J. Geophys. Res., 101, 2279-2295, 1996
(received March 20, 1995; revised August 18, 1995; accepted August 21, 1995.)
Copyright 1996 by the American Geophysical Union.
Paper number 95JA02587.
The ionosphere of an unmagnetized planet, such as Venus, is characterized by relatively high Pedersen conductivity in comparison to the terrestrial ionosphere because of the weak magnetic field. Collisional Joule dissipation of plasma waves might therefore be an important source of heat within the Venus ionosphere. However, any assessment of the importance of collisional Joule dissipation must take into account the cooling provided by electron heat conduction due to temperature gradients. Once heat conduction is included we find that collisional Joule dissipation is significant only in the bottomside ionosphere; waves observed at or near the dayside ionopause, or at higher altitudes (> 150 km) within the nightside ionosphere do not cause significant heating through collisional Joule dissipation. However, lightning-generated whistler mode waves propagate through the highly collisional bottomside ionosphere, and we have performed detailed wave propagation calculations where we self-consistently calculate the heating due to Joule dissipation and the cooling due to heat conduction. The heat conduction always exceeds the collisional cooling from elastic collisions. Because the high collision frequency at low-altitude results in a low thermal conductivity, a steep temperature gradient is required to provide the heat flux. However, this gradient thermally decouples the bottomside ionosphere from higher altitudes. Collisional Joule dissipation of lightning generated whistlers is not likely to have any consequences for the global ionospheric energy budget. Cooling by inelastic collisions, specifically the vibrational excitation of CO, further reduces the bottomside temperature. It is the inelastic cooling rate that determines the atmospheric heating rate, any excess heat again being carried away through heat conduction. We find that for typical wave field amplitudes of 10 mV/m the bottomside is heated to a few eV, while intense fields (100 mV/m) result in bottomside temperatures of a few tens of eV. This high a temperature may cause electronic excitation of the neutrals, which could result in optical or ultraviolet emissions from the ionosphere due to lightning. This possibility requires further investigation but requires the incorporation of additional inelastic cooling processes, such as electronic excitation of the neutral atmosphere.
Recently, Cole and Hoegy  suggested that collisional Joule dissipation of VLF waves is a significant source of heating within the ionosphere of Venus. Depending on the amount of collisional Joule dissipation, this could have important ramifications for the Venus ionosphere. For sufficiently high Joule dissipation the heating due to VLF waves would have to be included in the energy budget of the ionosphere. Even more significantly, if the heating rate is large enough to cause unrealistically high electron temperatures, then some of the earlier interpretations of the plasma waves observed at Venus may have to be revised, especially the identification of VLF bursts in the nightside ionosphere as being due to lightning-generated plasma waves.
There are two basic VLF wave phenomena observed within or near the Venus ionosphere for which collisional Joule dissipation may be important. The first of these is observed at the dayside ionopause, mainly in the 100-Hz channel of the Pioneer Venus Orbiter electric field detector (OEFD). The OEFD measures plasma waves at four frequencies, 100 Hz, 730 Hz, 5.4 kHz, and 30 kHz [Scarf et al., 1980c]. The 100-Hz waves observed at the dayside ionopause were initially identified as whistler mode waves generated at the bow shock, propagating through the magnetosheath to the ionopause [e.g., Scarf et al., 1980b], and it was argued that the waves were transporting energy, providing heat to the dayside ionopause though Landau damping.
More recently, Szego et al.  suggested that the waves were lower hybrid resonance waves generated by the relative drift between O ions of planetary origin and the shocked solar wind. As the waves are generated locally, Landau damping will act to inhibit wave growth, rather than provide a means for energy transport. Strangeway and Crawford  pointed out that Szego et al.  neglected Landau damping, and the damping of lower hybrid waves argues for an alternative wave mode and instability. Huba  suggested that the waves were instead an ion acoustic mode. Thus the mode identification of the waves observed at the dayside ionopause has not been resolved. Nevertheless, we should determine if collisional Joule dissipation is important for these waves, as that may provide an alternative means for transferring wave energy to the plasma, in addition to the dissipation associated with resonant wave-particle interactions.
In the opening paragraph we already alluded to the second wave phenomenon which may be subject to collisional Joule dissipation: the observation of VLF bursts at low altitudes in the nightside ionosphere. These waves have been attributed to lightning in the Venus atmosphere [e.g., Scarf et al., 1980a; Scarf and Russell, 1983; Russell, 1991]. If these waves are indeed due to lightning, then they must propagate through the most dense, and most highly collisional, region of the ionosphere. It is consequently very important that we determine how large the Joule dissipation for these waves may be. In particular, is the heating rate so large that the ionosphere cannot adequately absorb the energy? In which case, we might expect there to be detectable changes in the ionosphere due to lightning. We note that there have been no reports of lightning-related signatures within the ionosphere of Venus, apart from the VLF bursts. Small-scale irregularities were reported by Grebowsky et al. , who interpreted the irregularities as being due to a local instability. This instability was identified as a lower hybrid drift instability [Huba, 1992; Huba and Grebowsky, 1993]. Although initially suggested as an alternative to the lightning interpretation, the lower hybrid drift instability does not explain the majority of the VLF bursts [Strangeway, 1995a, b]. If we find that collisional Joule dissipation is significant for the nightside bursts, then perhaps we should revisit the issue of correlation between wave bursts and density fluctuations, as this may explain the 20% correlation between waves and density fluctuations found by Strangeway [1995b].
Within the nightside ionosphere two types of wave burst have been attributed to lightning. The first of these occurs at 100 Hz [Scarf et al., 1980a], the second is broad-banded in nature [Singh and Russell, 1986], being detected in the higher-frequency channels of the OEFD. Later work [Russell et al., 1988, 1989a], which did not contain telemetry errors apparently included in the analysis of Singh and Russell [Taylor and Cloutier, 1988; Russell and Singh, 1989] showed that the wideband bursts peaked at low altitudes and that the burst rate was maximum near the dusk terminator. It was consequently argued that lightning at Venus was generated mainly in the dusk local time sector [Russell, 1991]. We will not address the Joule dissipation of these wideband signals in this paper, mainly because the source of these high-frequency signals is not understood. They are observed in the propagation stop band between the electron gyrofrequency and the plasma frequency and so cannot be radiation from below the ionosphere [Strangeway, 1991a].
On the other hand, there is evidence that the 100-Hz bursts are propagating though the ionosphere from below. The wave burst rates are largest at low altitudes [Russell et al., 1988; Ho et al., 1991]. The waves are polarized perpendicularly to the ambient magnetic field, provided the data are restricted to signals that are within the whistler mode resonance cone under the assumption of vertical propagation [Sonwalkar et al., 1991, Strangeway, 1991b]. Furthermore, the burst rate decreases much less rapidly for increasing altitude if the data are restricted to within the resonance cone [Ho et al, 1992]. The resonance cone test, which only applies to waves that are propagating from below the ionosphere, is a strong indicator of a subionospheric source such as atmospheric lightning for the 100-Hz waves. Waves that propagate through the ionosphere are likely to be subject to collisional Joule dissipation.
It is the purpose of this paper to determine whether or not collisional Joule dissipation is important for plasma waves observed within the Venus ionosphere. In the next section we discuss the ionospheric heat budget, giving expressions for the different terms within the heat budget and the associated collision frequencies. In section 3 we compare orders of magnitude for the heating and cooling terms. We find that electron heat conduction can easily accommodate the heating due to collisional Joule dissipation, except at very low altitudes within the ionosphere. Thus waves observed near the dayside ionopause of Venus or at higher altitudes (> 150 km) in the nightside ionosphere do not supply any significant heating through collisional Joule dissipation. This finding does not preclude additional dissipation due to resonant wave-particle interactions, but a detailed estimation of this form of dissipation is beyond the scope of the present paper. In section 4 we perform detailed wave propagation calculations, following the methodology of Huba and Rowland  but with the added step of iteratively modifying the electron temperature altitude profile so as to balance the heat budget. In section 4 we artificially set the inelastic cooling rate through vibrational excitation of CO equal to zero. This facilitates the demonstration of how the electron temperature profile changes to accommodate the heating by Joule dissipation through heat conduction to higher altitudes. In section 5 we allow vibrational cooling to occur, and we find that this further reduces the electron heating. In addition, we point out that it is the inelastic collision cooling rate that determines how much heat enters the neutral atmosphere, not the Joule dissipation rate. Any excess heat is carried away through heat conduction. Last, we give some concluding remarks in section 6.
In (1) we chose the convention that a positive value for each of the terms corresponds to electron cooling. The heat flux q is parallel to the ambient magnetic field (unless otherwise stated, the subscripts "" and "||" indicate direction with respect to the ambient magnetic field). We assume that the electrons are isotropic, and q is the collisional heat flux associated with a temperature gradient. The quantity - Q is the heating due to Joule dissipation. Q is the cooling rate due to elastic collisions between electrons and ions and/or neutral particles, indicated by the subscript "m" as the cooling rate is determined by momentum transfer. As we shall see later, this is a relatively inefficient cooling process. Q is the cooling rate due to inelastic collisions. We shall also show later that the dominant inelastic cooling process at low altitudes is the vibrational excitation of CO, hence we denote inelastic cooling by the subscript "v".
The temperature gradient heat flux is given by [Huba, 1994]
In (2), K is the parallel heat conductivity, k is the Boltzmann constant, T is the electron temperature, n is the electron density, m is the electron mass, and v is the total electron collision frequency.
As written, (1) and (2) do not include any heat flux across the ambient magnetic field. At high altitudes, where the electron collision frequencies are low, this is justified since the parallel heat conductivity is much larger than the perpendicular heat conductivity. At low altitudes, where the collision frequencies are much higher, the better assumption is that only vertical gradients are present within the ionosphere. However, when the collision frequency is large with respect to the gyrofrequency, the thermal conductivity is independent of the orientation of the magnetic field, with magnitude given by K in (2). For high collision frequencies with respect to the electron gyrofrequency, we need no longer consider only the parallel temperature gradient, and we can replace by in (1) and (2).
In a cold plasma the collisional Joule dissipation rate due to a wave electric field which is perpendicular to the ambient magnetic field is given by [Cole and Hoegy, 1995]
assuming that the wave frequency () << or v. In (3), E is the perpendicular wave electric field amplitude, e is the electron charge, and is the electron gyrofrequency ( = eB/ m, where B is the ambient magnetic field strength). The Joule dissipation as given by (3) is the same as the Joule dissipation rate using the Pedersen conductivity, except for the factor of two which arises from the averaging over a wave cycle of the sinusoidally varying electric field. The Joule dissipation rate is smaller for increasing magnetic field strength, and is maximum when v = . Thus, as pointed out by Cole and Hoegy , we might expect the Joule dissipation to be larger in the weakly magnetized Venusian ionosphere than in the highly magnetized terrestrial ionosphere.
The collisional Joule dissipation associated with a wave field that is parallel to the ambient magnetic field is similar to (3), except that the electron gyrofrequency is replaced by the wave frequency:
In the introduction we noted that there is still some uncertainty in the wave mode identification for the waves observed near the dayside ionopause. Three wave modes have been considered: whistler mode; lower hybrid mode; and ion acoustic mode. Of these three, the first two are perpendicularly polarized, while the last is polarized along the magnetosheath flow velocity (i.e., independent of the ambient field direction). Given the uncertainties in mode identification we will use (3) to assess the relative importance of collisional Joule dissipation on the dayside.
In assessing the effect of collisional Joule dissipation we
will neglect any kinetic effects such as resonant wave-particle
interactions. Clearly, such processes are important in a warm
plasma, where wave phase speeds are comparable to electron
thermal speeds. For whistler mode waves, kinetic effects are
is the ratio of
thermal to magnetic
2 n k T / B ( is the permeability of free space). We will use the condition = 1 as an indicator of parameter regimes for which kinetic effects may have to be included.
On the nightside, the mode identification is much more clear cut. If the 100-Hz waves observed at low altitudes are due to atmospheric lightning, they must be whistler mode waves. Moreover, the data indicate that the waves are perpendicularly polarized with respect to the ambient magnetic field [Strangeway, 1991b]. Thus (3) is also appropriate for the nightside. With regard to kinetic effects, Strangeway et al. [1993a] and Strangeway [1995a, b] have shown that the low-altitude 100-Hz wave bursts tend to occur in regions of low . At the lowest altitudes we therefore expect collisional processes to be more important for wave dissipation than resonant wave-particle interactions.
The cooling rate due to elastic collisions is given by
In (5) the sum is over both ions and neutrals, T is the species temperature, and m is the species mass.
The cooling rate due to vibrational excitation of CO is derived in the appendix. At high temperatures (> 0.3 eV) the vibrational cooling rate, in W/m , is given by
where N is the neutral CO density, in cm, n is expressed in cm, and T is in eV.
The various terms in (1) depend on the electron-ion and electron-neutral collision frequencies. For electron-ion collisions
where is the Coulomb logarithm (~20), densities are in cm, and temperatures are again in eV. For electron-neutral collisions
where N is the neutral density in cm and is the collision cross section in cm.
The dominant neutrals in the nightside ionosphere of Venus are O and CO. We will assume the cross section for O is 2 10 cm, giving a collision frequency of ~ 8.4 10 N T , similar to the value given by Banks and Kockarts [1973, chap. 9], where N is the atomic oxygen density. For CO we use the results of Morrison and Greene , as discussed in the appendix. At higher temperatures (> 0.3 eV) the cross section has an asymptotic value of ~ 1.55 10 cm.
The electron-ion and electron-neutral collision frequencies are plotted as a function of electron temperature in Figure 1 . The solid lines give the electron-ion collision frequency for electron number densities of 10 and 10 cm. The dashed lines give the electron-neutral collision frequency for neutral densities of 10and 10 cm for O and 10 cm for CO. The low electron and O density curves are representative of the high-altitude ionosphere and ionopause, while the high electron density represents the low-altitude ionosphere. The CO curve is representative of the bottomside of the ionosphere (~ 130 km), while the N = 10 curve is representative of the middle (~ 150 km) altitude ionosphere (see, e.g., Theis et al.  and Hedin et al.  for altitude profiles of the electron and neutral densities respectively). Electron temperatures usually lie in the range 0.1 to 1 eV [Theis et al., 1980], although we have extended the temperature range beyond these limits in the figure. Figure 1 shows that at high altitudes, electron-ion collisions will dominate. At lower altitudes, where the electron temperature is lower, electron-ion collisions will still tend to dominate, because of the steep temperature dependence of the electron-ion collision frequency. Only at the very lowest altitudes, i.e., the bottomside ionosphere, do we expect electron-neutral collisions to be important. A similar result was found by Luhmann et al. , although they used a slightly lower electron-neutral collision frequency.
Figure 1. Electron collision frequencies as a function of electron temperature. The electron-ion collision frequency (solid line) is plotted for electron densities of 10 and 10 cm, while the electron-neutral collision frequency (dashed line) is plotted for neutral oxygen densities of 10 and 10 cm, and neutral CO density of 10 cm Ionospheric electron temperatures typically lie in the range 0.1 to 1 eV [Theis et al., 1980].
Figure 1 and (8) also show that for sufficiently high neutral densities and electron temperatures, the electron-neutral collision frequency can be large, > 10 s. Within the ionosphere, magnetic field strengths of 30 nT are well above the average, but this only corresponds to = 10 rad/s. Thus it is possible that v > , in which case (3) shows that the Joule dissipation is reduced, and increasing the electron temperature will further reduce the Joule dissipation.
The discussion in the previous section indicates that there are two limits which we can apply to the Venus ionosphere. The first is when electron-ion collisions dominate, while the second is when electron-neutral collisions dominate. As noted earlier, the former is more appropriate for high altitudes, while the latter is more appropriate for the bottomside ionosphere.
When electron-ion collisions dominate, we can rewrite the heating and cooling rates, expressed in W/m, as:
In (9), L is the scale length for heat conduction, as discussed below, given in kilometers. In (10), E is the wave electric field in volts per meter, B is the ambient magnetic field strength in nanoteslas, and we have assumed that the wave is polarized perpendicularly to the ambient magnetic field. The term in parentheses in (10) ~ 1 when v << , as is usually the case when electron-ion collisions dominate. In (11), m and m are the ion and proton masses respectively, and we have assumed that T>> T. In all these equations T is in eV, densities are in cm, and is again the Coulomb logarithm.
In specifying a scaling law for the heat conduction, from
- ( K T +K T). When electron-ion collisions dominate, K T and is independent of n. Hence the first term in parentheses will always be positive, and we therefore require T < 0 for · q > 0. This condition is not satisfied for a temperature dependence such as T exp(-x/L), but it is for T exp(-x /L) when x ~ 0. In deriving a scaling law we therefore assume that the temperature profile is such that ·q > 0, as required for the electron energy budget, and simply take - (K T )= K T/ L, where L is a heat conduction scale length.
When electron-neutral collisions dominate,
The symbols have the same meaning as in (9) to (11), and have the same units. In (9'), is the electron-neutral collisional cross section, and is the cross section for oxygen = 2 10 cm. When electron-neutral collisions dominate, the thermal conductivity tends to increase with altitude, and a negative temperature gradient will generally ensure that ·q > 0. In (10') the term in parentheses ~1 when v >> . In (11') m is the neutral mass, and we have assumed T >> T.
Turning first to the dayside ionopause, Figure 2 shows the various heating and cooling rates as a function of electron density (Figure 2a) and electron temperature (Figure 2b). We assume that the heat conduction scale (L) is 1000 km, the ambient magnetic field strength (B) is 50 nT, and the wave electric field amplitude (E) is 10 mV/m. Also, as noted above, we assume that electron-ion collisions dominate, i.e., we are using (9) to (11), and the ions are O [e.g., Brace and Kliore, 1991]. We choose a scale length of 1000 km since the ambient magnetic field is draped over the ionopause, and the important scales are horizontal. These scales will be much longer than vertical scales, and 1000 km is a reasonable order of magnitude estimate. Above the dayside ionopause, wave spectral amplitudes can be as large as 10 V/m/ Hz [Strangeway, 1991a]. We convert the spectral amplitude to a wave electric field amplitude by assuming that the spectral bandwidth is of order the wave frequency. Thus, at 100 Hz a spectral amplitude of 10 V/m/Hz corresponds roughly to an electric field amplitude of 10 mV/m.
Figure 2. Electron heating and cooling rate estimates for the dayside ionopause. Since electron-ion collisions dominate, the rates are given by (9) to (11). The rates are shown (a) as a function of density for T = 1 eV and (b) as a function of temperature for n = 1000 cm. It is also assumed that L = 1000 km, B = 50 nT, E = 10 mV/m, and the ions are O. As an indication of the relative importance of the heating and cooling rates, we include the energy density (nT). The dot on this line marks = 1, where we expect kinetic effects to be important.
In Figure 2a we assume that T = 1 eV, which is large for the ionosphere but is appropriate for the transitional region where the waves are observed. It is clear that for fixed temperature the Joule dissipation is always greater than the elastic collisional cooling. This is because both have the same dependence on the electron-ion collision frequency, and hence on electron density. More importantly, the electron heat conduction exceeds the Joule dissipation by a sufficiently large factor that we can increase the conduction scale length to 3000 km and still match the Joule dissipation, even for the highest densities shown. For densities of 1000 cm the conduction scale could be as large as 30,000 km, about 5 Venus radii.
We also show the energy density of the electrons, n T, to indicate how important the different heating and cooling terms are, since n T divided by the cooling or heating rate gives the approximate cooling/heating time constant. It is clear that except at the higher densities the Joule dissipation is relatively weak, with a time constant of more than 100 s. On the n T curve we mark where = 1, indicated by the dot. Thus at the higher densities we might expect kinetic effects to be more important, possibly enhancing the dissipation rate. However, heat conduction still plays a significant role in the electron heat budget.
Figure 2b shows how the rates depend on electron temperature. For Figure 2b we assume that n = 1000 cm. For very low temperatures the Joule dissipation can exceed the heat conduction. However, (9) shows that the heat conduction has a strong dependence on temperature, while (10) shows that the Joule dissipation decreases with increasing temperature. Thus even though the Joule dissipation may initially exceed the heat conduction, a small increase in temperature is sufficient to match Joule dissipation by heat conduction. Figures 2a and 2b demonstrate that Joule dissipation through electron-ion collisions is not an important source of heating for the dayside ionopause.
At higher altitudes within the nightside ionosphere electron-ion collisions will usually dominate, except for the lower plasma densities. The heating and cooling rates for this case are shown in Figure 3a. For Figure 3a we assume that L = 10 km, B = 30 nT, T = 0.1 eV, and E = 1 mV/m. These parameters have been chosen to correspond to an ionospheric hole [Brace and Kliore, 1991]. We have assumed that the ambient ions are O, which is usually the case at altitudes 150 km [Grebowsky et al., 1993], while we also assume that the neutrals are atomic oxygen with a density of 4 10. Since the magnetic field within a hole is generally radial, it is appropriate to consider vertical scales, and we choose a scale of 10 km. Also, since the waves tend to be somewhat weaker in amplitude, we assume a wave electric field of 1 mV/m. This wave amplitude corresponds to a Poynting flux of ~ 3 10 W/m for a refractive index ~ 1000, Russell et al. [1989b] reported a median Poynting flux of ~ 10 W/m, assuming a 30-Hz bandwidth.
Figure 3. Electron heating and cooling rate estimates for the nightside ionosphere. (a) The rates are shown for an ionospheric hole, altitude ~ 150 km. We assume that L = 10 km, B = 30 nT, T = 0.1 eV, E = 1 mV/m, the ions are O, and the neutrals are O with a density of 4 10 cm. (b) The rates are shown for the bottomside ionosphere, altitude ~ 130 km, where electron-neutral collisions dominate. We assume that L = 2 km, B = 5 nT, n = 1000 cm, E = 10 mV/m, the neutrals are CO and N = 10 cm. In Figure 3b we have also shown the cooling rate due to vibrational excitation of CO (6).
For the particular choice of wave and plasma parameters in Figure 3a , we again find that electron heat conduction can easily match the Joule dissipation within an ionospheric hole. Even though the heat conduction is of the same order as the Joule dissipation for densities of 10 cm, we do not expect the wave amplitudes to be as high as 1 mV/m for these densities. At these high densities the scaling laws are given by (9) to (11), while at lower electron densities (9') to (11') apply. Whistler mode waves tend not to be observed within the high-density regions at higher altitudes, since they are Landau damped and gyrodamped [Strangeway, 1992; 1995a]. The conduction scale length can be as large as 50 km for densities of 10 cm, and the conduction cooling will still exceed the Joule dissipation. Thus similarly to the dayside ionopause, we do not expect collisional Joule dissipation to be important at moderate altitudes ( 150 km) within the nightside ionosphere.
In Figure 3b we have chosen wave and plasma parameters corresponding to the bottomside ionosphere. We assume that L = 2 km, B = 5 nT, n = 1000 cm, E = 10 mV/m, the neutrals are CO and N = 10 cm. The wave amplitude corresponds to the wave intensities observed on the very low altitude passes during the Pioneer Venus Orbiter entry phase [Strangeway et al., 1993b]. We have chosen the very short scale length of 2 km, since this is of order the attenuation scale observed for the 100 Hz waves [Strangeway et al., 1993b], and is also of order the density scale height for CO [Kasprzak et al., 1993]. However, Figure 3b shows that even for this short a scale the Joule dissipation exceeds the conduction cooling, except for the higher temperatures. Thus we might expect that collisional Joule dissipation is an important heat source for the bottomside ionosphere.
In addition to the elastic cooling rate, we have also included the cooling rate through vibrational excitation of CO in Figure 3b. This cooling rate often exceeds that due to electron heat conduction, and may in fact be the means for balancing the Joule dissipation at the lowest altitudes. However, as discussed in the appendix, this cooling operates whether or not waves are present, and the cooling rate is so large that it may have important implications for the electron heat budget. In the absence of any other heat source, the vibrational cooling must be balanced by the heat conduction into the volume. Taking the heat conduction curve in Figure 3b as a guide, it is clear that the temperature gradient scale must be very short to supply sufficient heating, and as shown in the appendix, large topside temperatures may be required to provide the downward heat flux necessary to balance the vibrational cooling at the bottomside.
Last, in Figure 3b, > 1. However, since the collision frequency is >> and , it is by no means clear that resonant wave-particle effects are important. Electron motion is almost certainly dominated by collisions, and it is unlikely that electrons can remain in resonance with waves.
In concluding from Figure 3b that wave Joule dissipation is important for the bottomside we used a fixed wave amplitude and fixed neutral and plasma density. However, all these parameters are changing on very short vertical scales within the bottomside ionosphere. The Joule dissipation that causes electron heating also reduces the wave energy. If the heating rate is high, we would expect that very little wave energy would propagate into the ionosphere. It is therefore necessary to take into account the variation of the waves and the ambient neutrals and plasma if we are to assess realistically the importance of collisional Joule dissipation as a heat source for the ionosphere.
Recently, Huba and Rowland  presented an analysis of the VLF wave transmission characteristics of the nightside Venus ionosphere. In their analysis, Huba and Rowland performed a full wave calculation of the wave attenuation for the four frequencies sampled by the Pioneer Venus orbiter electric field detector. In this section we will use the methodology of Huba and Rowland  to calculate wave electric field altitude profiles for different ionospheric conditions. However, we will extend the work of Huba and Rowland by calculating the divergence of the Poynting flux (S) of the waves. Since the Joule dissipation is equivalent to minus the divergence of the Poynting flux (i.e., -Q = S), we can use the latter to determine the amount of heating caused by the waves, rather than the approximate form given by (3) . In passing, it should be noted that in our calculations Q evaluated using (3) and S differ by less than a factor of two. Also, although Huba and Rowland considered all four channels of the OEFD, we will only consider 100 Hz. The Joule dissipation is usually largest for the 100-Hz channel than for the higher frequencies.
The method of Huba and Rowland  is to numerically integrate the wave equation
In (12), z is altitude, E = E ± iE is the wave electric field for the cold plasma L (+) or R (-) mode, and k is the wave vector, given by c k = - / (-iv± , where is the electron plasma frequency. In this paper we will only solve for the R mode, as this corresponds to the whistler mode. It should be noted that for v >> the dispersion relation is essentially the same for both modes.
In order to calculate the transmission characteristics of the ionosphere we need to specify the ambient magnetic field, the electron density, the ion composition, and the neutral density and composition. We will allow the electron temperature to be a variable within the calculation, being adjusted self-consistently so as to balance the heat budget equation (1). The magnetic field is assumed to be vertical and constant, with the magnitude depending on the ionospheric conditions we wish to model.
The neutral density and composition are based on the Pioneer Venus entry phase results of Kasprzak et al. . We assume that the two dominant neutral species are O and CO, and we ignore all other neutral species. The O and CO scale height temperatures are 105 K and 109 K respectively [Kasprzak et al., 1993], giving a density scale height of ~ 6 km for O and ~ 2 km for CO. In (5) we assume that the neutral gas temperature is the scale height temperature.
The electron density profile is modeled using a density profile similar to that used by Huba and Rowland , with the modification that we allow the density to equal zero at the bottom of the altitude range under consideration (125 km). Thus the waves are free space modes at the bottom of the ionosphere. The density profile as a function of altitude is given by
where z = (z - z)/(z - z) , n is the density at z = z ~ the altitude of the density peak, z is the minimum altitude, where n(z) = 0, and z is a scale height. Note that since n(z) reaches a maximum at an altitude above z = z, n is not exactly the peak density, and z is not exactly the peak density altitude. Nevertheless, for convenience, we will refer to n as the peak density and z as the peak density altitude. In our calculations we assume z = 125 km, z = 140 km, and z = 20 km. The altitude minimum and peak density altitude are consistent with the occultation measurements from the Pioneer Venus Orbiter [Brace and Kliore, 1991], although the bottomside density profile is not well known. Indeed, the high vibrational cooling rate suggests that the bottom of the ionosphere could be higher than the assumed 125 km altitude. We vary n depending on the ionospheric conditions we wish to model. The ions have the same density profile as the electrons and are assumed to be O . For the purposes of calculating the electron cooling due to collisions we assume that the ion temperature is equal to the neutral gas temperature in (5).
At this stage we will arbitrarily set the vibrational cooling rate (Q) to zero in the heat budget (1). This is done for two reasons. First, as discussed in the appendix, including vibrational cooling requires a source of electron heating. As we shall see later, for an ionosphere which extends down to 125 km altitude, we find that the vibrational cooling always exceeds the heating due to wave dissipation. Thus, including the vibrational cooling in this case results in a temperature profile where the electrons supply heat to the bottomside to offset the cooling, rather than conduct heat away from the region of wave dissipation. Second, by artificially turning off the vibrational cooling, we can more clearly demonstrate how heat conduction acts to balance wave heating.
The first case we analyze models a deep ionospheric hole, with n = 1000 cm and the ambient magnetic field = 30 nT. The whistler mode waves detected at Venus are primarily detected in ionospheric holes [e.g., Strangeway, 1995b]. This is a consequence of both the reduced Landau damping and gyrodamping [Strangeway, 1992, 1995a], and the relative transparency of the bottomside ionosphere [Huba and Rowland, 1993]. The corresponding ionospheric parameters are shown in Figure 4a. As discussed above, the neutral densities are based on the Pioneer Venus entry phase observations, with the electron density given by (13). The electron temperature profile has been modified self consistently so that the heat budget equation (1) is satisfied, as we discuss below. The electron temperature at the bottom of the ionosphere is ~ 8 eV.
Figure 4b shows the associated characteristic frequencies. We have assumed an incident wave with frequency 100 Hz. Electron-ion collisions are relatively infrequent, even at 150 km, because of the low ambient density. At 150 km altitude, however, the electron-neutral collision frequency is also very low, ~ 10 s.
Figures 4a & 4b, Figures 4c & 4d. Wave propagation through the nightside Venus ionosphere for weakly attenuated 100-Hz signals. (a) Ionospheric parameters: The peak density is 1000 cm, and the ambient magnetic field is 30 nT, corresponding to a deep ionospheric hole. The electron temperature profile has been modified so that (1) is satisfied. (b) Characteristic frequencies: The electron collision frequencies, wave frequency, and electron gyrofrequency are shown. (c) Wave parameters: The wave electric field amplitude and Poynting flux are shown. The real and imaginary parts of the refractive index () are also shown. (d) Heat budget: The Joule dissipation rate, given by minus the divergence of the Poynting flux (-S), the divergence of the heat flux (q), and the elastic collision cooling rate (Q) are shown. Although not included in the heat budget, we have also included the vibrational cooling rate (Q) for reference.
Figure 4c shows the associated wave parameters. At the bottom of the ionosphere we assume a net applied wave field of 10 mV/m. This wave field is the sum of both incident and reflected waves. The actual incident wave field is ~ 0.3 V/m. Figure 4c shows that at the top of the model the wave amplitude is 0.5 mV/m, with a Poynting flux (S) of 6 10 W/m, similar to the values cited when discussing Figure 3a. For completeness, we also include the real and imaginary parts of the refractive index (). The imaginary part is largest at 132 km, and at this altitude the gradient in the Poynting flux is steepest.
The various terms that enter the heat budget are shown in Figure 4d. Because the elastic collision cooling ( Q) is so weak, the Joule dissipation, given by - S, is almost completely balanced by the divergence of the heat flux (q) , which cannot be resolved separately in the figure. Thermal balance has been achieved through iterative modification of the electron temperature profile. Initially, the temperature is assumed to be constant at 0.1 eV, and there is no heat flux. At the start of an iteration loop the wave amplitude is calculated as a function of altitude for the given ionospheric parameters. The q required for energy balance is then calculated from (1), with Q replaced by S. q is numerically integrated to specify a new temperature profile, subject to the constraints that T = 0 at 125 km, and T = 0.1 eV at 150 km altitude. The wave propagation and attenuation is then recalculated using the new temperature profile. This iterative procedure is repeated until the residual of the heat budget, summed over all altitudes, is < 10 of the root square sum of the constituent terms within the heat budget, giving the temperature profile shown in Figure 4a.
In Figure 4d we also plot the vibrational cooling rate, as given by (6), although this cooling has not been included in the heat budget at this stage. As noted earlier, Q generally exceeds the Joule heating rate. An additional electron heat source is required to offset this cooling.
Wave propagation through a moderately attenuating ionosphere is shown in Figure 5. For this case the peak density is 5000 cm, and the ambient magnetic field is 20 nT. These conditions correspond to a moderate ionospheric hole. Figure 5a shows that for this case the wave absorption again heats the bottomside ionosphere, resulting with a peak electron temperature ~ 8 eV. Because of the higher ambient density, v is close to v at 150 km (Figure 5b). The wave is more strongly attenuated in Figure 5c than in Figure 4c, with an amplitude of 0.03 mV/m, and a Poynting flux of 6 10 W/m at 150 km.
Figures 5a & 5b, Figure 5c & 5d. Wave propagation through the nightside Venus ionosphere for moderately attenuated 100-Hz signals. Similar in format to Figure 4. The peak density is 5000 cm , and the ambient magnetic field is 20 nT, corresponding to a moderate ionospheric hole.
Solutions for a strongly attenuated 100 Hz wave are shown in Figure 6 , where the peak density is 20,000 cm , and the ambient magnetic field is 5 nT. These parameters correspond to what is observed in the typical nightside ionosphere. Yet again the self-consistent bottomside temperature is ~ 8 eV (Figure 6a). In Figure 6b the electron-ion collision frequency is larger than the electron-neutral collision frequency at 150 km. It is clear from Figure 6c that the waves are strongly attenuated, the waves have essentially decayed to background by 135 km altitude. All of the Joule dissipation occurs below 135 km (Figure 6d). At higher altitudes the heat conduction changes sign, providing local heating to offset the elastic cooling.
Figures 6a & 6b, Figures 6c & 6d. Wave propagation through the nightside Venus ionosphere for strongly attenuated 100-Hz signals. Similar in format to Figure 4. The peak density is 20,000 cm, and the ambient magnetic field is 5 nT, corresponding to the typical ionosphere. In Figure 6d the Joule dissipation is so weak at higher altitudes that we also plot -q, as this balances the cooling due to ions.
In Figure 6a , > 1 throughout the altitude range. However, the collision frequency only drops below the wave frequency for altitudes > 135 km, and at this altitude the waves have essentially vanished. Thus we do not expect resonant wave particle interactions to significantly modify our conclusions.
Figures 4a, 4b, 4c, 4d, 5a, 5b, 5c, 5d, 6a, 6b, 6c & 6d show that irrespective of the relative transparency of the ionosphere the net amount of heating is roughly constant. For all three cases the bottomside temperature is increased to about 8 eV. From comparison of (2) and (3), under the assumption that v >> , we find that T ~ 400LE, where T is in eV, L is in kilometers and E is in volts per meter. If we assume that L is given by the neutral density scale height, which appears to be the case in Figures 4a - 6a(above), then for a net applied field of 10 mV/m and a scale length of 2 km we obtain a temperature of 8 eV, as found from the detailed calculations presented here. Although this temperature is large in comparison to typical ionospheric temperatures, the consequences for the total ionospheric heat budget are probably insignificant. In particular, because the electron-neutral collision frequency is high, the thermal conductivity is low. This allows the ionosphere to support a steep temperature gradient, and the upper ionosphere (> 140 km) is thermally decoupled from the heated region in the bottomside ionosphere. Additionally, we have not included vibrational cooling, which will reduce the electron heating. In the next section we show that the thermal decoupling is still present when we include vibrational cooling.
Figures 4a, 4b, 4c, 4d, 5a, 5b, 5c, 5d, 6a, 6b, 6c & 6d show that the electron cooling rate due to the vibrational excitation of is about 3 orders of magnitude larger than the cooling due to elastic collisions. As such this cooling should be included in the heat budget. However, as noted in the appendix, the vibrational cooling operates even in the absence of any wave heating, and a downward heat flux is required to offset this cooling. Consequently, unless Morrison and Greene  significantly overestimate the cooling rates, the bottom of the ionosphere may be as high as 130 km. In assessing the effect of vibrational cooling we will therefore assume that the z = 130 km, instead of 125 km as used in the previous section. Prior to presenting results of the wave attenuation calculations, however, we note that attempts to obtain solutions for z = 125 km resulted in unrealistically high topside temperatures, unless intense waves ( ~ 0.1 V/m amplitude) were applied at the bottomside of the ionosphere. These waves provide enough heat to offset the vibrational cooling, without requiring any additional downward heat flux. Less intense waves did not supply enough heat at the bottomside when z= 125 km.
In Figure 7 we plot solutions of the wave attenuation calculation for four cases: 10 mV/m applied field, weak attenuation (Figure 7a); 100 mV/m applied field, weak attenuation (Figure 7b); 10 mV/m applied field, strong attenuation (Figure 7c); and 100 mV/m applied field, strong attenuation (Figure 7d). For the weakly attenuated waves we assumed a peak electron density of 1000 cm and an ambient field strength of 30 nT. For strong attenuation we assumed a peak density of 20,000 cm and a field of 5 nT. The wave amplitudes have been chosen to reflect the average and extreme amplitudes expected at the bottomside. Since many of the features of the model have been shown in Figures 4a, 4b, 4c, 4d, 5a, 5b, 5c, 5d, 6a, 6b, 6c & 6d , we only show the heat budget terms, and the temperature and wave electric field profile in Figure 7.
Figure 7a shows the results for moderate amplitude, weakly attenuated wave field. In this case the vibrational cooling is large enough to offset the Joule dissipation, at least for some of the altitude range. The peak cooling rate is ~ 5 10 W/m ~ 3 10 eV cm s . For larger amplitude waves, Figure 7b shows that the vibrational cooling is insufficient to balance the Joule heating. The temperature is therefore elevated, in comparison to Figure 7a. Throughout the altitude range shown Joule heating is balanced by electron heat conduction. Thus only about 10% of the energy released through Joule dissipation actually causes vibrational excitation of CO , one cannot simply equate the Joule dissipation rate to a neutral atmosphere heating rate. The peak vibrational cooling rate is ~ 3 10 W/m, roughly an order of magnitude larger than that obtained for a 10 mV/m applied wave field. It should also be noted that the strongest cooling, i.e., the most rapid transfer of heat to the neutrals, does not occur where the electron temperature is highest.
Figures 7a & 7b, Figures 7c & 7d. Heat budget, electron temperature, and wave amplitude as a function of altitude, including vibrational cooling in the heat budget. (a) Weakly attenuated, moderate amplitude signal; (b) weakly attenuated, high amplitude signal; (c) strongly attenuated, moderate amplitude signal; (d) strongly attenuated, high amplitude signal.
An applied field of 100 mV/m is extremely large, corresponding to the very intense burst observed by the Pioneer Venus Orbiter, at ~ 128 km on Orbit 5055, assuming a 100 Hz bandwidth. Moreover, most of the incident wave electric field is reflected. For Figure 7b , the incident wave field is ~ 5 V/m, which is typical for electric fields due to terrestrial lightning at 100 km away from the lightning strike [Uman, 1987]. Since the spectral peak of lightning is usually in the few kHz range, we might expect the wave amplitude at 100 Hz to be typically about a factor of 10 less. Furthermore, while wave fields incident on the dayside or dusk ionosphere might be expected to be of this amplitude, since this is the local time range over which lightning appears to occur on Venus [Russell, 1991], in the nightside the waves are thought to have traveled some distance in the surface-ionosphere waveguide [Strangeway, 1995b], and we expect lower amplitudes. Last, if as we suggest here, the ionosphere is above 130 km altitude, it is possible that the spacecraft was below the ionosphere for the lowest-altitude measurements, and the wave fields measured at ~ 128 km may include vertical electric fields that are shielded from higher altitudes.
Figures 7c and 7d show the wave attenuation and heat budget for strongly attenuated signals. The decoupling of the bottomside from the topside is shown clearly in Figure 7c, where there is a temperature minimum at ~ 136 km. At higher altitudes, where the thermal conductivity is higher, a relatively weak positive temperature gradient provides the heat flux necessary to offset the vibrational cooling. At lower altitudes, where the Joule dissipation is occurring and the conductivity is lower, a stronger negative temperature gradient provides the upward heat flux. In Figure 7d, where the applied field is 100 mV/m, the topside electron temperature is slightly elevated. However, in Figures 7c and 7d the maximum vibrational cooling occurs at altitudes above the maximum Joule dissipation.
From Figure 7c one could perhaps come to the somewhat surprising conclusion that a modest amount of wave Joule dissipation is required to offset the vibrational cooling. In Figure A2, where there is no wave heating, we find a topside temperature of 0.5 eV is required to provide the necessary downward heat flux. Because of the wave heating in Figure 7c, less heat flux is required from above, and a more reasonable topside temperature of 0.1 eV is obtained.
We can use Figure 7c as a guide for how much heating of the neutral atmosphere is reasonable. The peak cooling rate in Figure 7c is ~ 10 W/m, where the electron density is ~ 20,000 cm , and T ~ 0.1 eV (again in a region where there is no wave Joule dissipation). Unless the vibrational cooling rate is severely overestimated, it seems reasonable to assume that the neutral atmosphere can readily absorb heat supplied at this rate, ~ 6 10 eV cm s , since the plasma parameters are consistent with in situ observations. At 140 km both O and CO have densities of ~ 2 10 cm, and assuming that the vibrational energy is ultimately converted to thermal energy [Cole and Hoegy, 1995], we get a heating rate of 0.04 K s atom, which for a neutral gas temperature of ~ 100 K, gives a doubling time of 2500 s, a little over 40 min.
Taking an electron cooling rate of 10 W/m as a rate that does not seriously perturb the atmosphere, only the strongly attenuated large amplitude waves ( Figure 7d) appear to be capable of perturbing the atmosphere when we incorporate vibrational cooling in our calculation. Even then, the net heating rate is only ~ 0.4 K s atom. However, we have not included other inelastic processes, specifically electronic excitation of CO and O, which are likely to be important when electron temperatures are a few eV. Enhanced ionization may even be possible. Such processes are thought to occur at the Earth [Taranenko et al., 1993a, b] in association with lightning, and it appears reasonable to expect similar effects at Venus for the most intense waves.
Through order of magnitude estimates of the relative importance of the different heating and cooling rates we find that collisional Joule dissipation of plasma waves is likely to be important only at low altitudes in the ionosphere of Venus. This conclusion arises from the inclusion of electron heat conduction in the heat budget equation. Except at the lowest altitudes, the heat flux associated with relatively small temperature gradients is sufficient to match the heating from Joule dissipation.
Near the dayside ionopause, temperature gradient scales > 1000 km can provide sufficient heat conduction to offset the Joule dissipation. Waves are mainly observed above the ionopause, where ambient plasma densities are of the order 100 cm [Crawford et al., 1993], and the scale lengths can be much longer, several planetary radii. At high altitudes in the nightside ( 150 km), temperature gradient scale lengths > 10 km are sufficient for heat conduction to balance Joule dissipation. Even longer scale lengths (> 50 km) are sufficient in the reduced density regions known as ionospheric holes, where the waves are usually detected.
Determining the relative significance of Joule dissipation in the bottomside ionosphere requires detailed wave propagation calculations, because the heating caused by Joule dissipation is a consequence of the attenuation of the wave fields. We have performed wave propagation calculations using the scheme of Huba and Rowland , modified to iteratively recalculate the temperature profile until the total heating rate is zero.
During the Pioneer Venus entry phase the OEFD measured 100 Hz waves around 130 km altitude [Strangeway et al., 1993b]. The waves decreased in amplitude with a scale height of the order 1 km, and with a peak amplitude of between 10 and 10 V m Hz, which corresponds to an electric field amplitude of a few tens of millivolts per meter assuming a bandwidth of 100 Hz. Thus the calculations presented here are consistent with the low altitude entry phase observations, and we might expect bottomside electron temperatures to be elevated to a few tens of eV for the most intense waves. As such, Joule heating by the most intense waves could possibly result in optical or ultraviolet emissions, or even enhanced ionization, which may in turn provide additional evidence for lightning on Venus.
However, while electron heating may be occurring, the high collision frequencies thermally decouple the bottomside ionosphere from higher altitudes, and we do not expect lightning generated heating to have any catastrophic consequences for the global energy budget of the Venus ionosphere and atmosphere. In particular, it is not the Joule dissipation rate, but the inelastic collision cooling rate that determines the amount of heat entering the neutral atmosphere. Electron heat conduction carries away any excess heat that cannot be absorbed by the neutral atmosphere. Since the inelastic cooling rate, which we have modeled by vibrational excitation of CO, is only weakly dependent on temperature above 0.2 eV [Morrison and Greene, 1978], the cooling rate is approximately independent of the amount of Joule dissipation, and we find electron cooling rates, and hence neutral atmosphere heating rates, of the order 10 W/m for typical wave field amplitudes. This rate appears to be well within the bounds of heating rates which can be accommodated by the neutral atmosphere.
As noted in the body of the text, CO is the dominant neutral at low altitudes in the Venus ionosphere. Morrison and Greene  have investigated the cooling of electrons through collisions with CO, and we use their work as a basis for parameterizing the collision cross-section, and vibrational cooling rate. Figure A1 shows the cooling rates from Morrison and Greene , as indicated by the symbols. It is assumed that the neutral gas temperature is 200 K. The solid lines in Figure A1 show least squares regression lines through the data. The functional forms for the regression were chosen to best fit the data, with the constraints that the curves approach an asymptotic form at high temperature, and that the functions are positive definite for all temperatures.
Figure A1. Cooling rates for electron-CO collisions [after Morrison and Greene, 1978]. The symbols give the cooling rates for vibrational (squares), electronic (diamonds), and rotational (triangles) excitation of CO, and for elastic, momentum transfer (circles), collisions. The solid lines give least squares fits used to parameterize the vibrational cooling rate and momentum transfer collision cross section.
Turning to the momentum transfer cooling rate first, from (5) and (7),
where temperatures are in eV, and densities are in cm. Thus the explicit temperature dependence in (A1) is removed prior to performing the fit. Note that (A1) has been expressed in eV cm s , for direct comparison with Figure A1, 1 eV cms = 1.6 10 erg cm s = 1.6 10 W/m.
The fit yields a collision cross section for momentum transfer of the form
The fit to the vibrational cooling rate curve is
Thus at high temperatures (> 0.3 eV) Q/Q ~ 1.34 10/T and the vibrational cooling generally exceeds the momentum transfer cooling by about 3 orders of magnitude.
However, while electrons lose most of their energy through inelastic collisions, they lose most of their momentum through elastic collisions. If we denote the effective collision frequency for momentum loss through vibrational excitation of CO as v then
and since m / m = 8 10, even though Q/Q ~ 1.34 10/T, the electron momentum loss collision frequency is essentially that due to elastic collisions. Thus it is the collision frequency for elastic collisions that gives the Joule dissipation rate, while the cooling due to inelastic collisions is the dominant collisional cooling term. This further emphasizes the relative inefficiency of cooling through elastic collisions.
Another consequence of the relative efficiency of vibrational cooling was alluded to in section 4, where we showed that the vibrational cooling rate exceeded the Joule dissipation rate at most altitudes. Even in the absence of waves, the vibrational cooling operates, and may require unrealistically high topside electron temperatures to provide the heat flux required to offset the cooling, depending on the assumed ionospheric density profile. This is shown in Figure A2 , where we plot solutions of the heat budget equation (1) , but with the Joule dissipation turned off, and cooling through vibrational excitation of CO (A3) turned on. The thick curves to the right of the figure show the temperature and cooling rate profile obtained for the same density profile shown in Figure 6. Since downward heat flux is the only source of heat in the model, (1) requires unrealistically high topside electron temperatures. When obtaining a solution of (1) in this case, we required that the electron temperature at lowest altitude be the same as the neutral gas, and allowed the topside temperature to float.
Figure A2. Electron temperature profiles and associated cooling rates for different ionosphere minimum altitudes. The temperature (thick solid line) and collisional cooling rate (thick dashed line) are plotted to the right for an ionospheric density profile that vanishes at 125 km altitude. The peak density is at 140 km, with a value of 20,000 cm. The thin lines show the temperature and cooling rate for the same peak density and altitude, but with the ionospheric minimum altitude at 130 km.
Because the vibrational cooling rate is proportional to N , while the thermal conductivity is inversely proportional to N (through the collision frequency), small changes in the minimum altitude of the ionosphere can have marked changes in the vibrational cooling rate. The thin curves to the left of Figure A2 show the temperature and cooling rate when we assume that the electron density vanishes at an altitude of 130 km. Clearly, while slightly elevated, the temperatures are much more reasonable. Thus in section 5, where we discuss wave transmission including vibrational cooling, we assume the bottom of the ionosphere is at 130 km.
As a last remark on inelastic cooling, we have not included other cooling processes, such as the fine structure excitation of atomic oxygen [Hoegy, 1976]. Except at higher altitudes (> 150 km), this cooling is relatively unimportant in comparison to the cooling by CO. However, a complete analysis of the electron heat budget should in the future include this and other cooling terms due to inelastic collisions with atomic oxygen.
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