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In-Depth Information
.
r
?
@
z
dž/
O
z
i!
ı
B
A
D
;
E
A
Dr
?
dž:
(5.81)
As can be seen from Eq. (
5.81
), the magnetic and electric fields of the shear Alfvén
mode are both perpendicular to the external magnetic field
B
0
. This conclusion
is consistent with the analysis made in Chap.
3
and is illustrated in Fig.
1.15
.
Nevertheless, the total field-aligned Alfvén current , j
z
A
, including the conduction
and displacement currents, is nonzero. Substituting Eq. (
5.81
)forı
B
A
into Eq. (
1.1
),
yields
i
r
@
z
dž
0
!
2
?
j
z
A
D
:
(5.82)
The FMS/compressional mode can be expressed by the potential ‰ as follows:
2
?
ı
B
F
Dr
?
@
z
‰
O
z
r
‰;
E
F
D
i!
r
?
.
O
z
‰/:
(5.83)
It follows from Eq. (
5.83
) that the electrical field of the compressional mode is
perpendicular to the external magnetic field as shown in Fig.
1.16
, while the parallel
current density j
z
C
D
0.
Fourier Transform over Space
As before, we assume that a local coordinate system has the
z
axis positive
parallel to the magnetic field
B
0
. The direct and inverse Fourier transforms of the
electromagnetic perturbations over the coordinates x and y perpendicular to
B
0
are
given by Eqs. (
5.3
) and (
5.4
). Applying the same Fourier transform to Eqs. (
5.78
)
and (
5.79
) gives the relationships (
5.7
) and (
5.8
) between the components,
b
and
e
,
of electromagnetic field and potential functions, A,dž, and ‰ in the .!;
k
?
/ space,
where ! is the frequency and
k
?
D
k
x
;k
y
stands for the perpendicular wave vector.
In the magnetosphere and ionosphere the potentials A and dž are related through
Eq. (
5.80
). Combining this equation and Eqs. (
5.7
) and (
5.8
), we come to the two
potential field representation
.
k
?
O
z
/
!
@
z
dž
C
k
2
b
D
i
k
?
@
z
‰
C
‰
O
z
;
(5.84)
?
and
e
D
i
k
?
dž
.
k
?
O
z
/!‰:
(5.85)
Here the potential dž describes the shear Alfvén while the potential ‰ corre-
sponds to the FMS mode.
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