Magnetospheric physics

Magnetospheric physics

PHYSICS REPORTS (Review Section of Physics Letters) 47, No. 2 (1978) 109-165. NORTH-HOLLAND PUBLISHING COMPANY MAGNETOSPHERIC PHYSICS K. SCHINDLER an...

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PHYSICS REPORTS (Review Section of Physics Letters) 47, No. 2 (1978) 109-165. NORTH-HOLLAND PUBLISHING COMPANY

MAGNETOSPHERIC PHYSICS K. SCHINDLER and J. BIRN Ruhr-Universitdt Bochum, 4630 Bochum, W. Germany Received April 1978

Contents: I. Introduction 2. Discontinuities 2.1. The bow shock wave 2.2. The magnetopause 2.3. The plasma sheet and the neutral sheet 3. Quasi-steady phenomena and structure of the geomagnetosphere 3.1. Basic equations 3.2. Topology of the geomagnetic field 3.3. Quasi-static structures 3.3.1. The dipole-dominated region 3.3.2. The geomagnetic tail 3.4. Convection 3.4.1. Time-dependent convection 3.4.2. Stationary convection 3.4.3. Reconnection 3.5. Departure from smooth convection

111 113 114 118 124 128 128 129 131 131 133 136 136 137 141 143

4. Dynamic phenomena 4.1. Magnetospheric storms and substorms 4.2. Substorm theory 4.2.1. The ion-tearing-mode 4.2.2. Interaction with the ionosphere: substormassociated field-aligned currents 4.2.3. On the role of the magnetopause 4.3. Dynamic phenomena on small scales 5. Other planetary magnetospheres Appendix References

Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 47, No. 2 (1978)109165. Copies of this issue may be obtained at the price given below. Au orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price DII. 23.00, postage included.

144 145 147 151 155 157 157 160 162 163

MAGNETOSPHERIC PHYSICS

K. SCHINDLER and J. BIRN Ruhr-Universität Bochum, 4630 Bochum, W. Germany

(~1c

NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

K. Schindler and J. Birn, Magnetospheric physics

1.

111

Introduction

During the last decade it has become apparent that quite different types of celestial bodies can possess magnetospheres. First of all, there is the Earth’s magnetosphere, which naturally was explored most extensively. Unpredicted properties were and are still being discovered. Jupiter and Mercury were shown to be surrounded by a region that has been interpreted as a magnetosphere. It is generally believed that pulsars are neutron stars with a surrounding magnetosphere and a class of models of radio galaxies involve magnetospheres as well. The term ‘magnetosphere’ is usually attributed to that region of space around a body (or galaxy) in which the magnetic flux connected to and generated in the body plays a significant role in determining the physical properties of that region of space. Regions where ionisation effects of neutral gas components are important (ionospheres) are conventionally excluded. A complete coverage of magnetospheric physics is clearly far outside the scope of a single review. We will confine our attention to a number of current problems, with emphasis on the case of the Earth’s magnetosphere. Our view point is essentially theoretical concentrating on macroscopic properties. It will not be possible to include details of the many measurements that have led to our present knowledge of magnetospheric physics. In this respect we should like to refer to the comprehensive documentation compiled by the American Geophysical Union in the July 1975 issue of the Review of Geophysics and Space Physics. It is characteristic of the present state of the art of magnetospheric physics that many of the important problems are yet unsolved. Thus, the appropriate method reviewing those aspects should be to formulate questions rather than to give unjustified speculations. If one accepts this hypothesis, one should even refrain from drawing a picture of the magnetosphere, because any drawing necessarily extrapolates known facts into regions which are not yet sufficiently explored. Therefore fig. 1.1 should be taken only as a schematic illustration of the various regions involved rather than as a realistic picture of the magnetosphere. Figure 1.la is a schematic sketch of the overall shape of the magnetosphere showing approximately correct ratios of length scales. The bow shock standing on the solar side of the magnetosphere is a consequence of the fact that, in the frame of the earth, the solar wind flow is supersonic. The presence of the solar wind plasma and of the interplanetary magnetic field (not shown) leads to the formation of a long tail. The conditions beyond the orbit of the moon are not yet very well explored. However, observations do suggest that a tail-like structure extends over several thousands of Earth radii (RE). Figure 1. lb illustrates the near-earth part of the magnetosphere. The plasmasphere is a co-rotating toroidal region of plasma with electron number density ranging roughly from 102 cm3 to i03 cm3. The plasmasphere is situated in the essentially dipolar region of low latitude magnetic flux. The magnetic flux issuing from the Earth at higher latitudes forms the tail lobes. Since the plasma is able to escape along the tail and is also convected towards the midplane, the lobes carry a plasma of extremely low density. The region between the two lobes is the plasma sheet with electron number density ranging from 10_I cm~3to several cm~3typical ion energies are 1—10 keV with electrons about one order of magnitude less energetic. Pronounced plasma regions adjacent to the solar wind are the plasma mantle and the boundary layer, both involving directed flow, the cusp and cleft and the entry layer. The magnetosphere boundary, the magnetopause, separates the magnetosphere from the magnetosheath, where the effects of the dtwiated solar wind flow dominate. The bow shock confines the magnetosheath on its subsolar side.

:

112

K. Schindler and .1. Birn, Magnetospheric physics

7~~1i1 (~ ~ 50 RE

_~

‘t4 SH0C~~~~~~GNET0PAUSE

-50

(a)

\ ~



—.—.— ~

~/_

—~

~

~

OBE open SBd hoes

ENTR LAYER

/

h~

~

PLASMA ShE~ c’osed

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—p..

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~

— —

BCUNDARY AYER

—~

SOLAR WGhD

/

(b)

MAGNET PG S

. — - —

.

.

Fig. 1.1. Schematic illustration of the Earth’s magnetosphere. (a) Overall view demonstrating the spatial extent of the magnetosphere. Although the exact structure of the magnetotail is not known beyond 100 RE, traces of the tail have been observed at distances of several 1000 RE. The figure shows the plane defined by the sun-earth line and the magnetic dipole axis of the Earth. For simplicity the tilt of the dipole axis and the magnetic flux through the boundary of the magnetosphere are neglected. (b) Structure of the near-earth part of the magnetosphere defining the various regions (see text).

K. Schindler and J. Bim, Magnetospheric physics

113

It must be emphasized that a unified theory which would predict the structure of the magnetosphere from suitable boundary and initial conditions does not exist. Therefore, our review necessarily will have a largely fragmentary structure. It is typical for known examples of magnetospheres that binary collisions between particles are negligible. Therefore, a fully satisfactory description requires the kinetic theory of collision-free plasmas (Vlasov theory): 8f0/3t+v 3f0/ôr+(e5/m~)(E+vxB/c).8f~/c9v—O,

(1.1)

VXE=—(l/c)0B/t9t, 4~~Jfsvd3v+~—-~I, VxB=

(1.2) (1.3)

V.E=41T~Jf~d3v~

(1.4)

V~B=0,

(1.5)

consisting of collision-free Boltzmann equations for the distribution functions f~of each particle species s coupled with Maxwell’s equations; cgs units will be used throughout. In fact, for the discussion of neutral sheets (section 2.3) or of collective dissipation based on micro-turbulence (section 4.3) a Vlasov treatment is indispensable. There are however also other processes for which a magnetohydrodynamic approach seems to be justifiable (e.g. the steady state structure of quiet magnetotails). In the latter cases the justification for using MHD models is not always available from first principles but is rather based on either ad hoc assumptions or empirical facts. For instance, the observed (although not yet fully understood) approximate pressure isotropy in the quiet plasma sheet of the Earth’s magnetosphere allows an MHD approach to the structure of the quiet magnetotail (section 3.3.2).

2. Discontinuities The first in-situ measurements in space have already clearly demonstrated the presence of distinct regions of strong spatial variation of the plasma and field quantities [e.g. 2.1, 2.2, 2.3]. These regions are often called ‘discontinuities’ although, of course, all quantities remain continuous when observed on a suitable scale. A collision-free fast shock wave (bow shock) forms as a consequence of the interaction of the supersonic solar wind with the magnetosphere. Location and shape of the bow shock can be understood from the picture of a blunt obstacle situated in a supersonic gas stream [2.4]. Further studies confirmed this picture and today’s discussion centres around the internal structure of the wave and around the nature of the dissipation mechanism (section 2.1). Unlike the bow shock, the magnetopause (fig. l.lb) is not understood, even in terms of a simplified magnetohydrodynamic picture, at least not for all locations and times. Except for extremely quiet conditions, when the magnetopause may possibly be approximated by a tangential discontinuity (vanishing normal components of the flow velocity and of the magnetic field), none of the standard five classes of MHD-discontinuities [2.5]seems to apply [2.6, 2.7]. In section 2.2 we shall indicate how this difficulty can probably be overcome in principle, although a detailed solution is not yet available. Further ‘discontinuities’ detected by early satellite measurements are the plasma sheet separating

114

K. Schindler and J. Bim, Magnezospheric physics

the two tail lobes with oppositely directed magnetic field in the tail and the neutral sheet (fig. l.lb). Also in this case the physical properties seem to depend strongly on the prevailing conditions. It seems possible that under quiet (undisturbed) conditions the concept of a neutral sheet has only geometrical but no particular physical significance (see section 2.3). On the other hand, for perturbed states, in which large fluctuations occur, as well as for the onset of those states, specific neutral sheet properties assume a key role (sections 2.3 and 4.2). Section 2.3 gives a condensed survey of the various possible neutral sheet effects. 2. 1. The bow shock wave Major aspects of the bow shock may be understood from a gas dynamic point of view, e.g. its existence, its overall shape and the necessity for an increase of entropy. Here we concentrate on several properties for which typical plasma effects are responsible. (a) Dependence of the down-stream state on the dissipation mechanism For a gas-dynamic shock wave the Rankine—Hugoniot jump relations specify the down-stream state (density, pressure, bulk velocity) uniquely in terms of the upstream gas properties. This is not necessarily true in the case of a collision-free shock wave, where pressure anisotropy may become important. In a Chew—Goldberger—Low model [2.8] there are 8 jump conditions (balance of mass, energy and three momentum components and continuity of the two tangential electric field components and of the normal magnetic field component) for 9 unknown down-stream quantities (mass density, two components of the pressure tensor associated with directions parallel and perpendicular to the magnetic field, three velocity components, three magnetic field components). This implies that the down-stream state must depend on the processes taking place inside the shock transition [e.g. 2.9]. Therefore, a detailed understanding of the internal shock structure is important also for an exact description of the global structure. Only in cases where it is possible to eliminate one of the down-stream quantities (e.g. by assuming pressure isotropy), are the external conditions independent of the processes occurring inside the shock. There is strong observational evidence indicating that in fact the question of the down-stream state and of dissipation are intimately connected. Frequently the down-stream state is grossly turbulent (fig. 2.la) so that even the dissipation may not always be limited to the shock transition itself (region of rapid change of average quantities). (b) Shock associated signals upstream Unlike a gas-dynamic shock, a collision-free shock wave is able to emit signals that propagate into the upstream medium. This possibility exists, for instance, for whistler waves with a group velocity Vg exceeding the upstream bulk velocity. For the relevant plasma conditions and assuming that the angle 0 between the direction of propagation and the magnetic field differs sufficiently from ~/2 one may use the cold plasma dispersion relation (see e.g. Fairfield [2.10]) c2k2/w’

= ~ü~e(C~e

cos 0— w’Y’.

(2.1)

Here w~and ~e are the electron plasma and cyclotron frequency, respectively, and w’ is the frequency measured in the rest frame of the plasma, such that the detected frequency w is given by

(2.2)

K. Schindler and I. Birn, Magnetospheric physics

X 500 00 40 1~

30

20

115

(to,)

5

10

7

I

(b)

120 40-

(a)



~:~

~ .

~

—110.31——

f (hz)

1326

07 6

~

ooo

-

..

~-_

800

1~ -

60

~

~

~600

;:~‘~

-

20

1325~30

2

COS 40

//

40

1325

/ /

H

Bx

—~-~-

00

10 0

\

/‘

~

400

lw~eCOSO/

.1 I

I

I

.1

.2

.3

I .4

5

.6

.7

.8

K

Fig. 2.1. (a) Magnetic field profile of the bow shock (laminar case) at 0.144 sec/sampler (Greenstadt et al. [2.15]),BT denoting the magnetic field 2. + B~+ B~.(b) The cold plasma dispersion relation (solid curve) and associated phase and group velocities (dashed curves) magnitude for typical B solar wind conditions. The shaded area indicates the region where left- and right-handed whistler waves could be observed upstream 1 = VB from the earth’s bow shock. (Figure taken from Fairfield [2.101.)

where ~ is the solar wind velocity (—400 km/sec) in the frame of the satellite. For sufficiently large phase velocities the whistlers would be detected as right-hand polarized waves. The fact that sometimes the observed polarization is left-handed indicates that in these cases the phase velocity Vph is smaller than v~,.As shown in fig. 2. lb this condition together with Vph> ~ is satisfied only in a narrow band corresponding to a wavelength centred at about 100 km. It is interesting to note that the shock width is of the same order of magnitude. The observed damping may be due to several reasons. Clearly, the waves could be damped by the presence of a turbulence field of smaller scale; a non-linear decay process of whistlers into waves of different k may also be relevant. Another type of plasma wave observed upstream of the bow shock lies in the frequency range from about 3 s’ up to more than 60 s~.In this case attempts to explain these waves as whistlers failed and

116

K. Schindler and J. Bim, Magnetospheric physics

it is generally believed that they are MHD-waves. But how can MHD-waves with phase and group velocity smaller than ~ appear upstream of the bow shock? The answer is simply that some fraction of the ions that are reflected from the shock wave excite the MHD-waves in the upstream region. Two excitation mechanisms have been discussed. Barnes [2.11] showed that ion-cyclotron resonance interaction between the reflected beam and the incoming plasma can excite MHD-waves. The wave number is determined by the resonance condition kJ)vb

f~,

(2.3)

where v~,is the velocity of the beam, k11 the wave number parallel to the beam and fI~the ion—cyclotron frequency. A typical amplitude of the turbulence was estimated and it seems that the attenuation length L ~ (vb/vA)k~’

(2.4)

(where VA is the Alfven-velocity) is consistent with the observed scale (——50 Earth radii). An interesting alternative suggestion is due to Fredericks [2.12],who assumed that the bow wave generates an ion beam with modulated gyro-phase and concluded that this effect also leads to MHD-waves even without the presence of an instability. (c) Dispersion In a gas-dynamic shock the steady state velocity profile may be understood as the result of the balance between steepening due to inertia and of the viscous forces. In plasmas the presence of dispersion complicates the matter because a suitable dispersive behaviour may also balance the inertial steepening. This leads to the well-known fact that even in the absence of dissipation, steady state structures (e.g. solitary waves, fig. 2.2b) may exist. If both dispersion and dissipation are present typical shock profiles are irreversible and non-monotonic. The combined effect of dispersion and of dissipation may be illustrated in terms of a simplified steady-state model, where the magnetic field B is perpendicular to the shock normal and a sufficiently small amount of resistivity (due to collective interactions, see section 4.3) is present. The pressure tensor is chosen to be isotropic. Note that this model is too simple to guarantee a realistic description of the bow shock since both upstream whistlers and anisotropy effects as discussed above are excluded. However, this is the simplest V/B) -

B a)

Fig. 2.2. Solitary wave and shock profile for propagation perpendicular to the magnetic field (schematic). (a) Effective potential (solid curve). (b) Associated solitary wave solution. (c) Solution for cases where a small amount of friction is added. The first oscillation on the left corresponds to the solid curve of (a); for the later oscillations the potential continuously deforms as indicated by the broken line in (a).

K. Schindler and I. Bim. Magnetospheric physics

117

example possible to illustrate the effect of dispersion on shock waves. For details we refer for instance to the review by Biskamp [2.13]. If resistivity is absent, the above-mentioned approximations provide a solitary wave solution (figure 2.2b), which can be understood in terms of a one-dimensional oscillator problem in an effective potential V(B): (2.5) d2B/dx2 = —dV(B)/dB, where the qualitative shape of V(B) is shown in fig. 2.2a. If resistivity is added, the oscillator will be damped and a shock profile as shown in fig. 2.2c will arise. It is important for the quantitative understanding of this process that the effect of the dissipation will not only be to damp but also to change the shape of the potential [2.9]. (The solid curve in figure 2.2a corresponds to the first oscillation on the upstream side, the broken curve to conditions further down-stream.) Without viscosity, continuous solutions of this type (fig. 2.2c), however, are available only for sufficiently small Mach numbers (subcritical shocks). Shock waves with larger Mach numbers (~2—3, the exact value depending on the details of the model) require viscosity [2.13]. If a magnetic field parallel to the shock normal is included, standing whistlers lead to an oscillatory structure also on the up-stream side. Further effects which are present in the actual bow shock but are ignored in such a simple model are time-dependence, three-dimensionality, particle reflection and trapping, pressure anisotropy and inhomogeneities in the upstream flow, which may complicate the actual situation enormously. If the magnetic field component parallel to the shock normal is sufficiently large, fluctuations dominate and a well-defined shock structure does not seem to exist. Laminar structures such as discussed above have been observed when the shock normal is nearly perpendicular to the magnetic field [2.15]. (d) Dissipation Our knowledge of the dissipation mechanism in the bow shock is still rather poor. It is generally accepted that the dominant dissipation processes are based on the interaction between a wave field (micro-turbulence) and the particles. A possible generation process for subcritical resistive shocks propagating perpendicular to the magnetic field seems to be current-driven ion-sound excitation (e.g. [2.13], [2.18]). If the critical current density is exceeded, ion-sound turbulence would lead to momentum exchange between the electrons and the ions parallel to the shock plane such that it has the effect of a resistivity. The effective collision frequency for this process is given by [2.13] P

OipeWturbIflTekAD,

(2.6)

where u~is the electron plasma frequency, AD the Debye length, k an average wave number, n particle density, Te electron thermal energy and Wturb the energy density of the turbulent field. For Wturb/flTe ~ 1 the steady state turbulent energy level Wturb will be the result of a balance between unstable growth of the linear instability and a non-linear damping mechanism (e.g. [2.16], see also section 4.3). It seems that for sufficiently small values of the relative velocity, (although larger than the critical value above which the linear mode is exited, e.g. [2.17])a steady state is not reached in a shock and the dominant damping process involved is linear Landau damping [2.18]. For higher values of the relative velocity between electrons and ions, non-linear processes appear which are not yet very well understood. Figure 2.3 shows an estimate of the effective collision frequency as a function of the relative velocity [2.19]. It seems that subcritical laminar shocks can

118

K. Schindler and J. Birn. Magnetospheric physics II 1~1 1/3

nP~

~

(n/rn) 1/4

(M/G 2) 1 ~

(M/m) 1/2

Vd/c

Effective collision frequency v,~as a function of the electron drift velocity Vd for the case of the ion-sound two-stream instability. fl~is the ion plasma frequency, c, the ion-sound velocity, A an empirical parameter varying between several lO~to 10°.M and m are ion and electron mass, respectively. (Taken from Galeev [2.18].) Fig. 2.3.

consistently be explained on the basis of current-driven ion-sound turbulence; however, other modes (such as Langmuir waves or cyclotron waves) may also play a significant role. Super-critical shock waves are probably dominated by collective interaction of the incoming ions with the reflected ion beam [2.19, 2.20]. Although the qualitative picture of a collective dissipation in the bow shock seems to be confirmed, many questions remain open. One of the major gaps in observations — the absence of unambiguous separation of space and time variations — will probably be filled in the near future by the use of two satellites at controlled separations. On the other hand, a full quantitative description of the complex properties of the bow shock is as hopeless (and as useless) as a quantitative grasp of all the little details of an overturning water wave on a given stretch of a beach. We leave it to the reader to judge whether or not fig. 2.4, which represents a bow shock crossing of the more complicated variety, supports this view point. 2.2. The magnetopause The magnetospheric discontinuity of which our understanding is poorest is the magnetopause. It is not even clear how the problem should be approached from the MHD point of view. We discuss several possibilities based on a generalized concept of stationary MHD discontinuities. In addition, there is the possibility of a grossly turbulent layer, where the picture of a stationary discontinuity breaks down. Possibily, the ‘entry layer’ [2.23](see fig. 1.1) is of this type. The urgency of a satisfactory understanding of the magnetopause is obvious from its crucial role in the particle, momentum and energy balance of the whole magnetosphere. For instance, the origin of large magnetospheric plasma populations such as the plasma sheet will probably remain in the dark, until the magnetopause problem is solved. The problem is complicated by the fact that it is fairly evident that different magnetopause conditions prevail at different times and at different locations. Note that this implies that terms like ‘closed magnetosphere’ or ‘open magnetosphere’ are inappropriate without further specification of spatial and temporal conditions. The appropriate description of the magnetopause will also be sensitive to the purpose for which it is used. If the magnetic field component normal to the magnetopause, which is one of the discriminating quantities, is sufficiently small, it may be ignored for purposes of momentum balance, whereas it may still be very important for the balance of particles. In this situation all we can do is to confine the discussion to a number of basic possibilities.

K. Schindler andI. Bim, Magnetospheric physics

119

UCLA, JPL SEARCH COILS

Log

[

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225 JPL FARADAY CUP 150 6cm2sec~)

NV(l0

TELEMETRY UNITS

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100

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0 10

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(mV/m)

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Fig. 2.4. Examples of pulsations in a non-laminar shock transition propagating nearly parallel to the magnetic field. The top frame is a measure of the fluctuations in several frequency ranges. NV is proportional to the mass flux, the XSH direction was chosen to coincide as closely as possible with the B-vector component in the shock plane. (Taken from Greenstadt em al. [2.42].)

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K. Schindler and J. Bim, Magnetospheric physics

(a) Tangential discontinuity A tangential discontinuity in MHD is a discontinuity in which both flow velocity and magnetic field are tangential to the plane of discontinuity. Tangential discontinuities are characterized by the continuity of the total pressure P1+B~/8rr= P2+B~/8ir. (2.7) Here, the pressure P may arise from a gas pressure p and/or from momentum transfer of a reflected beam of particles [2.20].Obviously, there are infinitely many profiles which obey this simple pressure balance. A hint towards the agent that determines the profile in an actual case is available from the kinetic description (Vlasov theory). There, the freedom lies in the arbitrary choice of the distribution function f = F(H, P~,P7), F> 0. (2.8) where H is the Hamiltonian and P~,P~,are the canonical momenta perpendicular to the direction of the normal n = (1,0,0). In this form f solves the collision-free Boltzmann equation (1.1) trivially for arbitrary F, such that only Maxwell’s equations (1.2—1.5) remain to be solved. Obviously, F is determined by the mechanisms that govern gain and loss of particles for the region considered. Since these effects are not contained in a simple one-dimensional approach to the discontinuity the transition profiles are not uniquely determined. One of the problems, one encounters in a more realistic consideration is that possibly the ionosphere might be able to short-circuit the magnetopause electric 2, field that AD is typically present to where is the Debye-length assure quasi-neutrality, i.e. charge neutrality in zeroth order of (AD/L) and L the characteristic length of the system. It was shown by Parker [2.24]and by Lerche [2.25]that there are conditions under which the short-circuit indeed leads to a difficulty associated with the effect of large neutralizing currents. On the other hand, Alpers [2.26] showed that solutions with E = 0 can be constructed with the aid of suitable particle distribution functions F(H, P,, P 5). As one expects intuitively, the minimum thickness of exactly charge-neutral profiles is of the order of an ion Larmor radius [2.26].In order for these solutions to apply to the magnetopause, it seems that some particle transport across field lines is necessary. Such transport could be provided by fluctuations (see section 4.3). A possible excitation mechanism for fluctuations was suggested by Eviatar and Wolf [2.27]based on a two-stream cyclotron instability. Collective transport would give rise to a diffusive entry of particles from the solar wind into the magnetosphere. Because of the loss of momentum experienced by the solar wind one can describe this effect as a ‘viscous interaction’ [2.22]between the solar wind and the magnetosphere. It should however be pointed out that a permanent viscous force is not necessary for a magnetotail to exist (see section 3.3.2). What is required is the presence of particles inside; they might have entered the magnetosphere during some process that happened at some earlier instant of time. (b) Other one-dimensional structures If the magnetopause is one-dimensional (dependence on other than normal coordinate ignorable) and if a tangential discontinuity is excluded, the magnetopause should be one of the discontinuities listed below (resulting from equations (2.10—2.15) with all r.h.s. set to zero): discontinuity contact discontinuity Alfvén discontinuity slow shock wave fast shock wave

violated ~ = 0, ~PI1= 0 ~BU= 0, IIPI = 0 ~ <0, ~S~> 0 ~SI> 0

(2.9)

K. Schindler and I. Birn, Magnetospheric physics

121

where S denotes entropy density. II ]j denotes the jump of the quantity inside the bracket across the magnetopause (magnetosphere value minus magnetosheath value). On the right hand side of (2.9) we have noted characteristic properties of these discontinuities (see e.g. [2.5])that one finds to be violated [2.7] upon analysis of the observations. At least in the high latitude tail regions, the magnetic field inside is considerably larger than outside with the opposite signature for the pressure. Thus ~B]1> 0, ~P~j<0. Assuming a flow inward from the solar wind (strongly suggested by the presence of the plasma mantle [2.28]) the entropy density S must increase in the case of the shock waves as the plasma enters the magnetosphere; however, since the pressure is observed to be smaller inside than outside one finds ~ <0. At first sight, these results seem to favour the tangential discontinuity as the only possibility. In fact, there seem to exist times and locations of the magnetopause, where this concept applies [2.29]. On the other hand, observations clearly show that there also exist cases where B0, the component normal to the magnetopause, does not vanish; which contradicts the notion of a tangential discontinuity. In fact, the value obtained in some of the examples exceeds about 10 times the error estimate [2.29].Fig. 2.5 shows such a case. (It should be pointed out that the determination of B0 from single-satellite data necessarily involve several assumptions, e.g. [2.29].) This dilemma exists both for the magnetopause and for the plasma sheet. In the much simpler case of the plasma sheet there is a fairly straightforward way-out. Although a detailed discussion is presented later (section 3.3.2) we anticipate the following result: the presence of a small normal component of the magnetic field is explained by a weak spatial dependence of the plasma and field 60~81

cT~

60~81

60r B2

602~ 83

(a)

301’ 81

3O~’Ri

3Om~

(b)

Fig. 2.5. Hodograms of the magnetic field corresponding to different magnetopause crossings. B1 and B2 lie in a plane approximately parallel to the magnetopause, B3 is the. component parallel to the inferred direction of the normal. (a) B3 essentially vanishes, suggesting that the magnetosphere was closed at time and location of the crossing. (b) Substantial Brcomponent, suggesting that the magnetosphere was open at the time and location of the crossing (taken from Sonnerup [2.29]).

K. Schindler and J. Bim, Magnetospheric physics

122

quantities along the tail axis (i.e. parallel to the ‘discontinuity’). As shown in the appendix, the quiet plasma sheet can in fact be understood as a generalization of a tangential discontinuity. It is tempting to approach the magnetopause problem from the same view angle. Although — to our knowledge — this concept has not yet been applied to the magnetopause it is easy to demonstrate, that it should be promising. Applying (scalar pressure) MHD to a two-dimensional steady state configuration (fig. 2.8 illustrates such a configuration for the case of the plasma sheet) one finds that the jump relations for one-dimensional discontinuities [2.5] generalize in the following way: ~Pl~z~

—~f-Jpvxdz,

i~p+f-~_+pv~u=

Upv~v~ — ~-

BXBJ

(2.10)

—-f_f = —--

f

(pv~v~—~—B1B~) dz,

(pv~+ p +

E—~)

dz,

1{(~1~-~)v~ 4J ~—BxBzvx}I=

(2.11)

(2.12)

_~BxBzvz]~~

[(~+tPj+~B~)vx

(2.13) (2.14)

~Bzfl~~JBxdz~ —

v~B~fl = 0.

(2.15)

The transition to a strictly one-dimensional structure may be carried out by the limit

a

——*0 0x

I

or by

dz—*0.

j

In particular, the tangential discontinuity is then recovered for B~= 0,

v~=

(p+fl=o,

0: (2.16)

which is the same as (2.7). The structure of the eqs. (2.10)—(2.15) suggest that B~and v~of order 2

of —

Ox j

j

dz

may be admitted as the tangential discontinuity is generalized to include a small x-dependence. This concept would suggest a simple conceptual solution of the above-mentioned difficulty. The magnetopause would appear as a two- (or even three-) dimensional generalization of a tangential

K. Schindler and J. Bim, Magnetospheric physics

123

discontinuity. Note that the closed magnetosphere is included as a limiting case, where the magnetopause is (locally) essentially one-dimensional (i.e. sufficiently thin for two-dimensional effects to be ignored). Observational support for the two-dimensional magnetopause is provided by the plasma mantle (fig. 1.lb). The observations indicate that the thickness of the plasma mantle at a given location is largest at times when the magnetosphere is expected to be open [2.30] (see also section 3.2). The presence of a normal component B0 through the magnetopause will generally imply that magnetic flux of terrestrial origin connects to interplanetary field lines. Strictly speaking, this property by itself does not necessarily imply non-ideal-MHD processes, as was pointed out by Roederer [2.31]. On the other hand, if we postulate in addition that plasma is transported from a region threaded by purely interplanetary field lines (not connected to the Earth) to a region where at least one end of the field lines connects to the Earth, ideal MHD breaks down and special processes are required. In the latter case one speaks of ‘reconnection’ [2.32]. If B~ 0 without such plasma displacement the term ‘interconnection’ seems adequate [2.31]. We shall demonstrate the necessity of non-ideal-MHD processes for the simple case of a two-dimensional configuration B

=

[B~(x,y, t), B5(x, y, t), 0],

(2.17)

E

=

[E~(x,y, t), E5(x, y, t), E2(x, y, t)].

(2.18)

In this case it is always possible to find a gauge such that B=VxA,

E=—(1/c)OA/ot—V4,

(2.19)

with =

4(x, y, t),

A

=

(0, 0, A(x, y, t)).

(2.20)

The curves of constant A are the magnetic field lines. We shall prove that in a reconnection situation the equation

(2.21)

E+(1/c)vxB=0 must be violated. Suppose (2.21) holds, then we obtain with the help of (2.19) or

—V~— -~~

e~+ -~VAe,,

— -~e~v VA

=0

OA/ot + v VA = 0,

(2.22)

—V4

(2.23)

+

(1/c)v~VA= 0.

Equation (2.22) implies that a fluid element always experiences the same value of A. On the other hand, if the fluid element moves into a region of different topology it has to move through a separatrix (dotted line in fig. 2.6) with an x-type neutral point. Our assumptions imply that the value of A on the separatrix has to change with time: dA/dtlsep.

0.

(2.24)

(The separatrix assumes the value of A attached to a particular volume element when the volume element passes the separatrix.) In particular, this must be true for the neutral point 3A/3tINp+vNp.VA~Np0,

(2.25)

K. Schindler and J. Birn, Magnetospheric physics

124

ti

sepo rot nv

Fig. 2.6. Schematic picture of plasma moving into region of different magnetic topology. The plasma element moves across the separatrix at time

where

is the velocity at the neutral point. Since VA( = B we have VA (NP = 0 such that OA/OtINp 0. On the other hand eq. (2.22) implies that dAIOtINp = 0. This means that our assumption (2.21) cannot be satisfied everywhere. It must certainly be violated in the vicinity of the neutral point. In other words, there is an electric field at the neutral point, which would not be admitted by ideal MHD. Note that in a steady state the vector potential assumes the form A = —Et; by a suitable gauge transformation the electric field E may also be derived from a potential çb = —Ey. Theoretical investigations of the plasma and field structure in the vicinity of a neutral point are discussed in section 3.4.3. It should be noted that the relative importance of ideal-MHD-processes, diffusion processes, and reconnection processes is not yet clear. VNP

2.3. The plasma sheet and the neutral sheet The surface where B~vanishes in the geomagnetic tail (z = 0 in fig. 2.8) and its neighbourhood is another layer of particular interest; it is usually refered to as the plasma sheet (see also fig. 1.1). In a simplified one-dimensional symmetric model of the tail the plasma sheet is a plane layer centered at z = 0, where the direction of the magnetic field reverses (fig. 2.7). In a somewhat more refined picture, the plasma sheet may be regarded as a two-dimensional generalization of a tangential discontinuity (see the appendix). The region around z = 0, where the magnetic field is small in a sense to be discussed later is called the neutral sheet. In this section we concentrate on the question: what are the characteristic physical processes associated with the neutral sheet? The overall structure of the tail and the quiet plasma sheet is considered in section 3.3.2 and in section 4. Much of the discussion given here applies not only to neutral sheets but also to neutral lines, although we do not refer to neutral lines explicitly. In its simplest form the plasma sheet is a one-dimensional structure with B~=B~=0, ~

E~~°E50,

To be specific, we assume furthermore that IB~Iapproaches a constant value as z(—*co. A suitable self-consistent solution which corresponds to local thermodynamic equilibrium is given by [2.33] f(r, v)

=

c exp(aH

+

f3P~),

B,,

=

B0 tgh(z/L~),

E,,

=

E~= 0,

E~0,

(2.26)

K. Schindler and J. Bim, Magnetosphenc physics

125

z

------

rr4r%r~r4r%r4r%rr4r~~4r4r4r4r41~ _ —--~~—~--~ --

- —~-~---~-...~---~--~------

-

-

---

-~-~ - ~-----~----------

--

---~---

---

~~----~-

---.---

---~--------~----

.-------—---~-

Fig. 2.7. One-dimensional neutral sheet configuration

where f(r, v) is the one-particle distribution function, H the Hamiltonian of the particle motion, P~the y-component of the generalized momentum and c, a, f3, B0, L~suitable constants. L~is the width of the plasma sheet; c,the a, ~3may be chosen arbitrarily. Since v,, and appear plasma (through the Hamiltonian) 2,, + v~ distribution function (2.26) corresponds to v~ isotropic pressure in the x, z in the form v plane. In the domain z(
v~= —E/B~,

(2.27)

such that v~(+~) = —v~(—~). Therefore, in the presence of an electric field E5 0, at least one of the above-mentioned simplifying assumptions has to be given up. The simple one-dimensional neutral

126

K. Schindler and J. Bim, Magnetospheric physics

sheet configuration (2.26) is insufficient also for other reasons. For instance, observations suggest that the Br-component cannot be ignored. To order the various generalizations of (2.26) we list the assumptions made (a)

B2=0,

(b) 8/Ox=0, (c) P~,= ~ (d) 8/Oy=O, (e) a/at=0.

P,,~= 0 (pressure ‘isotropy’),

(2.28)

Note that strictly speaking pressure isotropy was assumed only in the x, z-plane (fig. 2.8). We briefly discuss some of the generalizations that one obtains by dropping one or several of the conditions (2.28). In some of the cases we merely refer to other sections of this review where a more detailed discussion is given in a different context. We begin by considering two-dimensional steady states (retaining conditions (d) and (e)). Since observations suggest B~0 it seems reasonable to violate (a). It is easy to demonstrate that this implies that the system is either two-dimensional (violation of (b)) and/or anisotropic (violation of (c)), because otherwise the j x B force due to B~and j~cannot be compensated. Although (for quiet times) observations suggest that spatial variation along x is more important than pressure anisotropy (see section 3.3.2) we consider the combined effect of both possibilities. Choozing a model pressure tensor of the simplified form [2.8] P=P±l+BBB and assuming symmetry in the sense v,,(z) = v,,(—z),

v~(z)= —v~(—z),

B1(z) = —B~(—z),

B~(z)= B2(—z),

z

- —-—---—----—-----------

--

Fig. 2.8. Two-dimensional configuration (B, 0). The rectangular (broken line) indicates the region of integration as used to derive equations (2. l0)-.(2. 15).

K. Schindler andJ. Bim, Magnetospheric physics

127

we find under the assumptions discussed above that momentum balance along the x-direction takes the form z

=

0:

z0:

(i

f..(~+~-~)+pv~

-)f

~.3J

~

(2.29)

~

(2.30)

Equation (2.29) clearly shows that for isotropy and O/dx 0 no solution with B~0 exists. To achieve a steady state, either anisotropy (2.36), (2.37) or x-dependence (2.31)

0/t9x=O(B~)

or a suitable combination of both effects is required. It is tempting to conclude from (2.29) that for the realistic case of small B~a small anisotropy be sufficient for establishing a strictly one-dimensional steady state (0/Ox = 0). It should be noted, however, that this conclusion is valid only for a fairly thin layer around z = 0. Outside, where B2/4i~ and/or v~have assumed finite values, (2.30) yields for 0/Ox = 0 P

2/4i~— pB2v~v~/B,,B~, (2.32) 11 — P1 = B which is small only for very special conditions, which in general cannot be expected to be satisfied. I

/

/

//

/

If

/ /

//

B

/

®,b.

,

/

/

Ia

I

/

/

/

I,

/ /

I

/ /

I/I

/

/

I /

,

~

_~ELECTRON

~

BEAM

— —







_—~





_‘\

~

-—— —

\

‘- ——





—.-—.—..

~—-..

I

-.----

.1’

\

-.~

N.., \

\

i

\

-‘

I

‘I

\

\

‘I

\

\



‘I ii

0~~ Fig. 2.9. Electric potential distribution in a neutral sheet with vanishing normal magnetic field component (Cowley [2.39]).

K. Schindler and J. Bim, Magnetospheric physics

128

This means that a finite pressure aniso:ropy would be required for most cases with 3/Ox = 0. This applies, for instance, to sub-Alfvénic flow, as observed at quiet times. Furthermore, a strictly one-dimensional sheet with E5 0 would not exhibit qualitatively new electric field effects, because E5 can be transformed away by a suitable Galilei-transformation along x. (Note that in the strictly one-dimensional case both E5 and B~are constant.) As discussed in section 3.3.2 the opposite limit (isotropy and 0/Ox 0) gives more satisfactory results for the quiet tail. The corresponding model is even sufficiently simple to include y-dependence (condition (d) dropped). It should be noted that most of the steady state models discussed so far have not yet succeeded in taking into account the electric field E~.(Fortunately, it seems that E~is small in the steady state as discussed in section 3.4.) The case of y-dependent steady states was studied by Alfvén [2.38] and, more explicitely, by Cowley [2.39]. In these cases four restricting conditions were imposed, namely B~= 0, 8/Ox = 0, a/at 0, and the zero-temperature limit. Alfvén [2.38] combined mass conservation with Ampere’s law to show that the cross-tail electric potential 41 is related to the magnetic field B0 = B((z( —* co) by the relationship en041 = B~/4ir,

(2.33)

where n0 is the particle number density of the incoming particles assumed to be monoenergetic. Particles entering the neutral sheet IzI <\/aL are accelerated and gain maximum energy e41. Cowley [2.39] showed that the electric field is concentrated within a layer with thickness of the order of c/w~1at the ‘ion exit side’ (side of lower electric potential) of the tail (fig. 2.9). In the case of a warm plasma the current sheet width is of the order of the ion-Larmor radius [2.40].

3. Quasi-steady phenomena and structure of the geomagnetosphere 3.1. Basic equations

In this section we will deal with the structure and slow structural changes under quiet conditions considering time variations on a time scale much larger than the typical MHD time scale. (Linear MHD waves can, however, easily be included in this treatment by superposition.) In addition, only large scale spatial variations are considered such that the discontinuities treated in sections 2.1—2.3 cannot be resolved. Thus a macroscopic description seems adequate using the following set of MHD-equations with Maxwell’s equations (see for instance the discussion by Vasyliunas [3.1]): dp/dt°0p/0t+v~Vp—pV.v, p dv/dt

=

—Vp

+j x B/c,

(3.1) (3.2)

d(pp~)/dt= 0,

(3.3)

E+vxB/c0,

(3.4)

VxE

(3.5)

=

—(1/c)OB/Ot,

VxB=4lTj/c,

(3.6)

VB=0.

(3.7)

K. Schindler and J. Birn. Magnetospheric physics

129

The pressure is assumed isotropic which seems reasonable for quiet conditions, at least for the plasma sheet (see e.g. [3.2]).Anisotropy of the pressure tensor will be discussed in section 3.3.2. By the assumption of the idealized Ohm’s law (3.4) electric fields parallel to the magnetic field as well as electric fields at magnetic neutral points (as already discussed in section 2.3) are excluded. These effects will be discussed in sections 3.4.3 and 3.5 using a generalized form of Ohm’s law. In the case of small time variations, 8/at, v and E may be considered as of same order [2.7],which can be characterized by a common parameter of smallness, say 8, which represents the ratio of the characteristic time scales, i.e. the transverse travelling time of an MHD wave (order of 1 minute) divided by the time scale of interest, e.g. of the slow changes before the onset of a substorm (tens of minutes to hours). Introducting new quantities of order 1: r—&,

Vv/ô,

FE/ô,

equations (3.1)—(3.5) can be rewritten Op/Or+ VVp=—pV V, 82p(OV/Or+ V.VV)=—Vp+jxB/c,

(3.1’) (3.2’)

(3/Or + V V)pp~= 0,

(3.3’)

F+VxB/c=0,

(3.4’)

V X F = —(1/c)OB/Or. (3.5’) can be neglected compared to unity, equation (3.2’) reduces to the static balance of forces

j~

—Vp+jxB/c0,

(3.8)

while all other equations remain unchanged. The description of a quasi-static development of the magnetosphere thus requires the solution of eqs. (3.6)—(3.8) at any instant of time where the quasi-static solutions are connected through eqs. (3.1’), and (3.3’)—(3.5’), and through boundary conditions which in general may be (slowly) time-dependent. The boundary conditions consist of conditions at the magnetopause, for instance in form of a tangential electric field and normal magnetic field following from solar wind quantities in the magnetosheath, and of conditions at the ionosphere, which are usually expressed via Ohm’s law (see e.g. [3.3, 3.4]) jzzrty.(E+vXB/c) (3.9) Here a is the conductivity tensor including Pedersen and Hall components, and v 0 is the velocity of the neutral gas (usually assumed to corotate with the Earth). In section 3.3 we will deal with the quasi-static part, equations (3.6)—(3.8), which yield the quantities of zeroth order in 8 (p, B and j) including their first order corrections. The first order quantities E and v and the corresponding equations will be considered in section 3.4. 3.2. Topology of the geomagnetic field With a high degree of idealization one may distinguish two extreme cases of topological structures of the quiet magnetosphere (see e.g. [2.31], [3.5],and the discussion of the magnetopause in section 2.2) which seem to be correlated with the direction of the interplanetary magnetic field (IMF) and

130

K. Schindler and J. Bim, Magnetospheric physics

correspondingly with the strength of magnetospheric convection. (An appropriate indicator may be the occurrence and thickness of the plasma mantle [2.30].) (a) The closed magnetosphere, which seems to apply best to times of northward orientation of the IMF, with no connection of magnetospheric and interplanetary magnetic field lines (perhaps with the exception of the far end of the tail and of the cusp region). It must be emphasized again that the term ‘closed’ certainly cannot be applied to every boundary region of the magnetosphere. Indeed there seems to be no longer doubt that there exist open field lines at any time [3.6]. A closed magnetosphere may be visualized as an infinitely conducting closed body squeezing aside the external plasma flow and with it the interplanetary field lines (fig. 3.1). The corresponding discontinuity separating the magnetosphere from the interplanetary field, i.e. the magnetopause, must be a tangential discontinuity with no plasma flow across it (see section 2.2). In view of the 8-expansion for small electric field and slow convection speed this configuration represents the zeroth order in 6. (b) If, however, terms of first order in 6 and thus convection are included, the tangential discontinuity must be generalized as discussed in section 2.2 to allow for non-vanishing normal components of magnetic field and flow velocity at the boundary. This leads to the more general case of an open magnetosphere which seems to be the more adequate description for times of southward orientation of the IMF. Topological considerations as well as observations (e.g. [3.7], [2.291,[3.8], [3.1], [3.9]) have led to the following qualitative picture (see fig. 3.2): Interplanetary magnetic field lines become connected with magnetospheric field lines at a dayside x-type neutral line (for a non-laminar flow this reconnection may occur in a turbulent layer as well, with the possibility of more than one neutral line). The field lines are convected towards the night side where they become reconnected through another x-type neutral line far in the tail to form new closed field lines on the night side. The location of this neutral line (again, possibly more than one in the case of turbulence) is not known. Perhaps the lunar shadowing observations (e.g. [3.9]), which show that indeed particles get from open field lines onto closed field lines, are the best direct support of the

\

\

~

\\\~~..J

~

~

Fig. 3.1. Schematic picture of a closed magnetosphere

showing interplanetary magnetic field lines squeezed aside by a rigid body representing the magnetosphere bounded by the magnetopause.

~i

Fig. 3.2. Schematic picture of an open magnetosphere showing the topological connection of magnetospheric and interplanetary field lines. Thick arrows indicate the plasma flow directions especially in the neighbourhood of dayside and nightside x-type neutral points, where magnetic reconnection takes place.

K. Schindler and I. Bim, Magnetospheric physics

131

existence of reconnection. These results suggest that in the tail the reconnection process between interplanetary and magnetospheric field lines typically occurs beyond the orbit of the moon. In a three-dimensional picture with north-south symmetry the day side and night side neutral lines may form a closed loop in the equatorial plane [3.1]. It is obvious that quasi-steady convection in an open magnetosphere is connected with quasi-steady field line reconnection (see also sections 2.2 and 3.4). An indicator for an open magnetosphere and enhanced convection is provided by the occurrence and thickness of the plasma mantle, which as shown by Sckopke et al. [2.30] is related to southward orientation of the IMF. 3.3. Quasi-static structures A selfconsistent quasi-static description of the gross structure of the magnetosphere, which holds for quiet times, i.e. slow convection and small electric field (6 1), and isotropic pressure requires the solution of ‘~

Vp=jxB/c,

(3.8)

VxB4irj/c

(3.6)

V~B=0,

(3.7)

with and at any instant of time. A solution of these equations that holds for the entire magnetosphere is still not available. For the inner part, where the magnetic field is dipole dominated, and for the magnetotail, however, approximation techniques can be used to construct separate selfconsistent models as will be discussed in sections 3.3.1 and 3.3.2. The regions of validity of these solutions, unfortunately, do not overlap, so that it becomes non-trivial to fit them together. Such a fit has not yet been carried out. 3.3.1. The dipole-dominated region In solving equations (3.6)—(3.8) for the inner part of the magnetosphere up to distances of say about lORE, the fact can be used that during quiet times magnetospheric currents usually give only small contributions to the total magnetic field. This means that solutions can be gained by means of an expansion with respect to the small parameter /3~= 8irp~/B~, where p~and B~are characteristic values of plasma pressure and magnetic field in the region considered. In lowest order in f3~,,the magnetic field is a potential field following from V X B0 = 0 and V B0

0,

(3.10)

or equivalently 2U V 0=0 with B0=VU0.

(3.10’)

=

The magnetic field configuration in lowest order in f3~can thus be obtained by solving Laplace’s equation (3.10’) for appropriate boundary conditions which may be taken from observations. If a simple geometry for the boundary is used the solution can be gained even analytically from appropriate series of harmonic scalar potentials (see e.g. [3.101). A quantitatively more satisfactory agreement with the observations, however, can be obtained only if currents inside the boundary are included. These currents follow from the first order terms of the

K. Schindler and J. Bim, Magnetospheric physics

132

f30-expansion of equations (3.6)—(3.8), which read Vp1=j1xB0/c,

(3.11)

V x B1

(3.12)

=

4irj1/c,

V~B1=0,

(3.13)

or, if all first order quantities but Jt are eliminated Vx(.j1xB0)=O,

(3.14)

V.j1=0.

(3.15)

and Recent quantitative three-dimensional models ([3.1 l1—[3. 14]) unfortunately do not fulfill equation (3.14) (even (3.13) is fulfilled only approximately in some cases) such that they are unsatisfactory from a theoretical point of view. Although for an anisotropic pressure tensor there may be more freedom in choosing j1, it cannot be expected that an arbitrary current density j1 only fulfilling (3.15), will fit into a selfconsistent theory. In any case, these models are useful as analytic approximations of the measured field (which was the main purpose for constructing them). A selfconsistent solution of (3.11)—(3.l3) can be obtained in the following way: we assume that B0 = Va x Vj3 is expressed by Euler potentials (see e.g. [3.15])which are constant on field lines. As can be seen from (3.11) the pressure Pi is constant on field lines too and can thus be represented as a function of a and /3, i.e. Pt = P(a, /3, t). If the function P h~asbeen chosen (e.g. in accordance with plausible assumptions on plasma gain and loss on field lines or with observations) the component of j1 perpendicular to B0, j1, follows immediately from (3.11):

j± = (c/B~)B0x Vp~.

(3.16)

Equivalently j Va = —caP/0f3,

(3.17)

.

j.V~3c3P/3a

(3.18)

The parallel component of j~, J~can be derived from (3.15). If the cartesian coordinates x, y and z are substituted by curvilinear coordinates a, /3 and s, where s is the arc length along a field line, (3.15) can be rewritten: B{f

~

Va)

V13)

+~ ~

+-f-

(~i1Vs)} = 0.

(3.19)

Using (3.17) and (3.18) and integrating (3.19) along a field line we get 1 OP 3 (ds OP 3 (ds _J1.Vsz~rc__j__c___j_ .

or with

j1 ~Vs

~Jjj.OP

—jII+jJ~

3 Ids

B=’o/3lkj

(3.20)

Vs OP

a

fds

1

—~-—c~—~j -~-—~j1Vs.

(3.21)

It is easy to show that the expressions on the right hand side of equation (3.21) can be combined

K. Schindler and J. Bim, Magnetospheric physics

133

giving j11/B

=

—j1

V

J

(3.22)

ds/B.

The integration in (3.22) must be carried out along a field line. The lower integration limit must be chosen such that j~vanishes there, for instance at the equatorial plane for closed field lines in a symmetric configuration. The expression (3.22) is identical with a result given by Vasyliunas [3.16]; the factor 1/2 in Vasyliunas’ formula is due to the fact that there the integration extends along the entire field line from one ionospheric footpoint to the other one. From the current density j~the corresponding first order magnetic field B~follows via the Biot—Savart law 3r’. (3.23) B1(r) = 41T j1(r’) x (r— r’) d To complete the set of equations one must include the closure of the field-aligned currents at the ionosphere and the boundary conditions at the magnetopause. This, however, involves the electric field and the convection speed; this means that we have to defer the further discussion until we have considered these quantities in section 3.4. Nevertheless, we can already conclude at this point that the presence of the magnetospheric plasma will generally lead to field-aligned currents and therefore to an electric coupling between the magnetosphere and the ionosphere.

J

3.3.2. The geomagnetic tail Because of the balance of plasma pressure and magnetic pressure in the plasma sheet, the parameter f3~is of the order of 1 such that we cannot use a small f3~,expansion and the non-linear set of equations (3.6)—(3.8) must be considered. Eliminating p and j from these equations and representing B by Euler potentials a and /3 as already used in section 3.3.1 leads to Vx(B ~VB)=O,

(3.24)

B

(3.25)

=

Va X V/3.

An equivalent set of equations has already been given in section 3.3.1 for determining quantities of first order in f3~,.If i is expressed by a and /3 one gets in accordance with equations (3.17) and (3.18) V x (Va OP/0/3 = —V

X Vfl)

X (Va

.

V/3/4ir,

(3.26)

x V/3)~Va/4ir,

(3.27)

where P(a, /3) is again the plasma pressure which is constant along the field lines and thus a function of a and /3. For the magnetotail, these sets of equations can be simplified using the fact that, on the average, the gradients parallel to the plasma sheet are much smaller than the perpendicular gradients [3.17, 3.18]. This property can be expressed by an appropriate parameter 1, which was shown by Birn et al. [3.16] to correspond to the reciprocal value of the Mach number of the unperturbed solar wind. If terms of order ~2 are neglected compared to unity, equation (3.24) can be integrated [3.20] leading to ‘~

B VB,, = ‘I~r -j3(x, y), .

4.ir -f--~o(x,y).

B VB 5

=

(3.28)

134

K. Schindler and I. Bim, Magnetospheric physics

The third component of equation (3.24) follows from the two equations (3.28). These can be combined and integrated once more yielding the pressure balance in the z-direction (B~+ B~)/8ir+ P(a, /3) = ~5(x,y),

(3.29)

which shows that j3(x, y) represents the total pressure, while P(a, /3) can be identified as the plasma pressure. The two-dimensional case (y-dependence negligible) can be obtained from (3.26) and (3.27) by setting j3 = y, P = P(a), and identifying a with the y-component of the vector potential A~.In this case both sides of equation (3.27) vanish identically while (3.26) yields 2a/41T = dP/da. (3.30) —V Solutions of (3.30) have been given by Birn et al. [3.17] using an asymptotic method for small values of [3.18]. In a similar way the same authors found the general three-dimensional solution of equations (3.28) with (3.25) for the case of negligible field aligned currents [3.19, 3.20], which can be expressed by a single integration: z



z

f

2+ (8/3(x~Y)~2 da \ Oy / ~o(x.y) V8ir[fl(x, y) — P(a, /3)]

\ Ox Y)~ / 0(x, ~) = ~(O/3(x~

(331)

In equation (3.31), f3(x, y) and j9(x, y) must be chosen such that ~3= const. represent straight lines in the x, y plane, while j3 = const. must be the associated orthogonal trajectories. The simplest way to achieve this is to choose /3

=

C(y — y

0)/(x x0) (3.32) 2 = (x — x 2 + (y — y 2, (3.33) j5 = iS(r) with r 0) 0) with arbitrary constants C, x~and y 0. The limit x0 —~—co, C/x0 —* —1, y~= 0 corresponds to the two-dimensional case discussed above. The function ao(x, y) is defined by —

and

P(ao(x, y), f3(x, y)) = ~$(x,y).

(3.34)

In this theory the functions P(a, /3), z0(x, y), and j3(r) and the constants x0 and Yo remain arbitrary and can be used to fit the solution to the observations. Such a fit has been carried out by Birn et al. [3.19] (revised by Birn [3.20]; see fig. 3.3), using a power law for the high latitude B,,-component (equivalent to j3(r)) as obtained from references [3.2l]—[3.23].The position of the neutral sheet z0(x, y) has been chosen to be z0 = 0. The observed tail flaring in the x, y plane ([3.24], [3.35], see fig. 3.3b) determined the constant x0, while Yo = 0 because of the assumed symmetry. The experimental information about the width of the plasma sheet along the tail ([3.26]—[3.30], fig. 3.3a) has been used for choosing the functional dependence of p on a, while the dependence on /3 was chosen to demonstrate the possibility of including a thickening of the plasma sheet at the dawn and dusk sides which has been reported by several authors (e.g. [3.26],[3.29]; see fig. 3.3c). One of the consequences of the quantitative fit was that, for the data set considered, the tail flaring in the z-direction turned out to be slightly larger than in the y-direction (see fig. 3.3c), if there is no loss of magnetic flux through the magnetopause. Self-consistent quantitative models including field-aligned currents have not been carried out as yet. Birn et al. [3.19] discussed the problem qualitatively and showed that the existence of ~ is connected with a z-dependence of /3.

K. Schindler and I. Bim, Magnetospheric physics

30

135

(b)

50



Xsrn/RE

‘sm~E

30 (c)

ZsmIRE

~0

2

Y

5rn/RE

Fig. 3.3. Magnetic field configuration of the tail after Bun [3.20].(a) Field lines in the solar magnetospheric x, z-plane (solid lines). The dashed line represents the plasma sheet half width L(x). The symbols correspond to measurements of L mentioned in the text: hourglass, Bame et al. 13.261; circle, Bowling and Wolf (3.271; square, Nishida and Lion [3.28];cross, Meng and Mihalov [3.29];and error bar, Rich et al. [3.301.(b) Projections of magnetic field lines and position of the magnetopause (dashed line) in the solar-magnetospheric x, y plane. The symbols correspond to magnetopause measurements mentioned in the text: squares, Behannon (3.24]; and circle, Mihalov et al. [3.251.(C) Cross section of the tail at XSM —60 R~(large oval) resulting from an assumed circular cross section at XSM = —20 RE with y-dependenl isobars and plasma sheet half width (dashed line). The symbols correspond to those in fig. 3.3b.

The equilibrium solutions discussed in this section were derived using an isotropic pressure tensor. The effect of anisotropy can be easily discussed by use of equations (2.29) and (2.30) specialized for = (i — ~ ~ = (p1 + ~) for z =0, (2.29’) ~

vii— P±

—~

=

-

—~-~J [~+

2/4irB2,, +~—]dz for z 0.

(2.30’)

P1— P±—B

In the isotropic case (P 11 = P1) eq. (2.30’) shows that B~and 0/Ox must be of same order so that both sides of this equation could be2/4ir equal. is obvious from (2.30’) that outside z = and 0a canItproduce only small relative changes aofthin thelayer field around quantities small anisotropy P11 — P.j ~B their variation with x.

K. Schindler and J. Bim. Magnetospheric physics

136

Inside this layer, which has a thickness of order , a small anisotropy even of order 2 may in principle have a finite effect on the field quantities or on their x-dependence as can be seen from equation (2.29’). Such a possibility, however, is excluded if one assumes that the field quantities vary smoothly across this layer which seems to be a reasonable assumption for quiet quasi-static structures. For dynamic variations, however, a singular behaviour in this layer, which approximately coincides with the neutral sheet discussed in section 2.3, may be of importance (see also section 4). The results obtained in this section are correct to zeroth and first order in 6. As we demonstrated, this approximation is sufficient to provide a quantitative modelling of the tail. Note that our procedure guarantees that the result of all currents flowing inside or outside the magnetosphere is taken into account, because we considered the class of all possible ‘slender’ selfconsistent magnetotail configurations and fitted the remaining free parameters to the observed magnetic field. As follows from the general scheme discussed in section 3.1 the present results also constitute the first step for a selfconsistent theory of convection in the magnetotail. —



3.4. Convection 3.4.1. Time-dependent convection As already mentioned in section 3.1 the complete quasi-steady description of the magnetosphere must include boundary conditions at the magnetopause and the ionosphere. In view of the complexity of the magnetopause problem (sections 2.2 and 3.4.3) we are forced to introduce ad hoc assumptions for the boundary conditions at the magnetopause. In the case of the ionosphere we may argue in the following way. The closure of field-aligned currents at the ionosphere as well as the motion of the neutral gas causes an electric field which in general leads to changing boundary conditions for the magnetic field via Faraday’s law (3.5). Thus Ohm’s law for the ionospheric fields and currents (3.9) represents a ‘feed-back’ on the possible solution for the magnetic field. For stationary magnetic field configurations in the dipole-dominated region (where f3~,~ 1) this feed-back can be neglected such that the convection problem including electric fields and field-aligned currents can be treated separately from the determination of the magnetic field configuration. This approach will be discussed in more detail in section 3.4.2. Once the magnetic field and plasma configurations are known as a function of time, the corresponding electric field and convection velocity can be determined. We assume again that the magnetic field is represented by Euler potentials a and /3 such that B = Va X Vf3. Then it is always possible to represent the electric field by E=

-~

Va —

-~

~

V/I



Vq5,

(3.35)

with a scalar potential 41. From Ohm’s law (3.4) it is obvious that E~B —B Vçb = 0 such that 41 must be constant on field lines and thus a function of a and /3. If boundary values for the tangential component of E (and thus for 41, since a and /3 are known) are given, 41 can be mapped into the entire magnetosphere along the magnetic field lines such that the electric field is determined everywhere. The velocity component perpendicular to B follows immediately from Ohm’s law: 2)E x B. (3.36) = (c/B .

The parallel component can be determined from the adiabatic law (3.3) which can be rewritten using

K. Schindler and J. Bim. Magnetospheric physics

137

the continuity equation (3.1):

dp/dt = —ypV~v.

(3.37)

We introduce curvilinear coordinates a, /3, and s, where s measures the arc-length along a field line such that B~Vf=B~-F(a,f3,s,t)

(3.38)

for any quantity f(r, t) = F(a(r, t), /3(r, t), s(r, t), t). Expressing V. v in these coordinates, eq. (3.37) yields (3.39) Since the pressure p satisfies the quasi-static equation (3.8) it must be constant along field lines and thus a function of a and /3 at any instant of the time such that dp

OP(a,/I, t)~0P~

V~

Va)

(~+~

+~

V/3).

(3.40)

Equation (3.39) can be integrated along any field line giving the third component of v explicitly:

v~Vsidp B



yP dt

~ J B

1.~-d +-~-- 1’~d Oa J B 0/3 J B S~

(341

‘~

so

s 0

so

a

with = Oa/Ot and /3 = 0/3/at. The lower integration limit s0(r, t) = so(a(r, t), /3(r, t), t) has been chosen such that v Vs vanishes for s = s0. An appropriate choice for closed field lines is So = 0 at the equatorial plane if symmetry around this plane is assumed. Instead of v Vs the component of v parallel to B, v11 can be used, which is connected with v Vs through v Vs

=

Vu-Vs

+

v Vs -

=

v11+ v1 Vs.

(3.42)

Note that in general Vs and B are not parallel to each other. (In fact it can be shown easily that this is the case only for straight field lines and the current flowing perpendicular to B.) So the convection velocity and the electric field can be determined uniquely for appropriate boundary conditions if the magnetic field and the plasma pressure are known as functions of time. As can be seen from equation (3.41) with (3.42), v11 cannot be prescribed at both ends of a field line. Thus a smooth solution of the convection velocity and electric field is not possible everywhere for any given boundary conditions. This fact makes it plausible that singular layers may form in which one of the assumptions is violated (e.g. the smallness of v Vv). Such singular layers which may be represented for instance by the plasma mantle or by electrostatic double layers will be discussed in section 3.5. 3.4.2. Stationary convection Convection models of the magnetotail for small 6 require in the first step quasi-static solutions for p and B which include field-aligned currents. Since such models have not yet been worked out, we will concentrate in this section on the convection models of the dipole-dominated region. Some effort has been spent to model the convection electric field in the inner part of the

K. Schindler and J. Bim, Magnetospheric physics

138

magnetosphere where the magnetic field is predominantly current free (e.g. [3.16], [3.3l]—[3.35]). These models assume a known magnetic field which does not change with time. Thus the electric field may be described by a potential 41 which must be constant on field lines as can be seen from equation (3.4)*. A closed selfconsistent chain of equations including pressure anisotropy has been described e.g. by Vasyliunas [3.16], see fig. 3.4. For an isotropic plasma pressure this chain can be simplified using macroscopic quantities (i.e. the MHD-equations) only. A part of this chain has already been used in section 3.3.1 to calculate the field-aligned current from the pressure function, which was assumed to be known, and from the zeroth order magnetic field, which again may be assumed to be represented by Euler potentials a(r) and /3(r). The connection between the field-aligned currents and the electric field in the magnetosphere follows from the requirement that these currents are closed through a horizontally flowing current in the ionosphere. Continuity of current relates the inflowing magnetospheric current (which is assumed to flow parallel to the magnetic field and follows from equation (3.22)) to the (height integrated) °Parallelelectric fields which are excluded by eq. (3.4) will be discussed in section 3.5

Driving

MAGNETOSPEERIC

I ELECTRIC FIELD I

field

(Or

current)

IONOSPHERIC -~ .

Generalized Ohm’s law

ELECTRIC

FIELD

Ionospheric Ohm’s law

Kinetic

FIELD-ALIGNED

equation

CURRENT

Continuity of current

PARTICLE PRESSURE _________

___________

Momentum

I PERPENDICULAR CURRENT ___________________

conse rvat ion

Boundary

source

Fig. 3.4. Outline of the self-consistent calculation of magnetospheric convection after vasyliunas (3.161. The boxes contain the physical quantities to be calculated; the lines joining two boxes are labeled with the physical law connecting the corresponding two quantities with the arrows pointing to that one which can be calculated from the other. The open lines are labeled with the boundary conditions that must be specified.

K. Schindler and I. Birn, Magnetospheric physics

139

ionospheric current through V-J=jiicos~,

(3.43)

where x is the inclination of the magnetic field against the vertical line. By the assumption that J has horizontal components only, the vertical component of the electric field can be eliminated from Ohm’s law (3.9) yielding (see e.g. [3.36],[3.4]) for the case of zero neutral-wind velocity J=

~h

(3.44)

Eh,

with 1

— h — ~

+~

/ ~ ~H~ll cos x +(~+I~)sin2x ~~~X’\—IH~llcosx £~~iicos2~

(3 45)

Here a local coordinate system has been chosen such that z is in the vertical direction and the magnetic field vector lies in the x, z plane. Eh = (E,,, E~)represents the horizontal part of the electric field. ~, ~ and £H are the height integrated parallel, Pedersen and Hall conductivities, respectively. Fejer’s [3.36]expression differs slightly from (3.45) due to the fact that he uses the condition j~ = 0 to write Ohm’s law in the form Jh = Eh from which the height integrated formula (3.44) is gained. E can be expressed by the gradient of an electrostatic potential V; thus combining eqs. (3.43) and (3.44) leads to a differential equation for V:

V~h-VV—jl(cosX.

(3.46)

If expressed by spherical coordinates RE, 0, ~ (see e.g. [3.34]) ~ sin0x RE2 sin 0~ I. 00 f[cos

1— 0~~ [sin 0 O~1— j 0~O0cos x

0%” 00 j

—~--

~Y±

~H

+

0V~ ~H 00 0~cos xJ~=

j cos x

(3 47)

whereB~= 0 has been assumed. The solution of (3.47) requires boundary conditions for V or Eh which represent the effect of magnetic merging or some viscous drag at the boundary (the ‘driving field’, see fig. 3.4). However, since these interactions are largely unknown, they are usually taken into account by specifying V in some plausible ad hoc way on a circle in the ionosphere (e.g. [3.16], [3.31]—[3.35]). The solution for V at the ionosphere, following from (3.47) can be mapped into the magnetosphere as V is constant along the field lines. This yields the convection electric field and the corresponding perpendicular velocity given by eq. (3.36) =

(c/B2)E x B,

or using the Euler potentials a and /3 V-Va—OV/0/3,

v~Vf30V/0a.

(3.48)

The parallel velocity can be obtained in a similar way as in section 3.4.1 from the known pressure function with ~[rr

v Vp = v ~Va ~-+v

- V/3~=—ypV

v.

(3.49)

140

K. Schindler and J. Bim, Magnetospheric physics

One gets S

1 lOP 0V

S

OP 0V\ (ds

BBLo~yP(~.0a0/30/30a)J so

~~‘1~J

(ds so

—h-.

(3.50)

In the first attempts to calculate the convection electric field, Iii was simply neglected [3.16, 3.32] such that the problem was reduced to the calculation of V at the ionosphere from V~

. V V =0,

with boundary values given on some circle I at the edge of the polar cap. By mapping V along the magnetic field lines the solution could be gained inside a surface which was constituted by the circle I and the field lines which map this circle into the magnetosphere. Later models included a ring current (with anisotropic plasmas) either by including its effects in the quasi-steady equations (e.g. [3.31])or by calculating the time-dependent electric field starting from an initial configuration in which an arbitrary ring current has been added to a current-free electric field configuration (e.g. [3.33]). A common result is that the ring current shields the region inside the inner edge (Alfvén layer) from the convection electric field. Figs. 3.5 and 3.6 show an example.

-22

CURVE I

-22

-20

-20 -18 -16

tO

~~I21~I8_

______

-Is



(/~

~

-

_ -

-

-

-

I2~IOIi_

I 6

6-18-20-221 4 -2 I

~~~—1

12

I ~4~o22I 8

~

0

20

IO~-J

16 18

18

(a)

-10

(b)

20 22

22

Fig. 3.5. (a) Equipotentials in the equatorial plane after Jaggi and Wolf [3.33]assuming absence of field-aligned currents from the Alfvén layer. Corotation is not included. (b) The same as fig. 3.5a but including corotation.

K. Schindler and I. Bim, Magnetospheric physics -22

-22 -20

16

-Ia -

4

-16-

~

12

I 8

4I

_

-18-16

/ ~2~

________________

6

-2

-20

__

-14’

,~

141

-~ 2

4”68

_

I

5

0~—J

FVEN_LAYE~~.J I 0 12 -14 -16 -lB -20 ~22I

I9//II7III~~(N I1 2~’ ____________________________

_

2

s..-

-6 2~1

4

\3.~

~l3.34N~

-

l3,34~~~J

c ~-I5 6

(a)

6

(b)

18 20 22

~

-20

20 18 22

Fig. 3.6. (a) Equipotentials in the equatorial plane after Jaggi and Wolf [3.33]15 hours after an ion sheet started drifting inwards from the tail (corotation not included). (b) The same as fig. 3.6a but including corotation.

3.4.3. Reconnection As discussed in section 3.2, simple topological considerations make it plausible that an open magnetospheric model containing convection and electric fields must include x-type neutral points or lines, where reconnection of magnetic field lines takes place. In fact, there is considerable indirect evidence for reconnection in the magnetosphere [e.g. 2.29, 3.9]. It should, however, be emphasized that there is no direct observational evidence for the existence of such points. As shown in sections 2.2 and 2.3 this phenomenon cannot be described by ideal MHD, since Ohm’s law (3.4) requires that E = 0 at neutral points where v x B = 0. Thus Ohm’s law (3.4) must be replaced by its generalized form (see e.g. [4.31]): E+-~vxB c

nj+~j X BV nec ne

Pe+~(~+V

.(VJ+Jv)),

(3.51)

valid for a two-component plasma where the approximations me ~ m, and quasi-neutrality n,q 1 — flee 0 have been used and the collision term is assumed to be proportional to j thus defining the resistivity 77.

Stationary two-dimensional reconnection configurations representing the immediate environment of an x-type neutral point have been extensively studied. The concept was introduced by Sweet [3.37], Parker [3.38] and Petschek [3.39] and was refined in a number of subsequent papers (e.g. [3.401). Dungey [3.7] first suggested reconnection to occur in the magnetosphere. Quantitative approaches

K. Schindler and I. Bim, Magnetospheric physics

142

generally assume incompressibility and usually take into account the resistive term i~jon the r.h.s. of eq. (3.51) only. In his recent review Vasyliunas [3.40], however, generalized earlier models using Ohm’s law in the form (3.51) including both resistivity and inertia effects. (For collision-free reconnection based on inertia effects, see also [3.46].)The results obtained for collision-free plasmas where inertia effects dominate do not differ qualitatively from those where the resistive term plays the dominant role (which may be due to binary collisions as well as scattering of particles by small scale plasma turbulence). The quantitative difference can be expressed by the different length scales involved, the ‘resistive length’ 2/4lrvA, (3.52) = 77c where

vA = =

B/\/(4irp) is the Alfén speed, and the ‘electron inertia length’

c(me/4irne2)”2 =

(3.53)

c/wpe.

In the following we will briefly summarize some of the results obtained for stationary two-dimensional incompressible flow in the vicinity of an x-point. The magnetic field and flow pattern is shown schematically in fig. 3.7a: Oppositely directed magnetic fields of magnitude B 1~are transported against each other with a velocity vi,,. The major part of the magnetic field becomes annihilated in a diffusion region with the energy being transformed to kinetic energy of the outfiowing plasma. The diffusion region is the region, where the resistive or other terms of the r.h.s. of (3.51) play a significant role. The annihilation rate may become considerably enhanced by (slow) shock waves (dashed lines in fig. 3.7b), extending from the diffusion region (shaded area) to far outside (‘Petschek’s mechanism’ [3.39]). The speed v01,~of the outfiowing plasma turns out to be about the Alfvén speed VA = B,~/V4irpof the inflowing plasma, whereas the inflow speed v1~is essentially a free parameter determined by the boundary conditions. There may exist, however, an upper limit for v~~ in Petschek’s model this limit is about 0.1 VA (with no significant difference between the resistive and the inertia cases) whereas similarity solutions discussed by Sonnerup [3.41], Yeh and Axford [3.42] and Yeh [3.43], allow ~ VA. This limitation of t’~,,corresponds to the condition that for a moving plasma element magnetic energy has to be coverted into kinetic energy. This means essentially that for the similarity solutions the magnetic reconnection rate is controlled by boundary conditions and not by properties of the diffusion region. It is not yet clear whether an upper limit exists in the compressible case. It seems plausible that if the inflow speed exceeds the possible maximum value allowed in the incompressible case compression may enhance the Alfvén speed just outside the diffusion region and thus the ‘allowed’ reconnection speed, until it equals the inflow speed. The extension of Sonnerup’s solution to the compressible case by Yeh and Dryer [3.44] contains no limitation of the inflow speed. B,

:~

Fig. 3.7. (a) Schematic drawing showing the magnetic field pattern and the main flow directions (thick arrows) for reconnection in the vicinity of an x-type neutral point. The shaded area symbolizes the diffusion region where departures from ideal MHD become important. (b) The same as fig. 3.7a containing additionally slow shock discontinuities (dashed lines).

K. Schindler and I. Rim, Magnetospheric physics

143

Besides the fact that the existing stationary x-point solutions all contain some unsatisfying features, nothing is known about the stability of these solutions. So the question remains open whether stationary field line reconnection at x-point configurations actually occurs or not. A possible alternative is dynamic (time-dependent) reconnection; a process of this type will be discussed in section 4. 3.5. Departure from smooth convection By assuming the validity of ideal MHD with E + V x B/c = 0 several features of the magnetosphere are excluded which should be contained in a more complete description. One of them, steady state reconnection at x-type neutral points or lines, has already been discussed in the preceding section. Others will be discussed now using again Ohm’s law in its generalized form (3.51). As discussed by Vasyliunas [3.1, 3.40] for typical magnetosphere parameters, departures from idealized Ohm’s law (3.4) become important only if plasma and field quantities vary on a length scale which is small compared with the macroscopic length scale (order one or several Earth radii). This is the reason why the features discussed in this section appear as thin layers or discontinuities situated in the large scale structures. As already mentioned in section 3.4.1, a smooth solution for the convection velocity vu along a field line is in general not possible if boundary values at both ends are prescribed. Let us illustrate this for field lines connecting from the ionosphere to the magnetopause. The convection theory requires a given difference of the values of v11 at both ends (see equations (3.41), (3.42)). It seems unlikely that the conditions at the ionosphere and/or at the magnetopause are such as to accept a fixed difference of v11. A possible way out of this difficulty is to assume that thin layers form, where at least one of the assumptions of the smooth convection theory is violated. In this context it seems to be of interest that both at the magnetopause and in the inner magnetosphere there seem to exist layers where the assumptions of quasi-static ideal MHD are not valid. Inside the plasma mantle the balance of Lorentz force and pressure gradient is violated such that the inertia term, pv Vv, must be included in the balance of forces. In a large scale picture this layer may be identified with the generalized tangential discontinuity discussed in section 2.2. Similar consequences can be discussed for closed field lines with both ends at the ionosphere. For a smooth solution, vii at the ionosphere cannot be prescribed arbitrarily. This is obvious from configurations which are symmetric around the equatorial plane since v1 must vanish there so that v1 at the ionosphere is determined from eqs. (3.41) and (3.42). Thus conditions on v1 at the ionosphere may lead to the existence of layers where v1 changes significantly over short distances. Mass conservation determines the corresponding change of density p. The variation of v11 along the field lines leads to an inertia force pv1dvii/Os which must be compensated by a parallel component of the pressure tensor (V P)11. This component implies a parallel electric field component as is obvious from the generalized Ohm’s law (3.51). If the corresponding departure from quasi-neutrality is strong enough (as for structures on the scale of the Debye-length) a term (n,q, — nee)E must be included in the balance of forces which demonstrates the connection of the inertia term with E1 even more clearly. It is conceivable that this process is related to the formation of electrostatic double layers (such as suggested by Block [3.45]). An alternative explanation for the existence of parallel electric fields uses the resistive term ~ in Ohm’s law (3.51). Since, however, resistance due to Coulomb collisions is much too small to give a significant contribution, anomalous resistivity must be assumed which is due to turbulence created by

-

-

144

K. Schindler and J. Birn, Magnetospheric physics

instabilities. These instabilities, if current driven, require that the current density exceeds a certain threshold which is typically of order nevTe where v~0is the electron thermal velocity. This condition implies fairly large current densities. A simple estimate shows that the lateral extent of the regions of unidirectional flow of electric current should not exceed about 10 km because otherwise the associated magnetic field perturbation becomes too large. The flaring of the flux tubes and the corresponding decrease of the electric current density away from the Earth may limit the region along the flux tubes where the conditions for instability are satisfied. It is of interest to note that field-aligned currents in fact often show a sheet-like structure. The convection models discussed in section 3.4.2 show that the steady state convection and the corresponding electric field fall off abruptly at the Alfvén layer which constitutes the inner edge of the ring current. This layer can be easily understood as the boundary of the forbidden zone for the motion of single particles with a given energy. For two particle species these boundaries generally do not coincide. The resulting charge separation, however, leads to electrostatic fields which brings the boundaries closer to each other until a distance of the order of a Debye-length is reached. In Ohm’s law this electro-static field is balanced by the pressure gradient term. A similar mechanism seems to shield the electric field in the near Earth region which is driven by the corotation of plasma due to collisions with the corotating neutral gas. The corresponding discontinuity at the outer edge of the plasmasphere is the plasmapause. In this section we have shown that several important magnetospheric features cannot be described by a simple quasi-static MHD-approach. Nevertheless, the discussion given here is still highly idealized. In reality the various effects cannot be so clearly separated from each other as it might appear from our discussion.

4. Dynamic phenomena In this section we will give a brief description of selected aspects of the dynamics of the magnetosphere. By ‘dynamic’ we mean time-dependent processes that cannot be understood in the frame-work of the quasi-steady plasma approach described in section 3. We will emphasize large-scale processes concentrating on questions related to cause and effect rather than on the detailed morphological picture. For the latter we recommend the corresponding parts of the book by Akasofu [4.1]. We also neglect many aspects of wave-particle interaction; some microscopic processes are discussed in section 4.3. As for most physics phenomena, causality is an important distinguishing factor for dynamic magnetospheric processes. There exist phenomena that 1passive’. are the immediate consequence variations Mathematically, passiveofresponse is of external parameters. Such processes may be called described as an initial/boundary-value problem, where the initial and boundary conditions have to be carefully matched to the external perturbations. There is another class of dynamic processes that cannot so directly be related to external parameter variation. The time of their onset is not directly determined by a given external signal, or, at least, an external signal is not necessary. In this case, the external conditions change the state of the magnetosphere passively in such a way that it approaches some boundary of stability. When the boundary is crossed, the magnetosphere responds spontaneously without necessity for a direct external signal. In this case we may call the magnetosphere ‘active’, the corresponding mathematical

K. Schindler and J. Rim, Magnetospheric physics

145

problem is two-fold: An initial/boundary-value problem for the passive part and a stability analysis for the spontaneous part. 4.1. Magnetospheric storms and substorms A good example for a passive response of the magnetosphere seems to be the arrival of a sufficiently weak interplanetary shock wave. According to Akasofu [e.g. 4.1] ‘a weak interplanetary shock wave simply compresses the magnetosphere. After the solar wind returns to a pre-shock condition, the magnetosphere expands, returning to a quiet-time configuration’. In the case of a stronger shock wave the response is more complicated, giving rise to a magnetospheric storm as discussed further below. The most prominent example of active response is the magnetospheric substorm. Its probability of appearance seems to be tied to the negative polarity of the z-component of the interplanetary magnetic field [4.2, 4.3]. There is, however, not a direct causal relationship: The substorm does in general not appear when B~turns negative nor does it after a well-defined delay time. Rather, B~<0 seems to passively drive the magnetosphere towards some border-line of stability, the time of onset depending on details of magnetospheric conditions. Observations are consistent with the picture that a negative Be-polarity of the interplanetary magnetic field leads to energy influx into the magnetotail [4.4]. This is also suggested by the fact that a north-south pointing interplanetary field can topologically more easily reconnect with the south-north pointing geomagnetic field than a field of opposite polarity (fig. 3.2). Evidently [2.30], the strength of the interaction between the solar wind and the magnetosphere at the magnetopause — e.g. manifested by the thickness of the plasma mantle correlates well with the (negative) z-component of the interplanetary magnetic field. Although an external ‘trigger’ does not seem to be required, frequently an external non-linear perturbation appears to provide ‘the straw that breaks the camel’s back’. Nevertheless, the concept of the substorm as an active process may be applied even to substorm phenomena occurring just after the arrival of an interplanetary discontinuity [4.5]. The substorm involves characteristic phenomena in almost all regions of the magnetosphere. There are strong variations in the geomagnetic tail and acceleration of charged particles [4.6]. The qualitative nature of the magnetic field changes is not yet uniquely identified. Figure 4.1 shows a concept that is widely accepted [4.7, 4.8, 4.9], at least in its earlier stages. In this picture the most important aspect is the sudden change in the topology of the magnetic field associated with the formation of. at least one x-type neutral line. Frank et al. [4.101came to similar conclusions, although the picture varies in some details. It should however be noted that present observations do not prove the formation of the neutral line beyond any doubt [4.11]. It is therefore of interest to ask whether theory could help to answer this question (see section 4.2). Further substorm effects are the increase of the energy content of the ring current situated in the inner magnetosphere, enhanced activity in the auroral zones, microinstabilities leading to enhanced plasma transport, electric currents and fields parallel to the magnetic field. The substorm-associated electric currents lead to characteristic variations of the magnetic field as measured at a given latitude and universal time on the ground. For a comprehensive review of the morphology of substorms, see Akasofu [4.1]. In the region behind a strong interplanetary shock wave the plasma seems to be of significant variability. As one of the consequences, periods of pronounced negative values of B~are more frequent than during times when the solar wind is quiet. Thus, after the initial (passive) response of —

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146

\

.5-

R /-

u~~jI 7~II lr~,’z~i~

9

~

~

F -t-

Ill III

-4’ 5- 5)

~___‘~O//

/

-~ LU

0

U,5)

-

-

0

0

in

‘S B

K. Schindler and J. Birn, Magnetospheric physics

147

the magnetosphere to the shock wave, a period of enhanced substorm activity appears. Frequently, this activity injects plasma into the inner magnetosphere which leads to an enhancement of the ring current (main phase of a magnetospheric storm [4.12]). Passive response to external changes is predominantly a quantitative problem, its qualitative understanding does not seem to pose great problems. Thus, the key to the understanding of magnetospheric dynamics is to identify the modes of active response of the magnetosphere to external changes. A satisfactory description of these phenomena would solve the substorm problem and thereby answer the most pertinent questions of magnetospheric dynamics. Akasofu [4.1] classified the magnetospheric processes according to whether they are reversible or irreversible. This distinction is important for a discussion of energy balance; but it does not seem to be optimal from the view-point of cause-and-effect. The passive evolution occurring before the onset of a substorm does not necessarily coincide with the concept of a ‘growth phase’ [4.13]. In particular, it is not necessary that each substorm has its clearly identifiable passive phase. It is conceivable that the dynamic effects of a given substorm result in a magnetospheric state that is again not far from the stability boundary (which may depend on more than one parameter), such that the following substorm may occur only after relatively small changes. Substorms could even occur during periods of northward-pointing interplanetary fields, however, less frequently. The above picture is introduced here as a working hypothesis. Given the present observations it seems to be suggesting itself, although it is not proven beyond any doubt. 4.2. Substorm theory Adopting the working hypothesis discussed in the preceding section one is led to conclude that one of the most important aspects of magnetospheric dynamics is that of stability, with emphasis on large scale unstable processes. We emphasize that — at least from the view-point of cause and effect — it is of greatest importance to identify the substorm mode as an instability. There exist attempts to describe the substorm dynamics in terms of morphological concepts such as ‘current interruption’ or ‘expansion wave’ (for details see Akasofu [4.1]). These concepts undoubtedly have value e.g. as ordering schemes for a discussion of observed data. However, they cannot replace a stability analysis, answering the question whether a given perturbation actually undergoes unstable growth. Note that an expansion wave usually is a perfectly stable phenomenon. There must exist an instability as the direct cause of a substorm. Expansion waves, current interuption and other dynamic phenomena may well be the effect of the non-linear growth of an unstable mode. (The cause of the breaking of a dam is that its physical conditions have reached a critical border-line; one of the effects is an expansion wave travelling into the lake.) Present stability theory of the magnetosphere regarding substorms exists in two dimensions only, such that the y-dependence of both the equilibrium and the perturbation is ignored. Even under this simplification the problem is still very difficult such that it provides only partial answers. The main problem is that the tail of the magnetosphere involves a neutral sheet with the complicating factors discussed in section 2.3. In two dimensions the stability may be discussed in terms of variational principles, where the sign of the minimum of a functional 6W determines the stability 6 Wmin ~ 0 —+ stability, 6 Wmin <0 —* instability. (4.1) (Note that neutrality 6 Wmin = 0 is counted as stability.)

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148

Such variational principles hold for a considerable variety of plasma models, particularly for ideal magnetohydrodynamics [4.14],for resistive incompressible magnetohydrodynamics [4.15]and for the Vlasov theory [4.16, 4.17]. In all cases, the functional SW may be written as SW=SF+SG,

(4.2)

where SF always has the same form

f

d2r, ~ [(VA1)2 4rrA~d2Po/dA~] whereas SG depends on details of the model. Here, P SF

=

(4.3)



0(A0) is the equilibrium pressure (3.30) where the y-component of the vector-potential A0 takes the role of the Euler potential a; /3 can be set to y in two dimensions. A1 is the perturbation of the vector-potential. As shown below, for a large class of equilibria, covering most cases of practical interest, one finds that SG ~ 0 for all A1. Thus, (4.2) implies a sufficient stability criterion in terms of SF only SFmin

~

0

-+

stability.

(4.4)

For the case of ideal isotropic MHD, this criterion may be understood from thermodynamic arguments. As was shown by Grad [4.18]the minimum of the expression 2/8ir P] d2r (4.5) F = [B under isobaric variations (pressure kept constant for each displaced mass element) provides a sufficient stability criterion. In two dimensions, isobaric variations are easily described by choosing P = P 0(A0) where P0 is the equilibrium pressure. It is of interest that the first variation of (4.5) equated to zero provides the equilibrium condition 2A —V 0 = 4ir dP0(A0)/dA0, (4.6)

f



which coincides with (3.30). The second variation of (4.5) is identical with (4.3). These properties of F suggest an identification of F as the free energy of our system. Note that F — as a free energy should does not depend on details of the dynamic mechanism; it is the same for all three above-mentioned models. If the minimum of SF is positive, all states neighbouring the equilibrium have higher free energy than the equilibrium state, i.e. the equilibrium is stable; SFmjn <0 means that there exist neighbouring states that have less free energy than the equilibrium. In that case one has to find out whether these states are dynamically accessible by discussing the full SW, including 6G. For the above-mentioned models, SG has the following forms: ideal MHD [4.14]: 2))2 d2r, (4.7) = f ~yP(V~ (A1VA0/1VA01 resistive MHD, incompressible, resistivity ‘frozen’ into the plasma [4.15]: —

SG

=

f (d2P0/dA02)A~d2r,

(4.8)

where, for the present discussion we confine ourselves to the case d2P 0/dA~>0 (e.g. local thermal equilibrium).

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149

Vlasov Theory [4.16, 4.17]:

JI0Po/0~oI{((~)— {(* )})2} d2r,

=

(4.9)

where 0po/0~= ~ e2f(0F~/ OH

3v; F 0) d 0(H0, P0) is the particle distribution function for a given species as a function of the constants of the motion H0 (the Hamiltonian) and P~0(the generalized momentum conjugate to the y-coordinate), 0F0/0H0 is assumed to be negative; el/I1 is the perturbed Hamiltonian, el/I1 = — (e/c)v5A + e4~~ sums over the particle species. The variation of (4.9) is subject to the constraint

{~} {(l/i~)}= 0. symbols—, { }, ( ) denote averages defined in the following way: —

The

(Sds



/1

ds

°‘

(6) = f S d2r/

=

integration extended over arc-length s of a given field line (A0 = constant),

(4.11)

d~r D(H0,P50) region of accessibility of a particle with constants of the motion H0,P50 in phase space: el/i(r, P50) ~ H0, (4.12) 3v. (4.13) ~ e2JIF~ISd3v/~ e2J IF~Id

D

{S}

~

J

.

.

=

(4.10)

ID

In deriving (4.9) one has to assume that a given particle orbit actually covers (quasi-ergodically) its whole accessibility region D(H 0, P50). If there is an additional constant of the motion, or adiabatic invariant, the averages have to be redefined accordingly. It is also assumed that quasi-neutrality holds. For details we refer to the original literature [4.16, 4.17]. Clearly, under the present assumption t5G is always positive, i.e. stabilizing. Applying these stability concepts to the magnetosphere we have to approximate the relevant regions by two-dimensional structures. Fortunately, this is possible for the tail of the magnetosphere (see section 3.3.2) which turns out to be the most interesting region from the view point of the sub-storm instability. The first result that one finds for two-dimensional tail-like configurations is that free energy is in fact available [4.17].For small values of = L5/L~(i.e. small x-dependence) one concludes that the (unrealistic) case of a tail that converges with distance gives SFmin ~ 0 (stability) whereas the realistic case of a diverging tail (fig. 4.2a) always yields SFmjn <0. This result may be obtained in several ways. One possibility [3.17] is to determine for any cross-section of the tail (fixed value of x) the magnetopause location W that would correspond to marginal F-instability (i.e. SFmjn ~ 0): w

~.

B~ fdz 4~~j (4.14) B~8p(x,0)/0x Imposing the (realistic) condition that for the quiet tail p(x, 0) and B~(z0) vary monotonically with x and B~(x,0)> 0 we find that B~> 0(all z)

W> z

—*

B~changing sign

—~

-+

F-stability,

(existence of domain with z> W} —~F-instability.

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150

— (2)

)b)

Fig. 4.2. (a) shows the magnetic field lines of a two-dimensional self-consistent model of the magnetotail. This configuration is F-unstable (see text). (b) shows the stable equilibrium solution corresponding to the same function P(A) — exp(—2A) and the same boundary conditions as (a). (Field lines computed by W. Zwingmann, Ruhr-Universität Bochum.)

Here, F-stability is defined on the basis of the sign of SFmjn. Thus the configuration of fig. 4.2a—modelling the actual magnetotail—is F-unstable, because B. changes its sign. The F-instability of tail-like configurations also follows from the mathematical theory of the equilibrium equation (4.6). Suppose the following conditions hold for P0(A0) P~(A0)<0,

P~(A0)>0,

P~’(A0)<0,

(4.15)

where the prime denotes the derivative with respect to A0. Note that these conditions are indeed satisfied for the explicit model discussed in section 3.3.2. Then, mathematical theorems [4.19]

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151

N

Fig. 4.3. Typical behaviour of solutions of (4.6) with (4.15). P

x—x4~

0(A0) is chosen as A,3(A0) with fi(Ao) fixed. Note that for A
guarantee that always there exists either more than one solution of the Dirichlet boundary value problem or no solution at all. This property is illustrated in fig. 4.3 for a simple example, where the pressure P0 is multiplied by a continuous parameter A P0(A0, A)

=

AF(A0).

(4.16)

N is the maximum value of A0 A~lwhere A,, is the vacuum solution (A theory provides the result that 2Fmin >0 on the branch with smallest value of N, —

O t92Fmin <0

=

0). The mathematical (4 17)

on the other branch.

The configuration shown in fig. 4.2a was obtained by expanding with respect to (section 3.3.2). Figure 4.2b is another solution with the same boundary conditions and the same value of A. It is very close to the vacuum solution which indicates that it must lie on the lowest branch of fig. 4.3. (This identification is rigorous because for the case P(A 0) -~-exp(—2A0), as chosen here, there are exactly two solutions.) Thus, we can immediately conclude that the (realistic) configuration 4.2a is F-unstable, i.e. it stores free energy which may drive some instability. (The ‘tear-drop’ model of the early days of magnetospheric physics would be stable since B2 > 0 everywhere.) This leads us to the next question: Is there a mechanism which will actually release this free energy stored in the tail in some unstable way? 4.2.1. The ion-tearing-mode In order to answer this question we have to chose a specific plasma model. In this section we will treat the plasma as collision-free assuming that the Vlasov theory holds. It should be emphasized, however, that this is not the only possibility. If a sufficiently large level of micro-turbulence is present,

152

K. Schindler andJ. Bim. Magnetospheric physics

one might alternatively consider a resistive-fluid model. Within the Vlasov theory, we may start out from SW as given by (4.2) with SF and SG determined by (4.3) and (4.9). Alternatively, one may consider the following variational principle without additional constraints [4.16] ~

e2Jd3v

~-~j

(~ri)~],

(4.18)

where 3v j0(A0,~0)=~ eJ v5F0d denotes equilibrium charge and current densities, ~ summing over particle species. p0(A0,~0)=~eJFod3v,

The Euler—Lagrange equations of (4.18) take the form ~ e2J~frI(lpt) v~d3v

=

(4.19)

—AA 1,

~J

(4.20)

2 IOF 3v = 0. (4.21) 4i~Opo/O40 + A1 Opo/OA0 + e 0/OH0~(l/i1)d The eigenvalue A comes from normalizing ~5 A~d2r = constant, in order to exclude the trivial solution A 1 =0. We note that the variational approach cannot provide information about the growth rate. For evaluating the actual development of a given perturbation with time, one has to consider the dynamic equations, e.g. given by Biskamp and Schindler [4.20] 2A 1d~ OF V 1 1 010 A 1 010 1V 2 0 , c 0A0 ~ c O4~~‘ c ~ e j V OH0 vy \lPi 3v ~f~J ~ dt’~rI(t’)+(l/II)], (4.22) = ~ e~fd —



+



4~+

J



~J e2

d3v

(~fr~) = ~

e~Jd3v

~ J [iw

dt’~ 1(t’)+ (cfri)].

(4.23)

Here (cu1) may be interpreted either as the spatial average given in (4.12) or as a time average over the unperturbed orbit (note that we have assumed quasi-ergodic coverage of the accessibility domain D). After these more technical remarks we now continue to deal with the stability of the magnetospheric tail. Stability is governed by the variational principle (4.1) which according to (4.2) and (4.9) explicitly reads 2} d2r, (4.24) SW = SF + OpoIOçbo~{((‘/I~) {(l/’1)}) {(i4r~)}= {i/i~}. (4.25) —

Clearly, the average (l/i 1) vanishes for a strictly one-dimensional equilibrium (one-dimensional neutral sheet). In that case i/i1 may be Fourier-analyzed along x and (assuming k~ ~ 0) one easily finds (l/’1) = 0, because the integral with respect to x vanishes owing to terms —-exp(ik~x).A full quantitative analysis of (4.24) is not yet available. An important property may however be deduced in the following way.

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153

We start from the observational fact that at quiet times the plasma sheet tends to be fairly broad [4.21].This suggests that such states are stable. During times of negative polarity of the interplanetary B5-component, it seems that there is an enhanced influx of energy into the geomagnetic tail [4.22], possibly driven by day side reconnection processes. It is evident from explicit solutions of (4.6), e.g. [3.17], that enhanced plasma pressure implies a reduction of L, and of B5 i.e. the field lines become more and more stretched. Since for extreme stretching the second term in (4.24) vanishes and since SFmin<0, there must exist a critical degree of stretching at which SW changes its sign (positive—* negative). This critical state of the magnetosphere has been interpreted as the point of onset of a magnetospheric substorm [4.17].It is much more difficult to derive this condition quantitatively. Fairly simple conditions prevail if we can ignore the electric potential perturbation 4~.An ad-hoc justification for this assumption may try to rely on ionospheric shortcircuiting. We will, in a first step, in fact assume 4~= 0; the consequences of l/~= 0 will become clear when we include çb1 in the second step. For /~= 0 we may estimate the relative contributions of electrons and ions (protons) to the average term in (4.24) depending on the qualitative properties of the orbits. (Note that here we are dealing with typical neutral sheet effects such as mentioned in section 2.3.) The relevant parameters are the temperature ratio 0 = Te/Ti (usually about 0.1 in the magnetotail) and e = alL (c/oi0)IL where a is the Larmor radius of a typical particle of the species considered with respect to the external magnetic field B0 for realistic conditions 2 is considerably smaller than unity both for ions and electrons. For a rough estimate of the averages in (4.24) we may replace c(i/11) = (v~A1)by i35A1 where v~.is some typical value of v~.Since the averages ( ) and { } are defined quite differently, the typical values of v. to be used will in general not closely coincide so that we may write 2r = ~Je2 ~ i3~A~ d3v d2r. SG f ~ {z~A~} d With the aid of (kinetic) temperature (Boltzmann’s constant absorbed into T) T

(4.26) H 0 we

find

~J~

2r.

(4.27)

~—4~d

We may assume that in the equilibrium configuration the electrons have gyroscopic orbits (i.e. the gyrocentre theory holds). In fact, as we shall see, even the ions are close to gyroscopic in a normal field component B 5 of order 1 y. (In any case, if B5 is so small that both species are non-gyroscopic the strictly one-dimensional equilibrium is a good approximation [e.g. 4.16]; in that case one finds the electron tearing mode instability with a growth rate of order ~ vTeae L~ where VTe is the electron thermal velocity.) Thus, we may estimate ,

Vye

vTeae/Lz



cTt~/eB0L5,

(4.28)

where I’e is a typical value of the electron energy, such that the electron contribution SG is of order 2r. (4.29) f(nA~/fiL~) 0d Here ti is a typical value of the particle number density. This term, however, is smaller than the term 5 pgA~d2rin SW by a factor of 0 and can therefore be neglected. SGeiectrons

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154

2r such that stable If B, is sufficiently large, the ions will also be gyroscopic with 5G 5 p~A~ states seem possible. As conditions change in such a way that B,10~. and/or L, ddecreases the ions approach non-gyroscopic behaviour. Then the system must become unstable at the latest, when the ions are no longer turned around in the B, field during a typical growth time of the instability: —~



1,

(4.30)

where r, is the growth time of the ion-tearing mode [4.17] =

‘y~= v-fla~L~

(4.31)

and 1l~,,is the ion gyro-frequency with respect to the normal component B,(x, 0). At this stage the ions begin to behave in the same way as if B, was zero. The ions resonantly interact with tearing-type perturbations which grow exponentially with a growth rate given by (4.31). Interpreting (4.30) as the condition for the onset of a substorm one finds rough order-of-magnitude agreement with observed conditions [4.17,4.23]. (The earlier estimates differ by a factor 2ir which has been removed from (4.30) in view of recent results by Galeev and Zelenyi [4.24,4.25].) We now turn to the consequences of the omission of the electric potential 4~.Galeev and Zelenyi [4.24,4.25] studied the instability of a tail-like configuration including 4~,but neglecting twodimensionality. Discussing equations (4.22,4.23) they localized the integral terms by ignoring the z-dependence of the functions çb 1 and A1 in the time-integrals. This procedure seems justified in a strictly one-dimensional case, although strictly speaking a one-dimensional plasma sheet model depending on z only and with B, 0 is not a proper equilibrium (see section 2.3). Thus it is not yet clear to what extent the problem considered approximates a realistic configuration. Under these conditions the integro-differential equations (4.22) and (4.23) can be combined to a simple Schrodinger-type equation of the form 2A 2+ V(z, w, k —d 1ldz 2)A1 = 0, (4.32) —



where V(z, w, k~)may be computed from equilibrium properties. Clearly (cl4rr)A1(k~ V) is the electric current density. It turns out, that 4~contributes significantly to the current. (This is not immediately apparent from (4.32); note, however, that 4~is expressed by A1 via the quasi-neutrality condition (4.23).) The electron effect does not vanish in the limit Te 0. For sufficiently small Te the corresponding contribution to the current density is —

—~

j=



en0ik~çb1/B,,

(4.33)

which has a simple physical interpretation: Fig. 4.4 illustrates how the potential 4~arises. The fact that the electrons are tied to the magnetic field lines whereas the ions are not (assuming that the stability limit (4.30) was passed) leads to a tendency of separating charges. This effect requires an electric field along x to restore quasi-neutrality. This field is indicated in fig. 4.4b. Clearly, it leads to an electron E x B drift in the y-direction with electron current density (4.33). Note that the ions do not E x B drift, because they are not gyroscopic. Figure 4.5 shows the stability diagram that Galeev and Zelenyi obtain from the SchrodingerProblem (4.32). The upper curve corresponds to the onset of the ion-tearing mode just as described above without the effect of 4~.The electrons however give rise to a stabilizing effect described by the lower curve in fig. 4.5. Note that there still exists a considerable gap between the two curves where the ion-tearing mode may be active. Independently of the question of the presence of 4~,the perturbation of the field lines which is —



K. Schindler andI. Rim, Magnetospheric physics

155

z

resulting Fig. ° 4.4. ____ Schematic from difference illustration of electron the and ion regime. dynamics; of the instability. for suitable (a)ionospheric is the unperturbed conditions configuration; neutralizing(b) currents shows I~ thewillelectric occur;field (c) and pattern (d) extrapolate the the unstable mode ininto the development non-linear

1o_

3~

~

Q5

1.0

L~

Fig. 4.5. Stability diagram after Galeev and Zelenyi [4.25].The non-shaded region is unstable against the ion-tearing mode.

schematically illustrated in fig. 4.4, shows the tendency to form neutral points. The actual shape of the perturbation will depend on the initial perturbation. 4.2.2. Interaction with the ionosphere: substorm-associated field-aligned currents As discussed above, the answer to the stability problem depends significantly on the presence of the potential 4’~. As in the case of the magnetopause (section 2.2) it is of interest to discuss the possibility of the ionosphere to shortcircuit potential differences across magnetic field lines. Although there is no final answer to this question one may obtain a partial result in the following way. Suppose one included the ionosphere into the discussion. Although it seems extremely difficult to carry out such a procedure from first principles we can obtain some valuable information treating the frequency

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156

w of modes proportional to ei50z as an empirical parameter, chosing 1/IwI to coincide with the observed time constant (several minutes) of the tail dynamics. In that case it is sufficient to discuss the dynamics of the singular layer alone to obtain some relevant information. Using the simplest possible model (MHD model for electrons, Boltzmann-factor for the ions [4.25]) one may write the electron continuity equation in the form [4.26] OVez

Oz



/

iew T1

\

iek~wTe iek~w\ 2

~IenmeTi



/

ek~w

I

2

flenme!

\flencme

+

iek~wue\ 2 1A1. flencme/

(4.34)

Here, lien is the electron Larmor frequency with respect to the normal component B, and Ue the electron bulk velocity. Assuming k~L< I (necessary condition for tearing instability), me/mi —*0 and Tel T~~ 1 one finds approximately [4.26] Ovez/Oz

=



4~iew/ T, + A1 wkx/Bn.

If we ignored the ionosphere,

Vez

(4.35)

would vanish such that

—A1 ik~T1/eB,, (4.36) which may also be obtained from the theory of Galeev and Zelenyi [4.25] for Te*0. Thus one rediscovers the potential field introduced in fig. 4.4. On the other hand, if the ionosphere is able to reduce 4~significantly, more precisely for Icl’~I<(TileIwI)IOvezlOzI, we obtain for the electric current I~ associated with the electrons leaving the singular layer of width d, I~= i~enolwA1Id~L5lB,, (4.37) =

giving rise to sheets of field-aligned currents (fig. 4.6a) with alternating directions. The total current per sheet may be estimated [4.26] assuming 1A11 B~/k~, i.e. that the amplitude A1j becomes so large as to make neutral lines 2 VT~, (4.38) I — eno(Lylkx)(ailL) which gives ‘max 8 x l0~A/sheet for k~L— 1. It is of interest that such sheets are actually observed [4.27]. For each sheet the situation may be illustrated in terms of the circuit-diagram shown in fig. 4.6b. —

V

C ~Lngu~r

F Fig. 4.6a. Schematic view of the current system for two sheets linking the singular layer in the magnetotail with the auroral ionosphere.

R~

Lcyer

F

Fig. 4.6b. Equivalent circuit representing the interaction between the tail instability and the ionosphere including the inductive effects ofthe inner magnetosphere.

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157

Here the tearing mode appears as driving a voltage V = —ik~A1T1/eB~ at a capacitor with capacity 2noirL C = kT~le 5d1.

(4.39)

The ionosphere resistance R passively dissipates energy. We have also included 5A canmagnetospheric be expected to inductance L. For realistic values of R and L currents of the order of several 10 flow [4.26]. It should be emphasized that the present picture is still too crude to include all ionospheric effects. Two-diniensionality ignores all East-West effects. Thus, the important effect of deviation of a part of the tail current into the ionosphere [4.1] is not included. This feature could well be the effect of a three-dimensional tail instability interacting with the ionosphere. 4.2.3. On the role of the magnetopause In the entire discussion of tail instability we have largely ignored the presence of the magnetopause. The magnetosphere was either treated as infinitely extended along the z-direction (e.g. for computing y~in (4.31)) or the magnetopause was treated as a rigid boundary (e.g. in (4.14), where the boundary was located at z = W). At this point we briefly discuss the effects of a non-rigid magnetopause with the solar wind streaming along outside. We choose a simple one-dimensional equilibrium. For simplicity we use an MHD description with finite resistivity, which in the context of tearing behaves in many ways similar to the Vlasov system. For the low-frequency limit one finds a dispersion relation [4.28]that involves the magnetopause and the solar wind only through a parameter p~ L (w + ik~vo)2 (4.40) P’ v~ 2/Cs2m + k~ 1(~d)V(w + ~kxVo) Here VA and c~denote Alfvén and sound velocity, respectively; 1 and m correspond to lobe and magnetosheath values, respectively; v 0 is the speed of the solar wind and the magnetopause is located at z = ~d. Magnetopause deformation and the presence of the solar wind may be ignored if 11 is sufficiently large: =

(4.41) which is clearly satisfied for realistic magnetospheric conditions. Therefore, it is a good approximation to treat the magnetopause as a rigid boundary. The physical reason is two-fold; the tearing perturbation drops off rapidly with distance from the neutral sheet and the coupling between the tearing mode and surface modes is small because surface modes have a much higher frequency. 4.3. Dynamic phenomena on small scales In the previous section we dealt predominantly with the large scale properties of the magnetosphere. Here we will add a brief discussion of selected aspects of microscopic processes, e.g. occurring on the length scale of the Debye length. Microscopic processes cannot be fully separated from the macroscopic phenomena. An effective coupling between the two classes of processes takes place via enhanced transport due to microturbulence manifested by suitable transport coefficients on the macroscopic scale [e.g. 4.29]. Also

K. Schindler and J. Rim, Magnetospheric physics

158

small scale laminar structures (e.g. double layers, see section 3.5) may lead to significant macroscopic effects [4.30]. A complete coverage of this important field of magnetospheric physics would require a separate extensive review. Here we confine ourselves to mentioning some of the basic problems and examples involving small scale turbulence. To deal with the effect of small scale turbulence, the first step is generally to identify a linear instability. Because of the small scale this problem can be treated by discussing the dispersion relation for a spatially homogeneous background (e.g. [4.31]).Although this is in many cases a tedious task, linear instabilities of homogeneous plasmas are fairly well understood. Unstable modes relevant for the magnetosphere, are for instance the cyclotron modes driven by anisotropy and a cold plasma population or current driven instabilities such as the Buneman or ion sound mode. (Basic information on linear plasma instabilities may for instance be obtained from Krall and Trivelpiece [4.31].) The task of computing macroscopic transport-coefficients is solved only partially. Obviously, the transport-coefficients depend on the amplitude of the fluctuations such that a non-linear theory is required. Methods that have led to partial answers are available for the case of ‘weak turbulence’ where the energy density of the fluctuations is much larger than the fluctuation energy of the system in thermodynamic equilibrium but much smaller than the particle kinetic energy density. For longitudinal fluctuations resonant interaction tends to dominate non-resonant interactions. One distinguishes three types of resonant interactions [e.g. 4.32]: (a) linear wave-particle interactions (wk v = 0) (b) non-linear wave-particle interactions (w~ v) = 0, n ~ 2 (c) (non-linear) wave-wave interactions ~ w~1= 0, ~I k~= 0, n ~ 3. —

~



.

A similar classification may be applied to electromagnetic interactions; there however non-resonant interactions may also play an important role. The problem of collective transport may be broken down in the following way; case (a) is covered by quasi-linear theory where the average distribution function f satisfies a diffusion equation of the type Of/at

(0/Ov). [D. Of/0v].

=

(4.42)

The diffusion tensor D(v, t) is given by

J

D = 81r~!~2 Wk(t)

3k, —~Wk + ~

V ~7k

(4.43)

d

with I~= k/IkI and Wk(t) is the spectral energy density of the fluctuation, ~0k and 7k are real and imaginary parts of the frequency of the linear wave with the time-dependence of the form From (4.42) one may define an effective collision frequency Yeff~ 2fU~D~1V~

u=-~-Jvfd3v.

(4.44)

For the two-stream instability one finds 7eff~wpe(~ir)U2 W/kADnTC.

(4.45)

k is a suitable average wave-number and W the total energy density of the field fluctuations [4.32]. (Note that the Boltzmann constant is absorbed into Te.) Here it is assumed that the turbulence is spatially homogeneous, however, as discussed below, this assumption may be violated in important cases.

K. Schindler and J. Bim, Magnetospheric physics

159

However, the problem is not solved by equation (4.44), because W is not yet known. Furthermore, the procedure would not be able to yield a steady state with a finite fluctuation level, as follows immediately from (4.42). Thus, other non—linear processes, e.g. of type (b) or (c), are required to reach a steady state. At present, a number of such processes are discussed and it is not clear which one is dominant in each situation. For enhanced resistivity in the auroral zones Papadopoulos and Coffey [4.33] have suggested the following picture. A fast beam of electrons exites plasma oscillations via the two-stream instability. By non-linear interaction of type (c) (oscillating two-stream instability) the plasma oscillations give rise to low frequency ion fluctuations, which are subject to Landau damping. It seems, however, that in some cases the turbulence may be locally concentrated as was shown by Zakharov [4.34] and subsequently by other authors. The process of localization is referred to as a ‘collapse’. The consequences of these processes for ionospheric and magnetospheric conditions are not yet clear. Galeev [4.36] has pointed out that the collapse-phenomenon may be important for collective interactions in the ionosphere and magnetosphere; however, the consequences do not seem to be clear at the present time. Observationally, the situation is equally unclear. Several authors have suggested that the observed parallel electric fields are associated with a collective resistivity. In any case, there is one piece of strong direct evidence for the presence of turbulence, namely the terrestrial kilometric radiation (fig. 4.7).

Measurements made by Gurnett and Frank [4.37] have shown that there exists a broad band of quasi-electrostatic noise on auroral field lines at altitudes ranging from a few thousand km to more than 40 RE. 1~n addition, electromagnetic bursts exist in the whistler mode. Gurnett and Frank [4.37] conclude that the quasi-electrostatic noise is driven by some two-stream mechanism. Ashour-Abdalla and Thorne [4.38] demonstrated that the auroral zone plasma should be unstable to excitation of electrostatic ion-cyclotron harmonics driven by the presence of a loss cone. Since this instability IMP-B, ORBIT 82 FEB. 6, 972

L

I

I

I

io_12_~~ 78 kHz ~

o~

E

1012

-

-

:~HJ-~-~[H~H -~H _____

4

~

00 kHz



~

-

.

.

f f CUTOFF

LT

-



~~A~HERE

~~OOO

.

~

22.0 HR

PLASMAPAUSE -IS

.



-

800 UT

.

1~r--~

-

-

800

1900

2000

UT

PLASMAPAUSE

Fig. 4.7. The terrestrial kilometric radiation as observed by Gurnett [4.35].Although its emission mechanism is not yet clear, such observations constitute strong evidence for turbulent wave fields in the inner magnetosphere.

160

K. Schindler and J. Birn. Magnetospheric physics

would be quenched by Landau damping of 1—10 eV electrons this mechanism should be less effective at the day side and within the plasmasphere, which seems to be in qualitative agreement with observations. These are only a few examples of the non-equilibrium enhancement of the fluctuations in magnetospheric plasmas. It seems that via enhancement of the transport coefficients these fluctuations have important consequences for the particle balance of the magnetosphere. Important possibilities are the suggested [2.23] diffusive entry of plasma through the front side magnetopause and reconnection based on collective resistivity. Observations of electric field (E11) and current density (J~~) components parallel to the magnetic field are suggestive for postulating a turbulent resistivity supporting E11. It should be noted, however, that this is not the only possibility. Other possibilities such as double layers [4.301were briefly discussed in section 3.5.

5. Other planetary magnetospheres It is characteristic of the Earth’s case that it is the magnetosphere which primarily deflects the solar wind. Of the planets explored by in-situ measurements, Mercury, Earth and Jupiter have magnetospheres in this sense. Radio emissions suggest that Saturn, Uranus and Neptune fall into the same category [4.391. Venus has a rather weak magnetic moment such that the solar wind interacts directly with the ionosphere. In the case of Mars it is not yet clear whether the magnetic field or the ionosphere plays the dominant role in deflecting the solar wind, perhaps both effects are important. In any case, neither Venus nor Mars show well-defined magnetospheres so that we will exclude them from the discussion. Since Saturn, Uranus, Neptune (and Pluto) are insufficiently explored, we will confine our brief discussion of other planetary magnetospheres to Mercury and Jupiter. In several respects, Mercury’s magnetosphere resembles a miniature counterpart of the Earth’s magnetosphere [4.40]. There is a detached bow shock, a magnetopause and magnetospheric tail (fig. 5.1). Acceleration of particles and magnetic fluctuations (fig. 5.1) have been tentatively interpreted as the occurrence of a substorm [4.41]. If this interpretation together with the generally assumed absence of an ionosphere is correct this observation has an important consequence for substorm theories. On the one hand a substorm must be possible without an ionosphere (Mercury), on the other hand the presence of an ionosphere should give rise to a strong ionosphere-magnetosphere interaction. Note that the theoretical approach described in section 4.2 satisfies this condition. Jupiter has an enormous magnetosphere (stand-off distance is several 106 km) such that one might expect it to be a giant counterpart of the Earth’s magnetosphere. This is partly correct (see for instance Smith et al. [4.421or Michel [4.431 and references therein). There is a bow shock and a magnetopause. The external structure however is significantly different due to the large planetary rotation frequency and the correspondingly large centrifugal effects on co-rotating structures. One important consequence is the presence of a disc-shaped plasma reservoir (plasma disc) instead of the Earth’s plasma sheet. Accordingly, the quasi-steady theory has to take centrifugal effects into account [4.44, 4.45]. The dynamics of Jupiter’s magnetosphere is also much more involved than that of the Earth. It seems questionable whether a description in terms of a slow passive and a fast active phase applies. Possibly these phenomena are not clearly se~parableas their time constants become about equal.

K.

Schindler anti .1. Dim, Mag~slasphericphysics

161

NASA-GSFC MAGNETIC FIELD-MARINER 0 BS MP CA IOO[_ 8o~ 60

-

(~)

-~

-

+

.

~e-eow ~

‘I .

-

-~.

r

-~

L

-

~.

-,

-60

F4TTI~ -

-

-

~

~H-~’

~H- ~

~

~ • ~

o

OS

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______

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-

MP-MAGNETOPAUSE CA- CLOSEST APPROACH

~

-

~~O2O

.

2030

-

-

-

2040 29 MARCH 1974

i~2O~O

~TT~Tc0.

2100 UT

Fig. 5.1. Magntic field data for 1.2-s periods during encounter with Mercury in space-craft centered solar-ecliptic ct~rdinates.F denotes the magnetic field in4e~ssity,8 and qS are latitude and longitude, respei~tively;RMS represents a nieasui~eof the fie’d fluctuations (Ness et al. [4.40]).

DIURNAL TRAPPING PSEUDOTRAPPING

STABLE TRAPPING

DIURNAL ESCAPE

si~I;~5

SHOCK Fig. 5.2. A q~litativepicture of Jupitee’s magnetosphere suggested by Hill et at. (4.47]. The region of stable trapping i~also called the ‘plasma dsc’.

As in the case of the earth the origin of the trapped plasma is not yet clear. Reconnection processes may govern the topology of the magnetic field [4.46]. Figure 5.2 gives a sketch of the Jupiter magnetosphere due to Hill et at. (4.47], However, for Jupiter’s CH%C the 8tatcment made regarding the Earth (fig. 1.1, section 1) is even more true: a picture as fig. 5.2 irwolves a large amount of extrapolation. Much more observational and theoretical effort is required l~efor~ a reliable picture of the global structure of planetary magnetospheres can be drawn.

162

K. Schindler and J. Bim. Magnetospheric physics

Appendix It is instructive to show explicitly that the plasma sheet may be understood as a generalized tangential discontinuity. In section 3 the plasma sheet was treated in the limit of small = LZ/LX. Ignoring terms of order 2 and confining the discussion to two space dimensions (a/oy = 0), the vector potential A solves eq. (3.30). This equation may be readily obtained from the generalized jump relations (2.10—2.15) if specialized for static conditions (v = 0, E~ 0). Assuming a configuration as shown in fig. 2.8 we introduce the following scaling =

0(e),

=

0(1),

B~= 0(e),

B~= 0(1),

p

=

0(1),

(A.!)

and obtain from (2.10—2.15) to first order in e ~[p+ B~/8ir}J= 0,

(A.2) (A.3)

lIBJ=—fJB~dz.

(A.4)

(A.4) is satisfied identically if we introduce the vector potential A by B

=

VA X e5.

(A.5)

From (A.2) we obtain p

+

B~/81T= j3(x),

(A.6)

where j3(x) is the pressure at the mid-plane, j3(x) = p(x, 0). Equation (A.3) may be used to show that p(x, z) must have the form p(x, z) = P(A(x, z)). Without loss of generality we write p(x, z) = P*(A(x, z), x). Differentiating (A.3) with respect to the upper limit (z2) we find

of 1 B2\_O~’ 81T OA OA~oP Ox Dx 41 B~-~-~ x ( A7. ) ax\P Differentiating (A.6) with respect to z and eliminating OP*OA from (A.7) we find OP*/Ox = 0 which proves that p = P(A). Thus we obtain from (A.6) using (A.5) BB

4ir Oz



XI

~)



~.



P(A) + (OA/Dz)2/8rr

=

jJ(x),

(A.8)

which is the basic equation for the two-dimensional tail solutions discussed in section 3.3.2. In fact, differentiating (A.8) with respect to z we obtain (3.30) if a is identified with A. It is equally easy to show that any solution of (A.8) satisfies the jump relations (A.2)—(A.4).

K. Schindler and I. Bim, Magnetospheric physics

163

Acknowledgements This work was supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich “Plasmaphysik Bochum/Jülich”. The authors appreciate helpful discussions with P. Rosenau on generalized discontinuities and greatfu!ly acknowledge valuable comments by T.W. Speiser. References [2.11K.W. Behannon, J. Geophys. Res. 73(1968) 907. [2.21N.F. Ness, J. Geophys. Res. 70 (1965) 2989. [2.3] N.F. Ness, CS. Scearce and J.B. Seek, J. Geophys. Res. 69 (1964) 3531. [2.4] JR. Spreiter, Rev. Geophys. 7(1969)11. [2.51L.D. Landau and EM. Lifshitz, Electrodynamics of contimous media (Addison—Wesley, Reading, Mass., 1959). [2.61EN. Parker, Rev. Geophys. 7 (1969)3. [2.71K. Schindler, in: Magnetospheric particles and fields, ed. B.M. McCormac (D. Reidel Publ. Co., Dordrecht, Holland, 1976). [2.8] G.F. Chew, M.L. Goldberger and F.E. Low, Proc. Roy. Soc. A236 (1956) 112. [2.91P. Goldberg and K. Schindler, Phys. Fluids 13 (1970) 3056. [2.101D.H. Fairfield, J. Geophys. Res. 79 (1974) 1368. [2.11]A. Barnes, Cosmic Electrodyn. 1(1970)90. [2.12]R.W. Fredericks, J. Geophys. Rca. 80 (1975) 7. [2.13]D. Biskamp, NucI. Fusion 13(1973) 719. [2.14]CS. Gardner, H. Goertzel, H. Grad, C. S. Morawetz, M.H. Rose and H. Rubin, in: Proc. 2nd Intern. Coni. on the Peaceful uses of atomic energy (United Nations, Geneva) Vol. 31(1958) 230. [2.15]E.W. Greenstadt, C.T. Russell, F.L. Scarf, V. Formisano and M. Neugebauer, J. Geophys. Res. 80 (1975) 502. [2.16]R.C. Davidson, Methods in nonlinear plasma theory (Academic Press, New York and London, 1972). [2.17] B.D. Fried and R.W. Gould, Phys. Fluids 4 (1961)139. [2.18] A.A. Galeev, in: Physics of the solar planetary environments, ed. D.J. Williams (Boulder, Colorado, 1976) p. 464. [2.19]A.A. Galeev, in: Physics of the hot plasma in the magnetosphere, eds. B. Hultquist and L. Stenflo (Plenum PubI. Co., New York and London, 1975). [2.201S. Chapman and V.C.A. Ferraro, Terr. Magn. Atmos. Elec. 36 (1940) 77. [2.21]WI. Axford and CO. Hines, Can. J. Phys. 39 (1961)1433. [2.22] WI. Axford, H.E. Petschek and G.L. Siscoe, J. Geophys. Rca. 70 (1965)1231. [2.231G. Paschmann, G. Haerendel, N. Sckopke, H. Rosenbauer and P.C. Hedgecock, J. Geophys. Res. 81(1976) 2883. [2.24] E.N. Parker, J. Geophys. Res. 72(1976) 2315. [2.251I. Lerche, J. Geophys. Res. 72 (1967) 5295. [2.26] W. Alpers, Astrophys. Space Sci. 5(1969) 425. [2.27] A. Eviatar and R.A. Wolf, J. Geophys. Res. 73 (1968)5561. [2.28] H. R?senbauer, H. Grunwald, M.D. Montgomery, G. Paschmann and N. Sckopke, J. Geophys. Res. 80 (1975) 2723. [2.29] B.U.O. Sonnerup, in: Physics of the solar planetary environments, ed. D.J. Williams (American Geophys. Union, Boulder, Colorado, 1976) p. 541. [2.30] N. Sckopke, G. Paschmann, H. Rosenbauer and D.H. Fairfield, J. Geophys. Res. 81(1976) 2687. [2.31]J.G. Roederer, Space Sci. Rev. 21(1977) 23. [2.32] J.W. Dungey, Phys. Rev. Lett. 6 (1961) 47. [2.33] E.G. Harris, Nuovo Cimento 23(1962)115. [2.34] E.N. Parker, Phys. Rev. 107 (1957) 924. [2.35]T.W. Speiser, J. Geophys. Res. 70 (1965) 4219. [2.361J.W. Eastwood, Planet. Space Sci. 22 (1974)1641. [2.371G.H.A. Cole and K. Schindler, Cosmic electrodynamics 3 (1972) 275. [2.38] H. Alfvén, J. Geophys. Res. 73 (1968) 4373. [2.39] S.W.H. Cowley, Cosmic Electrodynamics 2 (1971) 90 and 3(1973) 448. [2.401M. Bornatici and K. Schindler, J. Geophys. Res. 79 (1974)529. [2.411P. Rosenau, private communication. [2.42]E.W. Greenstadt, CT. Russell, V. Formisano, P.C. Hedgecock, FL. Scarf, M. Neugebauer and R.E. Hoizer, J. Geophys. Res. 82 (1977) 651.

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p.

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