Geoscience Reference
In-Depth Information
6.2.4
The Mode Coupling
In this section we consider the field variations excited by the plasma perturbations
coming from the outer space into the magnetosphere. Such perturbations can be
resulted from the interaction between the solar wind and the Earth's magnetic field
at the magnetopause. The curvature of the magnetospheric boundary is ignored
since the perturbations wavelength is assumed to be much smaller than the size
of the magnetospheric cavity. The MHD box model of the medium is a reasonable
approximation at this point to proceed analytically and to treat the FLR structure.
The inner region of the magnetosphere is assumed to be free of the driv-
ing/external current so that J
y
D
0. To specify the problem, we assume that E
y
is a given function at the magnetospheric boundary x
D
l
2
. As one example, let
E
y
D
E
0
.
z
/ exp
i!t
C
ik
y
y
at the box surface x
D
l
2
while E
y
D
0 at
x
D
0. This function can describe the perturbation coming from outer space into
the magnetosphere or the surface wave generated at the magnetopause. Here ! is
the frequency of this wave. (In general ! is a complex value.) The inhomogeneous
boundary conditions at x
D
l
2
are important in the sense that they play a role of
source of field variations in the magnetosphere.
If k
y
D
0, then the shear Alfvén and compressional waves are described by
independent Eqs. (
6.31
)-(
6.32
). As before we seek for the solution for E
y
in terms
of the series (
6.43
) in eigenfunctions q
n
.
z
/ where the expansion coefficients A
n
.x/
satisfy the differential equation (
6.46
). If the boundary function E
0
.
z
/ can be
expressed as a series of eigenfunctions q
n
.
z
/, that is
E
0
.
z
/
D
X
n
d
n
q
n
.
z
/;
(6.50)
then the expansion coefficients are given by
Z
l
1
d
n
D
E
0
.
z
/q
n
.
z
/d
z
:
(6.51)
0
Whence it follows that the boundary conditions reduce to A
n
.0/
D
0 and
A
n
.l
1
/
D
d
n
.
We will study Eq. (
6.46
) by using a qualitative method since the explicit form
of the function V
A
.x/ is unknown. As is seen from Eq. (
6.33
), the Alfvén speed
depends on both the Earth magnetic field
B
0
and the plasma density. The plasma
density falls off more rapidly with distance x than
B
0
does, and hence the Alfvén
speed generally increases with distance. At the outer boundary of the magnetosphere
.x
D
l
1
/ the Alfvén speed is on one or two order of magnitude greater than that at
the conducting layer of the ionosphere .x
D
0/. Furthermore, if near the boundary
x
D
l
1
the wave frequency is so small that the inequality !=V
A
.x/
j
k
n
.!/
j
takes place, then the parameter in Eq. (
6.46
) is approximately constant. This means
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