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 , 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 . 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  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  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 .
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 . 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
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 = -nev, 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,
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)
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, jb = 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 . Second, the nonlinear jb 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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