Collisional Joule dissipation in the ionosphere of Venus: The importance of electron heat conduction

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.

3.       Joule Dissipation: Order of Magnitude Estimates

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:

(9)

(10)

(11)

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 (2) ·q - (K T) =
- ( 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,

(9')

(10')

(11')

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.

Text and figures by R.J. Strangeway
Converted to HTML by Chris Casler