Geoscience Reference
In-Depth Information
The electric field components can be found from Eq. (
6.3
)
E
r
D
B
V
'
;E
D
B
r
V
'
:
(6.14)
It is interesting to note that the scalar product of the electric field (
6.14
) with the
vector
B
0
D
.B
r
;B
;0/ equals zero, that is, the vector
E
is perpendicular to the
Earth's dipole magnetic field
B
0
. This implies that the electric field of the toroidal
mode is perpendicular to the Earth magnetic field shells.
Considering the toroidal mode, we are thus left with the set of Eqs. (
6.10
), (
6.13
),
and (
6.14
) for the functions ıB
'
, V
'
, E
r
, and E
. To study the standing waves, in
effect finding the normal modes of the axisymmetric magnetosphere, all perturbed
quantities are considered to vary as exp .
i!t/, where ! is the frequency. Replacing
the time-derivatives @
t
with the factor
i! and eliminating ıB
'
from Eqs. (
6.10
)
and (
6.13
) yields
0
r!
2
sin V
'
C
.
B
0
r
/
r
2
sin
2
.
B
0
r
/
V
'
r sin
D
0:
(6.15)
The only differential operator occurred at this equation is the operator .
B
0
r
/,
which defines in fact the directional derivative. In other words, this equation contains
only derivative along the magnetic field lines. This means that Eq. (
6.15
) describes
oscillations of the azimuthal velocity V
'
and magnetic shells that originate from the
rotation of the field lines about the symmetry axis. Each magnetic shell can vibrate
independently of each other. All the field lines belonging to the same magnetic shell
must vibrate synchronously, that is, the magnetic shell vibrates as a whole. To study
eigen oscillations of the shell, therefore, it is sufficient to consider the oscillations
of one of the magnetic field lines.
Equation (
6.15
) should be supplemented by the proper boundary conditions at
the ends of the magnetic field lines, that is at the points where the field lines
intersect the high conducting ionospheric E layer and the Earth's surface. The skin-
depth in the ionosphere at frequencies f
0:1 Hz exceeds the thickness of the
ionospheric conductive layer. In this notation, the E layer is usually treated in a
“thin” ionosphere approximation, while the Earth can be considered as a perfect
conductor, which reflects the electromagnetic waves totally. More usually we use
the boundary conditions of the impedance type in which the electric and magnetic
field components tangential to the ionosphere are related in a linear fashion.
In order to make our consideration as transparent as possible we, however, choose
a simplified approximation, considering the wave reflection off a perfect conductor
surface. In such a case, the boundary condition at the end of field line is
E
D
0.On
account of Eq. (
6.14
) one can derive the boundary condition V
'
D
0 at the ends of
the magnetic field line. It should be noted that these relations can serve as proper
boundary conditions rather for the sunlit ionosphere because of the high ionospheric
conductivity at daytime.
The field line shape of the dipole magnetic field is described by Eq. (
1.34
). This
equation relates the polar radius, r, to the magnetic latitude . If polar angle is
.
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