Geomagnetic Pulsations and Plasma Waves in the Earth's...

Geomagnetic Pulsations and Plasma Waves in the Earth's Magnetosphere

J. C. SAMSON

1 INTRODUCTION

Geomagnetic pulsations are the manifestation of ultra-low-frequency (ULF) plasma waves in the Earth's magnetosphere. These pulsations have frequencies ranging from approximately 1 mHz to greater than 10 Hz and appear as quasisinusoidal oscillations in magnetometer data recorded at the Earth's surface, in the ionosphere and in the magnetosphere. The waves with the lowest frequencies have wavelengths which are comparable to the size of the magnetosphere. Typically the lowest frequency pulsations have the largest amplitudes, with amplitudes of hundreds of nanotesla sometimes observed in the auroral zone. The upper limit of the spectral band for pulsations is determined by the hydrogen cyclotron frequency in the magnetosphere, which is generally less than 10Hz. These high-frequency pulsations seldom have amplitudes which are greater than several nanotesla in the magnetosphere, and typically have amplitudes much less than this on the Earth's surface. The magnetometer records in Fig. 1 illustrate only part of the wide variety of pulsations which are observed on the Earth's surface.

Like most magnetospheric phenomena, the energy for pulsations and plasma waves is ultimately derived from the solar wind. A major governing factor in the generation of pulsations is the convective flow of plasma in the magnetosphere. This flow is driven by either a viscous interaction with the solar wind (Axford and Hines, 1961) or magnetic merging and reconnection (Dungey, 1961; Cowley, 1984) at the magnetopause (Fig. 2). The reconnection process on the dayside is often transient and localized,

GEOMAGNETISM VOL. 4 ISBN 0-12-378674-6

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6 GEOMAGNETIC PULSATIONS AND PLASMA WAVES

RaÂ Smith H, January 27,1979

Universal b e

Fbrt McMurray D, December 13,1971

15.0 -1

Universal Time

Newcastle X, December 17, 1977

I

Universal Time

I ~niversal Time I

1 INTRODUCTION

Figure 2. A schematic of the two mechanisms driving convection in the magnetosphere (after Cowley, 1984). (a) Closed flux tubes are shown moving around the flanks of the magnetosphere, whereas reconnected flux tubes are swept over the poles. Viscous interaction takes place in the region of the boundary layer (dotted lines). (b) The equatorial projection of regions of convection due to viscous interaction are shown as hatched lines. Convection in other regions is driven by reconnection. (c) Same as (b) but for the ionosphere.

Figure 1. Some characteristic examples of geomagnetic pulsations recorded at ground-based stations. From top to bottom: continuous pulsations in the 1-lOmHz band (Pc5); giant pulsations in the 1-lOmHz band; structured continuous pulsations in the 0.1-lOHz frequency band; impulsive pulsations associated with the substorm expansive phase (Pi2).

484 6 GEOMAGNETIC PULSATIONS AND PLASMA WAVES

producing transient plasma waves in the magnetosphere. The convective flow can lead directly to coordinate space instabilities, such as the hydromagnetic Kelvin-Helmholtz instability near the magnetopause. The flow can also produce unstable distributions of plasmas in the magnetosphere. The collision frequencies are low enough to allow these unstable distributions to persist for intervals of time which are long compared with the time for the growth of a number of plasma instabilities.

In the magnetosphere, substantial amounts of plasma transport and convection take place within the boundary layers (Fig. 3). The flows within the low-latitude boundary layer have considerable velocity shear giving favourable conditions for the Kelvin-Helmholtz instability. Plasmas inside the boundary layers often have large gradients in pressure, density and magnetic fields, allowing the possibility of various coordinate space instabilities.

The plasma sheet, plasma sheet boundary layer and ring current are likely regions for the production of plasma waves because of the hot and often anisotropic plasmas found within these regions. During the substorm expansive phase, the enhanced earthward convection of plasmas leads to the injection of plasma sheet and plasma sheet boundary layer plasmas into the dipolar inner magnetosphere. Trapped particle fluxes increase, with the newly trapped protons drifting westward giving an enhanced ring current. The distributions of the energetic protons (10-lOOkeV) are often quite anisotropic giving favourable conditions for the bounce resonance, drift-mirror and ion cyclotron instabilities.

Figure 3. The primary transport regions of the magnetosphere (after Eastman, 1984).

1 INTRODUCTION 485

Abrupt temporal changes in convection within the magnetosphere are often associated with transient field-aligned currents. These changes in current and convection must propagate as Alfvin waves, giving transient sources for pulsations. Impulsive plasma waves in the 5-15 mHz frequency range are often recorded at the time of the onset of the substorm expansive phase. Both the expansive phase and these pulsations may be the manifestation of sudden changes in convection in the magnetotail.

In addition to sources within the magnetosphere, pulsations can derive energy from shocks and instabilities in the solar wind. Sudden impulses in the magnetosphere are caused by shocks in the solar wind flow, and these sudden impulses can produce damped pulsations which are detected at the Earth's surface. Protons which are reflected from the bow shock can produce ion cyclotron instabilities in the upstream solar wind. The ion cyclotron waves are then convected downstream to the magnetopause and couple energy into the magnetosphere.

A number of the energy sources and types of plasma waves in the magnetosphere are illustrated in Fig. 4. This figure is far from comprehensive, and there are many other types of plasma waves and pulsations, and possibly many other sources of energy.

w Latitude Boundary Layer ICW Solar Wind

Cleft Pulsations

Magnetotail

ICW Unstructured IPDP

Kelvin Helmholtz Instability

1 -,- - rift Mirror and Bounce Resonance Instability '-

HM Resonances Figure 4. The locations and sources of some plasma waves and pulsations in the magnetosphere. ICW: ion cyclotron waves; HM: hydromagnetic.


2 A BRIEF HISTORY

Historically, the first reported observations of geomagnetic pulsations are attributed to Stewart (1861) who noted pulsations of the geomagnetic field in records of a large geomagnetic storm recorded at Kew Observatory (Greenwich, England). Later observations of pulsations can be found in numerous reports including those by Angenheister (1920), Rolf (19311, and Sucksdorff (1939). The latter two authors studied features of large- amplitude7 highly monochromatic pulsations with frequencies near 10-2OmHz which are now classified by morphology as giant pulsations (Pg). An example is given in Fig. 1.

The International Geophysical Year (1958) (ICY) led to a substantial increase in the number of observations of geomagnetic pulsations (see Fig. 1 in Saito, 19691, and established the study of plasma waves and pulsations in the magnetosphere as a mature discipline. Another result of the ICY and the large increase in studies of pulsations was an associated and bewildering increase in the number, morphology and 'types' of pulsations. In order to add structure to this variety, a subcommittee of the Inter- national Association of Geomagnetism and Aeronomy (IAGA) suggested a classification scheme based on two main classes (Jacobs et a/., 1964). The first class includes pulsations of more or less continuous character and is denoted PC. This class is further broken into subclasses based on the pre- dominant frequency or period of the pulsations (Table 1). The second class includes more or less impulsive or irregular pulsations and is denoted Pi. A more complete set of classifications, including subclasses of the above7 is given by Saito (19691, Jacobs (1970) and Orr (1973).

The earliest satellite observations of pulsations are those of Coleman

Table 1. The IAGA classification of pulsations.

Notation Period range (s)

Continuous

Impulsive

3 CLASSIFICATION SCHEMES 487

et a/. (19601, Sonett et al. (1962), Judge and Coleman (1962), Nishida and CahilI(l964), and Pate1 and CahiIl(l964). Cummings et al. (1969) reported observations of relatively monochromatic magnetic pulsations recorded at synchronous orbit by the magnetometer on ATS 1. These pulsations were largely transverse, had peak-to-peak amplitudes of up to 20nT, and had frequencies in the range 3-2OmHz. They attributed these pulsations to hydromagnetic resonances in the magnetosphere.

While the number of experimental observations of pulsations was quite large before 1960, adequate theories were few and sparse. Perhaps the most fundamental, theoretical contribution before 1960 was Dungey's treatise on Alfvin waves in the outer atmosphere (Dungey, 1954). In this treatise he introduced the concept of hydromagnetic resonances and suggested that the Kelvin-Helmholtz instability at the surface of the Chapman-Ferraro cavity might be an important source of energy for geomagnetic pulsations. Both of these concepts, with some modifications, have played an extremely important role in modern theories of geomagnetic pulsations.

Contributions to early theories on pulsations were also made by Kato and Watanabe (1954), Dessler (1958), and Obayashi and Jacobs (1958). Complementary reading material for this chapter can be found in numerous earlier reviews of pulsations (e.g. Kato and Watanabe, 1957; Hultqvist, 1966; Troitskaya, 1967; Campbell, 1967; Dungey, 1967; Saito, 1969; Jacobs, 1970; Orr, 1973; Nishida, 1978; Rostoker, 1979).

3 CLASSIFICATION SCHEMES

Even though the morphological classification schemes in Table 1 have served well in the past, they are gradually being replaced by schemes based on genetical classifications (physical processes and generating mechanisms). This is a natural evolution in any field of science, and we shall attempt to follow genetical classifications if possible. However, it is convenient 'to divide the pulsations into three distinct frequency bands, low-frequency (1 mHz to 10 mHz) mid-frequency (10 mHz to 0.1 Hz) and high-frequency (0.1 Hz to lOHz), and two types of wavepackets, continuous and impulsive. Impulsive pulsations are caused by a variety of transient phenomena, including sudden impulses from the solar wind, flux transfer events, and rapid changes in magnetospheric convection. Continuous pulsations in the low-frequency band are caused predominantly by hydromagnetic instabilities such as the Kelvin-Helmholtz and drift-mirror instabilities. Many pulsations in the mid-frequency band are thought to originate from proton cyclotron instabilities in the solar wind, with propagation through the magnetopause. Continuous pulsations in the


Table 2. Classification of magnetospheric plasma waves and pulsations.

T Y P ~ Spectral band Sources

Continuous

Continuous

Continuous

Impulsive

Impulsive

Impulsive

1-10mHz (low freq.)

10-1OOmHz (mid-freq.)

0.1-1OHz (high freq.)

1-1OHz (low freq.)

10-1OOmHz (mid-freq.)

0.1-1OHz (high freq.)

Drift-mirror instability Bounce resonance Kelvin-Helmholtz

Kelvin-Helmholtz Proton cyclotron instabilities in the solar wind

Ion cyclotron waves due to proton cyclotron instabilities in the magnetosphere

Sudden impulses from the solar wind Abrupt changes in convection in the

magnetotail (in conjunction with substorm expansive phases)

Flux transfer events

Changes in convection in the magnetotail Field-aligned current-driven instabilities

Field aligned current driven instabilities

high-frequency band generally are caused by ion cyclotron instabilities in the magnetosphere. These frequency ranges are approximate of course, and there is considerable overlap in the spectral content and frequencies of the pulsations produced by various mechanisms. Table 2 gives a summary of this classification scheme, and the various sources of energy for the pulsations.

4 INSTRUMENTS AND OBSERVATIONAL TECHNIQUES

Pulsations have been detected and recorded by using a wide variety of magnetometers, including the older, magnetic variometers, proton preces- sion magnetometers, fluxgate magnetometers, rubidium vapour magnetometers and induction coil magnetometers (see Campbell, 1967, for a review). Geomagnetic pulsations also lead to induced earth currents, and these currents are easily detected using telluric electrodes to measure the horizontal electric fields near the Earth's surface (see e.g. Garland, 1960).

5 WAVES IN A MAGNETOPLASMA 489

Electric fields associated with pulsations can be detected in the ionosphere by using various radar systems including those measuring the intensity and Doppler velocity of ionospheric plasma waves (Greenwald et al., 1978; Allan and Poulter, 1984) and incoherent scatter radars. In the magnetosphere, the electric fields associated with pulsations have been seen by a variety of instruments including double probe (Mozer, 1973) and electron beam instruments (Melzner et al., 1978; Junginger et al., 1984).

Pulsations and plasma waves can also modulate energetic particle distributions in magnetospheric plasmas (Brown et al., 1968; Lanzerotti et al., 1969). A proper interpretation of the particle data can lead to a much clearer understanding of the plasma waves causing the pulsations (e.g. Kivelson and Southwood, 1983, 1985a).

5 WAVES IN A MAGNETOPLASMA

5.1 The dielectric tensor for a hot magnetoplasma

The various plasma environments in the ionosphere and magnetosphere allow a wide variety of plasma waves and instabilities. One way to add a measure of order to this variety is to introduce the conductivity tensor, or the effective dielectric tensor for the plasma. The sets of equations we use are the Maxwell equations

and the Boltzmann equations

9

collisions

where fm(r , v , t ) is the distribution function for species a (e.g. Montgomery and Tidman, 1964). The various parameters are defined in the Appendix. In general, we assume that only the Lorentz force

is important.


In practice, over much of the magnetosphere, but not the ionosphere, we can assume a collision-free plasma, and (a fa/at)co~~isions = 0. Consequently the Vlasov equation is appropriate.

To determine the current density j(r, t ) in equation (5.1.4) we note that the number density is:

and the current density is

For a magnetoplasma we have a zero-order magnetic flux density BO = [O,O, Bo] , and j and E are first-order quantities. Then to determine fa , we linearize the Vlasov equation to obtain

where the subscript 0 denotes zero-order and 1 denotes first order quantities. The first three terms on the left-hand side of equation (5.1.9) are just the total time derivative dfla/dt. Consequently we find f la(r , v , t ) by integrating with respect to time along a zero-order trajectory in space. That is

f ia(r, V , t ) - fla(ro, VO, to) = - (El + u x BI) . -

We shall assume that the various first-order parameters vary as exp [i(k r - u t ) ] , where both u and the components of k can be complex. then equations (5.1.2) and (5.1.4) give

We then introduce the effective dielectric tensor

where

The conductivity tensor u must be determined by solving equation (5.1.10)

5 WAVES IN A MAGNETOPLASMA 49 1

for the first-order distributions fla. Now equation (5.1.11) becomes

where C is the tensor representation of k x k x = kkT - k 2 / . Except for some specific conditions (e.g. a cold plasma), evaluation of

the integral in equation (5.1.10) is a tedious process, but the procedure can be found in a number of references (see e.g. Stix, 1962; Ichimaru, 1973; Cuperman, 1981; Oraevsky, 1983, and references therein). The results of the integration for a general hot magnetoplasma give the following components for the dielectric tensor

J,, is a ~ess-el function of the nth order and

The denominator in equation (5.1.15) is zero when w = kllvll - nuga. When n = 0, we have a Cherenkov resonance which leads to Landau damping. When n # 0, the cyclotron resonances occur.

5.2 Hydromagnetic waves

The dispersion relation is found in the usual way by finding solutions to


where

k2c2 N2=- k - B o and cos 6 = -.

03 k B o I

'The dielectic tensor for the hydromagnetic limit is determined by choosing the wavelength to be much larger than an ion gyroradius (kvÂ± \up 1 < ̂ I), and the frequency to be much less than the ion cyclotron frequency (a?/ lugs 1 4 1) (e.g. Barnes, 1966). When there is no relative drift between electrons and ions, the dispersion relation leads to the two uncoupled equations

and

Specific coefficients of the dielectric tensor are given by Hasegawa (1975, Chapter 2.4). In an isotropic plasma, equation (5.2.2) gives the shear Alfven wave, and equation (5.2.3) gives a magnetosonic, compressional Alfven wave, and an ion acoustic wave. If the plasma has a substantial component ofcold electrons then Â£3 is very large, and equation (5.2.3) becomes

which gives the compressional Alfven wave, and the non-oscillatory Weibel modes. The shear Alfven mode is the only one which is not damped. The ion acoustic wave is a longitudinal wave and is subject to Landau damping, while the compressional Alfven wave is subject to transit time damping (Stix, 1962).

The dispersion relation (without damping) for the three hydromagnetic modes can also be derived from the two-fluid equations (e.g. Stringer, 1963). The dispersion relations for these modes, and the modes near the ion cyclotron frequency are given in Table 3, and the curves are plotted in Fig. 5. A comprehensive review of the theory of hydromagnetic waves in the magnetosphere is given by Southwood and Hughes (1983).

Table 3. Dispersion curves and polarization for plasma waves near and below the ion cyclotron frequency (two fluid model, /3 =

Branch Mode Frequency range Dispersion relation Polarization of electrical field

Low frequency Slow magneto-acoustic or (slow) ion acoustic

Low Second ion cyclotron

Intermediate Shear Alfvkn

Intermediate First ion cyclotron

Intermediate Acoustic

High frequency Fast magneto-acoustic, (fast) compressional Alfvkn

High Helicon or Whistler k2v.$ cos 6

w > oils oil=

Longitudinal E \\ k

Longitudinal

Left-hand elliptical in plane -I- Bo

Left-hand circular

Longitudinal

Right-hand elliptical, in plane J- BO

Right-hand circular


5.3 Waves in a cold magnetoplasma

The dispersion relation for a cold magnetoplasma can be obtained from equations (5.1.15) and (5.2.1) by assuming that UT < ̂ UA, and making the suitable approximations in deriving the components of the dielectric tensor (e.g. Montgomery and Tidman, 1964). A simpler, and more direct approach is to solve the linearized equation of motion for a single particle, and use the equation j = Sn nag&, where ua is the velocity for species a (e.g. Stix, 1962, Chapter 1). The zeroth-order dielectric tensor is then

= [a" 'a i] (5.3.1)

where

and

In a cold plasma, the dispersion relation gives only two modes, the fast or compressional Alfvkn mode, and the shear Alfvkn mode. The fast right- hand polarized mode is not affected by the ion cyclotron resonances, whereas the shear Alfvkn mode becomes an ion cyclotron wave as u -+ <aga .

(e.g. Stix, 1962, Chapter 1). The shear mode has cutoffs (infinite phase velocity) when L = 0, and resonances (zero phase velocity) at u = (L + m). In a multispecies plasma, the two modes have a crossover (the dispersion curves cross) when k 11 BO and Â£1 = 0. The ion acoustic (low- frequency) and shear Alfvkn modes in a single ion species warm plasma also have crossovers at wga.

For propagation at an angle to Bo, the crossover is not present, but the dispersion curves separate as shown in Fig. 6. In this example, the crossover frequency is ucr = (1 + 1 5 f 1 2 u g ( ~ e + ) where 7 is the He"*" abundance

5 WAVES IN A MAGNETOPLASMA

Figure 5. Dispersion curves for waves in an ion-electron magnetoplasma. The curves were calculated using the dispersion relations for the two fluid model given by Stringer (1963). Propagation is at an angle of 45 . The low-frequency branch corresponds to the slow magneto-acoustic (ion acoustic) and second ion cyclotron modes (see Table 3). The intermediate-frequency branch corresponds to the shear Alfven and first ion cyclotron modes. The high-frequency branch corresponds to the fast magneto-acoustic or compressional Alfven mode, and helicons or whistlers.

ratio. As the frequency increases through the crossover frequency, the dispersion curve for the left-hand mode is connected continuously with the dispersion curve of the right-hand mode (class 11). Similarly, with increasing frequency, the dispersion curve of the right-hand mode is connected with the dispersion curve of the left-hand mode (class 111). For the example in Fig. 6, the cutoff frequency due to H z is ucf = (1 + 3q)ug(He+), (e.g. Roux et al., 1982). The resonances and the cutoffs lead to stop bands, between we. and wcf where the left-hand mode cannot propagate. For perpendicular propagation, the class I11 branch has a resonance at the bi-ion hybrid frequency, UM = [(I + 3q)/(l - 37/4)] 1'2ug(He4') (see also Stix, 1962, Chapter 2).

In the low-frequency, hydromagnetic limit (u <S \ uga 1) the dielectric tensor for a cold plasma is diagonal with

Displacement currents have been neglected in equation (5.3.2) and (5.3.3).


NORMALIZED WAVE NUMBER (K= k V ~ / f l H+)

Figure 6. Dispersion curves for plasma waves in a cold, multispecies (hydrogen, helium) plasma (after Rauch and Roux, 1982). In the diagram QH+ is the hydrogen cyclotron frequency. The subscripts 'bi', 'cf' and 'cr' stand for the bi-ion hybrid, cutoff, and crossover frequencies respectively. The dotted lines indicate RH polarization and the solid lines indicate LH. The angles of propagation are indicated on the curves. The ratio of the number density of He' to the total number density is 0.25.

The perpendicular components, EL, of the dielectric tensor are due to the perpendicular polarization currents carried by the ions, where

In addition, since the frequency is very low, e33 is large, and the parallel electric field E3 = 0 due to the mobility of the cold electrons along the magnetic field lines.

Propagation of low-frequency plasma waves in the ionosphere is somewhat more complicated because of the finite collision frequencies, and consequently we use the Boltzmann, rather than the Vlasov equation. However, in a cold plasma, we can assume that the collisions add an effective force mavava where urn is the collision frequency of species a with all

5 WAVES IN A MAGNETOPLASMA 497

other species (including neutrals). Then the linearized equation of motion for a single particle of species a is

Now the equivalent dielectric tensor is similar to equation (5.3.1), but

and

5.4 Reflection and transmission through the ionosphere

The ionosphere substantially alters the electric and magnetic fields of plasma waves which are incident from the magnetosphere. A proper interpretation of any ground-based observations of the magnetic fields of pulsations requires a reasonable model for the reflection and transmission of plasma waves by the ionosphere.

When a localized shear Alfvkn wave is incident on the ionosphere, the resultant ionospheric Pedersen currents are poloidal (cur1 free), whereas the Hall currents are rotational (divergence free). The ionospheric Pedersen currents lead to a reflected Alfvkn wave, whereas the Hall currents give a reflected fast magnetosonic or compressional wave which generally, is envanescent. The solenoidal Hall currents give the magnetic fields which are detected below the ionosphere, and contribute to the apparent 90' rotation of the horizontal magnetic field on the ground when compared with the magnetic field just above the ionosphere (Nishida, 1964; Hughes and Southwood, 1976). At the ground, the magnetic fields produced by the Pedersen currents tend to shield the magnetic fields produced by field aligned currents.

An incident guided compressional wave has no field-aligned current and its electric field is inductive or rotational. The ionospheric Pedersen currents are rotational or solenoidal and produce a reflected fast mode. The Hall currents produce a reflected Alfvkn wave.

It is also important to realize that in addition to the component due to


the Hall currents produced by the Alfvkn wave, the magnetic fields on the ground can have substantial components from currents on oblique field Sines. Even at 60Â geomagnetic latitude the field-aligned current's contribution is almost as large as the Hall current's contribution (Tamao, 1984).

To determine the reflection and transmission coefficients for the ionosphere we shall use a simple thin-sheet model for the ionosphere. In the low-frequency limit (a < va, 1 aga 1) the conductivity tensor can be determined from equation (5.3.6) by using suitable approximations, and has the components

where up, UH and 011 are respectively the Pedersen, Hall and parallel (field- aligned) conductivities. For the model we use the height integrated Pedersen and Hall conductivities Ep and EH.

We choose an incident Alfvkn wave with an electric field E exp i(kxx + kzz - a t ) , where E = [Ex, 0, 01 , the z-direction is along Bn (positive downward) and the ionosphere lies in the xy plane. The electric and magnetic fields, satisfy the compressional and shear ASfvkn wave equation above the ionosphere, whereas in the region between the ionosphere and the Earth's surface, the fields are quasistatic and have solutions of the form exp(1 kxlz) + exp(- [kx [z).

Using V x E = - aB/at, we have in the magnetosphere

and

where B = Bn + b and b is the first-order or perturbation field. The electric field consists of two parts, incident and reflected, and consequently

where I and R stand for incident and reflected respectively. Equation (5.1 -4) then gives:

where (air) and (mag) indicate the fields just below and above the ionosphere respectively. In the atmosphere, V x b = 0 and it follows that

6 PLASMA WAVES AND INSTABILITIES 499

by (air) = 0 giving a reflection coefficient

E X R R---EM-Ep (5.4.5) &I EM + Z p '

where EM = ( F ~ u ~ ) - l . Glassmeier (1984) has derived a more general formula for the reflection

coefficient, and has shown that if E IlVEp and E IIVEH in the ionosphere, then the reflection coefficient is still given by equation (5.4.51, with Ep replaced by the local height integrated Pedersen conductivity Ep(x, y).

On the ground, using the magnetostatic solution, the magnetic field is

(Nishida, 1964, 1978), where h is the height (positive) of the ionosphere. This equation shows the 90' rotation, and also indicates that waves which have large horizontal wavenumbers will be attenuated at the Earth's surface.

More complete solutions, including numerical solutions, for transmission of pulsation fields through the ionosphere can be found in Nishida (1964), Field and Greifinger (1965), Greifinger and Greifinger (19651, Inoue (1973) and Hughes and Southwood (1976).

6 PLASMA WAVES AND INSTABILITIES IN THE MAGNETOSPHERE

In general, we can divide the instabilities which occur in the magnetosphere (or any plasma) into two types, those occurring in uniform plasmas (velocity-space instabilities) and those occurring in non-uniform plasmas (coordinate space instabilities). The velocity-space instabilities occur because of anisotropic, or multihumped velocity distributions in a uniform plasma. These instabilities can be studied by using the general dielectric tensor, equation (5.1.15), for a uniform plasma. The coordinate space instabilities occur in a non-uniform plasma with gradients in density, pressure or other parameters. Certain of these instabilities require a specific stable geometry for the configuration of the zeroth-order magnetic fields and plasma parameters. For example, the magnetosphere is approximately axisymmetric (excluding the magnetotail), and allows mirroring of charged particles due to the dipolar configuration. This geometry leads to two further characteristic frequencies, the bounce frequency and the azimuthal drift frequency.

The variety of instabilities which occur in plasmas is immense, and we


shall briefly discuss only a few of the instabilities which are directly relevant to the observational features of pulsations. Hasegawa (1975) gives a comprehensive outline of the many instabilities which might occur within the magnetosphere. Southwood (1976) and Southwood and Hughes (1983) treat many of the hydromagnetic instabilities which are applicable to the magnetosphere.

6.1 Instabilities in uniform magnetoplasmas

In the hydromagnetic regime, we shall consider two velocity-space instabilities, mirror instabilities and non-neutral beam instabilities. At higher frequencies, velocity-space instabilities, including the various cyclotron instabilities, are likely the major source of energy for magnetospheric plasma waves and pulsations, particularly those in the 0.1-1OHz band.

In the hydromagnetic approximation, the components &22 and ~ 3 3 of the dielectric tensor both contain the integral

This term leads to damping or instability of the corresponding mode. For the ion acoustic mode we have Landau damping or inverse Landau damping. For the fast compressional mode, we have transit time damping (Stix, 1962).

The mirror instability occurs in a plasma with a large pressure anisotropy, specifically %- bll. This hydromagnetic instability follows from the dispersion relation in equation (5.2.4), where we have included a component of cold electrons. We assume that the zeroth-order distribution function of each species is bi-Maxwellian, and

and choose almost perpendicular propagation kL % kll. Then

where Z is the plasma dispersion function (Fried and Conte, 1961)

and the derivative Z r (x) = dZ/dx = - 2 [1 + xZ(x)] . We make the further approximations that u/kll + u~~~( ions ) , and u/k+ u ~ . Then using


Z ( x ) = i,h e x ( [ x 1 :̂ 1) we obtain the dispersion relation (Hasegawa, 1975)

a = - i k l l ~ ~ l l (ion)

For kI1 real, u is always purely imaginary, giving a non-oscillatory, decaying or growing mode. If the plasma is isotropic, <3Â = (il, then a is negative and imaginary, and we have transit time damping (Stix, 1962; Barnes, 1966). Instability and growth occur when

The magnetic fields in the mirror instability resemble those in Fig. 7. The pattern moves along Bo in time, and there is a resonance between the wave

-VpB FORCE

I PLASMA

DENSITY

Figure 7. A schematic of the mirror instability (after Hasagawa, 1969). The density of the dots is proportional to the plasma density.


and the particles travelling near the phase velocity m/ks along the field lines. The resonant particles experience a magnetic field which varies slowly with time, so that in the guiding centre coordinates, the first adiabatic invariant p,,, is conserved. Then the particle's guiding centre moves as if subject to a potential pmB. Consequently the motion of the guiding centres is similar to the motion of a particle in a periodic electrostatic potential leading to conditions analogous to those for Landau and inverse Landau damping.

The hydromagnetic mirror instability occurs when fl^, > Bs, or the perpendicular velocities exceed the longitudinal ones. Since the magnetic field has a mirror-like geometry, the particles are forced toward regions of weak magnetic field. This increase in particle density, and the large perpendicular velocities give diamagnetic fields which further reduce the magnetic field, and instability develops.

The remaining velocity-space instabilities we shall discusses are associated with transverse (electromagnetic) waves propagating parallel to Bo. Consequently k E = 0, and kL = 0. Then Â£1 = Â£2 and Â£1 = Â£2 = 0 in the dielectric tensor (equation (5.1.15)). We choose a new basis set composed of the three orthogonal vectors U L = 2 1 1 2 [1, i, 01, U R = 2 ' " [ I , -i, 01 and UT, = [O, 0,1] where the first and second vectors correspond to left- and right-hand polarization respectively. In the rotated coordinate system E' = [ER, EL, 0] where ER = &E and EL = u E . Similarly, the components of the rotated dielectric tensor become

where the plus and minus signs in the denominator correspond to the R and L modes respectively.

The dispersion relations for the two modes are

An Alfvkn wave for the L mode, trons, cold ions

instability can be found by using the dispersion relation and choosing a three-component plasma with cold elec- (the cold plasma is neutral) and a monoenergetic field-

aligned electron beam with particle velocities V B (Kimura and Matsumoto, 1968). Then integrating the terms in equation (6.1.7), the dispersion relation becomes


where 7 is the ratio of the number density of the streaming electrons to the number density of cold electrons. We assume that the total current carried by the electron beam is small enough so that the magnitude of the transverse magnetic fields generated by the beam is much less than Bo. In the limit 1 u 1-0 and 1 k \ -> 0, the dispersion relation becomes

The threshold for instability is VB > VA, and the frequency of the instability is vgi. Kimura and Matsumoto have shown that in the magnetosphere, typical frequencies for this instability can be in the 1-lOmHz band.

The ion cyclotron instabilities are electromagnetic instabilities which occur near the ion cyclotron frequencies wga, and satisfy the resonance condition w - kvl = Â wga (n = + 1 in equation (5.1.15)). We derive the dispersion mixture assume relation

relation by assuming that the plasma is quasineutral and has a of hot and cold ion species and cold electrons. We shall also that the proportion of hot ions is very small. The dispersion now has the form

where equation (5.3. l),

where c indicates a summation over cold species (including electrons), and ( C R L ) ~ ~ ~ must be evaluated using the integral in equation (6.1.7). We choose a bi-Maxwellian (equation (6.1.2)) for the hot ions, giving

2 1 Ci> Â Ugh (cR.L)~o~ = s 3 \ A h - [(Ah + l)(+fa)gh - W) Â ugh]

h I where h denotes a hot species, U = JUT and At, = T J . A / T ~ ~ ~ - 1 .

If 1 Re(^) 1 ^> 1 Im(w) I, then the growth rate is

where D is given in equation (6.1.11). To illustrate ion cyclotron instabilities further, consider a plasma with

cold electrons, cold protons and hot anisotropic protons. The ratio of the number density of hot protons to the plasma number density is 7 e 1. Then Re(D) is approximately ( C R , ~ ) ~ ~ , ~ . We find Im(D) by noting that the


argument of Z, ((os Â ug)/ Umk) %> 1 and consequently

where i denotes the ion species for the hot ions. For the R mode (fast or high-frequency mode), resonance occurs when

os - kvs = - q i and vll > m/k, indicating that particles moving faster than the phase velocity of the wave are resonant with the wave. The L mode has a resonance at u = osgi, and consequently is limited to frequencies less than ugi, in the vicinity of the resonance. Since os - kvIl = aim for the L mode, the particles must be travelling in the opposite direction to the phase velocity of the wave. From (6.1.10) we can see that the wave grows if

- w A > - (R mode) (6.1.15)

(agi + a )

and

03 A > - (L mode).

(wgi-w)

The R mode is unstable if the hot ions have Tll > TL, whereas the L mode is unstable if TJ. > TI,. Figure 8 shows some typical growth rates for the L mode in a plasma composed of cold electrons, cold protons and anisotropic hot protons.

The growth rates and convective growth rates for the cyclotron instability are modified considerably by the presence of both cold or hot heavy ions. The stop bands (see Fig. 6) for the L mode inhibit the instability in the spectral regions between osgm and ( ~ c t ) ~ . Just below the resonances at uga, the phase velocity of the wave is very low, allowing a large number of particles to interact with the wave, and giving sharp peaks in the convective growth rates (Gomberoff and Cuperman, 1982). An example is given in Fig. 9. where large convective growth rates occur in a sharp peak just below the oxygen cyclotron frequency. The presence of the He' has eliminated all growth above the He cyclotron frequency.

6.2 Instabilities in non-uniform magnetoplasmas

The drift wave or universal instability occurs in plasmas with spatial gradients in the distribution function (e.g. the density) (Mikhailovskii, 1967;

6 PLASMA WAVES AND INSTABILITIES

Figure 8. The growth rates of the ion cyclotron instability in a hydrogen plasma (see equation (6.1.14)). Solid curve: A = 1 .O, VA/ Ul = 0.1. Long dash: A = 1 .O, v J U . = 0.2. Short dash: A = 1.0, v f l l = 0.5.

Figure 9. Convective growth rate in a H4', He4' 0' plasma with 45% cold H + , 50% cold He"", 4% cold 04' , lÂ¡7 hot H4', A = 1.0 and T = 30keV.


Krall, 1968). The magnetosphere has a variety of regions with spatial gradients in the plasma parameters, and consequently it is not surprising that the possibility of drift instabilities in the magnetosphere has been considered by a number of authors, including Chamberlain (1963), Swift (1967), Tamao (1969), Coroniti and Kennel (1970) and Hasegawa (1971).

Most of the drift waves which are relevant to the magnetosphere are in the hydromagnetic regime (w < \ wga I) . Perhaps the simplest drift kinetic mode is that for a low (3 ((3 < ̂ m&) magnetoplasma with a density gradient perpendicular to a uniform magnetic field. The unstable mode is electrostatic, and propagates in a direction perpendicular to the ambient magnetic field. For a moderate-^ plasma (me/mi < ̂ (3 < ̂ I), the electrostatic wave couples with a shear Alfvkn wave, and for a high-fi plasma (0 = I), the fast compressional mode is coupled to the electrostatic wave.

To find the dispersion relations for these instabilities, we use the drift kinetic equation, which is derived from the Vlasov equation with the assumption u < ̂ \ wga \ and k < ̂ r; l . Then we average the Vlasov equation over time and space and introduce a guiding centre distribution function foa(r, vn, pm, t ) where pm = r n v ? / 2 ~ ~ is the first adiabatic invariant. Let u-L = (vod + vd) where vod and vd are the drifts due to the zero-order and first-order fields respectively, and 4 = VII . Once again we assume that there are no zero-order electric fields. Then

and

where we have assumed that the scale sizes of the zero-order magnetic fields are much larger than the scale sizes of the first-order magnetic fields, b. Then we have the drift kinetic equation

The dielectric coefficients and the dispersion relation are found by using a procedure essentially similar to that for an isotropic plasma, except that the linearized drift kinetic equation is used to determine the perturbed distribution functions. Details are given by Mikhailovskii (1967). A relatively


complete discussion of the various drift instabilities is given by Hasegawa (1975) and Mikhailovskii (1983).

To illustrate the drift kinetic dispersion relation for a Maxwellian plasma with moderate (3 =Â m^m\ and a density gradient perpendicular to Bo, we shall use the results of Mikhailovskii (1967) (see also Hasegawa, 1971). We assume the plasma is composed of electrons and protons with a cold, but finite temperature. We also assume that (vTi)coid 6 fai/kll < (~~e)cold and kÂ (fTi)cold/~~i s$ 1 . The three wave modes (Table 3 and Fig. 5), the fast compressional, Alfvin and ion acoustic, are coupled but we can simplify the problem by assuming that the propagation is dominated by the shear Alfvkn mode, and obtain a dispersion relation for the coupled ion acoustic and shear Alfvin modes.

Using the approximations for the plasma dispersion function Z, the dispersion relation becomes (Hasegawa, 1971)

where

* &v^-a fain. =- = kl VD (6.2.5)

faign.

is the drift wave frequency, x = Vln(n0) and VD is the diamagnetic drift velocity. The three real roots are (Hasegawa, 1975):

where the second equation gives the Alfvkn mode (+) and the ion drift mode (-).

We can combine the essential characteristics of the configurations required for the drift and mirror instabilities to produce a propagating instability. For example, if conditions are suitable for the mirror instability equations (6.1.5) and (6.1.6), then a spatially periodic structure will grow without oscillation. If a density or temperature gradient exists, then the diamagnetic drift velocity of the protons

will give oscillations with a frequency fai = v ~ k ~ = u*. In the magnetosphere, a hot plasma also experiences gradientlcurvature drift, and pressure gradient drifts, which considered in conjunction with mirror instability will lead to oscillatory fields. Hasegawa (1969) has shown that a more


rigorous derivation of the drift-mirror instability is a plasma with a density gradient gives the dispersion relation ( k ~ > k l )

which is just the dispersion relation for the mirror instability with the added term w?.

A more generalized treatment of these instabilities in magnetospheric plasmas, particularly ring-current plasmas, is given by Southwood (1976). Southwood found, for example, that the ring-current plasmas can be unstable before the conditions for drift-mirror instability occur.

Dungey (1954) first pointed out that the Kelvin-Helmholtz instability could be a likely source of energy for low-frequency pulsations, and since that time the Kelvin-Helmholtz instability at the magnetopause and low- latitude boundary layer has remained a viable and important mechanism for producing plasma waves and pulsations.

We consider two regions of uniform plasma, one at rest with velocity V I = 0 and mass density p i , the other with velocity v2 and density p i . The ambient geomagnetic field Bo, is also uniform. When the boundary is dis- placed by a plasma wave, the normal displacement and the total perturbed pressure are continuous across the boundary. This gives us the two boundary conditions:

where En is the normal component of the displacement vector and b is the perturbation magnetic field.

We look for solutions of the form exp[i(kSr - a t ) ] where k = kn + kt, and kn and kt are respectively the components of the k vector normal and tangential to the surface separating the fluids. From the equation of motion for a single fluid

we find


where w ' = w - k - v is the frequency of the wave in the frame of the moving fluid. The equation for En, equations (6.2.9) and (6.2.10) and the dispersion relation for the fast compressional mode (Table 3) give a 10- degree dispersion equation for the instability (Southwood, 1968).

If the plasma is incompressible (V- ? = O), the taking the divergence of both sides of equation (6.2.12) gives

Consequently kn = * i 1 kt 1 , and the instability produces a surface wave which decays exponentially in a direction normal to the surface separating the plasmas. Now we obtain the dispersion relation for the instability from equations (6.2.9), (6.2.10) and (6.2.12) giving

Noting that W [ = u(i.e. V I = O), u i = w - k t e v2, we have

and the mode is unstable when

The nearly dipole geometry of the inner magnetosphere, excluding the magnetotail and polar-cap field lines, leads to two additional periodicities in the motions of charged particles. Mirroring of the particles in the dipolar geometry leads to a characteristic bounce period TI), the time for the particle to travel from the mirror point in one hemisphere to that in the other hemisphere and return. The azimuthal drift of particles gives a characteristic drift period ra, the time required for the particle to drift once around the Earth. Expressions and approximations for q, and ~d are given by Hamlin et at. (1961).

The derivation of the dielectric tensor for an isotropic magnetoplasma leads to resonances and instabilities at frequencies given by u = k l l ~ l i - nuga, with n = 0 corresponding to Landau damping, and n + 0 corresponding to the cyclotron instabilities. In the dipolar geometry we are considering here, resonances occur when (Southwood et al., 1969)


where ub = 27i-/~b, ud = 27i-/~d, n is an integer and the wave field varies at exp [i(n<f> - ut)] , where 4 is azimuth or longitude. Since protons drift westward, resonance requires a westward propagating wave. Similarly electron resonance requires an eastward propagating wave. Details pertaining to the calculations of the requirements for instability, and the growth rates of these bounce resonance interactions, are given by Southwood et al. (1969) and Southwood and Hughes (1983).

Southwood and Kivelson (1982) have given a relatively simple graphical illustration of bounce resonances associated with charged particles and standing hydromagnetic wave structures. Figure 10 (from Southwood and Kivelson, 1982) is a simple representation of a second-harmonic wave, with the dipolar geometry of the Earth's magnetosphere mapped to a Cartesian

Figure 10. Trajectories of bounce resonant particles in a second harmonic standing wave. E, W, N, S, stand for east, west, north and south respectively (after Southwood and Kivelson, 1982).

7 PLASMA WAVES AND PULSATIONS IN THE LOW-FREQUENCY BAND 51 1

geometry. The boundaries at N and S correspond to the ionosphere in the northern and southern hemispheres respectively. The pattern is periodically repeating in an east-west (E-W) direction. The magnetic field is directed from south to north, and the electric field is indicated by the plus and minus signs (+, - ) with plus corresponding to a westward field. The heavy, straight lines show the paths of the guiding centres of two ions which both satisfy the resonance condition w = wb + mud. Both particles see only eastward directed fields of various amplitudes, and consequently suffer net acceleration. The first particle, indicated by the solid line, has a smaller pitch angle than the second particle, indicated by the dotted line. Consequently the first particle mirrors nearer the maximum in the electric field, and gains much more energy on each bounce.

7 CONTINUOUS PLASMA WAVES AND PULSATIONS IN THE LOW-FREQUENCY BAND (1-10 mHz)

7.1 Introduction

Geomagnetic pulsations in the low-frequency band are the magnetic signatures of plasma waves which have wavelengths which are comparable to the dimensions of the magnetosphere. In general, these low-frequency waves can be described by the hydromagnetic equations, since the scale sizes are much larger than a typical ion Larmor radius, and the frequencies are much lower than the gyrofrequencies of ionized hydrogen, helium and oxygen in the magnetosphere. Exceptions to the validity of the hydromagnetic approximation do occur, however, particularly in hydromagnetic resonances where the scale size transverse to the magnetic field can approach the Larmor radius. Then the wave modes must be described by using the equations for the kinetic Alfvkn wave (Hasegawa and Chen, 1 976).

The large wavelengths of the low-frequency plasma waves indicate that boundaries, including the ionosphere and magnetopause, and non- uniformities (e.g. the Earth's dipole field and the plasmapause) play an extremely important role in governing the configurations of the fields associated with the waves. Dungey (1954) was aware of this feature when he addressed the problem of resonant hydromagnetic modes or field-line resonances in a dipole field. Southwood and Hughes (1983) give a comprehensive review of the hydromagnetic theory appropriate for these low-frequency pulsations.

In the past, low-frequency continuous pulsations have been largely classified as Pc5. The large amplitudes of these pulsations (up to hundreds


of nanotesla in the auroral zone) and the low frequencies (less than lOmHz) allow easy identification and recording of these pulsations, even on standard magnetograms from mechanical variometers.

Low-frequency pulsations tend to have their maximum amplitudes in the auroral zones ( 6 5 - 7 0 geomagnetic latitude) (Jacobs and Sinno, 1960; Obertz and Raspopov, 1968). The region of the maximum intensity follows the approximate position of the auroral oval (Samson, 1972) except near local noon where the peak amplitudes can be slightly poleward of the average auroral oval. In the auroral zone, low-frequency pulsation activity peaks between 0600-1000local time (LT), with a smaller maximum near 1800-2200 LT (Saito, 1969). At mid-latitudes, these peaks are not evident, and low-frequency pulsations occur with equal probability over most of the day. The low-frequency pulsations are often most sinusoidal or narrow- band in the morning and early afternoon. In general the late-afternoon and night-time pulsations have short wavetrains with few cycles.

7.2 Hydromagnetic resonances

In an attempt to analyse the modes of the low-frequency hydromagnetic waves in the Earth's magnetosphere, Dungey (1954) developed the wave equations which are appropriate for a dipole field geometry. The equations for the two transverse magnetic field components are coupled except in a limited number of special cases. If we assume that the azimuthal or longitudinal dependence is given by eim$, then for m = 0, the toroidal (azimuthal) and poloidal components are uncoupled. In this case the toroidal or guided Alfvkn mode leads to standing waves along magnetic field lines because of the boundary conditions at the ionosphere and Earth. The resonant frequencies or eigenfrequencies of these standing waves depend on the length of the field line, and the mass density of the plasma.

For simplicity in our analysis of hydromagnetic or field line resonances we shall use equation (5.1.14) and the dielectric tensor for a cold plasma (equation (5.3.2)), which give

We have replaced C by V x V x and - iw by the time derivative a/d t . We have also explicitly included the field-aligned current. The perpendicular component of the equation is

7 PLASMA WAVES AND PULSATIONS IN THE LOW-FREQUENCY BAND 513

We have assumed that E = [El, E2, 01, since the parallel component of the electric field, E3 = 0 due to the low electron inertia. In generalized curvi- linear coordinates, xj (j = 1,2,3), we obtain the coupled set of equations

where dSj = hjdxj, and the hi are scale factors. For example, in a dipole field,

xi = - cos2 O/r (0 is latitude, r is radial distance), x2 = 4 (longitude) x3 = sin 0/r2 (in direction of Bo), hl = (r2/cos 0)(4 - 3 cos2 0), h2 = r cos 0 h3 = r3 (4 - 3 cos2 o) '^ .

If the waves are axisymmetric (ajaxz = O), then two uncoupled equations result:

Equation (7.2.5) describes the toroidal mode, and equation (7.2.6) describes the poloidal mode. Using Maxwell's equations we find that the magnetic field of the toroidal mode is polarized in the x2 direction, and the poloidal mode has components in both the XI and x3 (field-aligned) directions.

The Earth and ionosphere are good conductors and consequently the electric field of the resonances should be small near the ionosphere, and the reflection coefficient should be high. Then a WKB solution to the toroidal- mode equation in the magnetosphere leads to the eigenfrequencies

5 14 6 GEOMAGNETIC PULSATIONS AND PLASMA WAVES

where the integral is along a field line from northern to southern ionospheres. In general, except near the plasmapause, the eigenfrequencies decrease with latitude (see Fig. 11).

Other limiting cases which lead to decoupled equations are those with large azimuthal gradients or wavenumbers (d1dx-i is large). The wave then has magnetic fields which are polarized in the XI direction, and the equation is very similar to the toroidal wave equation. Radoski (1967) called this mode a guided poloidal mode.

Analytic solutions for the coupled set of equations can be found in systems which have a simple geometry, but still retain the essential feature that the Alfvkn velocity has a gradient perpendicular to Bo. Tamao (1966) used the equations for a dipolar system but looked for solutions valid near the equator, and assumed that U A had only radial gradients. Southwood

ALTITUDE k m L - V A L U E 500 1000 2000 4000 2 3 4 5 10

1000

500 0

Ill

(A 200

100

n 50

0 - a Ill 2 0 Q.

10

0' 10' 20' 30. 40' 50' 60' 70" GEOMAGNETIC LATITUDE

Figure 11. The fundamental period (n = 1, T = 27r/o)) for the toroidal mode (after Nishida, 1978).


(1974) used a model with Cartesian coordinates, and Chen and Hasegawa (1974a) used dipole coordinates, but assumed a WKB solution, limited to the region near the equatorial plane.

Tamao (1966) showed that, with the assumptions mentioned above, the coupled set of equations could be solved by separation of variables. He found that a region of resonant coupling occurs, where the amplitudes of the oscillations have singularities. In this resonance region, the isotropic mode couples to localized transverse oscillations. The field lines on which the coupling occur depend on the frequency and azimuthal wavenumber of the isotropic source mode. If dissipation or damping is included, the azimuthal (toroidal) magnetic oscillations show a narrow, radially localized peak. The azimuthal component of the magnetic field also has a characteristic 1 8 0 phase shift across the radial position of the resonance peak (see the example in Fig. 10 in Tamao, 1966). This phase shift in the azimuthal component, and the small phase change in the radial component lead to a polarization reversal across the resonance peak.

Southwood (1974), and Chen and Hasegawa (1974a) developed somewhat more comprehensive models of hydromagnetic resonances by including a source of energy, the Kelvin-Helmholtz instability at the magnetopause. They showed that quasimonochromatic surface waves associated with the Kelvin-Helmholtz instability, though evanescent, couple to toroidal mode standing waves in the inner magnetosphere. The most efficient coupling occurs at positions where the eigenfrequencies of the field lines match the frequencies in the surface waves.

In order to solve the coupled wave equations, we shall adopt the Cartesian geometry used by Southwood (1974). We assume solutions of the form

E(x, y, z, t ) = E(x)exp [i(k r - mt)] (7.2.8)

where x corresponds to the radial direction at the equator, y is the azimuthal direction, z is along Bo, E(x) = [Ex(x), Ey (x), 01 and k - r = kyy + kzz. In addition we assume that v~ is a function of x only. The boundary conditions are E = 0 if z = 0, ZL. Now equations (7.2.3) and (7.2.4) become

and


Solving for Ey gives the equation

where the dependence of Ey and VA on x is implied, and k2 = k$ + k\. The equation has two singularities. The first, occurring when (cn^/vi - k i ) = 0, corresponds to the position XR where the field lines resonate. Bearing in mind the boundary conditions, we have kz = nv1.Z~ (n an integer), and u = ~ T V A ( X R ) / ~ L . The second singularity, occurring when (fa^/vi - k2) = 0, corresponds to the position or turning point where the solutions change from oscillatory to spatially increasing or decreasing (Southwood, 1974).

Near the position of the resonance, x = XR, equation (7.2.1 1) becomes

If we assume that there are losses in the ionosphere, then kz is complex, and the equation can be written (Southwood, 1974)

where 7 = ~ e ( k ~ ) ~ r n ( k ~ ) / ( u ~ d(vi2)/dx)"'. The solution is

Ey (x) = AIo [ky (x - XR - iy)] + BKo [ky (x - XR - i-y)] (7.2.14)

where 10 and KO are modified Bessel functions. The coefficients A and B are determined from boundary conditions on the x coordinate. For example, we can assume that a source exists at the magnetopause, x = xm (perhaps a Kelvin-Helmholtz instability) and here Ey(xm) = Eoei*'. At the other end of the box, near the Earth, Ey = 0.

We shall not look at the solution in detail, but shall illustrate with a schematic of the general form of the solution (Fig. 12). Monochromatic energy from the magnetopause is coupled by the fast mode to the resonance at the position XK, leading to a localized maximum in the wave's electric field. The polarization of the perpendicular electric fields follows from equations (7.2.9) and (7.2.10), and

5- - iky -- l d E y E~ ( 2 - k ~ ) E~ d x '

The sense of polarization depends on the azimuthal direction of propagation, ky, and the logarithmic derivative E;'dfY/dx. The sense of polarization changes if the azimuthal (y component) direction of propagation


Distance Figure 12. The amplitude of the Ey component for a hydromagnetic resonance. Changes in the sense of polarization of the perpendicular electric fields are indicated by the dotted and solid lines.

changes. The sense of polarization also changes radially (x direction) or latitudinally on the ground near XR (solid line) and at the local minimum in Ey (dotted line). In Fig. 12, CC (counterclockwise) corresponds to LH polarization, which would be seen as CC on the ground (looking downward along field lines) in the northern hemisphere. This example is appropriate for westward propagation in the northern hemisphere.

Rather than solve for Ex and Ey using equations (7.2.9) and (7.2.10)' we can use Maxwell's equation to obtain

where b is the wave's magnetic field. Kivelson and Southwood (1985b, 1986) have shown that this equation is very similar to the equation which Budden (1961) used to describe tunnelling of radio waves (see also Southwood, 1974). The position of the resonance, where (w^/v\ - k i ) = 0, is analogous to the position of the plasma resonance for the radio waves. Kivelson and Southwood (1986) give further examples of solutions to this equation, as well as many references to earlier publications.

In analogy with Budden's solution, we can choose as boundary conditions a wave incident along the x-axis from oo, and an amplitude which falls to zero at - 00. The incident wave will be partially reflected from the region of the turning point, XT. If we further suppose that a wave disturbance is present at time t = 0, and a reflecting boundary exists at XB > XT, then the reflections at XB and XT can lead to cavity eigenmodes at


quantized frequencies. These cavity modes (the fast mode) lose energy to the transverse mode in the field-line resonances due to tunnelling.

Consequently in the magnetosphere we might expect the fast mode to produce a discrete set of standing waves corresponding to the eigenmodes of the cavity formed by the magnetopause and the magnetic shell at XT.

Allan et al. (1986) have found evidence for these cavity eigenmodes in numerical simulations of a hydromagnetic cavity with an impulsive energy input.

The monochromatic or sinusoidal nature of the low-frequency pulsations can be produced by a process which has a growth rate with a narrow-band spectrum, by the cavity modes previously discussed, and by a surface wave which occurs at a discontinuity in the density or magnetic field (Chen and Hasegawa, 1974b). For the surface wave, assume two regions of uniform but different densities pi and 02. The magnetic fields Bl and B2 are uniform in both regions, Bl \ B2, and Bl and B2 are tangential to the plane of discontinuity. This configuration is a reasonable approximation to the plasmapause. Then from equation (6.2.14) with v2 = 0, we have

where kt = [O, ky, kz]. Assuming a large azimuthal wavelength (I kY I 4 I kz I),

where the boundary conditions at the ionosphere require k^ = m r / Z ~ . In accord with the predictions of the theories of hydromagnetic

resonances, a number of statistical studies have indicated that the periods of low-frequency pulsations tend to increase with increasing geomagnetic latitude (Obayashi and Jacobs, 1958; Oguti, 1963; Ol', 1963). With the exception of a few special types of events a given pulsation event usually has the same frequency at all latitudes (Ellis, 1960; Obertz and Raspopov, 1968; Samson and Rostoker, 1972). The exceptions to this rule are the pul- .

sations with latitude dependent frequencies in the H component (Siebert, 1964; Voelker, 1968; Rostoker and Samson, 1972).

The general tendency for the latitude of the peak intensity of low- frequency pulsations to decrease with increasing frequency is illustrated in Fig. 13. These data are based on the data presented in Samson and Rostoker (1972), except that data points from events which peaked at the ends of the array have not been included. Except for the broad-band region of intensity peaks near 72-76Â¡~ the latitudes of the intensity peaks appear t o follow an almost linear trend (on this semilog plot).


Geomagnetic Latitude (degrees) Figure 13. The frequency of population events versus the latitude of the maximum power (H + D + Z components). This plot is based on the set of data in Samson and Rostoker (1972) for the interval 0730-1730 LGT.

Samson and Rostoker showed that the latitude-frequency dependence is adequately represented by the linear regression curve f - I =

- 58 + (37.4 Â 3.7)L, where f is the frequency in hertz. In general, the regression curve given by Samson and Rostoker tends to mark the upper frequency limit for the statistical studies of Obayashi and Jacobs (1958), 01' (1963), and Obertz and Raspopov (1968). This discrepancy may be due to the differences in the methods for analysis since the latter authors estimated the average frequency at a given latitude, whereas Samson and Rostoker estimated the latitude of the intensity maximum for a given frequency. The pulsation events with latitude-dependent frequencies (Voelker, 1968; Rostoker and Samson, 1972) give data which are in very close agreement with the regression curve.

Cummings et al. (1969) found that monochromatic pulsations observed by the geosynchronous satellite ATS-1 occupied two broad bands, one centred at 5mHz, the other at lOmHz (see Fig. 10 in Samson and Rostoker, 1972). They attributed these pulsations to second harmonics of field-line resonances.

The regression curve also appears to mark the upper frequency limit of low-frequency pulsations observed by OGO 5 (Kokubun et al., 1976) at various L values. In addition, the statistically averaged spectrum of


low-frequency electric field pulsations observed at L = 6.6 by GEOS 2 (Junginger et al., 1984) shows a wide spectral peak, with the upper limit of this peak close to the value predicted by the regression curve. Junginger etal. suggest that the electric field pulsations are resonances in the fundamental mode.

Singer et al. (1979, 1982) identified spatially limited pulsations in magnetometer data from the satellites ISEE 1 and 2. The oscillations were predominantly radially polarized, and tended to change structure at the same position in space rather than at the same time. They also occurred when the satellite was near the plasmapause. An example of these types of pulsations is given in Fig. 14. In this example, the frequency at L = 5 is approximately 17mHz, and tends to increase with decreasing L value. A second example given by Singer et al. (1982) has a frequency of approximately 5-8 mHz at L = 6.1. Singer et al. (1982) attributed these pulsations to a second harmonic standing wave. The radial polarization and the possibility that these waves are second harmonics suggests that the source of energy for the pulsations might be bounce resonance instabilities from hot ring-current plasmas (Southwood et al., 1969; Southwood and Kivelson, 1982; Hughes and Grard, 1984) (see also Fig. 10).

On the ground, low-frequency pulsations have polarizations which change with latitude and local time. The horizontal components of pulsations recorded at high latitudes show predominantly CC polarization in local morning, and CW polarization in the afternoon (Nagata etal., 1963; Kato and Utsumi, 1964; Troitskaya, 1967; Samson, 1972). The interval of CC polarization in the morning tends to be the clearest, whereas the afternoon sector is more confused with a mixture of CC and CW polarization. Pate1 and Cahill (1964) reported that irregular pulsation with frequencies between 5 and 8 mHz recorded by Explorer 12 tended to have CC polarization before 1100 LT and CW after. Dungey and Southwood (1970) found a similar pattern in pulsations recorded by Explorer 33. These diurnal changes, and the latitudinal changes in the sense of polarization (see below) are consistent with the theories which suggest that the Kelvin-Helmholtz instability at the magnetopause drives field-line resonances.

In addition to the diurnal changes mentioned above, low-frequency pulsations sometimes show latitudinal changes in their sense of polarization in the horizontal plane. Kaneda et al. (1964) and Obertz and Raspopov (1968) found that Pc5s at Sitka ( 6 0 . 0 ~ ~ ) and College (64.7ON) often had CC polarization when those at Barrow ( 6 8 . 6 ~ ) had CW. This latitudinal change in the sense of polarization appeared to be due to a large latitudinal phase change in the H component when the maximum amplitude of the pulsations was between College and Barrow.

Samson et at. (1971), Samson and Rostoker (1972) and Samson (1972)


OCTOBER 24, 1977 ISEE-1 6 2 Bx PERTURBATION MAGNETIC FIELD

FIELD ALIGNED COORDINATES

ISEE-1

ISEE-2

ISEE-1

1SEE-2

(ISEE-1) 2215 2225 2235 2245 UT (ISEE-2) 2222 2232 2242 2252 UT

L 5.8 5.4 5 . 0 4.7

Figure 14. Low- and mid-frequency pulsations recorded on the ISEE 1 and 2 satellites (after Singer et al., 1982). The Bx component is perpendicular to both the field- aligned and azimuthal components. The bottom two traces have been shifted so that they can be plotted as a function of the L value.

showed that latitudinal polarization reversals are commonly associated with peaks in the intensity, and that the polarization reversals follow a latitude-frequency dependence which is very similar to that for the intensity maxima. Samson (1978) showed that a latitudinal polarization reversal does not always occur, even though a pulsation train might have a pronounced latitudinal phase shift in H component near the latitude of intensity maximum. Samson (1972) also pointed out that there is probably a second polarization reversal poleward of the reversal at the intensity peak.


The pattern for the polarizations is depicted in Fig. 15. The polarization reversal near the maximum amplitude in Fig. 15 corresponds to the polarization reversal depicted by the solid line in Fig. 12 and the reversal depicted by the dotted line probably corresponds to the polarization reversal near the minimum, depicted by the dotted line in Fig. 12. Clearly, many of the features of the polarization pattern are consistent with a hydromagnetic or field-line resonance model. The change in the sense of polarization at 1100LGT also suggests that a Kelvin-Helmholtz instability may be the source of the energy which drives the resonances.

STARE radar measurements of ionospheric drifts and electric fields associated with low-frequency pulsations provide some of the most con- vincing examples of the existence of hydromagnetic resonances in the magnetosphere (Walker et al., 1979). This radar measures the intensity and Doppler velocity of plasma waves caused by two-stream instabilities in the ionosphere. The electric fields are calculated by assuming that the Doppler velocity is given by E X B. Walker et al. (1979) studied a number of monochromatic, low-frequency (Pc5) oscillations in the electric field data, and found many features which supported resonance models. Many of the events showed latitudinal peaks in the amplitude of the dominant spectral component of the N-S electric field. These peaks were about 1 - 2 wide,

Geomagnetic Time Figure 15. Polarizations of 5mHz pulsations at the Earth's surface in the northern hemisphere (based on the diagram in Samson, 1972).


and the N-S component showed an approximately 1 8 0 latitudinal phase change over this 1 - 2 interval (Fig. 16). Walker (1980) computed detailed numerical solutions for toroidal resonances and found that these solutions gave a very good model of the measured electric fields.

Greenwald and Walker (1980) showed that near the ionosphere, the currents in a hydromagnetic resonance consist of field-aligned currents closing through Pedersen currents in the ionosphere (Fig. 17). The zero-order wave electric field is very small near the ionosphere, and the electric field E, depicted in the figure, is the first-order field that is required to close the field-aligned currents through the Pedersen currents. Poulter et al. (1982) have compared STARE observations with TRIAD magnetometer data, and have found that this model is a reasonable approximation for the measured fields.

Latitude (GG) Figure 16. Latitude profiles of STARE radar observations of the north-south ionospheric electric fields (3.2-4.5mHz) in a hydromagnetic resonance (after Greenwald and Walker, 1980). The data are plotted in geographic coordinates.


'Ã‘Ã‘Ã‘ -4Ã‘Ã‘

(0 Figure 17. Field-aligned and ionospheric current model of a hydromagnetic resonance (after Greenwald and Walker, 1980). (a)-(d) show the currents and electric fields in steps of one quarter of the wave period. Electric fields are indicated by line arrows, and currents by large, open arrows. The ionosphere is at the bottom of each figure, between horizontal lines.


7.3 Hydromagnetic instabilities

The azimuthal propagation of low-frequency pulsations leads to a relatively simple method for determining the type of instability which may have caused the pulsations. If we assume that the pulsation or wave has an azimuthal dependence of the form eim^ (0 is longitude) over some interval of longitude, then we can classify the propagation by the measured value for m. In practice m might be determined by using the formula

where d>i and ~$2 are the longitudes of the positions where the pulsations were recorded, and 0, and 0 2 are the phases of the time series recorded at the two positions. Note, however, that the phases and direction of propagation are not unique, since the estimates may show spatial aliasing (i.e. the possible phases are (ai - 012) + 2 n ~ , where n is any integer). Further con- straints must be used in order to determine a unique estimate (e.g. Olson and Rostoker, 1978). This aliasing is seldom a problem with data from the STARE radar which has very high spatial resolution, but the aliasing can be a very severe problem in the analysis of data from satellites.

Models of Kelvin-Helmholtz instabilities at the magnetopause have a long history. Beginning with Dungey (1954), numerous authors have added to the list of publications on this topic (e.g. Atkinson and Watanabe, 1966; Southwood, 1968; McKenzie, 1970; Ong and Roderick, 1972; Walker, 1981; Pu and Kivelson, 1983; Yumoto, 1984; Miura, 1984). Recently, satellite observations have indicated the existence of a boundary layer adjacent to the magnetopause. The structure of this layer has been studied by Eastman et al. (1976), Eastman and Hones (1979), Haerendel et al. (1978) and Paschmann (1979). The existence of this boundary layer substantially modifies the characteristics of the Kelvin-Helmholtz instability.

While Ong and Roderick (1972) considered a finite thickness boundary layer, before the discovery of the low-latitude boundary layer, most theoretical work used a sharp boundary for the shear in velocities. From the dispersion relation, equation (6.2.19, the linear growth rate of the instability is given by

The growth rate has a maximum


when kt -LBi, Bi and kt \\ vz. This growth rate increases linearly with the tangential wavevector kt. If the boundary has finite thickness then there is a maximum growth rate for some value of kf Ong and Roderick (1972) find maximum growth at kt = 0.4/d, whereas Walker (1981) found kt = 0.6/d, where d is the thickness of the boundary layer.

In general, the Kelvin-Helmholtz instability at the magnetopause and low-latitude boundary layer gives rn values which are less than 10. If we assume that the k vector for the maximum growth rate of the Kelvin-Helmholtz instability is given by kd =s 0.6, and use the value d = 1 Re and a radial distance to the magnetopause 10-15 RE, then rn = 6-9. The direction of propagation should be antisunward with westward propagation in the local morning, and eastward propagation in the afternoon.

The polarizations for the Kelvin-Helmholtz instability follow from the characteristics of the surface wave where kn = k i 1 kt 1 . Then from V.b=O, knbn+kt*bt=O and

The Â indicates that the sense of polarization changes across the region of the velocity shear, whereas sign (kt) indicates that the polarization depends on the direction of propagation. Since maximum growth is typically for kt parallel to v i , we find just earthward of the region of shear that the waves are circularly polarized in an LH sense in local morning and RH in local afternoon. In the northern hemisphere, on the ground, this corresponds to CC and CW polarization respectively.

Instabilities in the ring-current plasmas are generally due to anisotropic distributions of hot protons or strong spatial inhomogeneities in the plasmas. The instabilities can generate compressional waves which then couple to the shear Alfvkn mode. Further complication is added by coupling these waves to hydromagnetic resonances or standing waves (Southwood, 1977) and then we must take into account the resonances between unstable particles and standing waves. In many cases, the instabilities are generated by westward-drifting energetic protons injected during the substorm expansive phase. A number of the instabilities, including the bounce resonance and some configurations of this drift-mirror instability will have westward phase velocities which are comparable to the energetic proton drift velocity ( - 10-20kms1 at 6- RE). The fact that hydromagnetic resonances typically have frequencies of 5-lOmHz on field lines threading the ring current plasmas, indicates that the m values should be greater than 20-40.

Many of the measurements of the polarizations of low-frequency pul-


sations, both on the ground and in the magnetosphere, are compatible with a Kelvin-Helmholtz instability (see Fig. 15) at least in the local morning, and early afternoon. However, after about 2000 LT, the polarizations indicate westward propagation for pulsations detected on the ground (Fig. 15). Consequently, these data suggest that on the average, late afternoon and evening pulsations may be due to instabilities generated by westward drifting protons.

Hughes et al. (1978) used magnetometer data from the ATS6, SMS1 and SMS2 satellites to measure the m values of pulsations in three spectral bands ranging from 5 to 40mHz. Pulsations in the 5-12mHz band generally had 1 m \ < 10, with predominantly negative values (westward propagation) before local noon, and positive values after.

Olson and Rostoker (1978) used data from high-latitude ground-based magnetometers to study the azimuthal propagation of low-frequency pulsations, and found that the phase velocities were largely independent of the frequency, having magnitudes of about 14 km s l. Mapped to the magnetopause, the phase velocities are approximately 140-210 k m s l . These values are compatible with observed antisunward plasma flow velocities in the low-latitude boundary layer. The sign of the m values changed approximately 1-2 h before local noon (Fig. 18) from negative m values (westward propagation) before local noon to positive m values after. The magnitudes of the m values were generally in the range 4 < 1 m \ < 10. This range of m values gives a thickness of 0.6 to 1.5 RE (kd = 0.6, radial distance 10 Re) for the boundary layer. These values are comparable to measured thick- nesses (Eastman and Hones, 1979).

The monochromatic low-frequency (Pc5) pulsations in the STARE electric-field data (Walker et al., 1979) typically have very low m values. Poulter (1982) estimated an m value of 5 for a monochromatic event. Villain (1982) showed that the polarizations and phases of these monochromatic events are consistent with a Kelvin-Helmholtz source, both in the local morning and afternoon.

Another type of pulsation that might also be produced by the Kelvin-Helmholtz instability in the low-latitude boundary layer is the low- frequency plasma vortex seen in the magnetotail (Hones etal. , 1978; Saunders etal. , 1981, 1983). Saunders et al. (1981, 1983) have shown that these vortices have characteristics of both the ion acoustic and shear Alfvkn modes (Table 3 and Fig. 5). The field-aligned component of the magnetic pulsations and the plasma pressure oscillations are 1 8 0 out of phase. The plasma flow velocity rotates with the same periodicity as the magnetic-field oscillations (see Fig. 19). Southwood and Saunders (1984) suggested that a coupling of the ion acoustic and Alfvin modes is caused by field line curvature.


Universal Time (hours) Figure 18. Azimuthal wave numbers m of low-frequency (Pc4, 5) pulsations recorded by the University of Alberta magnetometer array (based on the data in Olson and Rostoker, 1978). These data are the averages of the H- and D-component m values. Negative values of m indicates westward propagation, and positive values indicated eastward propagation. Local magnetic noon is at approximately 20 h UT.

Most low-frequency plasma waves and pulsations produced by instabilities in the energetic ring-current plasmas are likely to have m values greater than 20-40 (see the earlier discussion), and westward phase velocities. The prevalence of westward phase velocities for dusk-sector pulsations has already been suggested for the polarization data in Fig. 15. Two types of pulsations, the Pg or giant pulsations (see Fig. l), and the storm- time low-frequency (Pc5) pulsations often feature westward propagation and high m values.

Giant pulsations typically occur during relatively quiet geomagnetic

Figure 19. Plasma and magnetic field measurements of a vortex event recorded by the ISEE 2 satellite (after Southwood and Saunders, 1984). From top to bottom the plots are: the longitude of the plasma flow vector in GSE coordinates %; the field aligned component of the flow vz, the field-aligned magnetic perturbation bg the scalar proton pressure Pp. The satellite coordinates are given in terms of radial distance (R), GSM coordinate local time (LTGsM) and latitude (LATGsM).



intervals. The wavetrains are remarkably coherent and sinusoidal, and generally have frequencies of about 10 mHz (Rostoker et al., 1979; Green, 1979; Glassmeier, 1980). Most giant pulsations have m values in the range 20 < \ m \ < 40 (Rostoker et a/., 1979; Glassmeier, 1980; Poulter et al., 1983) and propagate westward. Glassmeier (1980) and Poulter et al. (1983) have suggested that the giant pulsations are produced by bounce resonance instabilities in the ring-current plasma.

Storm-time low frequency (Pc5) pulsations occur during the main phase of geomagnetic storms (Brown et al., 1968; Lanzerotti, et al., 1969; Barfield and Coleman, 1970; Barfield and McPherron, 1978). At synchronous orbit, the magnetic pulsations tend to be linearly polarized in the meridian plane, with field-aligned (compressional) and transverse components (Barfield and McPherron, 1972; Barfield et al., 1972). These pulsations generally occur near dusk and in the local afternoon. Higbie et al. (1982) and Takahashi et al. (1985) have also reported observations of compressional low-frequency pulsations at geosynchronous orbit, which occur during the recovery phase of substorms. These compressional waves have maximum occurrence frequencies at local noon.

Kremser et al. (1981) have studied 54 examples of storm-time pulsations in GEOS-2 data. They noted that the waves could be divided into two classes. In-phase waves show electron and ion fluxes which oscillate with the same phase. Out-of-phase waves show ion fluxes which are out of phase with electron-flux and magnetic-field oscillations. The in-phase events have maximum occurrence frequencies near 1700-1800LT. They attributed these waves to a drift-mirror instability.

Allan et al. (1982) analysed STARE data for one of the events in the study by Kremser et al. (1981) (27 October 1978). They found that the waves showed westward propagation with 1 m \ = 25-50. The phase velocity in the ionosphere was near 1.4-1.6 km s ' , which is compatible with westward drifting 40keV protons in the ring current. Allan et at. (1983) analysed four additional storm-time pulsation events in the STARE data and found westward propagation and large azimuthal m values for all four events.

Takahashi et al. (1985) found that eight examples of compressional pulsations occurring during the substorm recovery phase all showed westward propagation and large azimuthal wavenumbers with 1 m \ = 40-120. They suggested that the phase velocities of the waves (approximately 10 k m s l ) were compatible with a diamagnetic or guiding centre drift of 10 keV protons, and that the azimuthal wavenumber might be regulated by a field-line resonance or coupling of a drift-mirror wave and a standing Alfvin wave (Lin and Parks, 1978; Walker et al., 1982).

8 PLASMA WAVES AND PULSATIONS IN THE MID-FREQUENCY BAND 531

8 CONTINUOUS PLASMA WAVES AND PULSATIONS IN THE MID-FREQUENCY BAND (0.01-0.1 HZ)

8.1 Introduction

The majority of the plasma waves in the mid-frequency band appear to be generated by the solar wind at the magnetopause, or to originate from ion cyclotron instabilities in the upstream solar wind. Plasma waves generated by these instabilities are convected downstream to the bow shock, and couple through the magnetopuase to the inner magnetosphere.

Mid-frequency pulsations are commonly observed on the day side of the Earth (Jacobs and Sinno, 1960; Saito, 1969) and have amplitudes from fractions of a nanotesla to several nanotesla on the Earth's surface. Many of the characteristics of these pulsations appear to be regulated by the properties of the solar wind.

One of the earliest and most striking results from the studies of dayside mid-frequency (Pc3,4) pulsations shows that the frequency of the pulsations is regulated by the magnitude of the interplanetary magnetic field (IMF) (Fig. 20). Troitskaya et al. (1971,1972) and Gulelmi et al. (1973) found that f (mHz) s= 6 5 (nT), where B is the IMF magnitude. This relationship also appears to be true for some of the high-latitude pulsations (see Fig. 21) (Engebretson et al., 1986a). Russell and Hoppe (1981) found a similar relationship for upstream plasma waves in the solar wind.

Though the frequency of many mid-frequency band pulsations is regulated by the strength of the IMF, the amplitudes and occurrence frequency of the pulsations at ground-based stations seem to be influenced by the orientation of the IMF and the solar wind velocity. Bol'shakova and Troitskaya (1968) showed that an IMF direction perpendicular to the Sun-Earth line appears to suppress Pc3,4 activity.

The directional parameter of the IMF which seems to have the greatest influence on the dayside pulsations is the 'cone angle', Oxa, which is the angle between the direction of the IMF, and the Sun-Earth line. A para- metrisation based on ox^ was first used by Greenstadt and Olson (1976, 1977). They found that the pulsation activity at ground-based stations is significantly enhanced when ox^ is small. Further studies clearly supported the role of ox^ in regulating pulsation activity, both on the ground and in the magnetosphere (Takahashi et at., 1981; Russell et al., 1983; Troitskaya, 1984; Yumoto et al., 1985). Takahashi et al. showed that mid-frequency band pulsations recorded by the ATS 6 satellite had a maximum in the probability of occurrence near 1000LT (Fig. 22), and when 6xB < 30'.

The velocity of the solar wind also appears to play an important role in


Figure 20. The magnitude of the interplanetary magnetic field B plotted as a function of the period T of mid-frequency pulsations recorded at Borok (after Troitskaya et a / . , 1972).

regulating the amount of mid-frequency activity in the magnetosphere and on the ground. Greenstadt et al. (1979) found a substantial increase in the amplitudes of Pc4 on the ground when the solar-wind velocity exceeded 300-400 km s-l . Takahashi et al. (1981) found a very similar pattern in the power spectra of pulsations recorded at geostationary orbit by ATS 6, with the first detectable power occurring at velocities of about 300-440 km s", and with the power then following the functional relationship log (P) =

- 1 . 1 + 0.003 vsw where P is the power and vsw is the solar-wind speed. The multivariate analysis of Wolfe (1980) has emphasized that ox^ and

vsw seem to be the most important parameters influencing the occurrence and amplitudes of mid-frequency pulsations with little, if any, influence from the other parameters. In addition, vsw and ox^ show very little corre-


South Pole station

January 1983 - - 0 July1983 - -0

IMF magnitude (nT) Figure 21. The frequency of pulsations observed at South Pole Station plotted as a function of hourly averaged magnitudes of the interplanetary magnetic field (after Engebretson et al . , 1986a).

lation (Takahashi et al., 1981), indicating that these parameters probably influence the generation of the pulsations through different processes. Wolfe (1980) has shown that &B seems to play a strong role in regulating the highest frequencies in this spectral band ( - 15-30mHz), with less influence as the frequency of the pulsations decreases to - 4-8mHz. Con- versely, vsw has its greatest effect in the 4-8 mHz band, and least effect in the 15-30 mHz band. The strong influence of vsw on the 4-8 mHz band suggests that these pulsations may derive much of their energy from a Kelvin-Helmholtz instability in the low-latitude boundary layer.

8.2 Ion cyclotron waves and instabilities in the solar wind

Pulsations in the mid-frequency band probably derive their energy from two main sources. At the low-frequency end of the band, near 10 mHz, the


IMF ANGLE DEPENDENCE OF DIURNAL OCCURRENCE PROBABILITY FOR Pc3 MAGNETIC PULSATIONS

Figure 22. The dependence of Pc3 occurrence frequency on local time and S,B (after Takahashi et al., 1981).

UCLA Fluxgate Magnetometer ATS-6 June 74 - May 75

Kelvin-Helmholtz instability at the magnetopause and low-latitude boundary is a likely source. The second, and perhaps dominant, source of energy is the proton cyclotron instability in the solar wind. Large- amplitude (several nanotesla) mid-frequency (10-50 mHz) waves populate a large part of the upstream region of the solar wind where the field lines of the IMF map to the bow shock (Greenstadt et at., 1968; Fairfield, 1969).

Gosling et al. (1978) have found two distinct groups of energetic protons which are reflected from the bowshock, a lower energy (< 10keV) group which is strongly collimated along the IMF, and a higher energy (up to 40 keV) group of diffuse ions with broad angular distributions. Paschmann et a[. (1979) found that mid-frequency ( - 50mHz) plasma waves, often accompanied the diffuse ions but not the collimated beams. These plasma waves had large fluctuations in the density and magnetic field.

Hoppe et al. (1981) have shown that the most common ULF waves in the foreshock of the upstream solar wind are steepened shock-like wave packets which are associated with the diffuse ions. These waves are LH polarized in the spacecraft frame and propagate at substantial angles to the IMF. The quasimonochromatic waves observed by Greenstadt et a[. (1968) and Fairfield (1969) tend to be associated with reflected proton beams with a mixture of collimated and diffuse components. Both the shock-like and

90

- 0)

% 60- L

s 2 ' - CD x 30- 2

7 1 I 1

- - - -

-

-

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I 1 I I I I

0 4 8 I2 16 20 24 Loca l Time


quasimonochromatic modes appear to be magnetosonic waves which are RH polarized in the plasma rest frame. The observed LH polarizations occur because the waves are propagating upstream with velocities less than the solar wind velocity. Consequently the waves appear to be carried with or convected by the solar wind, away from the Sun.

Fairfield (1969) and Barnes (1970) suggested that an electromagnetic instability of the RH (magnetosonic) mode (see equations (6.1.1 1)-(6.1.13)) with resonance at u = k v b - us, where v b is the average velocity of the ion beam, might be the source of energy for these plasma waves and pulsations. Unfortunately this mode has maximum growth when k II B and cannot explain the obliquely propagating modes. However, Gary (1981) has shown that numerical solutions of the dispersion relation for a drifting Maxwellian distribution of protons indicates a second region of large growth rates when k is approximately perpendicular to B.

A schematic diagram depicting the characteristics of this instability for parallel propagating waves is given in Fig. 23. The instability occurs in the upstream region of the solar wind, on magnetic field lines which map to the bow shock. The waves are convected back to the bow shock, where they are transmitted through the magnetosheath and magnetopause to the inner magnetosphere.

Takahashi et at. (1984a) have presented a simple model of these pulsations which might explain some of their observed features. Following Sonnerup (1969), the guiding centre velocity of ions which are adiabatically reflected from the bow shock is

where usw is the solar-wind velocity and en, ex and e~ are respectively unit vectors parallel to the shock normal, the x direction (solar), and the interplanetary magnetic field. The Doppler-shifted frequency in the Earth's reference frame is

where v p h is the phase velocity of the waves and 4 is a unit vector in the direction of propagation. Here we have used the resonance condition 0 ) = k - U b - wgi.

From (8.2.1) and (8.2.2) we find

where we have assumed that v p h = V A . If we consider the direction of


Figure 23. A schematic of the generation of ion cyclotron waves in the upstream solar wind. The waves are convected downstream and couple through the magnetopause to be detected by satellites (indicated by the orbit of ATS-6) in the magnetosphere (after Takahashi et al., 1984a).

maximum growth, k \\ B, (Gary, 1981) and note that V A Â¥ vsw then 2

UE = - 4 ugi cos ox^, (8.2.4)

or, f~ (mHz) = 7 . 6 5 cos2 I ~ ~ B where B is the magnitude of the interplanetary magnetic field in nanotesla. This equation predicts frequencies which are in good agreement with the observations presented earlier.

When the ion cyclotron waves are convected downstream to the vicinity of the magnetopause, energy can be coupled to the inner magnetosphere by two somewhat different mechanisms. The first mechanism is a simple wave- transmission through the magnetopause. The second mechanism is Budden


tunnelling to field line resonances in the inner magnetosphere (see equation (7.2.16)).

The characteristics of the transmission depend very much on whether the magnetopause is open or closed. In the former case we have a rotational discontinuity, and in the latter case we have a tangential discontinuity (Levy et al., 1964; Yang and Sonnerup, 1977). A closed configuration is probably characteristic of the low-latitude regions, and an open configuration is probably characteristic of the high-latitude field lines and the polar- cap (Paschmann et al., 1976).

McKenzie (1970), Verzariu (1973) and Wolfe and Kaufmann (1975) have studied the transmission of hydromagnetic waves through a magnetopause with a tangential discontinuity, and have found that transmission coefficients are very small, except near normal incidence. Integrated over all incident angles, only a small percentage of the energy is transmitted through the boundary.

8.3 Hydromagnetic resonances and the propagation of mid-frequency pulsations in the magnetosphere

The polarizations and phase velocities of mid-frequency pulsations have been studied extensively at mid-latitude, ground-based stations (Lanzerotti et al., 1974; Green, 1976; Mier-Jedrzejowicz and Southwood, 1979, 1981; Orr and Hanson, 1981). Unfortunately, the results seem to be somewhat more complicated than those in the low-frequency band.

Green (1976) has found that pulsations recorded on the ground at mid- latitudes generally have very small azimuthal wave number (I m < 5), and largely eastward phase velocities over the whole of the dayside. Hughes et al. (1978) used magnetometer data from the ATS 6, SMS 1 and SMS 2 satellites to measure azimuthal wavenumbers in three spectral bands, 5-12 mHz, 12-25 mHz and 25-42 mHz. The two high-frequency bands exhibited predominantly eastward propagation near local noon and after, with a tendency to westward propagation in the local morning.

Further measurements of the phase velocities of the harmonics of mid- frequency pulsations observed by the ATS 6, SMS 1 and SMS 2 satellites (Takahashi et al., 1984b) indicate that these pulsations are propagating antisunward, but at very high azimuthal phase velocities, near 1700 km s-'. These phase velocities are much too high for a Kelvin-Helmholtz instability at the magnetopause, but are compatible with the local Alfvkn velocity. Consequently these data suggest that the pulsations' energy propagates from near local noon, toward the magnetotail. These propagation characteristics are compatible with the possibility that the energy for


the waves originates in the solar wind, and propagates through the magnetopause. Lanzerotti et al. (1976) have noted that ground-based observations of 15-27mHz pulsations show polarization reversals near local noon which are consistent with antisunward propagation.

The wavelengths in this frequency band are a sizeable fraction of the characteristic dimensions of the magnetosphere, and consequently these pulsations, like those in the low-frequency (1-lOmHz) band likely have electric and magnetic fields which are influenced by hydromagnetic resonance structures in the magnetosphere. For ground-based stations, the latitudinal position of the resonance can be identified by large latitudinal phase shifts in the H-component, along with a latitudinal peak in the amplitude of the pulsations. Often a change in the sense of polarization will occur near the latitude of the peak in amplitude. Fukunishi and Lanzerotti (1974) and Lanzerotti et al. (1974, 1976) have found evidence of resonances in mid-frequency pulsations recorded on the ground at L = 4 ( - 6 0 geomagnetic latitude). They inferred that many of these resonances and the polarization reversals were on field lines which passed near the plasmapause. Green (1978) also found evidence of resonances for pulsations recorded in the latitudinal range L = 2.5-4.0.

Studies using conjugate station pairs have tended to indicate that mid- frequency resonances seen on the ground are an odd mode (n = 1,3 ,5 ...) (Van-Chi et al., 1968; Lanzerotti et al., 1972; Fukunishi and Lanzerotti, 1974; Lanzerotti and Fukunishi, 1974) although the prevalent harmonic has not been established.

The magnetic fields of pulsations recorded by satellites at synchronous orbit are largely polarized transverse to the geomagnetic field (Arthur et al., 1977) and are almost always azimuthally (east-west) polarized (Takahashi and McPherron, 1983). The azimuthally polarized waves often have several spectral peaks which are harmonically related (Takahashi and McPherron, 1982; Takahashi et al., 1984b). The example in Fig. 24 shows that these spectral peaks can appear at a number of different satellites, and Takahashi et al. (1984b) have suggested that the peaks are due to the harmonics of hydromagnetic resonances or standing waves on field lines near the spacecraft. The large number of harmonics, six in this case, require a very broad-band source (- 10-30 mHz).

Figure 24. Power spectra of mid-frequency puslations recorded simultaneously by three satellites, SMS-1, SMS-2 and ATS-6 (after Takahashi et al. , 1984b).

MAY 1, 1975 1500-1600 UT I I I I I I I I I

Number of Points = 720 Degrees of Freedom = 18

SMS 1

SMS 2

Frequency (mHz)


9 CONTINUOUS PLASMA WAVES AND PULSATIONS IN THE HIGH-FREQUENCY BAND (0.1-10 HZ)

9.1 Introduction

Pulsations in this frequency band correspond to the Pcl and Pc2 classes (Table 1). On the Earth's surface, the maximum amplitudes of these pulsations are typically 0.1-1 nT in the auroral zones (Troitskaya, 1967; Kenney and Knaflich, 1967) with amplitudes an order of magnitude less at the equator. Ground-based recordings of these pulsations show two distinct groups. The first group contains pulsations with periodically modulated amplitudes, while the second group contains pulsations which show gradual, and sometimes random, changes in amplitude. Pulsations in the first group are often called structured pulsations or 'pearls' because of the similarity between the appearance of their time series and a string of pearls (Troitskaya, 1967; Saito, 1969; Jacobs, 1970). Conversely, pulsations in the second group are often called unstructured pulsations because of the lack of any periodicity in the modulation of the amplitudes. Further classes or subclasses of these pulsations follow from an inspection of the dynamic spectra (a three-dimensional plot of amplitude versus frequency and time) of the pulsations. Classification schemes based on the dynamic spectra are given by Kokubun (1970) and Fukunishi et al. (1981) (see also Table 4).

Table 4. Characteristic features of the dynamic spectra of continuous puslations in the 0.1-lOHz band.

Local time of Name Frequency (Hz) Characteristic structure occurrence (h)

--

H-M whistler 0.5-1 Periodic rising tones with 02-08 (pearl) fan structure

Periodic H-M 0.5-1 Periodic rising tones with 06-09 emissions parallel structure

H-M chorus 0.2-0.5 Mixture of structured and 09-18 unstructured pulsations

IPDP 0.1-0.5 Unstructured but with 16-01 frequency rising at a rate of - 0.5 Hz over an hour

Continuous 0.1-1 Unstructured 06- 18 emissions

Cleft pulsations 0.1-1 Unstructured 06-14 (IPRP)

H-M: hydromagnetic. IPDP: intervals of pulsations with diminishing periods. IPRP: intervals of pulsations with rising periods.

9 PLASMA WAVES AND PULSATIONS IN THE HIGH-FREQUENCY BAND 541

Structured pulsations typically have a modulation period of from 100 to 300 s, and frequencies from 0.2 to 5 Hz. These pulsations show two characteristic types of dynamic spectra. The first is a fan-shaped structure with the frequency rising with time. The rate of the frequency increase tends to decrease with each subsequent wavepacket. Because of the characteristic shape of the dynamic spectra these pulsations are sometimes called hydromagnetic whistlers (Jacobs and Watanabe, 1964; Obayashi, 1965; Kokubun, 1970). The dynamic spectra of the second type show a sequence of periodic intensification with rising frequencies forming a set of parallel lines, but no tendency to fan-shaped structure. These pulsations are often called hydromagnetic emissions following the suggestion of Jacobs and Watanabe (1 967).

When structured pulsations are recorded at conjugate stations, the modulation envelopes often have maxima in one hemisphere, while those in the opposite hemisphere have minima (Fig. 25). The dynamic spectra in both hemispheres show rising frequencies in the temporal interval in which the corresponding wavepacket occurs. These features emphasize that the structured Pcs are probably due to wavepackets which are guided along field lines, and reflect from the ionosphere to bounce from hemisphere to hemisphere. Amplification of the wavepackets must also occur in the magnetosphere, probably through proton cyclotron instabilities (Kennel and Petschek, 1966).

A variety of high-frequency pulsations are seen near local noon at stations which are on field lines which pass near the polar cusp (Matveyeva et al., 1976, 1978; Morris et al., 1982; Morris and Cole, 1985). The amplitudes of the pulsations range from fractions of a nanotesla to 1-2nT, and the frequencies range from 0.1 to 1 Hz. Some of the pulsation trains (IPRP, intervals of pulsations with rising periods) show gradual decreases in frequency, typically from 1 Hz to 0.1 Hz over an interval of several minutes. These plasma waves and pulsations possibly derive their energy from instabilities in plasmas situated in the boundary layer near the dayside magnetopause.

Even though a wealth of different structures is evident in the dynamic spectra of high-frequency pulsations observed on the ground, satellite observations show very few examples of structure pulsations (Bossen et al., 1976).

9.2 Ion cyclotron instabilities in the magnetosphere

Most of the theories for the origin of 0.1-10 Hz pulsations in the magnetosphere are based on the proton cyclotron instability of the L mode (equation (6.1.14)). The source of energy for the instability is provided by

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50"' 55"' ;Om 05" l o m 15'" O m 5 30"

Ogh L T l o h

Figure 25. Structured, high-frequency puslations (Pcl) recorded at conjugate stations (after Saito, 1969). The dynamic spectrum is a superposition of the spectra from Great Whale River (solid arrow) and Byrd Station (dotted arrow).


an energetic (energy > lOkeV), anisotopic (Ti. > Ti) distribution of protons (Kennel and Petschek, 1966; Gendrin, 1967; Cornwall et al., 1970; Gendrin et al., 1971). The growth rate curves (Fig. 8) indicate that the waves tend to grow in the frequency band from 0.05 to - 0.6 of the proton (hydrogen) cyclotron frequency. The growth rates are strongly dependent on the anisotropy A = ( T ~ f i ) - 1, and the ratio of the thermal velocity to the Alfven velocity (UI/v*). For very low anisotropies (A < 0.5), and low temperatures (v\lUi > 2), there is considerable absorption over much of the spectrum. Changes in the anisotropy have little effect on the frequency of maximum growth, whereas increasing thermal velocities shift the frequency with maximum growth to substantially lower values.

One of the main sources of the energetic and anisotropic ions are particles injected from the magnetotail during the substorm expansive phase (Frank, 1967; Kivelson and Southwood, 1975). The energetic protons drift westward to create the substorm enhanced ring current. In addition to creating conditions suitable for the growth of ion cyclotron waves, the hot and anisotropic protons may lead to the formation of drift-mirror like instabilities mentioned earlier. Cole et al. (1982) have also suggested that high-frequency cleft pulsations (including IPRP) might be caused by proton cyclotron instabilities due to the penetration of hot plasma from the solar wind into the boundary layer at the magnetopause.

In a cold plasma with electrons and protons, the dispersion relation for the ion cyclotron (L) mode ( a = wgik = k H ) is given by equation (6.1.11) and

For ions in resonance with the waves ( n = I), we have from the denominator in the dielectric tensor (5.1.15)

and consequently the resonance velocity is

The Alfven velocity has a minimum just inside the plasmapause, and consequently the resonant velocities are lowest, and the growth rates are hightest


in this region. Accordingly we might expect that the ion cyclotron instability would be most likely to occur in the region of overlap between the anisotropic and energetic protons of the ring current, and the dense cold plasma of the plasmasphere (Cornwall et al., 1970).

A number of different techniques have been used in order to determine the source regions of the pulsations recorded on the ground. Generally, the studies indicate that the source regions are on field lines which thread the plasmapause or outer plasmasphere (Heacock, 1971; Roth and Orr, 1975; Al'pert and Fligel, 1977; Baransky et al., 1981). Fraser et al. (1984), using a low-latitude magnetometer array (L = 1.8-2.8), have found that most of the structure (Pcl) pulsations originate on field lines which are located near or inside the average plasmapause (Fig. 26). Bossen et al. (1976) and Mauk and McPherron (1980) have found that the largely unstructured ion cyclotron waves observed on the ATS6 satellite typically occurred between

1 3 4 5 6 PROBABLE PLASMAPAUSE POSITION Lpp (Rel

Figure 26. Measured source positions of structured high-frequency pulsations (Pcl) plotted as a function of the position of the plasmapause (after Fraser et al., 1984). Sources on the dotted line would fall on the plasmapause.


10 and 2200LT, when the satellite is likely inside the plasmasphere. The Poynting fluxes indicated that the waves were propagating away from the equator.

9.3 The propagation of ion cyclotron waves in the ionosphere and magnetosphere

Even though the energy source of the high-frequency pulsations is probably a proton cyclotron instability, a number of other features of the waves require explanation. The repetitive nature of structured pulsations is probably due to the repetitive amplification of wavepackets, propagating along field lines in the L mode and reflection from conjugate ionospheres. The characteristics of the repetitive wavepackets and the dynamic spectra, with the rising frequencies, must be regulated both by the dispersion of the propagating wavepackets and the growth-rate spectrum.

Many of the features of the dynamic spectra can be explained by considering the dispersion of the wavepackets propagating in a cold plasma. In the schematic of the dynamic spectrum (Fig. 27) the delay ~ ( w ) between the amplitude maxima at frequency w , from one arrival to the next in time is

where u, = dw/Sk is the group velocity, and the integral is along a field line from the ionosphere in one hemisphere (0) to the other hemisphere ( - 0). If we assume that all frequencies are generated at the same time, near the equator, then the first packet in time has TI (w) = j aug(s , u)] ' ds and the nth packet has

If the magnetosphere is symmetric about the equator

and for large n

In the dynamic spectrum, the change in the rate of frequency rise will be proportional to n 1 giving the characteristic fan-shaped structure seen in hydromagnetic whistlers.

*o

Figure 27. A schmatic of the dynamic spectra of structured high-frequency (Pcl) pulsations (after Gendrin et at., 1971).


Other features of the dynamic spectra require some knowledge of the convective, and non-convective growth rates of the cyclotron instability. The spectral band of the waves is probably determined by the frequencies with maximum growth (Fig. 8) and will probably lie between 0.1 and 0.5wgi, where wgi is the equatorial proton cyclotron frequency. If heavy ions (He', 0') are present, then the spectra become much more complicated, with slots just above wgi (Fig. 9). Jacobs and Watanabe (1967) and Gendrin et al. (1971) have also shown that finite growth rates can substantially modify the shapes of the dynamic spectra. The parallel rising frequencies of the structured, periodic hydromagnetic emissions could be due to the influence of finite growth rates.

Criswell (1969) has calculated the amplification of L-mode wavepackets propagating along field lines within the plasmasphere. He calculated the bounce time T = r(w0) (equation (9.3.1) and Fig. 27) and the total amplification A(wo) integrated along the field line. The results are plotted in Fig. 28, where the period t = 21c/wo. The value R is the equatorial radius of the field line, and the amplification A is indicated by contours cutting the T versus t lines. The range of the distribution of the observed t and T should be enclosed by the lines outlining the higher growth rates. The scatter plot for observed values of T versus t in Fig. 29 show that this is often the case at low- and mid-latitude stations.

The characteristics of the pulsations observed on the ground are a consequence not only of propagation in the magnetosphere, but also of ducted propagation, and transmission through the ionosphere. Manchester (1966) and Greifinger and Greifinger (1968) have shown that the minimum in the Alfvkn speed in the F2 layer forms a waveguide for the propagation of the fast, R mode. The required fast mode is produced by coupling the L- polarized magnetospheric, guided mode with the fast mode in ionospheric regions with substantial gradients in the Alfven velocity.

The propagation of the ducted mode is illustrated in Fig. 30, which shows both the trapped mode, and the propagation through the ionosphere. In general, the ducted mode must have a wavelength which is less than the vertical thickness d of the waveguide. Consequently ducting is possible only when the frequency of the pulsations is greater than (v/^)min/ d, where ( v * ) ~ ~ , , is the minimum value of the Alfven velocity in the duct. Typically, waves with frequencies less than 0.5 Hz will not propagate in the duct. As the wave propagates horizontally in the duct, the non- uniform Alfven velocities lead to coupling to the L mode, attenuation of the wave, and transmission of energy through the ionosphere in regions far removed from the field lines which passed through the source in the magnetosphere.

The number of satellite observations of pulsations in the 0.1-10 Hz band

PATTERN REPETITION PERIOD VERSUS WAVE PERIOD (t)

- PLASMAPHERE

1 0 < A < 3 0 - a

5 < A < 10 --- I < A < 5 - 0

n

w

1 1 1 I I I I I I I 1.0 2 -0 3 .o 4 .O 5 .c

t (SECONDS) Figure 28. The pattern repetition period T versus the wave period t for proton cyclotron waves generated in the plasmasphere (after Criswell, 1969).

400' I I I , I I I I

- PAL0 ALTO . CALIF. ( R = 1.9. X.' 43.5' N) 0 - CHAMBCN LA FARET FRANCE (R= 2.4.X.= 50.4' N) - A - COLLEGE,ALASKA ( ~ = 5 . ' l , X o = 65.N)

A A

A

300- A -

0 > I 1 I I *

0 1 .O 2 .O 3.0 4.0 5.0

Figure 29. Observed pattern repetition periods versus observed wave periods for data from three observatories (see inset). The amplification contours from Fig. 28 are shown as solid lines (after Criswell, 1969).

HEIGHT r

DUCTED WAVE (E MODE) \ GION A

TRANSMITTED , /

-

Figure 30. The F-layer duct (after Manchester, 1966)

-z - REGION B

PHASE CONSTANT


has increased enormously since the mid 1970s. In all these observations, very few cases of structured pulsations have been reported, and most of the pulsations seem to be related to the unstructured pulsations seen on the ground. In a comprehensive analysis of the data from the geosynchronous satellite ATS 1, Bossen et al. (1976) found that almost all of the data showed unstructured pulsations, and they suggested that these pulsations might be connected with IPDPs. On the ground the IPDPs are unstructured, but exhibit a slowly rising frequency with time, typically 0.5 HZ h" (Fig. 31). Perraut et al. (1978) reported only two events with structured pulsations in 7 months of GEOS 1 satellite data. The data from GEOS 2 (Young et al., 1981) and ATS 6 (Fraser, 1982) have also yielded few structured events. However, detailed analyses of the ATS 6 and GEOS 1 and 2 wave and particle data have provided substantial evidence that the ion cyclotron resonance associated with anisotropic, hot protons is the source of energy for the pulsations (Mauk and McPherron, 1980; Roux et al., 1982).

Pulsations of the IPDP class usually occur after a substorm expansive phase, with delays of less than an hour (Fukunishi, 1969). Studies by Fraser and Wawrzyniak (1978) and Pikkarainen et al. (1983) show that the IPDPs drift westward at a velocity which is compatible with the velocities of westward-drifting hot protons injected from the magnetotail during the substorm expansive phase. A study by Maltseva et al. (1981) has shown that the maximum amplitude of the IPDPs moves to progressively lower latitudes as the frequency rises. Consequently, the rising frequencies are probably caused by the westward drift of the protons, with progressive earthward penetration of the protons. The earthward penetrations leads to growth in regions where the equatorial cyclotron frequency is greater, and growth occurs at a higher frequency.

The analyses of the spectra of 0.1-10Hz band pulsations recorded by satellites have clearly illustrated the important role of heavy ions, particularly He+ and 0 + , in the propagation and amplification of these pulsations. Most of the observations emphasizing the role of the heavy ions have come from three satellites, ATS 7 (Mauk and McPherron, 1980; Fraser and McPherron, 1982) and GEOS 1 and 2 (Young et al., 1981; Roux et at., 1982; Perraut, 1982). We should note, however, that the presence of heavy ions had already been deduced from ground based data (Dowden, 1966; Troitskaya and Gulelmi, 1967; Fraser, 1972).

A compilation of the spectra of the peak wave frequencies of the high- frequency pulsations or ion cyclotron waves recorded by ATS 6 shows a pronounced gap between u / u d (equator) = 0.2 and 0.3 ions (Fig. 32; Mauk, 1983). This gap, which is presumably due to the presence of He+,


AUG. 5, 1977 ! HU

UT (HOURS) Figure 31. The dynamic spectra of an IPDP event occurring at two conjugate stations (after Fukunishi and Toya, 1981).


u/^l (Equatorial) Figure 32. Statistics of the wave frequencies of pulsations recorded at two locations by ATS-6 (latitudes 0 and 1 2 ) (after Mauk, 1983). The expected cutoffs for various concentrations of He' and two values of the anisotropy ( A * ) are marked on the graph. The parameter OH+ is the equatorial hydrogen cyclotron frequency.

can also be seen in many of the GEOS 1 and 2 spectra. Occasionally a gap is seen near, and above the O + cyclotron frequency (Fraser, 1985).

The polarizations of the pulsations recorded at the equator are predominantly L-mode at frequencies below, and above the He+ gap, whereas off the equator, the polarizations are linear (i.e. have an R-mode component) above the gap, and L below (Fig. 33) (Young et al., 1981; Fraser and McPherron, 1982). These features are illustrated in Fig. 33. The data from GEOS 2 on the equator are LH polarized both below and above the local helium cyclotron frequency in the figure). In contrast waves at GEOS 1, which is off the equator, are LH polarized when the power is below the local helium cyclotron frequency and linear when above.

Many of the characteristics of these spectra and polarizations can be explained by an evaluation of the dispersion relation for a cold, multispecies plasma (Fig. 6). For parallel propagation, the R mode is only mar- ginally changed by the addition of He' ions, but the L mode has the added resonance at (oig)~e^ and the new cutoff at (o)cf)He^. This leads to the stop


band or gap between (wg)^ and ( M ~ ~ ) H ~ + in which the L mode cannot propagate. The change in polarization above the gap from L mode on the equator, to linear (L and R) off the equator can be explained by assuming that the ion cyclotron instability occurs near the equator. The instability generates L-mode waves on branch I11 (Fig. 6), below the equatorial ( w ~ ) H + . As these waves propagate away from the equator, w/(wg)~+ decreases and moves to lower normalized frequencies on the dispersion

L Equator -A-

FREQUENCY (Hz) FREQUENCY (Hz) Figure 33. Power spectra ofleft-hand (L) and right-hand (R) polarized pulsations recorded at three latitudes (0 , 11' and 12') by the GEOS 1 and 2 satellites. The diagram at the top left illustrates the propagation of L-polarized waves originating at the equator (after Gendrin, 1983).

GEOS 2 JAN

16.08 - 16 .I0 UT

0 1 2 3 5 6 7 8 9 10 11 12 FREQUENCY ( HZ

Figure 34. Power spectrum of a 44 s interval of GEOS 2 magnetic field data (after Perraut et a/. , 1982).


curve eventually passing through ucr, and the waves become partially R-mode. This is illustrated in the diagram in the top left of Fig. 33. The wave which begins at a frequency above ucr at the equator converts to R-mode as it travels away from the equator. The low-frequency edge of the gap in the spectra for pulsations recorded away from the equator will be shifted below the local helium cyclotron frequency.

Even though many of the characteristics of the spectra and polarizations can be explained by studying the cold plasmas dispersion curves, a more complete evaluation requires the computation of the growth rates for the cyclotron instability in a multispecies warm plasma. These growth rates have a controlling influence on the power spectra of the plasma waves, and also will contribute to the locations of the various spectral gaps, and the polarizations. In general, the presence of cold heavy ions will allow greatly enhanced convective growth rates at frequencies just below the heavy ion cyclotron frequencies. There is no amplification in the stop bands between the cyclotron and cutoff frequencies, but the upper edge of this gap in the growth rates is not necessarily at ucf, and the gaps are generally wider than the stop bands for L-mode propagation (see the example in Fig. 9) (Gomberoff and Cuperman, 1982; Kozyra et at., 1984).

In addition to the ion cyclotron waves with frequencies below the equatorial proton cyclotron frequency, data from the GEOS magnetometers have demonstrated the presence of pulsations and plasma waves with frequencies greater than the proton cyclotron frequency. In the equatorial magnetosphere, the waves have narrow-band, harmonically related spectral peaks (Perraut et al., 1982) (Fig. 34) which sometimes are multiples of the local proton cyclotron frequency. The waves are polarized in the direction of the geomagnetic field, and are likely magnetosonic waves propagating perpendicular to the geomagnetic field. These types of events appear to be correlated with energetic (5-30 keV) protons with a distribution maximized at a pitch angle of 90'.

10 IMPULSIVE PLASMA WAVES AND PULSATIONS

10.1 Introduction

The majority of impulsive pulsations are produced by direct coupling of energy from the solar wind, or by transient release of energy stored in the magnetotail. Shocks and discontinuities in the solar wind produce the sudden impulses (SI) which propagate as fast and Alfvkn modes in the magnetosphere. Transient and localized reconnection or flux transfer events on

10 IMPULSIVE PLASMA WAVES AND PULSATIONS 557

the dayside of the magnetopause (see Fig. 2a) propagate along field lines giving field-aligned current systems and transient pulsations which appear to be propagating in an antisunward direction. The substorm expansive phase marks sudden changes in convection and the release of energy in the magnetotail. Much of this energy is carried by transient field-aligned currents which close through Pedersen currents in the auroral ionosphere. The transient field-aligned currents propagate as shear Alfvkn waves which can be seen in the nightside magnetosphere and on the ground.

These impulsive injections of plasma-wave energy into the magnetosphere can eventually couple to cavity modes (equation 7.2.16)) and field- line resonances (equations (7.2.9, 7.2.10)), leading to transient quasimonochromatic oscillations at a wide variety of frequencies (Allan et al., 1986). Consequently we might expect that transient pulsations triggered by sudden pulses such as SIs will have polarization and phase characteristics which are similar to those found for low-frequency continuous pulsations.

10.2 Sudden commencements and sudden impulses

Sudden impulses are seen as steps or discontinuities in temporal recordings of the geomagnetic field. These transient and rapid (rise times about 1 to 5 min) changes in the magnetic field propagate as fast compressional mode and Alfven mode waves in the magnetosphere, and appear with less than 1 min delay over most of the Earth's surface. Normally the impulses have amplitudes of less than 50nT. At times, an SI will trigger a substorm sequence or storm sudden commencement (SSC). The SIs are the response of the magnetosphere to shocks and discontinuity in density, temperature and velocity propagating in the solar wind (Ogilvie and Burlaga, 1974).

The polarizations of the initial cycle of the SI tend to be elliptical in the horizontal plane. Over a wide range of latitudes in the northern hemisphere the impulses have CC polarization in the local morning, and switch at approximately 10-1200LT to CW polarization in the afternoon (Fig. 35) (Wilson and Sugiura, 1961). This pattern is reversed in the southern hemisphere. These polarizations are compatible with those expected for a surface wave propagating tailward on the magnetopause (see equation (7.3.4)) if no field-line resonances exist.

The SI are generally followed by pulsations in the low-, mid- and high- frequency bands (see Fig. 36). The high-frequency pulsations are sometimes seen within 1-2 min after the SI. Kokubun and Oguti (1968) have suggested that increased anisotropy of hot proton distributions due to the compression of the magnetosphere might lead to amplification of ion cyclotron waves. If the SI triggers a substorm sequence, then injection of hot


plasma into the ring current from the plasma sheet would further enhance ion cyclotron wave activity.

The damped low- and mid-frequency pulsations following the SI (bottom three traces in Fig. 36) are likely due to hydromagnetic waves and field-line resonances triggered by the SI. Saito and Matsushita (1967) found that the

TO SUN 4

Figure 35. The polarizations of sudden impulse magnetic fields at the Earth's surface (after Wilson and Sugiura, 1961). Solid circles indicate CC and open circles CW polarization.

Figure 36. Examples of pulsations triggered by sudden impulses and storm sudden commencements (after Saito, 1969). (a) High frequency pulsations (Pel). (b) Mid- frequency pulsations (Pc2, 3 band). (c) Mid-frequency pulsations (Pc4 band). (d) Low-frequency pulsations (Pc5 band).

CLASS

P s c I

P s c 4

P s c 5

T Y P I C A L M.AGNET0GRAM

ssc

REMARK

COLLEGE D E C . 4 . 1962

ONAGAWA

M A Y 24,1959

3 1 G DELTA

AUG.17, 1958


frequency of these waves tended to decrease with increasing latitude. Araki and Allen (1982) found that many ground-based transient pulsations associated with SIs showed latitudinal polarizations reversals between 64' and 7 2 north geomagnetic latitude.

Kaufmann and Walker (1974) and Nopper et al. (1982) studied low- and mid-frequency pulsations produced in the magnetosphere by SIs and suggested that the compressive fast mode caused by the shock in the solar wind couples to toroidal field-line resonances. Baumjohann et al. (1983) found that magnetic and plasma drift signatures of an SI recorded by the GEOS 2 satellite were compatible with an adiabatic compression of the magnetosphere. The SI was followed by a damped low-frequency, compressional magnetic-field oscillation. In a second study of low-frequency plasma waves triggered by an SI, Baumjohann et al. (1984) found that low- frequency oscillations following the SI had both compressional and transverse magnetic field components. They attributed the pulsations to tailward-moving localized surface waves on the magnetopause caused by a shock in the solar wind. The surface waves possibly coupled to field-line resonances in the magnetosphere.

10.3 Transient pulsations associated with flux transfer events

Many observations now point to the possibility that dayside reconnection at the magnetopause can occur as transient, spatially localized events (see Fig. 2), which are called flux transfer events (FTEs) (Haerendel et at., 1978; Russell and Elphic, 1979; Rijnbeek et al., 1984). Satellite measurements of the scale sizes of FTEs indicate sizes of about 2 / ? ~ tangential to the magnetopause (Elphic and Russell, 1979; Rijnbeek et al., 1984; Saunders et al., 1984). These scale sizes are just slightly larger than estimated wavelengths for the Kelvin-Helmholtz instability in the low-latitude boundary layer.

In an FTE, magnetic field lines which pass through the outer region of the boundary layer should map to the high-latitude ionosphere, near the convection reversal boundary. Cowley (1982) and Paschmann et al. (1982) suggested that the reconnected flux tube would have substantial field- aligned current. This field-aligned current is associated with vorticity in the plasma flow. Lee (1986) showed that the multiple x-line reconnection model of Lee and Fu (1985) predicts large plasma vortices in the flux tube following the propagation of a shear Alfven wave and the field-aligned current from the magnetopause to the ionosphere. Southwood (1985) suggested that the convection of the FTEs field lines would also generate two additional convection vortices in the ionosphere.


Very few ground-based observations of pulsations connected with FTEs have been reported in the literature. Goertz et al. (1985) identified a possible FTE in STARE data. This event had spatially localized flows across the convection boundary (Fig. 37). with scale sizes of about 50-300 km. Short- duration (approximately one cycle) low-frequency (1.5mHz) pulsations accompanied the localized convection event (Fig. 38). Lanzerotti et al. (1986) showed two examples of possible FTE pulsations in magnetometer data from stations in Antarctica.

IRREGULARITY DRIFT VELOCITY

Figure 37. Electron drift velocities observed by the STARE radar between 1241 UT and 1250 UT on 6 October 1979 (after Goertz et al., 1985). The sequence shows the evolution of an isolated region of large poleward and sunward flows.

Figure 38. Magnetic field observations covering the interval in Fig. 37 (after Goertz et at., 1985). The A component is approximately northward and the B component is westward.


10.4 Transient pulsations associated with polar magnetic substorms

10.4.1 Transient field-aligned currents and plasma waves

During intervals with magnetospheric substorms, substantial amounts of field-aligned current flow from the magnetotail to the auroral ionosphere. Magnetic-field measurements made on the ground and at synchronous orbit indicate that the current forms a loop with net downward current to the east, net upward current to the west, and closure through electrojet currents in the auroral ionosphere (see Fig. 39 and McPherron et al., 1973). The enhanced field-aligned currents of the substorm expansive phase are carried by shear Alfven waves, though there may be strong compressional waves near the region of the instability which leads to the expansive phase in the magnetotail.

In order to understand transient pulsations occurring during the substorm expansive phase it is necessary to have a relatively complete model of the propagation of localized electric fields and field-aligned currents in the magnetosphere, and the reflection of these electric fields from the ionosphere. The configurations of the electric fields and ionospheric

1 Expansion Phase Shor t Ci rcui t of Tai l Current


conductivities must be compatible with those found on field lines which thread the ionosphere near and within auroral arcs, since the transient plasma waves and pulsations are intrinsically related to the enhanced field- aligned currents which flow in or near the brightening auroral arcs.

Perhaps the simplest models which can add some insight into the formation of transient pulsations are the voltage or current 'generators' which launch a step in field-aligned current and a shear Alfvkn wave toward the ionosphere (Nishida, 1979). Examples of the temporal evolution of the

0

-1

0

-0.25

0

-2

0

- 0.5

0

-L,

I I I

-

- Genera tor

-

-

- -

-

-

- , I I

0 20 40 60 80

t /T ,

Figure 40. The electric and magnetic fields in the ionosphere due to an electric field generator in the magnetotail (bottom two traces) and due to a current generator (next two traces) (after Baumjohann and Glassmeier, 1984).


electric and magnetic fields these generators produce in the ionosphere are shown schematically in Fig. 40. In this example, the travel time from the generator to the ionosphere is TA, and the reflection coefficient of the uniform ionosphere (equation (5.4.5)) is 0.8. The procedures for calcu- lating these fields are given by Lysak and Durn (1983). The set of reflections from the ionosphere and source region lead to damped oscillating fields with a frequency (4TA)".

The propagation of localized electric fields and field aligned currents in a low-j8 plasma (18 < ̂ me/m;) can be analysed by using the cold-plasma dielectric tensor in equation (5.3.2) for the low-frequency hydromagnetic i t (u 4 1 ugi I or ~ T / T < ̂ 1 wgi I, where T is a characteristic timescale for the wave). We replace - kii by 9/9t and then

and

From equation (5.1.14) we have

(C- D)E=O (10.4.3)

where

C = V x V x , and D=diag

We use a coordinate system with x, y and z corresponding to directions 1, 2, and 3 respectively. We choose a/ay = 0 and Ey = 0, giving the two equations for Ex and Ez,

and

From equation (10.4.4) we see that when spatial gradients, QIQx, are large, the perpendicular and parallel (Ez) electric fields are coupled. Even though Ez is small it leads to the acceleration of electrons and the formation of field-aligned current. These field-aligned currents are fed by the perpendicular polarization currents jx = ( j n ~ v i ) 'dEX/dt. The shear Alfvkn wave is, in effect, coupled to the electron plasma oscillations. Note that as


9/9x-' 0, equation (10.4.4) gives the wave equation for the shear Alfven mode, and equation (10.4.5) gives Ez s= 0.

If we take the 9/9z of equation (10.4.5) and use equation (10.4.4) we obtain the equation

Equations (10.4.4) and (10.4.6) were used by Goertz and Boswell (1979) to describe propagation of localized electric fields, and field-aligned currents in a plasma where (3 < rne/mi (Fig. 41). Goertz and Boswell used a simple geometry in which the electric field reversed direction on a surface of magnetic field lines. Their solutions to the two equations indicate that the parallel electric field in the l o w 4 plasma points opposite to the direction of propagation. Consequently electrons are accelerated in the direction of the propagating wave, and behind the wavefront, field-aligned current flows antiparallel to the direction of the propagation. These field-aligned currents are fed by the polarization currents at the leading edge of the disturbance. At the time T, the wave reflects from the ionosphere, leading to a wave propagating upward with E a =s RExi (equation (5.4.5)). Since E,, ^> Em, E X R = -Exi, the upward propagating wave causes a decreased total electric field behind, but leaves an increased field-aligned current (the electrons are accelerated opposite the direction of propagation).

Figure 41. Sequences of pictures showing the temporal evolution of the localized electric fields associated with field aligned currents (based on the diagram in Goertz and Boswell, 1979). The dashed lines are the equipotential contours. The waves reflect from a uniform ionosphere (hatched region) at the time T .


Goertz and Boswell(1979) have also considered the propagation of localized electric fields and field-aligned currents in a high-/3 ( 0 > ~ e / m i )

plasma. In this case, the shear Alfven wave is coupled with the ion acoustic wave (low-frequency branch in Fig. 5).

Field-aligned currents and localized Alfven waves will also be produced if a more-or-less uniform Alfvkn wave reflects from an ionosphere with non-uniform conductivities (Ellis and Southwood, 1983; Glassmeier, 1984). In this case the enhanced field-aligned current grows upward along the field lines. In addition, rapid changes in ionospheric conductivities due to precipitating electrons, lead to a decrease in the ionospheric electric field and an upward propagating Alfvkn wave (Maltsev et al., 1974).

The processes involved in the formation of field-aligned current over auroral arcs are very complicated and require a self-consistent model which includes the propagation of localized Alfven waves in the magnetosphere, mechanisms for accelerating electrons to high energies (> lOkeV) and reflection of Alfven waves from non-uniform ionospheres with changing conductivities (due to electron precipitation). Lysak and Carlson (1981) and Lysak and Dum (1983) have considered models with microscopic turbulence driven by the field-aligned currents in the Alfven waves (kinetic Alfvkn waves in this case). Rothwell et al. (1984, 1986) have considered the problem of feedback due to changing ionospheric conductivities (see also Atkinson, 1970).

10.4.2 Transient pulsations marking the expansive phase

The expansive phase of polar magnetic substorms is marked by the brightening of a quiet, discrete auroral arc (Akasofu, 1968), by enhanced field-aligned currents of field lines threading the auroral oval, and by large increases in ULF magnetic field energy over the whole spectral band of geomagnetic pulsations (from 1 mHz to several hertz). One of the clearest and most consistent signatures of the expansive phase is the damped low- frequency (5-15mHz) pulsation train called Pi2s (Saito, 1961). The onset is also marked by higher frequency broad-band Pi1 or PiB pulsations (Heacock, 1967) (approximately 20mHz to several hertz). The Pils are often superposed on Pi2 wavetrains, particularly at mid-latitudes.

The high-frequency, broad-band bursts called PiB generally occur at the time of the brightening of an auroral arc (Bosinger et al., 1981), and are clearly correlated with the formation of enhanced, localized, upward field- aligned current (Kangas et al., 1979). The bursts can be as short as 1-2min and often show a fine-scale structure with repetitive bursts following irregularly in intervals of minutes.

The very board spectrum of the PiBs suggests that these pulsations do


not derive their energy from ion cyclotron instabilities in the magnetosphere or upper ionosphere. Bosinger et al. (1981) have noted that upward field-aligned currents are often carried by energetic, precipitating electrons, and consequently the mechanism producing PiBs may be intrinsically related to the mechanism for accelerating electrons. One possible mechanism is the formation of double layers at low altitudes ( I R E ) (Block, 1972) and the turbulence in these double-layer structures.

The transient Pi2 pulsations (see the example in Fig. 1) appear to be an inherent part of the enhanced field-aligned current of the substorm expansive phase. In fact, it probably makes little sense to separate these pulsations from the initial field-aligned currents in the expansive phase. The first onset of the Pi2 burst, and the expansive phase currents are coincident in time, and generally the first excursion the Pi2 wavetrain has the same polarity as the following magnetic perturbations from the substorm field- aligned currents (the 'bay' structure) (Rostoker, 1967). Figure 42 shows an example of Pi2 pulsations recorded by the ATS 6 satellite and on the ground at Fredricksburg.

Pi2 pulsations are correlated with a number of expansive phase phenomena, including energetic precipitating electrons. Pytte and Trefall (1972) showed that Bremsstrahlung x-ray events and Pi2s occurred almost simultaneously. Pytte and Trefall (1972) and Pytte et al. (1976) showed that riometer absorption spikes at auroral latitudes and Pi2s occurred at the same time. All these features lead to the possibility that the Pi2s, the enhanced field-aligned currents and the energetic electrons are part of a common mechanism for the initiation of the expansive phase.

In general, Pi2s have their maximum amplitudes near the substorm- enhanced westward electrojet (Olson and Rostoker, 1975; Rostoker and Samson, 1981; Samson, 1985). There is some evidence for a small secondary amplitude peak at mid-latitudes (Stuart, 1974; Fukunishi, 1975; Saito eta!., 1976). A number of authors have attributed this secondary maximum to the plasmapause.

Mid-latitude Pi2s tend to be very monochromatic, and have relatively simple narrow-band spectra (Stuart and Booth, 1974). Conversely, high- latitude Pi2s have very complicated spectra (Olson and Rostoker, 1975) particularly near the region of the substorm-enhanced westward electrojet. Despite the differences in the spectra, Pi2s have frequencies which appear to be independent of latitude (Saito and Sakurai, 1970; Stuart, 1972). The complicated high-latitude spectra are probably due to spatial structure and

Figure 42. Transient, low frequency pulsations (Pi2) occurring at the time of the substorm expansive phase at geosynchronous orbit and on the ground at Fredricksburg (after McPherron, 1981).

GROU

ND P

i2 S

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OR

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ag.

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Ma

gn

et

og

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m

D-COM

P NO

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very dynamic motions in the field-aligned currents and ionospheric electrojets (Pashin et al., 1982).

The polarizations of Pi2s on the ground show distinct patterns in the orientation of the major axis of the horizontal polarization ellipse. At subauroral latitudes and mid-latitudes the orientation is NE-SW before 2300 LT, and NW-SE after (Saito, 1961; Rostoker, 1967; Baransky et al., 1970; Bjornsson et al., 1971). Figure 43 from Bjornsson et af. shows this effect quite clearly. Note that the directions of the vectors have been determined by the direction of the initial excursion in the Pi2. South of 60" geomagnetic latitude, these orientations are compatible with a substorm current wedge (Fig. 39) with net upward current before 2300LT and net downward current after. The latitudinal change in the direction of the H component at - 60" geomagnetic latitude probably marks the latitude where there is a transition from magnetic fields which are predominantly due to field-aligned currents, to those due to the westward electrojet. These patterns in the orientation of the major axis of the polarization ellipses have also been confirmed for individual events (Baransky et al., 1980; Lester et af., 1983), and for polarizations plotted in substorm-centred coordinates (Lester etal. , 1983; Samson and Harrold, 1983).

local time

Figure 43. The direction of polarization of Pi2 events (HD coordinates) (based on the data in Bjornsson et at., 1971).


The sense of polarization of mid-latitude Pi2s tend to be predominantly CC in the northern hemisphere (Rostoker, 1967) or CW in the southern hemisphere (Christoffel and Linford, 1966). These polarizations are compatible with a westward moving magnetospheric plasma wave which is localized at high latitudes (see equation (7.2.15)). At auroral and subauroral latitudes, the pattern of the polarizations is far more complicated. Bjornsson et al. (1971) also found CC polarization south of 6 0 - 6 5 ~ . However, poleward of 60" -65"~ they found that the polarizations were mostly CW. Fukunishi (1975) found CC polarization for all local times at a station near 6 0 ~ . At higher latitudes, however, the data show a transition from CC before 2100-2400LT to CW thereafter.

Other studies of the Pi2 polarizations have compared the polarization maps with the position of the substorm breakup (Kuwashima, 1978; Rostoker and Samson, 1981, Pashin et al., 1982; Samson and Rostoker, 1983). Kuwashima (1978) and Pashin et at. (1982) found four quad- rants of different polarization. The approximate boundaries of the quad- rants are marked by the latitude of the breakup (Kuwashima, 1978; Rostoker and Samson, 1981), and by the longitude of the head of the westward travelling surge (Pashin et al., 1982). Samson and Harrold (1983) were able to show that the variety of polarizations could be fitted to a relatively simple pattern. Near the longitude of the breakup (within Â 10" of the centre) there are two latitudinal polarization reversals, with CC polarization for regions more than 4" equatorward of the latitude of the onset, CW polarization between 4 equatorward and the approximate latitude of the onset, and CC poleward of the latitude of the onset. Outside this longitudinal region there is only one latitudinal polarization reversal, with CC polarization equatorward of the latitude of the onset, and CW poleward. Samson (1985) was able to show that most of these polarizations are compatible with a westward-moving current wedge similar to that in Fig. 39.

Measurements of the mid-latitude phase velocities and m values of Pi2s indicate that most of the pulsations exhibit predominantly westward propagation with phase velocities ranging from 1 to 4 longitude s 1 (Mier- Jedrzejowicz and Southwood, 1979; Baransky etul., 1980; Lester etal. , 1983; Samson et ul., 1985). Lester et at. (1984) and Samson etal . (1985) have found that there are indications of eastward propagation far to the east of the centre of the substorm current wedge (Fig. 44). At high latitudes, the Pi2s appear to propagate away from the centre of the substorm current wedge (Samson and Harrold, 1985).

Satellite observations of Pi2s in the magnetosphere show many of the features of a transient field-aligned current system. In Fig. 42, for example, the first excursion of the D component on ATS 6 is in the same direction


Substorm Centered Long. Figure 44. Inverse phase velocities of the H-component of mid-latitude ( 5 5 ~ geomagnetic) Pi2s (after Samson and Harrold, 1985). The data are plotted in substorm centred coordinates (longitude relative to the centre of the substorm current wedge). At the longitudinal centre of the wedge the H-component of the expansive phase is maximum and the D-component is zero. Negative values indicate westward propagation.

as the overall polarity of the substorm field-aligned current (shaded region). Gelpi et al. (1985) found that Pi2s at geosynchronous orbit occupy a very limited longitudinal extent and tend to be localized within the region of the substorm current wedge.

A relatively complete model of the oscillating field-aligned and ionospheric currents associated with Pi2s will likely require a generalization of the transient current models in equations (10.4.4) and (10.4.5), and Fig. 41. Unfortunately, the model is made much more complicated by the requirements for azimuthal propagation, and reflection from an ionosphere with the non-uniform conductivities which are typical of auroral arcs. The problem is essentially transient in time, and three-dimensional in spatial coordinates. However, a number of simpler models have attempted to explain specific features of the Pi2s wavetrains or polarizations (e.g. Mallinckrodt and Carlson, 1978; Nishida, 1979; Kan et al., 1982; Southwood and Hughes, 1985).


10.4.3 Transient pulsations during the substorm

During the evolution of the substorm very-low-frequency impulsive pulsations, called Pi3, are sometimes seen in magnetometer records at auroral latitudes (Saito, 1978). Kiselev and Raspopov (1976) further divided Pi3 into Pip pulsations which occur in the local evening and Ps6 which occur in the local morning. The Ps6 pulsations have very low frequencies (< 1 to -3mHz) and appear to be associated with auroral omega bands (Gustafsson et al., 198 1; Andre and Baumjohann, 1982; Opgenoorth et a/., 1983). The strongest signature of the Ps6 is typically found in the D component, which can reach hundreds of nanotesla in the auroral region. Often the wavetrains are very short, lasting little more than one cycle. Most Ps6s show eastward propagation with velocities from about 0.1 to 2 km s" in the auroral ionosphere (Kawasaki and Rostoker, 1979; Gustafsson et al., 1981; Andre and Baumjohann, 1982; Opgenoorth et al., 1983; Rajaram et al., 1986).

A number of different current models have proposed for Ps6 dis- turbances. Saito (1978) proposed a meandering electrojet model. Kawasaki and Rostoker (1979) and Rostoker and Apps (1981) suggested that Ps6s were caused by eastward-moving sequences of N-S ionospheric currents fed by field aligned currents. Gustafsson et al. (1981), Andre and Baumjohann (1982), Nielson and Sofko (1982) and Opgenoorth et al. (1983) found that the ionospheric electric fields measured by the STARE system and the ground-based magnetic fields of Ps6s were compatible with a sequence of eastward-drifting current loops with alternating upward and downward field-aligned currents at approximately the same latitude. They suggested that magnetic perturbations on the ground were due to circular Hall-current loops produced by the electric fields associated with the field- aligned currents. Andre and Baumjohann (1982) and Opgenoorth etal. (1983) found that upward currents were associated with the bright areas in the omega bands, and downward currents were associated with the dark areas.

At present, there is no complete theory or model which explains the bulk of the characteristics of Ps6s. Opgenoorth et a/. (1983) and Rajaram et a/. (1986) have pointed out that Ps6s occur on field lines which might thread the plasma sheet boundary layer, or low-latitude boundary layer. Rajaram etal. (1986) considered the possibility that Ps6s are due to Kelvin-Helmholtz instabilities in the low-latitude boundary layer of the magnetotail.

The Pi1 class of impulsive pulsations includes, in addition to the PiBs discussed earlier, a type of pulsation, sometimes called Pic, which occurs in the morning sector, after the substorm onset. These pulsations, which


typically have frequencies between 0.03 and 0.2 Hz, are generally irregular, and impulsive but unlike the PiBs, they persist for intervals of tens of minutes. These pulsations are predominantly auroral zone phenomena, and have been correlated with pulsating aurora (Campbell and Rees, 1961; Campbell, 1970).

Much evidence now suggests that these pulsations are produced locally in the ionosphere through modulation of ionospheric conductivities by the precipitation of energetic electrons (energy > 35 keV) (Johnstone, 1978; Oguti et al., 1984; Oguti and Hayashi, 1984; Engebretson et al., 1986b). If large ambient electric fields exist in the ionosphere, then the precipitating energetic electrons will modulate the ionosphere conductivities and currents. This leads in turn to localized shear Alfven waves propagating upward along the field lines and the formation of field-aligned currents (see equations (10.4.4) and (10.4.5)). Engebretson et al. (1983) have shown that the waveforms of many pulsations accompanying pulsating aurora are not sinusoidal, and have sharp onsets with an exponential decay of each pulse. These features argue against plasma instabilities such as ion cyclotron instabilities, but are compatible with impulsive changes of conductivity in the ionosphere.

At times, pulsating aurora and the associated geomagnetic pulsations show distinct periodicities. These periodicities are also evident in rocket measurements of precipitating electrons (Bryant et al., 1971, 1975). The rocket measurements often show considerable energy dispersion, with higher energy electrons detected before the lower energy electrons. Bryant et at. (1971) have shown that the dispersion is compatible with an equatorial source for the electrons. Unfortunately, there does not yet seem to be any clear understanding of the mechanism for producing these modulations in the energetic electron fluxes (Southwood and Hughes, 1983).

11 EPILOGUE

Almost a century elapsed between the first published observations of geomagnetic pulsations and the development of adequate theories to explain the origin of these fluctuations in the geomagnetic field. The rapid progress over the last three decades has been largely due to a careful blending of the technology of space science with recent advances in the theory of waves in magnetoplasmas. Though considerable progress has been made, many questions remain unanswered. Continuous plasma waves and pulsations now have a reasonably firm experimental and theoretical foundation, at least for small-amplitude, linear plasma waves. Conversely, transient pul-

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sations and plasma waves still require considerably more sophisticated theories and possibly more observations before we can assume that an acceptable level of understanding has been achieved. These transient and impulsive waves pose difficult problems for the theoretician because of the generally complex geometry and rapid temporal evolution of the processes. In addition, the short timescales and spatial localization of some of the transient plasma waves lead to extreme difficulties in interpreting data from satellites which are moving at appreciable velocities.

Further progress and discoveries in phenomena associated with ULF plasma waves in the magnetosphere will undoubtedly depend on improved experimental techniques and measurements. However, it is also likely that numerical or computer experiments will play an increasingly important role, particularly in the studies of transient waves, non-linear phenomena, and wave-particle interactions. The Earth's magnetosphere gives us a splendid laboratory for the study of plasma waves. We have not yet exhausted the enormous potential of this laboratory.

ACKNOWLEDGEMENTS

I would like to thank W. Allan, B. J. Fraser, K.-H. Glassmeier, M. G. Kivelson, R. L. McPherron, G . Rostoker and H. J. Singer for many helpful and informative discussions. Research for this project was supported in part by the Natural Sciences and Engineering Research Council of Canada.

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APPENDIX

APPENDIX. LIST OF SYMBOLS

A B cc

cw c c (subscript) D E e (subscript) F f fa

H h (subscript) 1 IMF I m i (subscript) 1 k kn kt L LH LGT LT m ma n n

P Qa

RE Re RH r rg T t u

anisotropy ( Ti/ Ti - 1) magnetic field counterclockwise polarization viewed downward on the Earth clockwise polarization velocity of light cold species magnetic east electric field electron force frequency distribution function of species a magnetic north hot species identity matrix interplanetary magnetic field imaginary part ion current density wavevector normal component of the wavevector tangential component of the wavevector McIlwain's L parameter left-hand polarization local geomagnetic time local time azimuthal wavenumber mass of species a number density of species a pressure charge of species a (with sign) Earth's radius real part right-hand polarization spatial position gyro- or Larmor radius temperature time J U T


u A

Ud

UD

us u s w

VT z z

(subscript) 13 5 &

Â£

6 ~ X B

u u b

u b i

Wcf

u c r

w Wg

OJ P

11 (subscript) -L (subscript) t (superscript)

T (superscript)

Alfvkn velocity drift velocity diamagnetic drift velocity sound velocity velocity of the solar wind thermal velocity plasma dispersion function vertical magnetic field (positive downward) species ratio of plasma to magnetic pressure Kronecker delta dielectric tensor dielectric constant of free space wave normal angle (between k and Bo) angle between the Sun-Earth line and the

interplanetary magnetic field first adiabatic invariant permeability of free space collision frequency fluid displacement mass density charge density height integrated Hall conductivity height integrated Pedersen conductivity Hall conductivity Pedersen conductivity bounce period drift period gyro- or Larmor frequency angular frequency bounce frequency bi-ion hybrid frequency cutoff frequency crossover frequency azimuthal drift frequency gyro- or Larmor frequency (with sign) plasma frequency component of vector parallel to Bo component of vector perpendicular to BO Hermitean adjoint; for example atb is the inner product of the complex vectors a and b transpose; for example, abT is the outer product of the vectors a and b

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