Geoscience Reference
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wave equations in the inhomogeneous plasma is required. Nevertheless, the
dispersion equations obtained for the homogeneous or the weakly inhomo-
geneous plasma allow us to estimate spatial and temporal scales of plasma
disturbances.
Let a plane wave be propagated in a homogeneous plasma. It is convenient,
as before, to choose the z -axis in the direction of the undisturbed magnetic
field B 0 . Let all wave components be dependent on ω and x as
iωt +
exp(
ik x x ) and independent on y . Hence we may symbolically write
∂t =
∂x = ik x ,
∂y =0 .
iω,
(1.107)
Equation (1.107) shows that all field quantities contain the factor exp(
iωt +
ik x x ) . This factor is assumed to be omitted in the same way as the time
factor exp(
iωt ) is omitted. Then the remaining terms are functions of z and
Maxwell's equations (1.48), (1.49) and (1.54)-(1.56) reduce to
d E y
d z
= ik 0 b x ,
(1.108)
d E x
d z
= ik 0 b y + ik x E z ,
(1.109)
ik x E y = ik 0 b z ,
(1.110)
d b y
d z
=
ik 0 ( ε E x + ε E y ) ,
(1.111)
d b x
d z
=
ik 0 (
ε E x + ε E y )+ ik x b z ,
(1.112)
ik x b y =
ik 0 ε E z ,
(1.113)
where ε = ε xx , ε = ε xy . Elimination of E z and b z from (1.108)-(1.113)
gives
d s
d z
= ik 0 Ts ,
(1.114)
where
k 2
k 0 ε
0
0
0
1
E x
0
0
1
0
E y
b x
b y
T =
,
s =
.
(1.115)
k 2
k 0
ε
ε
0
0
ε
ε
0
0
We put here k x = k 2
because of the axial symmetry with respect to the
magnetic field.
In a homogeneous medium the coecients of (1.115) are independent
of the coordinate z and (1.114) can be solved by finding four eigenvalues
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