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expressed through the magnetic latitude (northern hemisphere) via
D
=2,
the equation for magnetic field lines can be written as r
D
LR
e
sin
2
, where R
e
is
the mean Earth radius, L is the McIllwain parameter. In this notation the operator
B
0
r
taken along the field line can thus be rewritten as
0
M
e
4L
4
R
e
sin
7
d
d
:
B
0
rD
(6.16)
Substituting Eq. (
6.16
)for
B
0
r
into Eq. (
6.15
) and rearranging yields (Dungey
1954
,
1963
; Cummings et al.
1969
)
4R
e
2
L
8
!
2
sin
10
0
M
e
1
sin
d
d
d
d
V
'
sin
3
C
V
'
D
0:
(6.17)
We recall that the plasma velocity V
'
in Eq. (
6.17
) must be equal to zero
at two intersection points where the corresponding field line crosses the bottom
of the ionosphere. Substituting r
D
R
e
into Eq. (
1.34
) one can find the angles
corresponding to these intersection points. We are thus left with the equation
sin
2
0
D
cos
2
0
D
L
1
for the sought polar angles
0
and magnetic latitude
0
.
Finally, the proper boundary conditions for Eq. (
6.17
) take the form V
'
.
0
/
D
V
'
.
0
/
D
0, where
0
D
arcsin L
1=2
and
0
is the angle corresponding to
the conjugate intersection point. The atmospheric depth is disregarded here.
Owing to the complexity of Eq. (
6.17
) the general analysis of this differential
equation encounters some difficulty. As would be expected, considering the finite
length of the field line segment bounded these two interception points, the given
boundary problem has periodic solutions. The fundamental toroidal mode of
Eq. (
6.17
) has numerically been studied by Dungey (
1954
,
1963
). For example,
according to this calculation made at the plasma density
D
10
18
kg=m
3
,the
period of the fundamental mode can be approximated by the formula
0:6
sin
8
0
T
; (in second).
(6.18)
Taking the numerical values of the magnetic latitude
0
D
45
ı
, 55
ı
, 65
ı
and 70
ı
;
the typical periods of the fundamental mode are as follows: T
D
10 s, 54 s, 11 min
and 55 min, correspondingly, while the corresponding eigenfrequencies lie much
below the IAR and Schumann resonances.
Thus, Eq. (
6.17
) describes the toroidal field oscillation in the magnetosphere
or the standing shear quasi-Alfvén waves in the dipole approximation of the
geomagnetic field. In this case the plasma velocity has only an azimuthal component
and the “frozen in” magnetic field lines therefore vibrate within the resonance
shell. The toroidal (twisting) oscillations manifest themselves through the azimuthal
magnetic component and through the electric component orthogonal to the magnetic
shell. Such quasi-Alfvén modes are referred to as the class of the FLRs.
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