Geoscience Reference
In-Depth Information
6.2.3
Cavity Mode
In this section we consider as before k
y
D
0 in order to treat each wave mode sepa-
rately. In a cold homogeneous plasma the phase velocity of the compressional/FMS
wave is independent of the angle between the wave vector
k
and the external
magnetic field
B
0
. In the MHD box model the properties of this mode are defined
by Eq. (
6.32
), which is an ordinary 2D (two dimensional) wave equation for an
inhomogeneous plasma. This means that in contrast to the Alfvén waves which are
guided by the field lines, the compressional waves can propagate in all directions
and fill the whole resonance cavity. In the magnetosphere these waves are therefore
referred to as the class of cavity modes.
We will seek for the solution of Eq. (
6.32
) in terms of the series
E
y
D
X
n
A
n
.x/q
n
.
z
/;
(6.43)
where the orthonormal eigenfunctions q
n
.
z
/ of the problem are given by
Eq. (
6.126
). These eigenfunctions satisfy the boundary conditions (
6.34
)atthe
southern .
z
D
0/ and at the northern ionospheres .
z
D
l
1
/. Substituting Eq. (
6.43
)
for E
y
and Eq. (
6.126
)forq
n
into Eq. (
6.32
) yields
X
ǚ
A
0
m
C
k
A
k
m
A
m
q
m
D
i
0
!J
y
;
(6.44)
mD1
where k
A
.x/
D
!=V
A
.x/, and the prime denotes derivative with respect to x.
The eigenvalues k
n
.!/ are the roots of Eq. (
6.123
). In two extreme cases of the
non-conducting ionosphere
†
P
D
0
and of the perfect conducting ionosphere
†
P
!1
there are only real roots
k
n
D
n=l
1
; where n
D
1;2;3:::;
(6.45)
which are independent of the frequency !. Overall, if the Pedersen conductivities
†
P
are finite and nonzero values, the eigenvalues are complex.
Consider first the problem of free oscillations assuming for the moment that
J
y
D
0. Multiplying both sides of Eq. (
6.44
)byq
n
.
z
/, integrating these equations
over
z
from 0 to l
1
and using the condition (
6.127
) of orthogonality of the
eigenfunctions q
n
.
z
/, we come to
A
0
n
C
2
A
n
D
0;
(6.46)
where
2
.x/
D
k
A
.x/
k
n
.!/:
(6.47)
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