Geoscience Reference
In-Depth Information
6.1.3
Poloidal Mode
Now we consider another important mode of the magnetospheric oscillations, that
is, the poloidal (breathing) mode, which contains the set of field components ıB
r
,
ıB
, V
r
, V
, and E
'
. In contrast to the toroidal mode, which is mainly due to the
shear Alfvén waves the poloidal mode is caused by the compressional waves, which
propagate isotropically. To treat the basic properties of this mode we first take the
azimuthal component of Eq. (
6.7
)
i
0
!E
'
D
r
1
B
0
Œ@
r
.rıB
/
@
ıB
r
:
(6.19)
As before all the functions are assumed to vary as exp .
i!t/.
Maxwell equation
r
E
D
i!ı
B
in spherical coordinates now is reduced to
@
sin E
'
D
i!ıB
r
r sin ;
(6.20)
r
1
@
r
rE
'
D
i!ıB
:
(6.21)
The set of Eqs. (
6.19
)-(
6.21
) can be solved for E
'
to yield the equation for poloidal
mode
r
@
1
sin
@
sin E
'
!
2
r
V
A
E
'
C
@
r
rE
'
C
1
D
0:
(6.22)
Moreover the azimuthal plasma velocity V
'
D
0 while the components V
r
and V
can be expressed through E
'
as follows:
B
B
0
B
r
B
0
V
r
D
E
'
;V
D
E
'
:
(6.23)
It follows from Eq. (
6.23
) that the scalar product of the poloidal mode velocity
V
p
D
.V
r
;V
;0/ with
B
0
is equal to zero so that the plasma velocity
V
p
is
perpendicular to the magnetic shell in contrast to the quasi-Alfvén oscillations.
The poloidal mode is not guided by the field lines and can cover the whole
magnetosphere or the large part of that. These modes are referred to as the
class of cavity modes, which can propagate via FMS/compressional waves. In a
homogeneous plasma, the phase velocity of the compressional waves is independent
of the angle included between the plasma velocity vector and the geomagnetic field.
Not surprisingly, the cavity oscillations due to compressional waves can fill the
whole magnetosphere and the cavity mode spectrum is dependent on conditions
at the outer boundary of the magnetosphere, that is, at the magnetopause.
Notice that the FMS/poloidal mode results in considerable variations of the field-
aligned magnetic field, whereas the field-aligned electric current is small. On the
contrary, the field-aligned current of the toroidal quasi-Alfvén mode has a finite
value while the longitudinal magnetic field is small.
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