WHISTLER-MODE PROPAGATION IN THE COLLISIONAL
IONOSPHERE OF VENUS
R. J. Strangeway
Institute of Geophysics and Planetary Physics
University of California, Los Angeles, CA 90095, USA
ABSTRACT
Waves identified as whistler-mode waves have been observed in the nightside ionosphere of Venus by the
Pioneer Venus Orbiter. These waves are propagating in a collisional, weakly magnetized ionosphere, and
it has been argued that as such they should be damped through collisions, and the wave should not retain
the characteristics of the whistler-mode because of non-linear modifications to the dispersion relation. We
show that non-linearities do not modify the wave dispersion since the non-linear force causes a
longitudinal current, and the associated charge separation electric field acts to balance the non-linear
force. The quasi-longitudinal approximation is therefore appropriate for whistler-mode waves in the
nightside ionosphere of Venus. Further, we find that while the waves will heat the bottomside ionosphere,
little effect is found at altitudes where in situ observations were made by the Pioneer Venus Orbiter.
INTRODUCTION
It has been argued in a series of papers (Cole and Hoegy, 1996a,b, 1997) that the co llisional Joule
dissipation of low frequency waves in the ionosphere of Venus is so large as to preclude their existence as
naturally occurring waves. More specifically, Cole and Hoegy argue that the 100 Hz waves observed in
the nightside ionosphere of Venus cannot be electromagnetic waves caused by atmospheric discharges
(lightning). Strangeway (1996, 1997a,b) has pointed out, however, that the Joule dispersion can be easily
absorbed by the plasma and carried away from the heating region as heat flux, except at the lowest
altitudes (< 150 km) in the nightside ionosphere.
An additional point raised by Cole and Hoegy (1996b, 1997) is that the reported wave electric field
amplitudes are sufficiently large that, under the assumption that these waves are whistler-mode waves, the
wave magnetic field can be of comparable magnitude to the ambient magnetic field. They argue that the
waves must be non-linear in their dispersive properties. Strangeway (1997b) noted, however, that for the
whistler-mode waves the wave current and magnetic field are nearly parallel and the j × b force is small.
Since this is the primary non-linear term in the dispersion relation, he argued that the dispersion relation is
given by the quasi-longitudinal approximation (eg. Stix, 1992).
In this paper we will reiterate some of the points raised in the earlier comment (Strangeway, 1997b),
which was written without knowledge of Cole and Hoegy (1997). Our emphasis is therefore directed to
pointing out why Strangeway (1997b) and Cole and Hoegy (1997) differ in their conclusions. We will
also re-analyze the electron temperatures associated with Joule dispersion taking into account cooling by
electronic excitation of CO2.
CIRCULARLY POLARIZED WAVE
Before discussing in detail the theoretical arguments for neglecting non-linear effects in the whistlermode dispersion relation, it is useful to consider the idealized case of a right-hand circularly polarized
wave propagating without dissipation. This example is chosen as it corresponds to the parallel
propagating whistler-mode. Letting the wave vector define the z-axis of a Cartesian coordinate system,
then the wave magnetic field varies as
b = b[cos(ω t – kz), sin( ω t – kz), 0]
(1)
where the terms in braces correspond to the x, y, and z components of the wave field. From Ampere’s Law
and Faraday’s Law it can be shown that the wave current density is given by

ω2 
µ 0 j =  1 − 2 2  kb
 k c
(2)
i.e., j × b = 0.
NON-LINEAR FORCES FOR THE WHISTLER-MODE
To understand why non-linear forces can be neglected for the whistler-mode, in addition to the case
discussed above, it is necessary to go back to first principles. From Maxwell’s equations
∇× j=
− 1 2 1 ∂ 2 
∇ − 2 2 B
µ0 
c ∂t 
∇ ⋅ j = −ε 0
(3)
∂
∂ρ
∇⋅E = −
∂t
∂t
(4)
where j is the current density, B is the magnetic field, E is the electric field, ρ is the charge density, and
the other symbols have their usual meaning.
For the whistler-mode we assume that only electrons carry current, and from the Lorentz force law
including collisions
∂j
j
j
e
+ νj − ( j ⋅ ∇) − (∇ ⋅ j) = ω pe 2 ε0 E −
j×B
∂t
ne ne
me
(5)
In Eq. 5 ν is the collision frequency for momentum transfer, n is the electron density, e is the magnitude
of the electron charge, me is the electron mass and ωpe is the electron plasma frequency. The last two
terms on the left hand side of Eq. 5 can be neglected provided δn/n 1. Thus Eq. 5 can be rewritten as
∂j
e
+ νj = ω pe 2 ε0 E −
j×B
∂t
me
(6)
From the divergence of Eq. 6, together with Eq. 4
e
∂
 ∂ρ
= −ω pe 2 ρ +
∇ ⋅ ( j × B)
 + ν
 ∂t
 ∂t
me
(7)
Hence
 Ω B 
ρ ≈ O e2
kj
 ω pe B0 
(8)
for a plane wave, where Ω e is the gyro-frequency (eB0/me), B0 is the ambient magnetic field strength, and
k is the wave number.
Defining the longitudinal current (jL) as that parallel to the wave vector, then kjL = ωρ, and
 ωΩ e B 
jL

≈ O
2
jT
 ω pe B0 
where jT is the transverse current (≈ j).
(9)
1, not b/B0
1. This is the
Eq. 9 gives the condition for non-linearity in the wave dispersion: jL/jT
major difference between our analysis and that presented by Cole and Hoegy (1997b) and arises because
the longitudinal current must result in a parallel electric field, which in turn must be included in Eq. 6.
The force due to this electric field, which is ignored by Cole and Hoegy, balances the non-linear j × B
force. From Eq. 9 non-linear effects are important if (b/B0) ≈ O(ωpe2/ωΩ e) ≈ 10 3n for 100 Hz waves,
where n is the density in cm-3 and B0 ≈ 30 nT. Since n
1 in the Venus ionosphere, jL/jT
1 even when
b ≈ B0.
On taking the curl of the force law Eq. 6 it can be shown that the only non-linear terms are due to the
longitudinal current:
∇×(j×B) = (B0⋅∇
∇)j – (jL⋅∇
∇)b – B(∇⋅jL)
(10)
since ∇⋅b = 0, i.e., bL = 0. From Eq. 9 we can neglect the longitudinal current, i.e., ∇×(j×B) = (B0⋅∇
∇)jT .
Hence, using Eq. 3,
∂2 
e
∂

2 ∂b
+
 + ν   c 2 ∇ 2 − 2  b = ω pe
(B ⋅ ∇)j T
 ∂t

∂t me ε0 0
∂t 
(11)
This equation is linear in the wave fields, again emphasizing the point that Eq. 9 gives the condition for
non-linear dispersion.
From Eq. 11 and Maxwell's equations it can further be shown that
µ2 = 1−
ω pe 2
ω (ω + iν − Ω e cosθ )
(12)
for a plane wave, where θ is the angle between the wave vector and the ambient magnetic field. Thus
neglecting the longitudinal current results in the quasi-longitudinal approximation of the whistler-mode
dispersion relation.
In deriving Eq. 12 it should be emphasized that we have only assumed that electrons carry the current, and
that the longitudinal current (i.e. parallel to the wave vector) can be neglected. Both assumptions are
reasonable for the ionosphere of Venus. Because these assumptions yield the quasi-longitudinal
approximation to the whistler-mode, we find that the wave dispersion only depends on the component of
the magnetic field parallel to the wave vector (Ω ecosθ), and it can be shown that the wave fields are
circularly polarized in this case.
WAVE TRANSMISSION THROUGH THE IONOSPHERE
Since the quasi-longitudinal approximation applies for the whistler-mode, we can revisit the wave
transmission calculations of Strangeway (1996). In his earlier calculations he did not include electronic
excitation of CO2, and as a consequence he obtained artificially high bottomside temperatures. Here we
present calculations that include electronic excitation of CO2, based on cross-sections given by Fox and
Dalgarno (1979).
(c)
Density
Wave Parameters
150
150
145
145
Alt (km)
Alt (km)
(a)
140
E
140
b
135
130
1
10
100
Ne (cm-3)
1000
1.0
10.0
E (mV/m), b (nT), S (µW/m2)
(d)
Temperature
150
145
145
140
Joule
Conduction
Vibrational
140
135
135
130
0.1
130
10-13
10.0
100.0
Heat Budget
150
1.0
Te (eV)
S
130
0.1
10000
Alt (km)
Alt (km)
(b)
135
Electronic
10-12
10-11
10-10
10-9
Heating/Cooling Rate (W/m3)
10-8
10-7
Fig. 1. Results from the wave propagation calculation for a profile corresponding to an ionospheric
hole. Shown are: (a) Electron density; (b) Resultant electron temperature profile; (c) Wave parameters
(electric field, E, wave magnetic field, b and Poynting flux, S); and (d) The heat budget, with electron
heating shown as dashed lines, and electron cooling shown as solid lines.
Figure 1 shows the results of a calculation for a moderately attenuated wave (see Strangeway (1996) for
details of the method). For simplicity, we have assumed a vertical magnetic field. Because of refraction
the wave vector will be vertical (Sonwalker et al., 1991, Strangeway, 1991) and our assumption of a
vertical magnetic field is equivalent to assuming parallel propagation. As we shall see below, the wave
amplitude chosen (30 mV/m below the ionosphere) is significantly larger than we would expect for
typical wave observations. The density profile in Figure 1a peaks at 140 km, with a density of 5000 cm-3.
The ambient magnetic field strength (B0) is 30 nT and the wave frequency is assumed to be 100 Hz,
corresponding to Pioneer Venus observations.
The resultant temperature profile (Figure 1b) has a peak of ≈ 5eV at 132 km, just below the altitude where
the Joule dissipation rate (Figure 1d) is maximum. By 150 km the temperature has decreased to ≈ 0.35
eV. Theis et al. (1984) give a median temperature of ≈ 0.07 eV at this altitude in the nightside ionosphere.
However, we do not expect the wave-heated ionosphere to be average. Furthermore, the Pioneer Venus
Orbiter rarely sampled below 150 km. The elevated temperatures shown in Figure 1b would be difficult to
detect.
The Poynting flux (S) at 150 km in Figure 1c is ≈ 4 µW/m2. Russell et al. (1989) report a Poynting flux of
≈ 0⋅1 µW/m2. Taking into account that Russell et al. assumed a 30 Hz bandwidth, we could increase this
value by a factor of 3. Even then, however, the observed Poynting Flux is at least an order of magnitude
lower than that shown in Figure 1c. Thus the assumed wave is intense in comparison to observed waves.
We have repeated the calculation with a lower amplitude wave (10 mV/m at 130 km). In this case the
bottomside temperature was ≈ 2 eV, while the topside temperature was 0⋅1 eV, our assumed asymptotic
value. Because the collision frequency is lower for lower temperatures, the attenuation is not as strong and
the topside Poynting flux is ≈ 1 µW/m2, again much larger than that reported by Russell et al. (1989). It
should also be noted that for a more strongly attenuating ionosphere (n = 20,000 cm-3 at 140 km, B0 = 5
nT) there is no signature of wave heating at 150 km.
The heat budget in Figure 1d indicates that the electronic excitation of CO2 acts to control the bottomside
temperature. At the lowest altitudes the very high electron-neutral collision frequency (νen) results in a
low thermal conductivity, and heat conduction is a relatively inefficient cooling term. The altitude at
which the Joule dissipation is maximum is where νen ≈ Ω e. Since the dissipation corresponds to
absorption, the wave amplitude decreases rapidly at this altitude. At higher altitudes the Joule dissipation
rate decreases because of attenuation and the rapidly falling electron-neutral collision frequency. The
temperature at intermediate altitudes is determined by balancing Joule dissipation with cooling through
vibrational excitation of CO2. Heat conduction associated with the strong temperature gradient also affects
the temperature profile. At different altitudes the heat conduction can be either a heating or cooling term
in the energy budget.
Returning to the bottomside temperature profile, the cooling due to electronic excitation increases rapidly
for temperatures above 1 eV. Indeed for such high cooling rates as shown in Figure 1 we might expect
both enhanced ionization and UV emission. On the other hand, for the lower amplitude waves (10 mV/m
at 130 km), the electronic cooling decreases by two orders of magnitude. Thus only the most intense
waves might cause detectable signatures within the bottomside ionosphere.
CONCLUSIONS
Contrary to the opinions expressed by Cole and Hoegy (1996b, 1997), the dispersion relation for the
whistler-mode wave is not significantly modified by non-linear effects in the nightside ionosphere of
Venus. There is no reason for rejecting the identification of the 100 Hz waves in the nightside ionosphere
of Venus as whistler-mode waves. This is because the non-linear force manifests itself as a longitudinal
current, and the associated charge separation electric field in fact balances the j × b force of the wave. The
force due to this electric field was ignored in the analysis of Cole and Hoegy (1997). From our analysis,
the condition for non-linearity in the dispersion of the waves is (b/B0) ≈ O(ωpe2/ωΩ e).
It should be emphasized that our conclusions do not imply that there are no non-linear effects. Indeed the
wave propagation analysis presented in the previous section is implicitly non-linear. Joule dissipation is a
non-linear effect, depending on j⋅E, and modifying the collision frequency to take into account the wave
heating incorporates that non-linear effect. It is our opinion that Eq. 12 adequately describes the wave
dispersion, and any non-linear effects can be included by modifying the plasma parameters to reflect the
effects of wave heating. Urrutia and Stenzel (1996) came to a similar conclusion in analyzing laboratory
experiments. They state (p. 2597) “… whistler pulses with Bwave ≤ B0 remain linear in collisionless
plasmas … Nonlinearities arise from electron heating in collisional plasmas through modification of the
wave damping.” The wave propagation analysis in the previous section follows this approach.
We have determined the heating due to Joule dissipation for a wave of sufficient intensity to carry at least
ten times the Poynting flux observed in the nightside ionosphere of Venus, propagating through a plasma
with the properties of an ionospheric hole. We find that this wave may cause electronic excitation of CO2
at lowest altitudes, and slightly elevated temperatures at altitudes where in situ measurements were made
by the Pioneer Venus Orbiter (near 150 km). However, no effects of wave heating are apparent at 150 km
for either a more strongly absorbing ionosphere, or a weaker signal. While collisional Joule dissipation
may be important in the bottomside ionosphere, such dissipation does not preclude the generation of the
observed waves by atmospheric lightning.
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