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.
5. Cooling Through Inelastic Collisions
Previous: 3. Joule Dissipation: Order of Magnitude Estimates
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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.
5. Cooling Through Inelastic Collisions
Previous: 3. Joule Dissipation: Order of Magnitude Estimates
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Text and figures by R.J. Strangeway
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Last modified: Feb. 10,1996