On the nature of electric fields in the Earth's inner magnetosphere (a review)

E. E. Antonova and B. A. Tverskoy

Nuclear Physics Research Institute, Moscow State University, Moscow, Russia

Contents

Abstract

Introduction

The development of the quantitative theory of magnetosphere-ionosphere coupling (first works in this direction were Tverskoy [1969, 1972], Vasyliunas [1972]) and the experimental observation technique has allowed us to explain many specific features of large-scale magnetospheric convection and to highlight the problems which still remain unresolved. The goal of this paper is to consider the most significant aspects of the available theoretical approaches to this problem, including the nature of the dawn-dusk electric field and mechanisms of field screening in the inner magnetosphere. Major attention will be paid to those research directions which have not been discussed in the literature thoroughly enough, but which allow us to eliminate the discrepancy between the existing theoretical descriptions and the experimental data. At first we discuss the role of azimuthal pressure gradients in formation of the field-aligned current system and magnetospheric convection. Then we consider peculiarities of solution of the stationary and nonstationary linear problems of magnetospheric convection and their applicability to the description of the experimental data. Attention will also be given to the role of diffusion processes in formation of the quasi-stationary plasma distribution and to the problem of the convection crisis.

Distribution of the Hot Plasma Pressure in the Magnetosphere

Though satellite measurements in the near-Earth space plasma have been performed for a long time, there is still no sufficiently complete picture of distribution of the plasma pressure in the magnetospheric cavity. It has been shown experimentally (see references in the review of Tsyganenko [1990]) that across the magnetotail, the condition p + B_x²/8p = !  const, where p  is the hot magnetospheric plasma pressure in the central regions of the plasma sheet close to the isotropic region [ Christon et al., 1991; Lui and Hamilton, 1992; Stiles et al., 1978] and B_x  is the field-aligned (along the tail) magnetic field component, is satisfied with a high degree of accuracy. According to the data of AMPTE/CCE [ Lui and Hamilton, 1992], in magnetically quiet conditions, the pressure at the geocentric distances (2.5-9) R_E (where R_E  is the Earth's radius) only slightly depends on the azimuthal angle, peaks at L 3 , and at 3 < L < 8  is well approximated by the dependence p L^-S , where S = -3.24 0.22 . At L < 5 , a strong anisotropy of pressure is observed ( p > p_|| , where p  and p_||  are the transverse and field-aligned components of the plasma pressure). During geomagnetic storms, variations in the radial profile of the pressure associated with the substorm injections of particles into the inner magnetosphere were observed [ Lui et al., 1987]. On the whole, the pressure profile corresponds to the integral westward ring current. Experimental information about the distribution of azimuthal pressure gradients is not available.

It has long been believed that, solving the problem of flowing around of the magnetosphere by the solar wind plasma, one can neglect in the first approximation the presence of hot plasma in the cavity and, hence, the nondipole sources of magnetic field. In this case the existence of the geomagnetic tail was qualitatively attributed to frozen-in magnetic field into plasma and to stretching of the field lines in the antisolar direction. In the numerical models of the field [ Tsyganenko, 1990], the existence of the plasma sheet and current in the tail was postulated. The numerical MHD simulations [ Walker and Ogino, 1989; Watanabe and Sato, 1990] of the process of magnetosphere formation and its dynamics have shown that the plasma sheet formation can be due to the presence of the rarefied plasma around the dipole before the flow. After the flow begins, the plasma gathers in the plasma sheet, until magnetostatic equilibrium between the pressure of the heated plasma in the sheet and the magnetic field in the parts of the tail being formed is reached. At the lobe point, the magnetic field pressure balances the dynamic pressure of the solar wind plasma, and the hot plasma pressure is much less than the magnetic pressure.

However, near the dayside magnetopause, the point on the field line where the field strength is minimum does not lie in the equatorial plane, but at high latitudes (this effect is typically considered in connection with splitting of particle drift shells [ Shabanskiy, 1972]), where the hot plasma pressure is already comparable to the magnetic pressure and dynamic pressure of the solar wind (the near-cusp region). Therefore, in solving the problem of flow, taking into account filling of the cavity with plasma can prove to be a key factor in estimating the effect of the solar wind magnetic field on the distribution of fields and currents in the magnetosphere and, hence, on the geomagnetic activity. For instance, the studies carried out by Sibeck et al. [1991] and Tsyganenko and Sibeck [1994] showed that changes in the position of the magnetopause during variation of the interplanetary magnetic field (IMF) result from changes in the magnetic field of the intramagnetospheric sources (field-aligned current and tail current), and not by processes of the reconnection type.

The Role of Plasma Gradients in Formation of Field-Aligned Currents

The role of azimuthal pressure gradients of hot magnetospheric plasma in the formation of the region II currents of Iijima and Potemra [1976] is well known. The picture is not that clear for the region I currents. The Iijima and Potemra [1976] picture of the region I current densities gave maxima at 1000 and 1400 MLT. According to the magnetic field models such regions can be projected onto the magnetosphere boundary layers, whereas the other main part of the region I currents is projected onto the plasma sheet. The results of such projection lead Stern [1983\link, 1983] to a suggestion that the region I currents have two sources: one in the boundary layers and the other in the plasma sheet. The latter can be connected with the azimuthal plasma pressure gradients. Karty et al. [1984] and Yang et al. [1994] modeled formation of the region I currents postulating the existence of pressure minimum in the center of plasma sheet and fixed dawn-dusk potential drop. The role of the plasma pressure gradients in the magnetospheric boundary layers during northward IMF orientation was analyzed by Song et al. [1992, 1993]. Foster et al. [1989] restored the picture of field-aligned currents on the basis of radar measurements. They showed that the integrated field-aligned currents have maxima about dawn and dusk (0600 and 1800 MLT) but not about 1000 and 1400 MLT. The latter can be seen from the Iijima and Potemra [1976] pictures if one takes into consideration that the region I current belt width in these hours is lower than during night hours. So it is necessary to suggest that the main contribution to the formation of the region I currents is introduced by the source lying deep in the magnetosphere. Since on the ionospheric altitudes transverse currents are supported by transverse electric fields (apart from the wind dynamo currents), an inner magnetospheric source of the region I currents means an inner magnetospheric source of dawn-dusk electric field. This possibility was discussed by Antonova and Tverskoy [1990]. The generation mechanism of the dawn-dusk electric field in the inner magnetosphere [ Antonova and Ganyushkina, 1995a, b\link8] do not require action of the MHD generator or quasi-viscous interaction in the magnetospheric boundary layers or creation of pressure gradients in these layers (the latter possibility was analyzed by Johnson [1978] and discussed in many works). The connection of boundary layers with inner magnetospheric processes may be not so simple as has been originally suggested (see reviews by Lundin [1988], Saunders [1990]) and requires additional consideration. Note that direct measurements of the electric fields near the magnetopause [ Lindqvist and Mozer, 1990; Sonnerup et al., 1990] do not give a direct answer to the question about penetration of the solar wind electric field into the magnetosphere or about the action of quasi-viscous processes. In the majority of measurements, the field fluctuation amplitudes are well above the regular field amplitudes, which hardly agrees with the commonly used assumption of equipotentiality of magnetic field lines. The main proof of the action of the reconnection mechanism is therefore considered to be the dependence of geomagnetic activity on the IMF orientation [ Lundin, 1988].

The major difficulty encountered by the concept of generation of the dawn-dusk potential difference in the magnetosphere by the processes occurring in the magnetospheric boundary layers is the localization of these fields. According to the available magnetospheric models [ Tsyganenko, 1990], the magnetospheric boundary layers are projected onto the near-cusp regions, and the field-aligned currents of region I are distributed over the polar edge of the auroral oval. Feldstein and Galperin [1985], Galperin and Feldstein [1989], and Elphinstone et al. [1991] showed that the auroral oval is projected onto the inner regions of the plasma sheet. Correspondingly, the maximum potential drop between the morning and evening sides of the magnetosphere takes place deep inside the plasma sheet, rather than at the polar cap boundary, which is hardly consistent with the field penetration from the solar wind.

The mechanism of creation of the dawn-dusk field by the azimuthal pressure gradients does not require the assumption of equipotentiality of magnetic field lines, which apparently can be violated elsewhere than in the region of strong field-aligned currents flowing from the ionosphere. The mechanism is based on the assumption of the fulfillment of the condition of magnetostatic equilibrium for the velocities of plasma motion much less than the sound and Alfv\'en velocities. In the case of isotropy of plasma pressure this condition is given by

(1)

where j is the transverse current density, B is the magnetic field, and c  is the velocity of light (in case of anisotropic plasma pressure)

Since in the magnetospheric regions of major interest for the problem considered the pressure is close to isotropic [ Christon et al., 1991; Lui and Hamilton, 1992; Stiles et al., 1978], in what follows we shall use (1). The consequence of (1) is the expression for the field-aligned current density at ionospheric altitudes [ Tverskoy, 1982; Vasyliunas, 1970]

(2)

where n is the vector of the normal to ionosphere, W = dl/B  is the magnetic field tube volume, dl  is the element of the magnetic field line length, and   is the two-dimensional gradient. Both W  and P  are analyzed as functions of latitude and longitude in the ionosphere. The dependence of j_||  on W  allows us to regard W  as a physically isolated value which can be used as a coordinate in description of the high-latitude processes. The urgent need for the choice of an adequate coordinate system is due to violation of the adiabatic invariance of particle motion at geocentric distances exceeding (7-8) R_E , where the dipole field is appreciably distorted by the magnetospheric current systems, and that is why the use of the L  parameter of McIlwain becomes inconvenient and not physically justified. The calculation of positions of isolines W =!  const in the projection on ionospheric altitudes performed on the basis of Tsyganenko 87 models [ Tsyganenko, 1987] and discussion of the choice of the coordinate system for the description of high-latitude processes have been given elsewhere [ Antonova and Ganyushkina, 1994; Antonova et al., 1993].

Equation (2), together with the equation for current continuity in the ionosphere (see Appendix A), gives the basic equation for the low-frequency magnetosphere-ionosphere coupling in the high-latitude approximation [ Tverskoy, 1982, 1983]

(3)

where S  is the tensor of the ionospheric conductivity integrated over the dynamo layer thickness, and Fⁱ  is the potential in the ionosphere. It follows from the latter that the magnetostatically equilibrium field-aligned currents and the electric field resulting from their closing can exist only where the conductivity in the ionosphere differs from 0 and  . In the first case, the field-aligned current is zero and the condition of the magnetostatic equilibrium reduces to the condition that constant-pressure surfaces coincide with constant volume surfaces, an assumption that is commonly used in the physics of laboratory plasma. In the second case, when conductivity tends to  , the equilibrium electric fields become zero.

It follows from (1) and (3) that the presence of the magnetostatic equilibrium field-aligned currents in the cavity can be revealed not only by determining the projection of W  on the magnetic field line, which is beyond the accuracy limits of the available magnetic field models [ Tsyganenko, 1990] and presents difficulties for experimental measurements, but also by analyzing the distribution of lines of constant W  and current lines in the equatorial plane (Figure 1). Since ( j p) = 0 , the noncoincidence of the current lines and lines of constant W  leads, according to (3), to the existence of field-aligned currents. The case of generation of field-aligned currents corresponding in sign to the current system I of Iijima and Potemra [1976] (the current enters from the morningside and exits from the eveningside) corresponds to the larger curvature of isolines W =!  const than of the current lines in the equatorial plane, as shown in Figure 1. Antonova and Ganyushkina [1995a] showed that the semiempirical magnetic field models Tsyganenko 87 and Tsyganenko 87W [ Peredo et al., 1993] conform to just this qualitative picture.

By closing in the ionosphere, the field-aligned currents create the large-scale dawn-dusk field which requires for its support only the presence of corresponding azimuthal pressure gradients or, equivalently, a definite geometry of the W = !  const surfaces. The effect of the IMF on the dawn-dusk electric field is attributable to changes in the configuration of the magnetospheric current systems and the magnitudes of currents in them. Antonova and Ganyushkina [1995b] showed that for southward IMF, the angle between the isolines W =!  const and current lines increases, and, hence, the region I currents increase. Therefore in solving the problem of the IMF effect on magnetospheric activity it is not the electric field penetration into the magnetosphere, but changes in the currents in the tail and on the magnetopause that should be explained. However, as noted above, this cannot be explained in the framework of the model of flow of an empty cavity. From the information about the distribution of field-aligned currents at ionospheric altitudes [ Iijima and Potemra, 1976] and calculations of the volumes of magnetic field tubes and their gradients [ Antonova and Ganyushkina, 1994; Antonova et al., 1993], Antonova and Ganyushkina [1995c] estimated the pressure gradients needed to sustain field-aligned currents. It was shown that the magnetospheric cavity contains enough plasma to maintain the calculated pressure gradients both for currents of region II and for region I currents (because of appreciably lower p , relatively larger pressure gradients are required for region II currents). The obtained day-night pressure differences proved to be comparable with pressure magnitudes at corresponding geocentric distances measured in the AMPTE/CCE experiment [ Lui and Hamilton, 1992], which apparently hampers the use of the linear approximation for description of the magnetospheric convection.

The question arises as to what are the conditions for which isolines W =!  const will coincide with the current lines and the field-aligned current will not be generated. First of all, this will occur for the cavity with the azimuthal symmetry (the region of dipole field lines in the Earth's magnetosphere), where isolines W = !  const and current lines in the equatorial plane are concentric circles. Field-aligned currents will also not be generated in fields with translational invariance where W =!  const and current lines are parallel straight lines. The plasma sheet regions remote from the Earth are close to the latter configuration. There still remains the question of whether isolines W =!  const and current lines can coincide in a strongly asymmetrical part of the magnetospheric cavity where the field lines transform from a dipole configuration to one extending into a tail. Note that in the absence of the electric field in the magnetosphere the plasma sheet should have disappeared due to the ion drift through the morningside to the eveningside (on the order of several hours). To sustain the plasma sheet in steady state, particles should be constantly injected into the magnetosphere from the morningside and ejected into the evening part of the low-latitude boundary layer from the eveningside. In this case plasma should have been heated at first from the temperature of the boundary layer to that of the plasma sheet (approximately by an order of magnitude) and then cooled again, which seems to be highly improbable. The existence of a constant earthward directed plasma flow caused by the presence of the dawn-dusk field is therefore the necessary condition for the existence of the plasma sheet in quasi-equilibrium. Ashour-Abdalla et al. [1993, 1994] studied in detail the processes of such a formation in the one-particle approximation for a given distribution of the electric and magnetic fields.

During magnetospheric substorms, the cavity is quickly replenished by heated and accelerated particles. Since substorms mainly occur for the southward directed IMF, in the case of negative IMF a source acts periodically in the inner magnetospheric regions (at L = 7-10 ) increasing the magnetospheric plasma pressure. The ensuing state is not magnetostatically balanced in the majority of cases. The excess of the formed particles is ejected in the form of a plasmoid into the distant regions of the magnetospheric tail.

Observations also reveal a destruction of the plasma sheet during long periods of the northward orientation of the IMF and reduced geomagnetic activity. For positive IMF, compression of the auroral oval occurs, and arcs are formed in the polar cap, which Frank et al. [1986] called the theta aurora. According to the observations [ Frank et al., 1986; Huang et al., 1989a], the appearance of the theta aurora is associated with penetration of the plasma of the plasma sheet into the tail parts, which is quite naturally interpreted as bifurcation of the plasma sheet (formation of the plasma structure of the Maltese-cross type). The disappearance of the plasma sheet during a long-lasting positive IMF was detected by the ISEE 3 satellite [ Fairfield, 1993]. The observed phenomena are most probably associated with violation of equilibrium between the plasma pressure and magnetic pressure due to the current weakening in the tail. In this case the theta aurora current system (the outflow of the field-aligned current on the morningside and inflow on the eveningside) and the presence of the electric field directed from dusk toward dawn in the polar cap can be explained quite easily [ Antonova and Ganyushkina, 1995b]. Note that the plasma system has a minimum energy if it is without power, i.e., pressure gradients are absent in it, and the destruction of the plasma sheet for positive IMF can therefore be regarded as a natural process of transition of the magnetosphere into the state with the lowest energy.

The discussed approach to the description of the process of formation of large-scale magnetospheric convection makes it possible to relate the processes in the magnetospheric boundary layers to the disturbances of the plasma motion in the regions of projection of boundary layers at the ionospheric altitudes, i.e., in daytime hours. Thus the difficulties of the theory of large-scale reconnection are obviated. The solar wind energy is transmitted to the magnetosphere through its current systems. The most complicated point of the gradient mechanism of convection is maintenance of the quasi-stationary state in the conditions of regular plasma motion, at which isosurfaces W = const intersect. This problem will be discussed below when considering the problem of "convection crisis."

Large-Scale Magnetosphere-Ionosphere Coupling at a Low Plasma Pressure

It is possible to perform the analytical consideration of the problem of plasma stability in the cavity for b = 8pp/B² 1 , which is valid for the processes in the region of dipole field lines and, in part, at the inner edge of the plasma sheet. Since the properties of the obtained solutions allow us to explain many specific features of intramagnetospheric processes, we discuss in detail the assumptions used for solving this problem and properties of the solutions.

In solving the problem of large-scale magnetosphere-ionosphere coupling, two questions arise which have no final solution at present. It is necessary to know the relation between the potential variations in the magnetosphere and ionosphere and to find the relation between the pressure changes and changes of the magnetospheric potential. It is typically assumed in solving the problem of magnetosphere-ionosphere coupling that because of a high plasma conductivity the magnetic field lines can be considered to be equipotential. The difficulty encountered in checking this assumption is connected with the small number of simultaneous measurements of the electric field at one and the same field line. Weimer et al. [1985] reports the results of four measurements of this type by DE 1 and DE 2 satellites. It was shown that at invariant latitudes of 55-60^o, the identical electric field variations were detected at the altitudes of 800 km and 12,000 km. However, at latitudes of 65-70^o, variations in the magnetospheric potential appreciably exceeded the ionospheric potential variations. To elucidate the dependence of potential variation on the scale of disturbance, the Fourier analysis of fluctuations was performed. It was shown that for the scale  1000  km, in the projection on the ionospheric altitudes, the potential fluctuations in the magnetosphere and ionosphere coincide, and for the scale of less than 200 km the field lines cannot be considered to be equipotential. In this case, the dependence

(4)

obtained by Knight [1973] and Antonova and Tverskoy [1975] and experimentally verified by Lyons [1981] and Bosqued et al. [1986] for the inverted V  type structures "works" well. Here, F^m  is the potential in the magnetosphere, j^*= e n_e T_e^1/2/(2 pm_e)^1/2  is the electron current corresponding to the free gas dynamic outflow of electrons from the magnetic field tube, e  and m_e  are the charge and mass of an electron, n_e  is the electron concentration in the region of the field line top, and T_e  is the electron temperature. Note that dependence (4) was obtained on the assumption of conservation of the magnetic moment and electron energy during acceleration in a laminar magnetic and field-aligned electric field in the case where the electron distribution is isotropicy and merely Maxwellian at the "top" of the field line, i.e. above the region of field-aligned electric field. Its experimental verification which also includes the results of observations of shell structures in the electron distribution functions with a cut in the region of the source cone [ Antonova, 1984] points to a small contribution of the nonadiabatic processes during electron acceleration (according to Bosqued et al. [1986], it is not more than 20%). However the isotropic distribution outside the source cone points to the action of powerful processes of isotropization which are still incompletely studied during the time less than the time of one bounce oscillation for particles that reach the top of the field line.

Such an isotropization can be the consequence of both the conventionally studied processes of quasi-linear relaxation of anisotropic functions of particle distribution (the studies were started for the magnetosphere by Kennel [1966] and significantly advanced by Bespalov and Trachtengertz [1986] and Lyons and Williams [1984]) and violation of the adiabaticity of motion due to the nonlinear potential disturbances comparable with the Larmor radius of electrons. Note that under the condition of developed turbulence, appreciable field-aligned electric fields and relevant potential variations giving, on the whole, a zero contribution to the acceleration process can exist at the field line above the region of acceleration of electrons giving rise to the inverted V  type structures. This altitude region is still little studied. The existence of irregularities in the electric potential must lead to the nonadiabaticity of the ion motion as well. So far stochastization of the ion motion has been studied in detail only for the case when the Larmor radius of an ion is comparable with the scale of the magnetic field inhomogeneity [ Buchner and Zelenyi, 1989; Chen and Palmadesso, 1986]. Note that observations of the fields in the plasma sheet reveal strong fluctuations in both the electric and magnetic components [ Petersen et al., 1982], and the electric field fluctuations are often not accompanied by the magnetic field fluctuations. On the whole, it is most likely that the condition of equipotentiality of magnetic field lines is satisfied neither for the regions with fairly strong field-aligned currents flowing from the ionosphere and exceeding the limit provided by a free gas dynamic outflow of electrons from the magnetic field tube into the ionosphere nor for sufficiently long magnetic field lines extended into a tail. However, the equipotential approximation describes fairly well the physics of the process for sufficiently large-scale disturbances at the inner edge of the plasma sheet. This simplification makes it possible to reduce the three-dimensional problem to the two-dimensional one.

In the linear approximation (see Appendix B and C), it is assumed that the undisturbed plasma distribution is independent of the azimuthal angle (the pressure is constant on W =!  const surfaces for the hydrodynamic description of processes) and the disturbance which depends on the azimuthal angle in the stationary problem and on the azimuthal angle and time in the nonstationary problem is small compared with the nondisturbed distribution. The potential disturbance is regarded as being of small magnitude at first order. These assumptions allow us to relate the pressure disturbance to the potential disturbance and to transform the basic equation for the magnetosphere-ionosphere coupling into the equation predicting the potential distribution on the sphere with a given conductivity distribution.

For the first time, the nonstationary problem in the drift approximation was solved by Tverskoy [1969, 1972], and the stationary problem was solved by Vasyliunas [1972] who used significant simplifying assumptions about the properties of plasma (a sharp inner plasma sheet edge and invariability in the number of particles in the magnetic field tube inside the sheet). The results of these works were reviewed and generalized by Pellat and Laval [1972]. Since then, a large number of papers concerned with the Alfv\'en screening of the inner magnetosphere have been published. Among recent publications, the most detailed analysis and review of the earlier obtained results was given by Blanc and Caudal [1985] and Del Pozo and Blanc [1994]. The problem of Alfv\'en screening in the stationary case was reduced to solving the equation for current continuity in the ionosphere for a given potential distribution at the polar cap boundary. In the problem the "magnetospheric conductivity" (exceeding by more than an order of magnitude the ionospheric Hall conductivity) was added to the latter. Therefore, many publications concerning the "electrotechnical" description of the magnetospheric processes appeared (see the book of Lyatskiy [1978] and the references therein).

The electrotechnical approach made it possible to describe a noticeable weakening of the field equatorward from the auroral oval in the case of a proper choice of the magnitude of the "magnetospheric conductivity" and gave the field amplitudes in the auroral oval close to those experimentally observed. However, the azimuthal field distribution did not agree with the experimental data. Indeed, in the stationary case the engendered field-aligned current with which the inner magnetosphere screening is associated is proportional to  F/ j  and therefore, as noted by Pellat and Laval [1972], the maximum field-aligned currents must be observed at night and noon hours. In the experiments, currents of region II are the highest at the evening and morning hours. Attempts were made to solve the problem of the "phase shift" by taking into account the nonuniformity of conductivity and particle injection (see Peymiat and Fontaine [1994] and the references therein). For instance, Peymiat and Fontaine [1994] showed that taking into account injection in the regime of a strong pitch angular diffusion can lead to rotation of the initial electric field distribution by 3 hours, but this does not give an adequate description of the east-west component of the electric field. The dawn-dusk maxima of the field-aligned currents of regions I and II are given by the solution of the nonstationary problem of convection where the field-aligned current density is proportional to ² F/ j² .

This result indicates that when the disturbance reaches the steady state nonlinear regime, the localization of the original disturbance is preserved and no transition of this disturbance (through rotation of the azimuthal field component and field-aligned current) to the configuration predicted by the solution of the stationary linear problem occurs. The problem of the phase shift does not arise in numerical models of convection in which particles are injected into the inner magnetosphere and then travel in the generated self-consistent field [ Chen et al., 1982; Harel et al., 1981a, b\link35; Pudovkin and Zakharov, 1992; Spiro et al., 1981; Wolf et al., 1982]. In these models, the pressure maximum is formed near midnight and hence the field-aligned currents of the Alfv\'en screening are zero in this region. The disadvantage of these models is the assumption of an empty inner magnetosphere (there are no particles there before injection), and therefore they can give a good description of the processes only during strong disturbances. At the nonlinear phase of disturbance development, the assumption of smallness of the azimuthal pressure gradients is violated and they can become comparable with the pressure itself, consistent with the calculations of azimuthal gradients by Antonova and Ganyushkina [1995c].

Another aspect of the problem is associated with the radial pressure profile observed by AMPTE/CCE [ Lui and Hamilton, 1992]. As noted above, at L > 3  the magnetospheric pressure decreases monotonically with increasing radial distance. The solid line in Figure 2 shows schematically the dependence of pressure on W  (in the dipole field W L⁴ ). During magnetospheric substorms, injection of particles into the inner magnetosphere occurs, and the profile becomes nonmonotonic (the dot-and-dash line in Figure 2). Let us recall that for the azimuthally symmetrical distribution of the low-pressure plasma to be stable, the criterion of Kadomtsev [1963]

(5)

must be satisfied (for the kinetic description ( pL⁷)' > 0 [ Tverskoy, 1969, 1972]), and therefore a sufficiently slowly decaying monotonic pressure profile is stable with respect to the electric field excitation. In this case the assumption of the localization of currents of region II at L < 3  made by Peymiat and Fontaine [1994] surely contradicts the experimental data. The quasi-stationary azimuthal plasma asymmetry seems to be more probable. We discussed this asymmetry as a source of the current system of region I associated with the magnetic cavity geometry. For the currents of region II, the geomagnetic field is close to that of the dipole and the pressure asymmetry is due to the action of the dawn-dusk field leading to formation of the pressure maximum near midnight. In this situation, the pressure distribution can be maintained monotonic in radius if the p(W)  dependence for the daytime (the dashed line in Figure 2) is more abrupt than for the night hours. The data of Lui and Hamilton [1992] do not contradict to this assumption. In order to confirm it, it is desirable to have simultaneous measurements of the radial pressure profile for two local times.

The injection-induced region of pressure growing with increasing geocentric distance is the source of nonsteady state convection. Tverskoy [1969, 1972] solved the nonstationary linear problem that allows description of the system of electric fields and currents generated by such a profile (the dot-and-dash line in Figure 2). In the case of a power dependence of pressure on L , the problem can be solved analytically and is of the form of expansion in terms of Bessel functions if the symmetric part of pressure p(L) = p₀(L/L₀)^-7  for L > L₀  and p(L) = p₀(L/L₀)ⁿ  for L L₀ , where L₀ = ²q₀  is the location of the radial pressure profile maximum. For small injections, the conditions of the type of |p'| p₀  are satisfied. As shown by Tverskoy [1969, 1972], even a small deviation from uniform equilibrium gives a picture which appreciably differs from the observed one. For the harmonics of the type (-imj) , the eigen functions are in the form U_m = A(q /q₀)^m  for q q₀  and U_n^m= B T_m [q₀ l_n^m(q₀/q )ⁿ⁺⁵/ (n +5)]  for q q₀ , where A  and B  are functions of time. The spectrum of magnitudes of l^m_n  is determined from the condition of continuity of U^m_n  and U^m'_n  at q = q₀ . For m = 1 , l₀ =4(n+5)/q₀²  and l_n =[(n+5)² /q₀²] x_n , where x_n  are close to the roots of T₁(x) . The behavior of functions U¹₀  and U¹₁ [ Tverskoy, 1972] is shown in Figure 3. The characteristic times of variations of harmonics t_n l_n , and hence the characteristic time of decay or growth of a two-vortex harmonic U¹₀ j , are less than the characteristic time of decay or growth of a four-vortex harmonic U¹₁ j  by approximately a factor of 50.

The solution of the problem of penetration of the electric field applied from the outside into the depth of the magnetosphere shows that this field can penetrate only if it sufficiently rapidly varies with time (the situation inverse to the skin effect). The charge separation resulting from the magnetic drift of particles leads to formation of the current system of region II and efficient screening of the inner magnetosphere. The development of instability must lead to disappearance of the disturbance which caused it, i.e., of the region of pressure growing with increasing radial distance. By creating the quasi-stationary azimuthally asymmetric distribution, the quasi-stationary dawn-dusk field will maintain the current system of region II which effectively screens its influence on the inner magnetosphere.

The azimuthally asymmetric plasma distribution itself can be unstable. Ivanov and Pokhotelov [1987] analyzed the stability of such a distribution with respect to development of short-wave flute modes. It was shown that if the criterion for an ordinary flute instability (5) is not satisfied, then for a flute instability with a variable pressure on the surfaces of equal specific volumes under the condition of maximum increment to arise, it is necessary that the magnitude of the field-aligned current, i.e.,  p₀/ j , exceed a certain threshold value. Antonova [1993] analyzed the development of this instability taking into account a possible nonequipotentiality of magnetic field lines. It was shown that taking due account of nonequipotentiality owing to which the field amplitudes in the magnetosphere can significantly exceed the field amplitudes at ionospheric altitudes leads to an appreciable increase in increments in the regions of outflowing field-aligned currents. The maximum increments in this case will correspond to the regions of maximum outflowing field-aligned current. Unfortunately, the analysis of stability of the hot magnetospheric plasma distribution is mostly restricted at present to the regions of low plasma parameters b . The development of ballooning modes at finite b  in the short-wave approximation was analyzed by Ivanov et al. [1992].

Magnetospheric Turbulence

The possibility of generation of small-scale harmonics and nonequipotentiality of magnetic field lines for the scale of less than 200 km in the projection on ionospheric altitudes noticeably complicates the analysis of magnetosphere-ionosphere coupling because the nonequipotentiality of disturbances leads to the exchange of particles between magnetic field tubes. As a result of interaction between harmonics and energy exchange, a cascade process can arise in the turbulence spectrum and a quasi-stationary equilibrium turbulence spectrum will be formed, for which the dawn-dusk field will be the major large-scale energy source and dissipation will occur due to both heating of the ionospheric plasma during the flow of transverse currents which close the field-aligned currents in the ionosphere and absorption of the wave with transverse scales close to the Larmor radius of a hot magnetospheric ion. Plasma heating in the plasma sheet up to the temperatures exceeding the temperatures of particles in boundary layers can be associated with the latter process. These processes are still incompletely studied theoretically and experimentally. Weimer et al. [1985] and Basu et al. [1988] showed that expansion of the electric field measurements at low altitudes in the Fourier series give the Kolmogorov spectrum of the transverse electric field fluctuations

(6)

Measurements by the DE 1 satellite at an altitude of 12,000 km yielded

(7)

i.e., a growth of the fluctuation amplitude with decreasing wavelength to the scales of the order of the Larmor radius of a hot magnetospheric ion (  10  km in the projection onto the ionospheric altitudes). Determination of the plasma velocities in the plasma sheet at geocentric distances from 16 to 19 R_E  from measurements of particle fluxes by ISEE and AMPTE/IRM satellites [ Angelopoulos et al., 1993] showed that the stochastic velocity component is an order of magnitude greater than the regular component. The existence of electric field fluctuations in the tail exceeding by more than an order of magnitude the field fluctuations at ionospheric altitudes (in the case of corresponding projection) is well known from measurements in the tail [ Scarf et al., 1984] and on auroral field lines [ Mozer et al., 1980]. On the whole, the flow pattern corresponds to the superposition of vortices of different scales (Figure 4). Thus, as shown by Antonova [1985], the experimental observations speak in favor of the action of effective mechanisms of diffusion intermixing. Indirect evidence of the action of these mechanisms can be provided by a relatively weak dependence of the plasma concentration and temperature on the radial distance. For instance, checking of the relation between the field-aligned current and field-aligned pressure drop in the inverted V  type structures indicates that j^*  is constant. In the AMPTE/CCE experiment [ Lui and Hamilton, 1992], the plasma concentration was independent of radial distance for 6 < L < 9 . Invariability of the ionic temperature with latitude was noticed by Antonova et al. [1991] while analyzing the Intercosmos-Bulgaria 1300 satellite measurements. The data of Huang and Frank [1994] also point to the isothermal nature of the central part of the plasma sheet. The question about the magnitude of the polytropic index g  which is used in the hydrodynamic description of the large-scale convection is closely connected with the problem of intermixing in the plasma sheet. The attempts to infer its magnitude from the experimental measurements did not give an unambiguous result. For instance, while studying the correlation between concentration and pressure, Baumjohann and Pachmann [1989] obtained g_p  close to 5/3 , and Huang et al. [1989b] obtained g_T < 1  in the studies of correlation between temperature and concentration. Zhu [1990] reported that

Since g = 1  corresponds to the isothermal distribution, the results of determination of g  are also likely to provide evidence of the action of intermixing processes. The attempt to take into account theoretically the influence of irregular particle fluxes on the solution of the large-scale convection problem was made by Antonova [1987]; however, this aspect of the problem requires further investigation.

The Problem of Convection Crisis

The possibility of the existence of quasi-equilibrium convection flows in the Earth's magnetosphere encounters the problem of "convection crisis" raised by Erickson and Wolf [1980]. The fact is that if the number of particles in the magnetic field tube is conserved and the tube moves toward the Earth from distant tail regions, the plasma pressure in the tube must increase extremely rapidly, which is not consistent with the experimental data. Consideration of the equilibrium plasma distribution in the magnetospheric tail led Schindler and Birn [1982], Birn and Schindler [1983], and Schindler and Birn [1986] to the conclusion that time-independent solutions of the equilibrium equation (in the two-dimensional case, the Grad-Shafranov equation) cannot exist. The time-dependent solutions with W = W(t)  due to a decrease in the vertical field component in the plasma sheet during convection were obtained. This evolution of the configuration leads to thinning of the plasma sheet, with the final stage being formation of the tearing-type instability. However, in the experiment, the steady state convection regime is detected, though rarely [ Sergeev et al., 1994]. When solving the problem, Kivelson and Spence [1988] and Ashour-Abdalla et al. [1994] assumed that the number of particles in the tube is not conserved due to magnetic drift. In view of development of turbulent fluctuations, the problem of the "convection crisis" loses its importance because the concentration gradients arising during convection will decrease due to diffusion. Since the motion in the plasma sheet is directed toward the Earth, i.e., toward concentration increase, equilibrium between regular and diffusion flows is possible

(8)

where V is the regular velocity, and D  is the diffusion coefficient. If condition (8) is satisfied, the steady state plasma distribution is possible. The problems of particle heating during turbulence energy dissipation require separate consideration.

Conclusions

Analysis of the experimental observational data and theoretical approaches to the description of the magnetospheric convection shows that the generation of large-scale field-aligned currents and electric fields in the magnetospheric cavity results from the presence of pressure gradients along the equal volume isosurfaces of the magnetic tube. The action of this mechanism manifests itself in the geometry of the magnetospheric cavity at high latitudes and can be revealed by analyzing the profiles of isolines of equal volume and current lines in the equatorial plane. For the Earth's magnetosphere, the former have a larger curvature than the latter, which corresponds to the generation of currents entering the magnetosphere on the dawnside and flowing from it on the duskside and to the appearance of the large-scale electric field directed from the dawn toward dusk.

The presence of this field sustains the existence of the plasma sheet in the magnetospheric tail. The effect of the IMF on the field magnitude in this mechanism is due to changes in the currents on the magnetopause and inside the magnetosphere due to variations in IMF. The morning-evening field gives rise to an azimuthal pressure gradient in the region of dipole field lines with the maximum near the midnight, which provides the appearance of the currents of region II and of effective screening of the inner magnetosphere. The solution of the linear stationary problem of convection does not describe the location of maxima of currents of region II and the azimuthal electric field distribution at low latitudes.

The solution of the nonstationary linear problem allows us to determine the positions of the currents of region II being formed and localization of the convection electric fields and to estimate the characteristic times of growth of the morning-evening field and Alfv\'en screening.

In addition to large-scale fields, medium- and small-scale fields can be generated, and the spectrum of low-frequency turbulence leads to diffusion and variability in the number of particles in the magnetic field tube due to nonequipotential field lines. Taking into account the processes discussed leads to reconsideration of a number of commonly accepted concepts in the physics of magnetospheric plasma and helps in better explaining the experimental observational data.

Appendix A: Equation for the Current Continuity in the Ionosphere

The equation for continuity of the current integrated over the dynamo layer thickness is given by

(A1)

where I is the magnetic declination,  j_||  is the field-aligned current density, and   is the two-dimensional gradient (  j_|| > 0  if the current flows from the ionosphere). If we consider the transverse scale d , for which |dFⁱ| |Eⁱ|d |dF_||ⁱ| |E_||| h , where h  is the dynamo layer thickness, dF  and dF_||  are the transverse and longitudinal variations of the potential, and  E_||  is the field-aligned magnetic field, (A1) can be written as

(A2)

where

In the spherical coordinate system ( q  is the colatitude and j  is the azimuthal angle)

(A3)

In the high-latitude approximation, q q, I = 1 

(A4)

If S_p, S_H =   const

(A5)

At the equatorial boundary, in the case of symmetry of hemispheres, the boundary condition is J_q = 0  (the condition of zero current).

Appendix B: The Drift Kinetic Description

In solving the problem of magnetosphere-ionosphere coupling, it is convenient to use Eulerian potentials ( a, , b )

(B1)

In the dipole magnetic field a = - B₀ R_E²sin² q /r , b = R_E j , where B₀ = -0.308  G is the field at the equator, and r  is the geocentric distance.

In case of conservation of the first and second adiabatic invariant, the dependence of the distribution function f(a, b, m, J, t ) on the Hamiltonian H(a, b, m, J, t ) is given by [ Pellat and Laval, 1972]

(B2)

For equipotential magnetic field lines

(B3)

The equation for continuity of cold electrons of the magnetosphere on the assumption of nondivergence of the ion current is

(B4)

which gives on integration over the magnetic field tube

(B5)

where N_e (a, b, t) = ne di/B =N  is the number of electrons in the magnetic field tube with a unit flux and B_i  is the magnetic field in the ionosphere. The factor of 2 is associated with the existence of two conjugated hemispheres. From (B2) and (B5) we obtain

(B6)

If eF^m  is regarded as a small disturbance of the Hamiltonian, the linear solutions for (B2) can be found

By designating the azimuthal drift velocity through w_d (a, m, J) = (1/eR_E)( K/ a) , we obtain

(B7)

From (B7), for  / t=0 , it follows that

(B8)

and for the characteristic times of the process t w^-1_d ,

(B9)

Appendix C: Two-Fluid Hydrodynamic Description of Slowly Moving Plasma

On the assumption that the main contribution to the magnetospheric pressure comes from slowly moving ions, i.e., p_e p_i = p  (p_e 0.1p_i  in the plasma sheet) and the plasma velocity is | v| v_A, v_s where v_A  and v_s  are the Alfv\'en and sound velocities, respectively, the velocities of electrons  v_e and ions  v_i  are determined from (quasi-neutrality is assumed)

(C1)

where F^m  is the potential in the magnetosphere. In what follows we use F^m = F (a, b, t) , i.e., we assume that the magnetic field lines are equipotential. The equation for energy transfer is written as

(C2)

where g = 5/3  is the index of the adiabat, W = di/B  is the magnetic field tube volume, and the term in the right-hand part sums up the sources and "sinks" of the plasma pressure (injection of ions from the ionosphere and ejection into the ionosphere, heat flows, etc.). For simplicity, we take (dp/ dt) = 0 , and take into account the possibility of nonadiabatic processes by introducing g different from 5/3 .

Using the Eulerian potentials, we take W = W(a) , and then

(C3)

where B_i  is the magnetic field at the ionospheric altitudes. In the linear approximation

  p = p₀ (a) + p' (a, b, t), |p'| p₀  

From (C1) and (C2) with the right-hand side equal to zero, it follows that

(C4)

By introducing the hydrodynamic analog of the magnetic drift frequency

we obtain

(C5)

Equations (C3) and (C5) predict the relation of the field-aligned current with the disturbance of the potential. In the stationary case  / t = 0 , by introducing the number of particles in the magnetic field tube N₀ = p₀W/T₀ , we obtain

(C6)

In the nonstationary case, for the characteristic times of the process t (w_d ^MHD)^-1 

(C7)

In the dipole field in the high-latitude approximation at g = 1 , by equating (A4) and (C6) at Fⁱ = F^m = F , we obtain the equation for convection in the form

(C8)

where S^* = (1/2) e N₀  is the "magnetospheric conductivity".

Acknowledgments

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