Geoscience Reference
In-Depth Information
In a similar fashion we may obtain the Fourier transform of the parallel electric
current density produced by the shear Alfvén wave
ik
2
?
@
z
dž
0
!
j
z
A
D
:
(5.86)
As we have noted above, the field representation through the vector and scalar
potentials satisfies the Faraday law given by Eq. (
1.2
). It is useful to demonstrate,
additionally, that Eqs. (
5.84
) and (
5.85
) satisfy a Fourier transform of the Faraday
equation given by Eq. (
5.6
). In other words, we now show that substituting of
Eqs. (
5.7
) and (
5.8
)for
b
and
e
into Eq. (
5.6
) gives an identity. To verify this
statement one should take into account that
k
?
.
O
z
k
?
/
D
k
2
?
O
z
;
(5.87)
and
O
z
.
k
?
O
z
/
D
k
?
:
(5.88)
In this notation the first term on the right-hand side of Eq. (
5.6
) is reduced to
i .
k
?
e
/
D
i!k
2
‰
O
z
C
i .i!A
@
z
dž/.
k
?
O
z
/:
(5.89)
?
The second term of Eq. (
5.6
) can be converted to
O
z
@
z
e
D
i@
z
dž.
k
?
O
z
/
!@
z
‰
k
?
:
(5.90)
Combining Eqs. (
5.89
) and (
5.90
) and rearranging we come to the following
equation
i .
k
?
e
/
CO
z
@
z
e
D
!A.
k
?
O
z
/
!@
z
‰
k
?
C
i!k
2
?
‰
O
z
D
i!
b
; (5.91)
that coincides with Eq. (
5.6
), which is required to be proved.
Cylindrical Coordinates
In the course of the main text, some of the phenomena are considered in the cylin-
drical coordinates r;', and
z
. On account of the representation of the perpendicular
divergence operator in the cylindrical coordinates, the calibration equation (
5.75
)
reduces to
1
r
@
r
.rA
r
/
C
1
r
@
'
A
'
D
0:
(5.92)
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