Geoscience Reference
In-Depth Information
The plasma velocity normal to the boundaries of equatorial ionosphere and
magnetopause is assumed to be equal to zero, that is V
x
D
0 at x
D
0 and
x
D
l
2
.FromEq.(
6.30
) it follows that E
y
and A
n
are both zero at x
D
0
and x
D
l
2
. Equation (
6.46
) under this homogeneous boundary condition is a
special case of the so-called Sturm-Liouville problem for determination of the
resonance frequencies of the cavity mode. To estimate the fundamental frequency
of the normal oscillations, consider the case of constant Alfvén speed when the
parameter in Eq. (
6.47
) is independent of x. The solution of Eq. (
6.46
) can be
thus written as A
n
D
C
n
sin x, where C
n
is the undetermined constant. Under
boundary conditions alluded to above the parameter is given by an equation similar
to Eq. (
6.45
), that is
m
D
m=l
2
;
(6.48)
where m is integer. For simplicity, the eigenvalues k
n
.!/ isassumedtobegivenby
Eq. (
6.45
). Combining this equation with Eqs. (
6.47
) and (
6.48
) we obtain the set of
resonance frequencies
!
n;m
D
V
A
n
2
1=2
m
2
l
2
l
1
C
:
(6.49)
In the framework of the MHD box model the curvature of Earth magnetic
field is ignored. To give a numerical estimate of the fundamental eigenfrequency
.n
D
m
D
1/, it is necessary at this point to find a suitable estimate of the
parameters appearing in Eq. (
6.49
). We recall that the x axis approximates the
radial direction. If the outer boundary of the magnetosphere l
2
D
LR
e
corresponds
to the McIllwain parameter L
5, the corresponding length of the field line
l
1
7:7R
e
. Substituting the Earth radius R
e
D
6:4
10
3
km and the Alfvén
speed V
A
D
10
3
km/s into Eq. (
6.49
), we get f
11
D
!
11
=.2/
0:02 Hz. The
cavity resonance period of the fundamental harmonic is about T
11
D
f
1
11
50 s.
It should be noted that we have obtained only the rough estimate of the period and
frequency of the fundamental harmonic.
As would be expected, an FMS-wave in the magnetosphere may increase in
amplitude as the wave frequency is close to the frequencies of the global resonances.
This kind of oscillations can cover a significant part of the magnetosphere. In the
framework of the MHD box model the properties of the cavity mode are similar to
that of the TE mode excited in the inhomogeneous resonator. Since the y direction
corresponds to the azimuthal coordinate of the magnetosphere, the transverse
electric field E
y
corresponds to the azimuthal component E
. In some sense, the
cavity mode is identical in its properties to the poloidal mode in the curved magnetic
field. Some concerns about the mode coupling and the energy dissipation are found
in the next sections.
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