Geoscience Reference
In-Depth Information
5.1.4
Solution of Wave Equations in the Magnetosphere
We recall that in the framework of our model shown in Fig.
5.4
the Alfvén velocity
is the constant value, V
A
D
V
AI
, within the resonance cavity .0<
z
<L/, while
V
A
D
V
AM
in outer space. In the region
z
>Lone should seek for the solution of
Eq. (
5.12
) in the form of upward propagating Alfvén wave, that is
dž
D
C
1
exp
i!
z
V
AM
;
(5.14)
where C
1
is undetermined coefficient. Indeed, on account of the factor exp .
i!t/
we can reduce Eq. (
5.14
) to the following form
dž
D
C
1
exp
i!
z
V
AM
t
;
(5.15)
which describes the Alfvén wave propagating upward at the velocity V
AM
.
The implication of solution (
5.15
) is that the IAR upper boundary is transparent,
in part, for the shear Alfvén waves that causes the leakage of the wave energy from
the resonant cavity into the magnetosphere. This effect along with the energy loss
due to Joule heating predominantly in the conducting E-layer results in the energy
dissipation of the waves trapped in the resonance cavity, so that only several first
IAR resonances can be detectable on the ground.
Actually the magnetospheric Alfvén waves can propagate along the magnetic
field lines in both directions due to the wave reflection from the conjugate
hemispheres (see Fig.
5.3
) thereby exciting the field-line Alfvén resonances in the
Earth's magnetosphere. In the next section we shall consider these resonances in
more detail along with other kind of fundamental oscillations in the magnetosphere.
As before we seek for the solution of Eq. (
5.13
) in the form of upward
propagating FMS wave .
z
>L/, that is
‰
D
C
2
exp
M
z
L
;
(5.16)
where C
2
is undetermined coefficient. For convenience we have introduced the
dimensionless function
M
and dimensionless frequency x
0
via
!L
V
AI
:
2
M
D
k
2
L
2
2
x
0
;
0
D
(5.17)
?
The function
M
has two bifurcation points, !
D˙
k
?
V
AM
, in the complex plane of
!. The sign of
M
in Eq. (
5.17
) should be chosen in such a way to satisfy the wave
radiation condition at the magnetospheric end, whence it follows that the imaginary
part of
M
must be positive when !>
j
k
?
j
V
AM
(for the real k
?
and !). In the
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