Geoscience Reference
In-Depth Information
Let us simplify the mathematics, using the quasi-neutrality condition
N
=
N
e
≈
N
i
.
The quasi-neutrality condition only holds for
|
∇
N
|
N/r
D
. In this case, it
is possible to replace (14.6) and (14.7) with
∂N
∂t
+
∇
·
J
=0
.
(14.11)
This equation is equivalent to (14.6) and (14.7) under the condition of
∇
·
J
e
=
∇
·
J
i
=
∇
·
J
.
(14.12)
Substitution of (14.9) and (14.10) into (14.12) for the electric field
E
p
,
gives
(
σ
E
0
)
,
∇
·
(
σ
E
p
)=
−
∇
·
(14.13)
where
σ
=
σ
e
+
σ
i
.
The MHD-signal travels in the ionosphere at Alfven velocity
c
A
100 km/s.
It therefore takes the signal much less than the process scale-time to cover
the newly created
1 km inhomogeneity. Hence there is almost no phase delay
across the MHD-signal and the quasi-stationary approximation can be used
(see Section 9.1). In this approximation, neglecting any induced magnetic
fields, it is possible to assume that
E
p
is curl-free and therefore
E
p
=
∼
−
∇
ϕ
p
.
Then (14.13) becomes
∇
·
σ
ϕ
p
=
(
σ
E
0
)
.
∇
−
∇
·
(14.14)
The potential
ϕ
p
obtained from the solution of (14.14) allows us to find
N
e
−
N
i
from
1
4
πe
∇
2
ϕ
p
.
N
e
−
N
i
=
−
Since,
∇
·
J
e
=
∇
·
J
i
,
∇
·
J
in (14.11) can be written as
J
=
1
∇
·
2
∇
·
(
J
e
+
J
i
)
.
Substituting this relation into (14.11), we obtain
N
v
n
+
σ
i
−
E
0
=0
.
∂N
∂t
σ
e
2
Ne
+
∇
·
(14.15)
One can divide into three stages of electron/ion dynamics. At the first stage
when the neutral concentration is large, the collision frequency of electrons
with neutrals is higher than the corresponding cyclotron frequency, the same
is true for ions, albeit for a different collision frequency (
ν
en
ω
ce
,ν
in
ω
ci
). And therefore the Hall conductivity is small. The charged particles move
together with the neutral component and the flux caused by the dynamo-field
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