Geoscience Reference
In-Depth Information
The solution of Eq. (
6.133
) depends on the boundary conditions at x
D
0
and x
D
l
2
. Without specifying these boundary conditions, we now will study
the differential equation (
6.133
) to find some features which are common to all
solutions. This equation exhibits strong singularities found in the denominator of
its second term. Assuming for the moment that k
A
.x/
D
!=V
A
.x/ is a monotonic
function of x, then Eq. (
6.133
) may have two regular singular points, say x
D
;
where
k
A
./
D
k
n
;
(6.134)
and x
D
where
k
A
./
D
k
n
C
k
y
:
(6.135)
Following Southwood (
1974
), we suppose that the value of in Eq. (
6.135
)is
real and consider a small neighborhood of the singular point x
D
where the
function k
A
can be expanded in a power series of x
, that is, k
A
k
n
C
k
y
C
k
A
0
.x
/
C
o.x
/. Here the derivative
k
A
0
is taken at x
D
. In the first
approximation Eq. (
6.133
) is thus reduced to
x
C
k
A
0
.x
/b
n
D
0:
b
0
n
b
0
n
(6.136)
Southwood (
1974
) has shown that two independent solutions of Eq. (
6.136
)are
finite at the singular point x
D
. The implication of this singularity is that x
D
corresponds to a turning point, where the solutions change in character. We cannot
come close to exploring this problem in any detail, but we need to note that this point
divides a space of MHD box into two regions. In the first region the solutions are
quasioscillatory in nature, whereas in the next one the solutions are monotonically
increasing or decreasing functions of distance. In other words, the point x
D
corresponds to a turning point where solutions change from being oscillatory in
nature to characteristically growing or decaying with coordinate x.
More importantly, the solution can be infinite at the next singular point, x
D
,
which corresponds to the FLR conditions. Indeed, substituting k
n
D
k
A
into
Eqs. (
6.123
) and (
6.124
) we come to Eq. (
6.37
), which describes the resonance
frequencies of the Alfvén oscillations. The implication here is that if k
A
./
D
k
n
then the field line at x
D
will resonate with the shear Alfvén wave since the wave
frequency ! equals to one of the Alfvén resonance frequencies.
If the energy dissipation in the conjugate ionospheres is neglected, the eigen-
values are given by k
n
D
n=l
1
, that is, they are real whence it follows that the
resonant point is real as well. In the dissipative case the parameter k
n
is a complex
value so that the roots of Eq. (
6.134
) are in the complex plane x. Decomposing the
roots x
n
into its real and imaginary parts, we obtain x
n
D
n
C
iı
n
. In what follows
we do not specify the value and sign of ı
n
.
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