Geoscience Reference
In-Depth Information
It is instructive and useful to solve (2.28) for the ion and electron velocities
in terms of the driving forces. For simplicity we consider a single ion species
of mass
M
, using the symbol
m
for the electron mass and the symbol
e
for the
elemental charge, which is taken to be positive. For spatially uniform ion and
electron temperatures, we have the two equations
(
)
−
nMv
in
(
)
0
=−
k
B
T
i
∇
n
+
nM
g
+
ne
E
+
V
i
×
B
V
i
−
U
(2.30a)
=−
k
B
T
e
∇
+
−
(
+
V
e
×
)
−
nmv
en
(
V
e
−
)
0
n
nm
g
ne
E
B
U
(2.30b)
where we have used the fact that
n
i
=
n
, the plasma density. The electric field
in this equation is the one that would be measured in an earth-fixed coordinate
system. This is usually the electric field that is measured in ionospheric experi-
ments. It is nevertheless instructive to express these equations in a reference frame
moving with the neutral flow velocity
U
. Transformation between two coordi-
nate systems moving at a relative velocity
U
does not leave the electric field
invariant even if
n
e
=
c
, where
c
is the speed of light. Jackson (1975) discusses
transformation of the electromagnetic fields
E
and
B
between two coordinate
systems and shows that in the moving frame,
|
U
|
1
c
2
1
/
2
E
=
(
U
2
E
+
U
×
B
)
−
/
(2.31a)
B
c
2
1
c
2
1
/
2
B
=
U
2
−
U
×
E
/
−
/
(2.31b)
where the primed variables are those measured in the moving frame and the
unprimed variables are measured in the earth-fixed frame.
It is easy to show that the
U
c
2
term in (2.31b) is small compared to
B
for any reasonable values of
U
and
E
in the earth's atmosphere or ionosphere.
However,
U
×
E
/
×
B
is the same order of magnitude as
E
and must be retained. Thus,
for
|
U
|
c
,
E
=
E
+
U
×
B
(2.32a)
B
=
B
(2.32b)
Another way to interpret these equations is that in a nonrelativistic transfor-
mation the current density is not significantly changed,
J
=
J
, but the charge
ρ
c
=
ρ
c
. The explanation for this “asymmetry” between the electric
and magnetic fields lies in the fact that very small charge densities can produce
significant electric fields in a plasma.
Following the notation of Haerendel (personal communication, 1973), we
may now transform the terms in (2.28) to a reference frame moving with the
density is,
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