Published by the American Geophysical Union

Vol. 1, No. 1, April 1998

*E. E. Antonova and B. A. Tverskoy*

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

- Abstract
- Introduction
- Distribution of the Hot Plasma Pressure in the Magnetosphere
- The Role of Plasma Gradients in Formation of Field-Aligned Currents
- Large-Scale Magnetosphere-Ionosphere Coupling at a Low Plasma Pressure
- Magnetospheric Turbulence
- The Problem of Convection Crisis
- Conclusions
- Appendix A: Equation for the Current Continuity in the Ionosphere
- Appendix B: The Drift Kinetic Description
- Appendix C: Two-Fluid Hydrodynamic Description of Slowly Moving Plasma
- Acknowledgments
- References

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.

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*_{x}^{2}/8*p* = ! 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 * 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 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* 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\link* 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**F*^{i} 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."

It is possible to perform the analytical consideration of the
problem of plasma stability in the cavity for *b* = 8*p**p*/*B*^{2} 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 *p**m*_{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*/ *j*^{2} .

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\link* 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* *L*^{4} ). 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 ( *pL*^{7})' > 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*) = *p*_{0}(*L*/*L*_{0})^{-7} for *L* > *L*_{0} and *p*(*L*) = *p*_{0}(*L*/*L*_{0})^{n} for *L* *L*_{0} ,
where *L*_{0} = ^{2}*q*_{0} is the location of the radial
pressure profile maximum. For small injections, the conditions of
the type of |*p*'| *p*_{0} 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 (-*im**j*) , the eigen functions are in the
form *U*_{m} = *A*(*q* /*q*_{0})^{m} for *q* *q*_{0} and *U*_{n}^{m}= *B* *T*_{m} [*q*_{0} *l*_{n}^{m}*q*_{0}/*q* )^{n+5}/ (*n* +5)] for *q* *q*_{0} , 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* = *q*_{0} . For *m* = 1 , *l*_{0} =4(*n*+5)/*q*_{0}^{2} and *l*_{n} =[(*n*+5)^{2} /*q*_{0}^{2}] *x*_{n} ,
where *x*_{n} are close to the roots of *T*_{1}(*x*) . The behavior of functions *U*^{1}_{0} and *U*^{1}_{1} [* 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 *U*^{1}_{0} *j* , are
less than the characteristic time of decay or growth of a
four-vortex harmonic *U*^{1}_{1} *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., *p*_{0}/ *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].

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 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.

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.

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 |*d**F*_{}^{i}| |*E*_{}^{i}|*d*_{} |*d**F*_{||}^{i}| |*E*_{||}| *h* ,
where *h* is the
dynamo layer thickness, *d**F*_{} and *d**F*_{||} 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).

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

(B1) |

In the dipole magnetic field *a* = - *B*_{0} *R*_{E}^{2}sin^{2} *q* /*r* , *b* = *R*_{E} *j* , where *B*_{0} = -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 *e**F*^{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) |

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} 0.1*p*_{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 (*d**p*/ *d**t*) = 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

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 *N*_{0} = *p*_{0}*W*/*T*_{0} , 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 *F*^{i} = *F*^{m} = *F* , we
obtain the equation for convection in the form

(C8) |

where *S*^{*} = (1/2) *e* *N*_{0} is the "magnetospheric conductivity".

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