Geoscience Reference
In-Depth Information
15.1 Foundation of the Theory
Basic Equations
In order to describe the propagation of a neutral wave in an inhomogeneous
partially ionized atmosphere and interaction of this wave with the ionized part
is to use the MHD-equations with the gravity force.
Let us introduce a mean velocity,
v
, the total pressure,
p
, and density,
ρ
of the ionospheric plasma
v
=
m
e
N
e
v
e
+
m
i
N
i
v
i
+
m
n
N
n
v
n
m
e
N
e
+
m
n
(
N
i
+
N
n
)
,
p
=
p
e
+
p
i
+
p
n
,ρ
=
m
e
N
e
+
m
n
(
N
i
+
N
n
)
Summing equations of momentum with gravity force for the electron (1.44),
ion (1.45) and neutral (1.46) components, yields
m
e
N
e
d
v
e
d
t
+
m
i
N
i
d
v
i
d
t
+
m
n
N
n
d
v
n
d
t
p
+
ρ
g
+
1
=
−
∇
c
[
j
×
B
]+
e
(
N
i
−
N
e
)
.
Suppose first that
v
e
∼
v
n
and condition of quasi-neutrality
N
i
=
N
e
is
fulfilled, then the equation of the magnetohydrodynamics become
v
i
∼
ρ
d
v
d
t
p
+
ρ
g
+
1
=
−
∇
c
[
j
×
B
]
.
(15.1)
Multiplying (1.44) by
m
e
, (1.45) by
m
i
and (1.46) by
m
n
and supposing that
m
i
≈
m
n
and summing them, we obtain the equation of the mass conservation
∂ρ
∂t
+
∇
(
·
v
)=0
.
(15.2)
Combining (1.44) and (1.45) supplemented by the gravity force and omit-
ting non-linear terms, we obtain ([14], [15])
m
e
ν
en
e
j
+
1
m
i
N
i
ν
in
(
v
n
−
v
i
)
−
c
[
j
×
B
0
]
−
∇
(
p
e
+
p
i
)+
m
i
N
i
g
=0
,
(15.3)
B
0
]=
N
e
e
2
ω
ce
B
0
[
j
E
+
1
e
m
e
∇
ν
e
j
−
×
m
e
{
c
[
v
i
×
B
0
]
}
+
eN
e
ν
en
(
v
n
−
v
i
)
−
p
e
.
(15.4)
In deriving these equations we have used the inequalities
m
e
m
n
,
|
∂
j
/∂t
|
(
m
n
/m
e
)
1
/
2
ν
in
. The gravitation force can not produce essen-
tial currents [14]. Besides,
ν
e
|
j
|
and
ν
en
∼
∇
p
e
and
∇
p
e
can be neglected for the large-scale
perturbations.
It follows from the definition of the mean velocity that it is equal approx-
imately to the velocity of the neutral molecules:
v
≈
v
n
+(
N/N
n
)
v
i
≈
v
n
.
Search WWH ::
Custom Search