Geoscience Reference
In-Depth Information
Plasma oscillations can be excited by sources situated both inside and outside
the box. In order to take into account the internal source of excitation of the normal
modes we now add the driven current,
J
d
, to the conduction current on the right-
hand side of Maxwell equation (
1.5
). The driven current is assumed to be a given
function. Hence, Eq. (
6.5
) for the plasma motion is reduced to
0
@
t
V
D
.
r
ı
B
/
B
0
0
.
J
d
B
0
/:
(6.24)
As is seen from Eqs. (
6.3
) and (
6.24
) the electric field
E
and the plasma
acceleration @
t
V
are perpendicular to the unperturbed magnetic field
B
0
. So we seek
for the solution of the problem in the form
E
D
E
x
;E
y
;0
and
V
D
V
x
;V
y
;0
.
All perturbed quantities are considered to vary as exp
ik
y
y
, where k
y
is the
perpendicular wave number. Thus Eq. (
6.24
) is reduced to
0
@
t
V
x
D
@
z
ıB
x
@
x
ıB
z
0
J
y
B
0
;
(6.25)
0
@
t
V
y
D
@
z
ıB
y
ik
y
ıB
z
C
0
J
x
B
0
;
(6.26)
where J
x
and J
y
are the projections of the driven current
J
d
.
In this approach the Faraday's law (
1.2
) reads
@
z
E
y
D
@
t
ıB
x
;
(6.27)
@
z
E
x
D
@
t
ıB
y
;
(6.28)
ik
y
E
x
@
x
E
y
D
@
t
ıB
z
:
(6.29)
The plasma velocity is related to the electric field through Eq. (
6.3
), that is
V
y
D
E
x
=B
0
;V
x
D
E
y
=B
0
:
(6.30)
Substituting V
x
and V
y
into Eqs. (
6.25
) and (
6.26
) we come to the set of equations
for the electromagnetic fields. If k
y
D
0, this set is split into two uncoupled
sets of equation describing the shear Alfvén
E
x
;ıB
y
;V
y
and FMS waves
E
y
;ıB
x
;ıB
z
;V
x
. A close analogy exists with axisymmetric magnetic field, in
which, as we have noted, both the modes are uncoupled in an extreme case of
azimuthal harmonics with m
D
0. It should be noted that the first mode .E
x
/
corresponds to the toroidal field in the axisymmetric magnetosphere, whereas the
second mode
E
y
corresponds to the poloidal field.
Assuming that all the perturbed quantities vary in time as exp .
i!t/ and solving
this set of equations for E
x
and E
y
we come to the following wave equations:
@
z
C
!
2
=V
A
E
x
D
i
0
!J
x
;
(6.31)
@
x
C
@
z
C
!
2
=V
A
E
y
D
i
0
!J
y
;
(6.32)
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