Geoscience Reference
In-Depth Information
Fig. 1.16
Thesamefield
components as shown in
Fig.
1.15
but for fast and slow
magnetosonic (FMS and
SMS) waves
z
E
y
ʴ
B
V
ʸ
B
0
k
x
Substituting Eq. (
1.62
)forı
B
into Eq. (
1.63
), taking into account the vector identity
(
1.55
), and rearranging we arrive at
!
2
k
2
V
A
V
D
c
s
.
k
V
/
k
k
2
.
V
V
A
/
V
A
;
(1.64)
where V
A
and
V
A
are given by Eqs. (
1.60
) and (
1.61
). Projections of this vector
equation onto x and y axes can be written as
a
xx
V
x
C
a
xy
V
y
D
0;
(1.65)
a
yx
V
x
C
a
yy
V
y
D
0;
(1.66)
where the matrix coefficients are given by
a
xx;yy
D
!
2
k
2
V
A
c
s
k
x;y
C
k
2
V
Ax;y
;
(1.67)
a
xy
D
a
yx
D
k
2
V
Ax
V
Ay
c
s
k
x
k
y
;
(1.68)
and V
Ax
and V
Ay
denotes the projections of the vector
V
A
onto the coordinate axes
x and y.
The set of Eqs. (
1.65
)-(
1.66
) has a nontrivial solution for V
x
and V
y
only if the
determinant of the set is equal to zero. Hence we come to the following dispersion
relation
n
V
A
C
c
s
˙
h
V
A
C
c
s
2
4c
s
.
O
x
V
A
/
2
io
1=2
!
2
k
2
D
1
2
;
(1.69)
where
O
x
stands for the unit vector parallel to
k
. This equation describes two different
modes. The sign plus in Eq. (
1.69
) determines the mode, which is referred to as the
fast magnetosonic (FMS) wave while the sign minus corresponds to the so-called
slow magnetosonic (SMS) wave. Below we show that the fast wave propagates
almost isotropically, while the slow wave is a strongly anisotropic mode. Both the
magnetosonic modes readily carry a field-aligned magnetic perturbation; that is,
they describe a magnetically compressive mode.
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