Geoscience Reference
In-Depth Information
For an electrostatic field
E
=−∇
φ
. Substituting this expression for
E
into (2.45),
taking the divergence, setting
∇·
J
=
0, and taking
B
to be in the
z
direction
yields
2
x
2
2
2
y
2
−
(σ
P
) ∂
φ/∂
−
(σ
H
) ∂
φ/∂
x
∂
y
−
(σ
P
) ∂
φ/∂
2
2
z
2
+
(σ
H
) ∂
φ/∂
y
∂
x
−
(σ
0
) ∂
φ/∂
=
0
The terms containing
σ
H
cancel, leaving
2
x
2
2
y
2
2
z
2
∂
φ/∂
+
∂
φ/∂
+
(σ
0
/σ
P
) ∂
φ/∂
=
0
(2.46)
Making the change of variables
dz
=
(σ
P
/σ
0
)
1
/
2
dz
dx
=
dx
dy
=
dy
converts (2.46) to
∇
2
φ
=
0
(2.47)
which is Laplace's equation in the “reduced” coordinate system. That is, the sub-
stitution has transformed the real medium into an equivalent isotropic medium
with a greatly reduced depth parallel to the magnetic field (the
z
direction in
the calculation). The ratio (
2
is plotted in Fig. 2.7 for a typical iono-
spheric profile. Above 130 km the ratio exceeds 100, reaching 1000 at 300 km.
At high altitudes,
1
/
σ
0
/σ
P
)
σ
0
/σ
P
continues
to increase as the ion-neutral collision frequency and the plasma density, which
determine
σ
0
becomes independent of density. The ratio
σ
P
, continue to decrease. One of the basic approximations of magne-
tohydrodynamics (MHD) is that if the conductivity parallel to the magnetic field
10
3
10
2
10
100
150
200
250
300
Altitude (km)
1
/
2
plotted as a function of height for a typical
Figure 2.7
The mapping ratio (
σ
0
/σ
P
)
mid- to high-latitude ionosphere.
Search WWH ::
Custom Search