Geoscience Reference
In-Depth Information
The same approximations can be applied to Eq. (
1.35
) to yield
E
D
B
0
V
:
(6.3)
Here
E
denotes the perturbation of the electric field since a constant electric field
is assumed to be absent. Returning to the equation of motion (
6.1
) and substituting
Eq. (
1.5
)for
j
into this equation, we have
0
d
V
=dt
D
.
r
B
/
B
:
(6.4)
In what follows we restrict our analysis to the case of a dipole approximation of the
geomagnetic field
B
0
, given by Eq. (
1.30
). Substituting
B
D
B
0
C
ı
B
into Eq. (
6.4
),
taking into account that
r
B
0
D
0, and considering the small amplitude waves, so
that d
V
=dt
@
t
V
, the equation of plasma motion is reduced to
0
@
t
V
D
.
r
ı
B
/
B
0
:
(6.5)
Taking the cross product of both sides of Eq. (
6.5
) with
B
0
and substituting
Eq. (
6.3
)for
B
0
V
into this equation yields
0
@
t
E
D
B
0
Œ.
r
ı
B
/
B
0
:
(6.6)
Now we should use Eq. (
1.55
) for triple cross product with
A
1
D
A
3
D
B
0
and
A
2
D
ı
B
. Applying this equation to the right-hand side of Eq. (
6.6
) yields
0
@
t
E
D
B
0
.
r
ı
B
/
B
0
Œ
B
0
.
r
ı
B
/:
(6.7)
The set of Eqs. (
6.2
), (
6.3
), and (
6.7
) constitutes the suitable single-fluid descrip-
tion of dynamics of a magnetized plasma. A general analytical solution of the
plasma dynamics problem is not yet at hand although the numerical solutions, which
can be applied to the actual magnetosphere, have been studied in detail (e.g., see Lee
and Lysak
1989
,
1990
; Alperovich and Fedorov
2007
).
As we have noted above, to the first order the Earth magnetic field is described
through the dipole approximation. If the polar
z
axis is positive parallel to the Earth's
magnetic moment
M
e
and the origin of the coordinate system is in the Earth center,
then the Earth's dipole magnetic field is the axially symmetrical one and has only
the components B
r
and B
given by Eq. (
1.33
) while B
'
D
0.
In this approximation we consider the axially symmetrical problem, in which
all the values are independent on '. As we shall see, in this case the equation
set is split into two independent parts: the first one contains the components of
electromagnetic perturbations ıB
'
, E
r
, E
and azimuthal velocity V
'
, and the
second one contains the components ıB
r
, ıB
, E
'
, V
r
, and V
. The first mode is
referred to as the shear Alfvén wave and the next one is the FMS/compressional
wave. According to geophysical terminology, the standing quasi-Alfvén wave which
contains the azimuthal magnetic field ıB
'
is termed the toroidal mode, while the
standing compressional wave .ıB
r
;ıB
/ is referred to as the poloidal mode.
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