*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.

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3. Joule Dissipation:
Order of Magnitude Estimates

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1. Introduction

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Title and Abstract

(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]

(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]

(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:

(4)

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

(5)

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

(6)

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

(7)

where is the Coulomb logarithm (~20), densities are in cm, and temperatures are again in eV. For electron-neutral collisions

(8)

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.

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3. Joule Dissipation:
Order of Magnitude Estimates

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1. Introduction

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Text and figures by R.J. Strangeway

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Last modified: Feb. 10,1996