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that occurs in the bottomside of the F layer, while larger plume separations
(100-600 km) may require gravity wave seeding. We turn now to nonlinear
calculations.
4.3 Nonlinear Theories of Convective Ionospheric Storms
4.3.1 Two-Dimensional Computer Simulations
Considerable effort has gone into the development of computer simulations of
the Rayleigh-Taylor instability. The set of equations, which are solved in the two
dimensions perpendicular to
B
, is a subset of the full governing equations. They
consist of two continuity equations for the electrons and ions,
∂
t
+∇·
n
j
V
j
=
n
j
/∂
0
(4.21a)
the electron velocity equation with
κ
e
1,
E
׈
a
y
V
e
=
(4.21b)
B
the ion velocity equation for intermediate
κ
i
,
eB
2
V
i
=
(
1
/
B
)
[
(
M
/
e
)
g
+
E
]׈
a
y
+
ν
in
M
/
[
(
M
/
e
)
g
+
E
]
(4.21c)
the charge continuity equation,
∇·
J
=
0
=∇·
(
n
i
e
V
i
−
n
e
e
V
e
)
(4.21d)
which is a form of the quasi-neutrality condition and
E
=−∇
φ
(4.21e)
In obtaining these equations we have assumed there is no neutral wind, we
have neglected the inertial terms and the pressure-driven terms in the equations
for conservation of momentum, and we have assumed that the
g
×
B
electron
velocity term is small due to the small electron-ion mass ratio.
The electrostatic potential
φ
is divided into a zero-order term,
φ
0
, and a per-
turbation term,
δφ
. If we require the zero-order ion velocity to be zero, then we
∇
φ
0
=
/
must have
e
. This zero-order electric field is the order of a micro-
volt/meter and is somewhat artificial, but it creates an equilibrium about which
to perturb the system. In reality, a larger zero-order electric field generally exists.
The electron continuity equation evaluated in a frame of reference moving with
the (
E
0
×
M
g
B
2
) velocity is
B
/
)
∇
δφ
׈
a
y
·∇
∂
n
/∂
t
−
(
1
/
B
n
=
0
(4.22)
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