Geoscience Reference
In-Depth Information
terms of a scalar potential
Ψ
(
x, y, ζ
):
ξ
=
∇
×
Ψ
z
.
⊥
Due to the plasma being frozen into the magnetic field
B
0
, transverse dis-
placements result in the appearance of the transverse magnetic component
b
,
given by
b
B
0
d
dζ
ξ
+
(
x, y, ζ
)
.
=
(4.30)
The electric field
E
in the Alfven wave is
c
A
c
E
=
−
[
z
×
b
⊥
]
.
(4.31)
As an example, consider two kinds of Alfven waves. The first is a linearly
polarized Alfven wave of frequency
ω
with displacements, say, in plane (
x, z
),
the potential
Ψ
is independent on
x
and
ξ
x
=
∂Ψ
(
y, ς
)
∂y
,
y
=0
.
Non-zero components of displacement, velocity, magnetic and electric fields
and field-aligned current are, correspondingly,
ξ
x
=
ξ
0
(
y
)cos
ω
ωt
,
x
=
ωξ
0
(
y
)sin
ω
ωt
,
c
A
z
−
c
A
z
−
c
A
ξ
0
(
y
)sin
ω
ωt
,
ωξ
0
(
y
)sin
ω
ωt
,
b
x
B
0
ω
c
E
y
B
0
−
c
A
z
−
−
c
A
z
−
=
=
c
A
ξ
0
(
y
)sin
ω
ωt
,
j
B
0
4
π
c
ω
=
−
c
A
z
−
(4.32)
where
ξ
0
is a given distribution of displacements
ξ
along
y
. The field-lines
disturbed by a linearly polarized Alfven wave are shown in Fig. 4.1a.
Let
be the cylindrical coordinate system with
z
axis directed
along an external magnetic field
B
0
. For the torsional wave, the potential
Ψ
is axially symmetric and it depends only on the radial distance
. Then the
displacement is pure azimuthal
ξ
ϕ
(
)=
{
, ϕ, z
}
∂Ψ/∂
and the harmonics are
ξ
ϕ
(
)=
ξ
0
(
)cos
ω
−
ωt
.
c
A
z
−
The disturbance of the magnetic field is also purely azimuthal
ξ
0
(
)sin
ω
c
A
ωt
.
b
ϕ
B
0
ω
c
A
−
z
−
=
The field-lines lie on the coaxial cylindrical surfaces (see Fig. 4.1b).
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