Geoscience Reference
In-Depth Information
The magnitude of ionospheric structure and its time evolution are related by
the equations of continuity and momentum, both of which contain the plasma
velocity. If we consider an ionospheric structure not subject to any production,
then the continuity equation for the ions may be written as
∂
n
/∂
t
=
(
−
L
)
−∇·
(
nV
i
⊥
)
−∇·
(
nV
i
||
)
Here the ion velocity has been expressed in terms of its components perpendic-
ular (
⊥
) and parallel (||) to the magnetic field. If we further simplify the problem
by considering electric fields produced by the structure itself and neglect any
parallel ion motion produced by the structure, then the ion velocity is given by
(8.15) and the continuity equation can be rewritten in the form
∂
n
/∂
t
=−
L
−∇·
(σ
Pi
/
e
)
E
+∇·
(
D
i
⊥
∇
n
)
At this point it is a common practice to “linearize” the equations by expressing
each of the plasma properties as the sum of a background value and a small
perturbation value due to the existence of the structure. Then all terms containing
products of perturbation values are ignored, since they are much smaller than
the other terms. If we denote all background values with the superscript “o,”
assume a horizontally stratified background ionosphere and ignore losses, then
the continuity equation may be written finally as
=−
σ
e
∇
x
E
o
Pi
D
i
⊥
∇
2
∂
n
∂
t
/
+
x
n
(10.18)
Now consider two extreme situations. First, suppose the local electric field
associated with the plasma structure in the F region is completely shorted (
E
0)
by mapping to a highly conducting E region. Investigation of (10.18) shows that
for
E
=
=
0 the classical diffusion equation results with
D
⊥
=
D
i
⊥
, the ion diffusion
coefficient. This is the maximum possible value of
D
under the circumstances
considered. In the other limit, suppose that the structure cannot drive a Pedersen
current because it maps to an E region that is an insulator. Then we may solve for
the electric field obtained by setting the current in (10.17) to zero and substitute
the result in (10.18). In the F region, where
R
⊥
<<
1, the electric field from (10.17)
becomes
e
D
i
⊥
−
D
e
⊥
E
x
=
∇
x
n
o
iP
o
eP
σ
+
σ
Substituting this result into (10.18) yields
D
i
⊥
−
σ
D
i
⊥
−
D
e
⊥
∂
n
∂
o
iP
2
=
∇
x
n
o
iP
o
eP
σ
+
σ
t
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