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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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