to amplification of the waves. The resulting waves are ducted from the equatorial interaction region to the ground.

The injected electrons travel eastward under the influence of gradient-curvature and E ×B

drifts. The energy-independent E × B drift has both radial and azimuthal components, and

in the midnight-dawn quadrant is directed earthward and eastward. This results in motion

towards smaller L, and consequently an elevated resonance frequency accompanied by an

increase in the local electron gyrofrequency, which raises the maximum allowed frequency for

ducted propagation. The electrons are also subject to an energy-dependent drift due to the

spatial gradient and curvature of the Earth’s magnetic field. This drift carries them eastward,

the more energetic particles and those located further from the Earth having higher angular

drift velocity. The dispersive drift leads to a temporal variation in both the anisotropy and

the number of electrons in resonance at a given frequency [Collier and Hughes, 2004b]. The

highest energy electrons drift most rapidly and thus arrive first within an observer’s field of

view. The condition (3.5) indicates that these high energy electrons are resonant with low

frequency waves. Resonance at progressively ascending frequencies occurs as particles of lower

energy drift around into the field of view.

4 ULF Waves

ULF waves occur at frequencies below 300 Hz, although in a magnetospheric context the ULF band extends from roughly 1 mHz to 1 Hz, the upper limit corresponding approximately to the various ion gyrofrequencies. ULF wavelengths are comparable to the scale size of the magnetosphere. The nature of ULF waves in the magnetosphere may be understood in terms of magnetohydrodynamic (MHD) theory.

4.1 Magnetohydrodynamics

The MHD model is a single fluid description of the macroscopic behaviour of a plasma in the long wavelength, low frequency limit. The medium is treated as a continuous fluid subject to forces arising from electric and magnetic fields. The properties of the plasma are averaged over volumes which are larger than the typical inter-particle distances, yet smaller than the dimensions of the system.

MHD makes a few assumptions about the plasma: it should be collision dominated, the characteristic time and length scales should be larger than those associated with ion Larmor gyration and, despite high collisionality, the resistivity should be low [Freidberg, 1987]. The plasma in the magnetosphere is an extremely tenuous medium and collisions are exceedingly rare. The magnetic field, however, inhibits the transverse motion of the charged particles and thus allows for a fluid description. The restrictions on the dimension of the system imply that MHD should hold everywhere in the magnetosphere except in thin structures like the magnetopause or the neutral sheet.

The MHD equations, relating the motion of the plasma to the electromagnetic fields, are

∂ρ

∂t + ∇ · ρv = 0 (4.1a)

ρ dv

dt = j × B − ∇p (4.1b)

∇ × E = − ∂B

∂t (4.1c)

∇ × B = µ

j (4.1d)

dp dρ = γp

ρ . (4.1e)

where the polytropic index, γ, is the ratio of specific heats. The continuity equation (4.1a) dictates that there are no sources or sinks of matter in the system. The momentum equation (4.1b) is an expression of Newton’s second law and relates the acceleration of the medium to the Lorentz force and pressure gradient. The effects of collisions between the electron and ion constituents does not enter into (4.1b) since the associated forces are equal and opposite. Faraday’s law and the low-frequency form of Amp`ere’s law are (4.1c) and (4.1d).

The displacement current has been neglected since the characteristic speeds in MHD are much smaller than the speed of light. The adiabatic equation of state (4.1e) implies that changes

21

in pressure emerge from mechanical work done on the fluid. The momentum equations for the two species may be used to derive a generalised form of Ohm’s law which, with a few simplifying assumptions, can be reduced to

E + v × B = 0. (4.1f)

The motion of the plasma is often described by the displacement, ξ, which is defined by ξ =

Z

v dt (4.2)

whence v = ∂ξ/∂t = −jωξ.

A reduced set of MHD equations can be obtained by eliminating variables within (4.1).

The current density may be removed from (4.1b) using (4.1d), ρ dv

dt = −∇



p + B ² 2µ



+ B · ∇B

µ

(4.3a)

yielding an expression for the rate of change of momentum density in terms of the pressure gradient and magnetic tension. For a cold plasma in a dipole field the magnetic pressure gra- dient is balanced by the magnetic tension. Consistent with the cold plasma conditions which generally obtain in the outer magnetosphere, the kinetic pressure may often be neglected in (4.3a). This may be inappropriate in the ring current region during geomagnetically disturbed periods. The continuity equation can be written in terms of the convective derivative of density and combined with (4.1e) to give

dp

dt = −γp∇ · v. (4.3b)

Finally, (4.1f) substituted for E in (4.1c) gives

∂B

∂t = (B · ∇)v − B∇ · v − (v · ∇)B. (4.3c) In a uniform magnetic field the last term in (4.3c) vanishes, while in a non-uniform field it introduces coupling between different modes.

The existence of MHD waves was discovered by Alfv´en in 1942. Small amplitude MHD waves may be treated by linearising the governing equations. This entails expressing the wave amplitudes in density, velocity, pressure, magnetic and electric fields as small perturbations and neglecting all resulting second-order terms. The magnetic field is written as B → B+b, where B is the equilibrium field and b is the perturbation. Since the equilibrium velocity of the fluid is zero there is no zero-order electric field, and E thus denotes the electric field perturbation.

The length and time scales appropriate to MHD waves are such that the electrostatic and magnetostatic conditions

∇ × E = 0 (4.4a)

∇ × b = 0 (4.4b)

often apply [Walker, 2004, Section 16.3.3].

The linearised forms of the reduced MHD equations (4.3) may be used to derive a wave equation. If a plane wave of the form

exp[j(k · r − ωt)] (4.5)

23

is assumed, then the wave equation in a uniform medium becomes [Walker, 2004, p. 130]

ω ² − (k · v ^A ) ²  v − (v A ² + v _S ² )(k · v)k + (k · v ^A ) [(v A · v)k + (k · v)v ^A ] = 0, (4.6) where

v _A = B

√ µ

ρ (4.7)

is the Alfv´en velocity and

v _S = r γp

ρ (4.8)

is the speed of sound.

The vector equation (4.6) constitutes a set of simultaneous, homogeneous algebraic equa- tions which only has a non-trivial solution if the determinant of the coefficient matrix is zero.

This prescription leads to two independent dispersion relations:

ω ² − k k ² v _A ² = 0 (4.9a) and

ω ⁴ − k ² (v _A ² + v ² _S )ω ² + k ² k ² _k v _A ² v ² _S = 0, (4.9b) where k k = k cos θ is the component of the wave vector parallel to the background magnetic field. These relations describe shear (4.9a) and compressional (4.9b) Alfv´en waves. Although the two modes propagate independently in a homogeneous medium, they become coupled in the vicinity of inhomogeneities.

The equations (4.9) can be manipulated to give expressions for the phase speed:

v _p ² = v _A ² cos ² θ (4.10a)

and

v _p ⁴ − (v A ² + v _S ² )v ² _p + v _A ² v _S ² cos ² θ = 0.

The latter is quadratic in v _p ² and has solutions v _p ² = 1

2



(v _A ² + v ² _S ) ± q

(v _A ² + v _S ² ) ² − 4v ² A v _S ² cos ² θ



. (4.10b)

It is apparent from (4.10) that MHD waves are anisotropic but non-dispersive: their phase speed depends on the direction of propagation, but is independent of frequency.

Various techniques exist for treating wave propagation in inhomogeneous media. If the properties of the medium vary slowly then phase integral methods may be appropriate. Alter- natively, a smoothly-varying medium may be approximated by a stratified medium in which the physical properties are uniform within each stratum.

The phase integral method [Walker, 2004, Chapter 11] assumes that the phase of the wave evolves more rapidly that the wave amplitude or the characteristics of the medium. For a plane wave the spatial gradients in the wave equation have the form

∇e ^jk · ^r = je ^jk · ^r ∇(k · r) = je ^jk · ^r [k + (r · ∇)k].

If the properties of the medium vary slowly over the scale of a single wavelength, then the gradient of k can be neglected and the medium may be treated as locally uniform. The phase along the propagation path is given by

δ = Z

k (r) · dr (4.11)

where k is determined by the dispersion relation at r.

A more sophisticated technique for treating propagation in slowly varying media is the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) approximation [Hassani, 1999; Walker, 2004], in which the phase variation k · r is replaced by the phase integral (4.11) and the amplitude is a slowly varying function of position.

In a stratified medium the germane parameters are the reflection and transmission coef- ficients at the interface between two disparate regions. The Fresnel reflection coefficient for normal incidence on a boundary between media of refractive index n 1 and n 2 is

R = n 1 − n ² n 1 + n 2

. (4.12)

The sign of R indicates whether or not there is a phase reversal for the reflected wave.

4.1.1 Magnetosonic Waves

The dispersion relation (4.9b) describes two compressional modes. Since the sound speed enters into the dispersion relation, these are known as magnetosonic waves. The magnetic perturbation lies in the plane containing k and B. The induced E × B drift has a component in the direction of propagation and thus leads to compression and rarefaction of the medium.

The phase speeds of the magnetosonic waves are given by (4.10b). It is apparent that one root is always greater than the other. The former is labelled the fast mode, while the latter is called the slow mode.

The inequality v A ≫ v ^S applies widely in the magnetosphere where, in general, β ≪ 1.

Under these conditions (4.10b) reduces to ω ²

v _A ² = k _k ² + k ² _⊥ = k ² (4.13a)

and

ω ²

v ² _S = k _k ² (4.13b)

where

k _⊥ = − ˆ B × ( ˆ B × k)

is the component of the wave vector perpendicular to the background magnetic field. The fast wave (4.13a) is isotropic and has a phase speed of v A irrespective of direction, while the slow wave (4.13b) is a sound wave propagating parallel to the magnetic field. The latter mode is rapidly damped and will not be considered further.

The components of the fast mode satisfy the conditions [Cross, 1988]

∇ · E = ∇ · b = ∇ · j = E · b = 0 (4.14a)

∇ · v 6= 0. (4.14b)

The compression of the medium is reflected in (4.14b).

25

4.1.2 Shear Alfv´ en Waves

The shear Alfv´en wave involves the distortion of the field lines perpendicular to the plane containing the wave vector and background magnetic field. The magnetic field fluctuation induces an oscillating electric field which leads to an E × B drift of the plasma, causing it to follow the magnetic field perturbation. The mechanism is thus analogous to a wave on a stretched string since the plasma oscillates as if it were fastened to the magnetic field lines. Since the plasma drift is perpendicular to the direction of propagation the wave is not compressional. The inertia of the ions causes them to lag behind the electrons, which leads to a polarisation current.

The dispersion relation (4.9a) only depends on the parallel component of the wave vector and k ⊥ is thus unrestricted. Although k may possess an arbitrary orientation, the group velocity is strictly aligned along the magnetic field direction. From (4.1c) the phase velocity is related to the amplitude of the transverse electric and magnetic fields, v A = E/b, and this relationship can be used to estimate the plasma density.

The components of the shear Alfv´en wave are subject to the conditions [Cross, 1988]

∇ · v = ∇ · b = ∇ · j = E · b = 0 (4.15a)

∇ · E 6= 0. (4.15b)

Electrostatic waves have E k k so that ∇ × E = 0. Since k · E 6= 0 from (4.15b) the shear Alfv´en wave is partly electrostatic. Equivalently, because the wave is non-compressional the component of ∇ × E parallel to the background magnetic field is zero and the perpendicular component of E is thus given by the gradient of a scalar potential.

4.1.3 Poynting Flux

The flux of energy carried by an electromagnetic wave is described by the Poynting vector, Π = E × b

µ

. (4.16)

The Poynting vector is parallel to B for the shear Alfv´en wave, but parallel to k for the fast wave [Dungey, 1967].

The magnitude of the Poynting flux depends not only on the amplitudes of the electric and magnetic fields but also on their relative phase. Suppose that b ∼ cos ωt and E ∼ cos(ωt + ψ), then the associated Poynting flux is

Π ∼ cos ψ + cos(2ωt + ψ),

which oscillates at twice the frequency of the underlying wave but averages to

hΠi ∼ cos ψ. (4.17)

Now if b and E are in phase quadrature then ψ = ±90 ^◦ → cos ψ = 0, which corresponds to a standing wave. If b and E are in phase or antiphase then cos ψ = ±1 and energy is travelling either parallel or antiparallel to the magnetic field.

The time-averaged Poynting flux is [Chi and Russell, 1998]

hΠi = 1 T

Z T 0

E × b

µ

dt. (4.18)

The integration interval, T , should be at least as long as the period of the lowest frequency

component in the spectrum.

4.2 ULF Pulsations

The Earth’s magnetic field varies on time scales extending from a fraction of a second to millions of years [Jacobs et al., 1964]. Variations with periods less than a few hundred sec- onds are known as geomagnetic pulsations or micropulsations, although the latter term is an anachronism and something of a misnomer. Such pulsations are an ubiquitous feature of the terrestrial magnetosphere.

Stewart [1861] made the first documented observations of fluctuations in the Earth’s mag- netic field with periods in the ULF range. However, almost a century elapsed before Dungey [1954] posited MHD waves in near-Earth space as the origin of these pulsations. A number of concise overviews of the subject exist in the literature [Saito, 1969; Orr, 1973; Yumoto, 1986;

Glassmeier, 1995; McPherron, 2002], while a succinct review of various theories is given by Walker et al. [1992].

Pulsations are observed on the ground as variations in the amplitude and direction of the magnetic field. In the ionosphere they are revealed as oscillations in the E × B plasma drift detected by ionospheric radars. Compressional ULF waves may modulate energetic electron precipitation [Coroniti and Kennel, 1970], leading to fluctuations in ionospheric electron den- sity which can be detected in riometer data [Brown, 1964]. Modulated electron precipitation can locally modify ionospheric conductivity and regulate currents at comparable frequencies, thereby producing magnetic fluctuations.

Both shear Alfv´en and fast mode waves can lead to standing oscillations in the magneto- sphere: field line resonances (FLRs) in the former case and cavity or fast mode resonances (FMRs) in the latter. The quality factor for these resonances is determined by the reflection coefficients at the appropriate boundaries. In the case of the FLR the ionospheric conduc- tivity regulates the efficiency of reflection [Newton et al., 1978], while for the FMR both the ionospheric conductivity and Alfv´en velocity gradient are operative [Fujita and Glassmeier, 1995].

4.2.1 Classification

ULF pulsations may be classified according to their origin [Yumoto, 1986]: exogenic pulsations originate outside the magnetosphere and are thought to be driven by the solar wind; endogenic pulsations are excited within the magnetosphere either by transient reconfigurations of the field and plasma or by the liberation of free energy through an instability.

Pulsations may either be regular variations of a continuous nature or irregular fluctuations.

Irregular pulsations, Pi, are most often observed in association with geomagnetic activity and are correlated with other upper atmospheric phenomena. They constitute the microstructure on magnetic records of storms and substorms. Continuous pulsations, Pc, have been attributed to the Kelvin-Helmholtz instability due to the velocity shear across the magnetopause between the solar wind and magnetospheric plasma [Southwood, 1974; Chen and Hasegawa, 1974a]. Pi’s tend to be observed mostly in the night sector, while Pc’s are generally a daytime phenomenon.

These two groups are further divided on the basis of frequency, yielding the classification

scheme presented in Table 4.1.