Geoscience Reference
In-Depth Information
Equations (15.3) and (15.4) can be used to eliminate
v
i
and to get relation-
ship between the current density
j
and
E
. Quasi-neutrality demands that
N
e
=
N
i
=
N
. The final result is
⎛
⎞
σ
P
σ
H
0
E
=
1
⎝
⎠
,
j
=
σ
E
where
c
[
v
×
B
0
]
σ
=
−
σ
H
σ
P
0
0
0
σ
0
with
N
0
e
2
m
e
ν
e
,
1+
β
e
β
i
1+
β
e
2
β
i
2
+
β
e
2
,
β
e
1+
β
e
2
β
i
2
+
β
e
2
σ
0
=
P
=
σ
0
H
=
σ
0
For perturbed values of the velocity (
v
) , density (
ρ
) , pressure (
p
)and
electrical current (
j
) taking into account the gravity force and pressure, the
equation
system
of
one-fluid
magnetohydrodynamics
(4.17a)-(4.17d)
becomes
ρ
0
∂
v
∂t
p
+
ρ
g
+
1
−
∇
×
B
0
]
,
=
c
[
j
∂ρ
∂t
=
−
ρ
0
∇
·
v
−
v
∇
ρ
0
,
∂p
∂t
s
=
γ
p
0
c
s
ρ
0
∇
·
=
−
v
∇
p
−
v
,
.
(15.5)
ρ
0
Let
p
0
and
ρ
0
depend on the height
z
as exp (
z/H
)
,
where
H
=
kT/
(
m
n
g
)
is the atmospheric scale height. Then from (15.5) we get
−
∂
j
∂t
×
B
0
∇
vg
+
c
s
∇
·
v
+
g
(
γ
∂
2
v
∂t
2
1
ρ
0
c
−
∇
·
=
1)
v
+
(15.6)
with
c
s
=
γgH
, where
c
s
is the sound velocity. Maxwell's equations together
with (15.6) describe the interaction of neutral gas with plasma.
Plane Acoustic Waves
Let us consider low-frequency waves emitted by a large-scale ground-based
source and propagating over ionized atmosphere with equilibrium pressure
p
0
and density
ρ
0
. For the sake of simplicity, let both parameters are functions
of the vertical coordinate
z
.
Let the horizontal magnetic field
B
0
be along the
y
-axis and the wave
propagates along the
z
-axis of the right-handed Cartesian coordinate system
{
with the
z
-axis pointed upwards from the Earth. All variables depend
only on
z
. It can be written as
∂
x
=0
,∂
y
= 0. Then the projections of
Ampere's and Faraday's laws on
z
-axis give
x, y, z
}
j
z
=0
,
z
=0
.
(15.7)
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