Geoscience Reference
In-Depth Information
in space and laboratory plasmas cannot be studied in the ray approxima-
tion. Full-wave equations must be used in order to comprehend these effects.
We shall concentrate on the analysis of the full-wave equation mainly on
one-dimensional problems, where suciently simple and obvious results can
be obtained by using elementary mathematical methods. In Chapter 5 a 2D
problem will be considered, in the elementary case of straight field-lines.
The
z
-axis of the Cartesian coordinate system
{
x, y, z
}
is chosen along
the external magnetic field
B
0
=
B
0
z
.
At
β
1and
B
0
(
x, y
) = const, the
equations (4.22), (4.23) for an inhomogeneous low-pressure plasma become
L
ξ
=
−
∇
⊥
(
∇
·
ξ
)
,
(4.42a)
⊥
⊥
∂z
c
s
∇
·
)
.
1
ρ
0
c
s
∂
L
s
ξ
=
−
(
ρ
0
ξ
(4.42b)
⊥
Solutions of (4.42a) for an inhomogeneous plasma are coupling Alfven and
FMS-waves, and (4.42b) are ion sound waves. Just as in the homogeneous case,
transversal compression of plasma in FMS-wave results in a small longitudinal
displacement
ξ
which is found as a forced solution to inhomogeneous equation
(4.42b). Since
β
1, the ion sound wave propagation velocity is much less
than Alfven velocity, and from (4.42b) it follows that
ξ
∝
β.
The longitudinal displacement
ξ
in Alfven and FMS-waves vanishes.
1D Case
Let Alfven velocity depends only on
x
-coordinate:
c
A
=
c
A
(
x
)
.
Consider unbounded plasma. Turn to the Fourier presentation both in time
t
and in the
{
y, z
}
coordinates. Fourier harmonics of transverse displacement is
ξ
⊥
(
x
)exp(
−
iωt
+
ik
y
y
+
ik
||
z
)
and longitudinal perturbation of the magnetic field is
b
(
x
)exp(
−
iωt
+
ik
y
y
+
ik
||
z
)
.
Equation (4.42a) becomes
=
B
0
k
A
(
x
)
ξ
x
,
db
||
dx
k
2
−
(4.43)
b
(
x
)
B
0
dξ
x
(
x
)
dx
=
−
ik
y
ξ
y
(
x
)
−
,
(4.44)
k
A
(
x
)
ξ
y
(
x
)=
ik
y
b
(
x
)
B
0
k
2
−
,
(4.45)
where
k
A
(
x
)=
ω
2
/c
2
A
.
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