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.

Next: 3. Joule Dissipation: Order of Magnitude Estimates
Previous: 1. Introduction
Top: Title and Abstract

2.       Heat Budget

      The equations given below governing collisions and transport are as given in the work by Huba, [1994, pp. 31-39]. A more detailed discussion on collision frequencies and the electron heat budget can be found for example in the work of Banks and Kockarts [1973, chaps. 9 and 22], albeit with slightly different numerical constants. Under the assumption of steady state and no relative flow between species the electron heat budget is given by

      equ.1       (1)

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]

      equ.2       (2)

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]

      eq3       (3)

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 [1995], 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 important for ~1, where is the ratio of thermal to magnetic pressure =
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 [1978], 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. [1980] and Hedin et al. [1983] 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. [1984], 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.

Next: 3. Joule Dissipation: Order of Magnitude Estimates
Previous: 1. Introduction
Top: Title and Abstract

Text and figures by R.J. Strangeway
Converted to HTML by Chris Casler
Last modified: Feb. 10,1996