*J. Atmos. Terr. Phys., 57*, 537-556, 1995

(Received in final form 19 May 1994; accepted 27 June 1994)

Copyright © 1995, Elsevier Science Ltd

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The lower hybrid drift instability is generated by the relative drift between electrons and ions caused by a density gradient. The magnitude of the particle drift is determined by the gradient perpendicular to the ambient magnetic field; a parallel gradient does not cause a drift. The drift is perpendicular to both the magnetic field and the gradient, and the waves propagate in the direction of the drift. For a particular species the density gradient drift is given by

(1)

where *v*
is the thermal velocity of the species,
*L*
is the scale length of the density
gradient and
is the species gyro-frequency. For a magnetic field gradient the
drift is given by

(2)

where
*L*
is the scale length for the change in magnetic field magnitude.

If the plasma is in pressure balance then it can be shown that

(3)

where the subscript e and i denote electrons and ions respectively. If the ion and
electron temperatures are equal then
*L*
= -*L*. Thus, as
increases, the
magnetic field gradient drift becomes comparable to the pressure gradient drift,
but is in the opposite direction, and the instability is quenched for
> 1. Low
is therefore a necessary condition for both the lower hybrid drift instability and
whistler-mode propagation in the nightside ionosphere of Venus.

However, unlike the whistler-mode, low
is not the only requirement for
the lower hybrid instability. As noted above, the lower hybrid resonance
frequency is only a few Hz, and the electron collision frequency can be large
enough to damp the instability, especially for high densities and weak magnetic
fields. The electron collision frequency
*v*
= 2.91
10*nT*
s,
where is the
Coulomb logarithm
15, the density is expressed in
cm
and the temperature is
in eV [*Huba and Grebowsky*, 1993]. For a density of
10
cm
and a temperature
of 0.1 eV, *v* = 15
s,
while the lower hybrid resonance frequency
()
30 rads
when *B* = 30 nT.
Thus, the collision frequency can be comparable with the wave
frequency.

Another condition that applies to the 100 Hz waves, but not to the density
fluctuations, is the requirement that the wavelength be
100 m, so that the wave
can be Doppler-shifted to 100 Hz through spacecraft motion. *Huba and
Grebowsky* [1993] found maximum growth occurred for
*k*
2, although the
actual value depended on the choice of the plasma parameters, and the gradient
scale length. As noted above, this implies an electron Larmor radius
35 m,
which for
*T*
= 0.1 eV requires a magnetic field strength of 30 nT. If the field
strength is smaller than this, then the
*k*
required for Doppler-shift to 100 Hz
becomes too large, and the waves tend to be damped.

Thus, the lower hybrid drift instability requires low
,
low collision
frequencies, and small electron Larmor radii to generate short wavelength waves
that can be Doppler-shifted to 100 Hz. In
Figure 11
we show where the burst
intervals used in determining the burst rates discussed above occur as a function
of electron density and magnetic field strength. In order to explore whether or not
these bursts correspond to lower hybrid drift waves, we have plotted several
reference curves. Above, we discussed the various parameters relevant to the
lower hybrid instability assuming an electron temperature of 0.1 eV. However, the
temperature is not constant, and we find that
*T* =
0.188 (*n*/2.45
10)
for the
burst intervals, using a least squares regression, where
*T*
is in eV, and *n* is in
cm.
The correlation coefficient is 0.686, with 1234 points. This regression line
allows us to specify the temperature for a given density, and so determine the
following reference curves as a function of density and magnetic field strength:
= 1,
*v*/
= 0.25, and
*k*
= 3. As an approximate rule of thumb, we expect the
lower hybrid drift instability to generate Doppler-shifted 100 Hz waves in regions
for smaller values of these parameters.

Fig. 11. Scatter plot of burst occurrence as a function of electron density and magnetic field strength. Various limiting curves are also shown. The lower hybrid drift instability is most likely to occur for low , lowv/, and lowk. The whistler-mode requires low only. At any particular density, lower values of ,v/, andklie above the limiting curves shown.

Figure 11
indicates that there are large regions of the
*B*-*n*
parameter space
where bursts occur, but the approximate conditions for lower hybrid instability are
not satisfied. However, most of the bursts occur in the region where
< 1, as we
expect for the whistler-mode. A word of caution is in order when interpreting
Figure 11.
The various limiting curves are indicative of the likely region of lower
hybrid drift instability, but we have not performed an instability analysis. *Huba
and Grebowsky* [1993] present instability limits in a similar format. They find that
for sufficiently high drift speeds, corresponding to short gradient scales, the
collision frequency and
constraint can be relaxed for high densities, while the
Larmor radius restriction is less important for low densities. The maximum drift
speed used is twice the ion thermal velocity, i.e.,
*v*
2 kms.
From (1)
*v*/*v*
=
/2*L*,
implying that
*L*
0.25.
The ion Larmor radius is
10 km for *B*
30
nT, and the high drifts invoked by *Huba and Grebowsky* [1993] correspond to
gradients scale lengths
2.5 km. This is an extremely short scale length; with this
scale length, the density changes by two orders of magnitude in
12 km which,
for a spacecraft velocity of 10 kms,
would correspond to a two order of
magnitude change in density in just over 1 s. Another possible restriction
of the applicability of the lower hybrid drift instability at Venus is the extremely
narrow propagation angle because the plasma composition is mainly
O.
*Huba and Grebowsky*
[1993] note that electron Landau damping will become important for angles ~
0.33° away from perpendicular propagation.

In summary, it is possible that the lower hybrid drift instability can operate
in the Venus nightside ionosphere. However, the gradients required are extremely
steep, and electron Landau damping is likely to be important. In addition, we
noted earlier that the association between Langmuir probe anomalies and 100 Hz
bursts is low, about 20%. Thus, the lower hybrid drift instability may explain the
anomalies reported by *Grebowsky* [1991], but it only explains a small fraction of
the 100 Hz bursts observed at Venus. The 100 Hz waves are more likely to be
whistler-mode waves.

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