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5
Hydromagnetic Resonators
5.1 Model and Basic Equations
'Box' Model
In the previous chapter we have been mainly concerned with the behavior of
MHD-waves in inhomogeneous plasma for the special case of an unbounded
medium. This led to important simplification in the theory and enabled us to
study some peculiarities of the electromagnetic field and plasma displacements
in the vicinity of a resonance shell. A substantial feature of the real magne-
tosphere, disregarded in Chapter 4, is the finiteness of the magnetospheric
system in the direction of the magnetic field-lines. In this chapter we aban-
don the assumption of medium unboundedness and consider a model of a
bounded MHD-medium. We follow ([17], [19]) who proposed a simple model
of an MHD-box to study the interaction between MHD-waves of different
kinds. We shall consider the waves within the MHD-box in Cartesian coordi-
nates, with significant reduction in algebraic complexity. The model enables
us to reveal new important peculiarities of hydrodynamic perturbations in
magnetospheric plasma. Particularly, it gives the simplest way to understand
the principal features of field-line resonance (FLR).
In the box model, the dipole geomagnetic field is replaced by a uniform
field. Thus, we treat it as a straightened geomagnetic field. The transfer
from the dipole field to the hydromagnetic box is shown on two panels of
Fig. 5.1. Consider then the rectangular coordinate system
with
z
directed along the uniform external magnetic field
B
0
. Axis
x
corresponds
to the radial direction and
y
corresponds to the azimuthal direction. Cold
magnetized plasma (
β
{
x, y, z
}
1) is inside a parallelepiped with ribs
l
x
,l
y
,
and
l
z
. Plasma density and Alfven velocity depend only on the
x
coordinate, i.e.
ρ
0
=
ρ
0
(
x
)and
c
A
=
c
A
(
x
)
.
Coordinate
x
corresponds to the radial coor-
dinate in the equatorial plane of the magnetosphere, the face
x
= 0 is the
equatorial region of the ionosphere, the face
x
=
l
x
is the outer boundary of
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