J. Geophys. Res., 102, 22,279-22,281, 1997

(Received January 7, 1997; revised May 7, 1997; accepted June 18, 1997.)

Copyright 1997 by the American Geophysical Union.

Paper number 97JA01815

In a recent series of papers, Cole and Hoegy [1996a], Strangeway [1996], and Cole and Hoegy [1996b] have discussed the pros and cons of the whistler-mode identification for the waves observed in the nightside ionosphere by the Pioneer Venus Orbiter, based on the recognition that the waves would undergo significant collisional Joule dissipation at low altitudes. To further their case against the whistler-mode identification, Cole and Hoegy [1996b] (hereinafter referred to as CH) make two new points. The first is that the electron heating causes a significant upward transport of the ionosphere. The second is that the wave electric field amplitudes reported are so large that the wave magnetic field at times exceeds the ambient magnetic field, and consequently, the wave dispersion could not be described adequately by standard quasi-longitudinal wave dispersion [e.g., Stix, 1992]. However, under closer scrutiny, neither of these conclusions stand.

In arguing against the whistler-mode identification and the hypothesis that atmospheric lightning is a source for the waves, CH ignore other evidence. Such evidence includes the wave polarization, which is perpendicular to the ambient magnetic field [Strangeway, 1991]; the dependence on magnetic field orientation, where the burst rate is highest for vertical magnetic fields [Ho et al., 1992]; and the association of the waves with low electron beta, as expected for whistler-mode waves [Strangeway, 1992; Strangeway, 1995a; Strangeway, 1995b]. It is not clear how one can explain these observed properties of the waves if it is assumed that the waves are generated by the spacecraft, as argued by CH.

While CH argue that wave Joule dissipation may cause significant ionospheric outflow, it must be recognized that the heating occurs mainly in the bottomside ionosphere, and the resultant electron pressure gradient force is still downward below the density peak. Only the upward pressure forces are relevant to the issue of ionospheric depletion. Table 1 shows the altitude at which the pressure gradient changes sign and the associated electron temperature at this altitude for the wave dissipation as modeled by Strangeway [1996]. The high-amplitude cases (100 mV/m) were chosen as an upper limit to the range of wave amplitudes expected, while the moderate amplitude cases (10 mV/m) were more representative of typical wave fields. For the moderate amplitude cases the electron temperatures at the altitude where the pressure force changes sign are similar to the observed temperatures. Hartle and Grebowsky [1990] investigated the upward transport of ions for typical observed temperature profiles within ionospheric holes. In their study they required other sources of ionization in addition to the chemical sources, and as such, any mismatch between outward flux and ionization sources is a general question for the nightside ionosphere of Venus, not specific to the case for or against lightning.

Determining if the outflow is excessive requires an estimate of the ionization sources. CH estimate the source strength by equating it with the loss rate due to recombination, although the nature of the source is not specified. Without knowing the details of the ionization sources and sinks (including outflow), this rate should be treated as a rough source-rate estimate at best. To this end, we assume an ambient density of 20,000 cm and a temperature of 0.1 eV as representative of the quiet ionosphere at 140 km (i.e., excluding holes, where Hartle and Grebowsky [1990] indicate that there is outflow). From CH this gives a potential ionization source of ~40 cm s. The pressure gradient scale length is ~10 km (Table 1), while the ion-neutral collision frequency is ~5 s at 140 km. Thus, following the reasoning given by CH, unless the electron temperature near 140 km is >>0.35 eV for density = 20,000 cm or >>7 eV for 1000 cm, there are insufficient grounds for considering the upward flux excessive. The data in Table 1 are sufficiently close to these limits to preclude excessive outflow.Table 1. Altitudes at Which the Pressure Gradient Changes Sign and the Associated Electron Temperature for the Wave Attenuation Cases Modeled in Figure 7 of Strangeway [1996].

Before addressing the second major point of this comment, I wish to again make note that electron heat flux is the dominant means for balancing Joule dissipation, especially at higher altitudes, as discussed previously by Strangeway [1996]. CH (p. 21,787) attempt to refute this by making the assumption that no heat flux can flow out of the topside of the ionosphere, because otherwise "we would be heating the whole ionosphere and eventually the solar wind." This is a most curious assertion, requiring no further comment other than to note that the thermal capacity of the solar wind is essentially infinite.

CH are correct in pointing out that at times
the wave magnetic field can be large, of the
order a few nanoteslas in magnitude, if the
waves are whistler-mode waves. However, it
should be noted that the wave field (**b**) being
comparable to the ambient field
(**B**) is a
necessary, but not sufficient, condition for
nonlinear effects to be important.

With regard to wave dispersion, while the total magnetic field may be changing magnitude and direction, as noted by CH, this does not invalidate the use of the quasi-longitudinal approximation. For wave fields varying as , the quasi-longitudinal (QL) approximation to the collisional Appleton-Hartree dispersion relation is

(1)

where
is the refractive index,
is the
electron plasma frequency,
is the electron
collision frequency,
is the angle between the
wave vector (**k**) and the magnetic field, and
is the electron gyrofrequency equal to
*eB*/*m*,
where *e* is the magnitude of the electron
charge, *B* is the total magnetic field
magnitude, and *m*
is the electron mass.

In (1), cos can be replaced by
.
From Maxwell's equations,
= 0.
Hence =
=
= constant,
and (1) gives the
dispersion for a plane wave even when **b** is
large. The QL dispersion relation only depends
on the component of the magnetic field along
the wave vector. Note that because the electron
plasma frequency is generally much larger than
any of the other characteristic frequencies,
the QL approximation is valid over a large
range of propagation angles. The QL
approximation applies provided cos >>
10 for a 100-Hz wave at 150-km
altitude in a nightside ionospheric hole.

Even though (1) is generally a robust
description of the wave dispersion, there is
still the question of whether or not a large **b**
requires the inclusion of nonlinear effects in
the governing equations used to derive (1). In
other words, in addition to the a posteriori
argument given above, can we present an a
priori proof that large **b** does not necessarily
require the inclusion of nonlinear terms in the
wave dispersion relation?

The ultimate goal in this discussion is the
derivation of a wave dispersion relation for a
plane wave. I will therefore assume that the
spatial gradient is in one direction and
defines the *x* axis of a Cartesian coordinate
system; that is,
.
Furthermore, since
the wave frequency is large in comparison to
the ion gyrofrequency, I will assume that only
electrons carry current and that these
electrons are cold; that is, the Lorentz force
for electrons can be used to determine the
current density **j** =
-*n**e***v**,
where *n* is the
electron density and **v** is the velocity of the
electrons. If we further separate vectors into
components parallel (||) and perpendicular ()
to the *x* axis, then
=
,
,
and ,
where **B**
is the ambient magnetic field, which
makes an angle with respect to
in the *x-z*
plane, **E** is the wave electric field, **j** is the
wave current density, and the wave magnetic
field (**b**) is perpendicular to
since = 0.

Considering the parallel and perpendicular components of the Lorentz force equation, together with Maxwell's equations,

(2)

(3)

(4)

(5)

(6)

where
is the permittivity of free space.
Note that these equations are complete within
our assumptions and have not been linearized.
However, only (3) and (5) contain nonlinear
terms, and it is the parallel current density
in (5) which introduces nonlinearity into the
perpendicular wave fields. This is shown
explicitly on the right-hand side of (5), while
*j*
enters on the left through charge
conservation
()
and advection
().
From charge conservation
it follows that *j*,
and the resultant
nonlinearities associated with *j*,
are
directly related to the amount of
compressibility of the wave (remembering that
we have neglected ion motion), and
whistler-mode waves are usually characterized as being
highly noncompressive.

For a wave varying in time as , . From (2) and (3)

(7)

Thus, provided
>>
max(*B*,*b*)*e*/*m*,
|*j*| <<
|*j*|. In this case, we
can neglect *j* in (5),
regardless of whether
or not *j*
is given by linear or nonlinear
forces, and the transverse () wave fields are
decoupled from the longitudinal (||) fields.
That |*j*| is <<
|*j*| does not depend on any
assumptions regarding the wave refractive
index, but it is this condition that results in
the low compressibility of the whistler mode.

If we take the limit *j* =
0 in (5), then
(4), (5), and (6) form a closed set of linear
equations. This set of equations can be used to
derive (1), assuming wave fields varying as
,
with **k** =
*k*;
that is,
.
As a
consequence, even when the wave magnetic field
is large, the nonlinear forces provide only a
small correction to (5), and the wave
dispersion is well described by the QL
dispersion relation (1). Again, it should be
noted that (1) arises from consideration of
(7), not vice versa.

It can further be shown that under the limit
*j* = 0,
the wave magnetic field is circularly
polarized, and hence so is **j**.
From Maxwell's
equations,
,
where
is the permeability of free space. For a right-hand
circularly polarized wave, *b* =
-*ib* and
hence *j* =
-*ij*.
Furthermore, for a circularly
polarized wave, **j**
and **b** are parallel and in
phase, provided *k* is real. Therefore, for such
a wave,
**j****b** = 0, and there is no nonlinear
force associated with the wave magnetic field,
even though **b** is large. Thus, while (7) shows
that |*j*| <<
|*j*|
and **j** is only weakly
nonlinear, we find that this is also the case
for *j*
itself at higher altitudes in the Venus
ionosphere (>140 km), where the collision
frequencies are low and the imaginary part of
the wave vector is very small. At lower
altitudes, where collision frequencies are higher,
**j**
and **b** are no longer in phase,
and *j* will
include nonlinear terms, but as already noted,
*j*
can be neglected in deriving the wave
dispersion relation.

To conclude, the case against the
whistler-mode hypothesis as presented by CH is weak.
Because the wave heating mainly occurs in the
bottomside, where there is a steep upward
density gradient, the electron pressure force
is still downward below the ionospheric density
peak. At higher altitudes, some outflow may
occur, but this outflow will be similar to that
predicted by Hartle and Grebowsky [1990].
Second, the nonlinear **j****b**
force associated with
the large wave fields only causes weak currents
and then only in the more highly collisional
low-altitude ionosphere. As a consequence, the
wave dispersion is described well by the
quasi-longitudinal approximation, even though the
wave magnetic field is large. The arguments
presented by CH are insufficient reason for
rejecting the whistler-mode hypothesis for the
waves observed in the nightside ionosphere of
Venus, and furthermore, lightning within the
atmosphere of Venus still remains the most
likely source for these waves.

Cole, K. D., and W. R. Hoegy, The 100-Hz
electric fields observed on Pioneer Venus
orbiter and a case against the whistler
hypothesis for them, *J. Geophys. Res., 101,*
21,785-21,793, 1996b.

Hartle, R. E., and J. M. Grebowsky, Upward ion
flow in ionospheric holes on Venus, *J.
Geophys. Res., 95,* 31-37, 1990.

Ho, C.-M., R. J. Strangeway, and C. T. Russell,
Control of VLF burst activity in the
nightside ionosphere of Venus by the magnetic
field orientation, *J. Geophys. Res., 97,*
11,673-11,680, 1992.

Stix, T. H., *Waves in Plasmas*, Am. Inst. of
Phys., College Park, Md., 1992.

Strangeway, R. J., Polarization of the
impulsive signals observed in the nightside
ionosphere of Venus, *J. Geophys. Res, 96,*
22,741-22,752, 1991.

Strangeway, R. J., An assessment of lightning
or in situ instabilities as a source for
whistler mode waves in the night ionosphere
of Venus, *J. Geophys. Res., 97,* 12,203-12,215, 1992.

Strangeway, R. J., An assessment of plasma
instabilities or planetary lightning as a
source for the VLF bursts detected at Venus,
*Adv. Space Res., 15,* 4(89)-4(92), 1995a.

Strangeway, R. J.,
Plasma wave evidence for
lightning on Venus, *J. Atmos. Terr. Phys.,
57*, 537-556, 1995b.

Strangeway, R. J.,
Collisional Joule
dissipation in the ionosphere of Venus: The
importance of electron heat conduction,
*J. Geophys. Res., 101*, 2279-2295, 1996.

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