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B
E
0
an
d
J
5
p
E
0
J
5
nM
g
3
B
B
B
2
g
E
3
B
n
1
E
2
2
2
1
1
1
2
2
2
=
n
0
1
1
n
2
5 0
E
E
3
B
Figure 4.9a
Schematic diagram of the plasma analog of the Rayleigh-Taylor instability
in the equatorial geometry.
The density is equal to
n
1
above the interface and zero below. The gravitational
force is downward, antiparallel to the density gradient, and the magnetic field is
horizontal, into the paper. An initial small sinusoidal perturbation is also illus-
trated, and we assume that the plasma is nearly collisionless—that is, that
κ
i
and
κ
e
are large. From (2.36c) we can determine the electrical current by considering
the ion and electron velocities due to the pressure gradients and gravity. First, we
note that the pressure-driven current does not create any perturbation electric
fields, since the current is everywhere perpendicular to the density gradient. The
pressure-driven current thus flows parallel to the modulated density pattern and
has no divergence.
Turning to the gravitational term in (2.36c), the species velocity is proportional
to its mass, so the ion term dominates. A net current flows in the
x
direction with
magnitude
J
x
=
nMg
/
B
Since the current is in the
g
B
direction, which is strictly horizontal,
J
x
will
be large when
n
is large and small when
n
is small. There is thus a divergence,
and charge will pile up on the edges of the small initial perturbation. As a result,
perturbation electric fields (
×
δ
E
) build up in the directions shown. These fields
in turn cause an upward
B
drift of the plasma in the region of plasma
depletion and a downward drift in the region where the density is high. Lower
(higher) density plasma is therefore advected upward (downward), creating a
larger perturbation, and the system is unstable. An analogous hydrodynamic
phenomenon is illustrated in the series of sketches in Fig. 4.9b. These have been
derived fromphotographs of the hydrodynamic Rayleigh-Taylor instabilitywhen
a light fluid supports a heavier fluid against gravity. Initial small oscillations in
δ
E
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