A collisional shear instability in a magnetized plasma is described and evaluated. The instability is related to electrostatic Kelvin Helmholtz but operates in inhomogeneous plasmas in the collisional regime. Boundary value analysis predicts that the linear growth rate for the instability could be comparable to that of the collisional interchange instability in the equatorial F region ionosphere under ideal conditions. An initial value simulation of a nonlinear model of the instability run under realistic conditions produces growing waves with a relatively long growth time (50 min) and with an initial wavelength of about 30 km. The simulation results are consistent with recent radar observations showing large-scale plasma waves in the bottomside equatorial ionosphere at sunset prior to the onset of spread F conditions. The role of shear instability in preconditioning the F region for interchange instabilities to occur after sunset is discussed.
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 This paper examines the role of shear instabilities in triggering equatorial spread F. It is well known that strong shear exists in the bottomside equatorial F region ionosphere around twilight, where the plasma flow in the bottomside reverses from westward to eastward with increasing altitude [Kudeki et al., 1981; Tsunoda et al., 1981]. A number of factors are thought to contribute to shear flow, including E region dynamo winds, vertical winds and horizontal electric fields on flux tubes with significant Hall conductivity, and vertical currents sourced in the electrojet region near the solar terminator (see Haerendel et al.  and Haerendel and Eccles  and for recent reviews). Which of these factors is most important remains unknown, in part because they are difficult to measure directly using in situ or remote sensing.
 It is also widely known that layers of plasma irregularities (called bottom-type layers) exist near the region of shear flow [Woodman and La Hoz, 1976; Hysell, 2000]. These layers, which often form at sunset and can exist until as late as about 23 LT, serve as precursors for more dynamic equatorial spread F events. Kudeki and Bhattacharyya  argued that the irregularities in bottom-type layers are generated by wind-driven gradient drift instabilities which are readily excited in regions of retrograde plasma drift (where the local plasma velocity and neutral wind are antiparallel). They also showed that the layers arise out of the evening vortex produced by a combination of bottomside shear and the evening enhancement and subsequent reversal of the zonal electric field. The layers are readily observed using low-power coherent scatter radars and offer a means of monitoring the shear flow. An example of shear flow in a scattering layer observed by the JULIA (Jicamarca unattended long-term observations of the ionosphere and atmosphere) radar using interferometry is shown in Figure 1.
 Recently, Hysell and Chau  presented radar imagery supporting the Kudeki and Bhattacharyya  hypothesis regarding wind driven gradient drift instabilities and also showing that bottom-type irregularities sometimes occur in patches arranged periodically in the zonal direction. They surmised that the periodicity was caused by large-scale (λ ∼ 30 km) waves, the different phases of which being alternately stable and unstable to wind-driven instability. Such waves, if present, could serve as seeds for large-scale interchange instabilities and full-blown spread F. If so, then not only the occurrence of the layers but also their periodic structure could be utilized for forecasting spread F.
 In this paper, a causal relationship between shear flow and the large-scale seed waves involving a new collisional shear instability is investigated. Such a relationship could form the basis of a longer-range forecast strategy, since the shear emerges several hours before sunset and can be measured directly with incoherent scatter radars.
 The connection between bottomside shear and equatorial spread F has been investigated before, but mainly from the point of view of shear stabilization of otherwise growing waves. A series of nonlocal boundary value analyses by Guzdar et al. , Huba and Lee , and Satyanarayana et al.  all pointed to the stabilization of interchange instabilities by shear flow and the movement of the most unstable modes to longer wavelengths. Such findings seemed to explain why large-scale waves in the equatorial electrojet and equatorial F region dominate intermediate-scale waves with higher local growth rates. However, shear stabilization has been challenged by Fu et al. , Ronchi et al. , and Flaherty et al. , who pointed out the limitations of boundary value analyses, which neglect the so-called transient response that can predominate in sheared flows and which overestimate the wavelength of the most unstable mode in the early stages of the flow. The issue of shear flow stabilization remains unresolved, and a number of recent theoretical and computational studies continue to support the premise [Hassam, 1992; Sekar and Kelley, 1998; Chakrabarti and Lakhina, 2003]. Studies have also investigated the effects of shear on parallel electron dynamics important at high latitudes [e.g., Satyanarayana et al., 1987b; Shukla and Rahman, 1998] as well as the generation of kilometric plasma irregularities by parallel shear flow near auroral arcs [e.g., Basu et al., 1984; Basu and Coppi, 1988, 1989; Willig et al., 1997].
 Transverse shear-driven instabilities in the ionosphere have received less attention by comparison. Satyanarayana et al.  and Keskinen et al.  examined the growth of electrostatic Kelvin Helmholtz instabilities in the F region ionosphere, including the effects of collisions and inhomogeneous plasma density. They found that collisions and inhomogeneity each reduce the growth rate of the inertial regime instability independently and identified the collision frequencies necessary for marginal stability. However, they did not explore parameter space in the collisional, strongly inhomogeneous limit. In this paper, it is shown that fast growing waves associated with a collisional branch of the instability exist for appropriately chosen velocity, density, and collision frequency profiles, including profiles broadly representative of the equatorial F region.
 This paper is organized as follows. A fluid model general enough to contain both electrostatic Kelvin Helmholtz instability and collisional and inhomogeneity effects is developed and analyzed as a linear boundary value problem. Fast growing solutions in the collisional regime are found using idealized plasma number density, drift, and collision frequency profiles. Ionospheric stability is then analyzed as an initial value problem using a fully nonlinear two-dimensional numerical simulation. The simulation shows that shear instability occurs under realistic conditions and also provides insights into the physical mechanism at work. Finally, the role of shear instability in the ionosphere is discussed.
 A nonlocal, two-dimensional fluid model describing a shear instability in a magnetized plasma in the plane perpendicular to the geomagnetic field is developed. The model includes the effects of ion inertia and ion-neutral collisions as well as density and collision frequency gradients. The ions obey the continuity equation
where n and v are the plasma number density and velocity, respectively, and where the latter is assumed to be dominated by the E × B drift velocity. The current is density assumed to be dominated by the ion Pedersen and polarization currents:
where Ωi and νin are the ion gyrofrequency and ion-neutral collision frequency, respectively, u is the zonal wind speed (taken here to be constant), E = −∇ϕ is the electrostatic field, and where the other terms have the usual meaning. In addition to the ion continuity equation, the plasma must obey the quasineutrality condition
The coordinate system adopted is on in which is horizontal and perpendicular to the geomagnetic field at the equator, is horizontal and parallel to the magnetic field, and is vertical. Linearization of equation (1) and equation (3) proceeds according to the following scheme:
where a plane wave dependence is assumed in the direction but where the vertical variation in the fields is retained explicitly. The subscripts indicate the order of the variables; n∘ represents the background density gradient, and embodies the background zonal drift velocity profile v∘ = × ∇/B. Note that both gravity and a background zonal electric field have been excluded from our model in the interest of isolating shear instabilities from Rayleigh Taylor and E × B instabilities that would otherwise be present.
Likewise, substitution of equations (4) and (5) into equation (3) produces the following linearized relationship between the perturbed number density and electrostatic potential:
where, for the sake of comparison, the ion-neutral collision frequency has momentarily been treated like a constant. Equations (6) and (7) and are then identical to equations (13) and (14) derived by Keskinen et al.  for the particular density and velocity profiles they chose to analyze but are otherwise somewhat more general. Upon combining equation (6) with equation (7), we arrive at the following dispersion relation:
In the collisionless limit where the plasma density is also taken to be constant, this reduces to
which is the standard dispersion relation for the transverse Kelvin Helmholtz instability [e.g., Mikhailovskii, 1974; Satyanarayana et al., 1987a]. This instability has been studied exhaustively and is known to emerge in regions where has a vanishing second derivative. The linear growth rate peaks at about 0.18 V∘/L for wave numbers with kL ∼ 0.5, where V∘ and L here are the amplitude and length scale of the shear flow described by a hyperbolic tangent function (see below). Comprehensive reviews of Kelvin Helmholtz instabilities in plasmas can be found in Chandrasekhar , D'Angelo , Mikhailovskii , and Treumann and Baumjohann .
 In the bottomside equatorial F region ionosphere, both density and collision frequency gradients must be retained in any realistic nonlocal transverse instability model. The generalized form of equation (8) can readily be shown to be:
which is the model equation for the potential analyzed below.
3. Nonlocal Analysis
 The nonlocal model expressed by equation (10) is an eigenvalue problem for the complex frequency ω = ωr + iγ. We solve for the growth rate γ as a function of wave number k by finding the eigenvalue ω and eigenfunction ϕ1 that satisfy equation (10) along with the boundary conditions. In this case, the boundary conditions derive from our expectation that the solution to equation (10) will be localized around the vicinity of the region of shear flow and density inhomogeneity and vanish elsewhere. Far from the shear region, the right side of equation (10) must become negligibly small, and so the solution for the eigenfunction can be approximated by ϕ1 ∝ e±kz. We apply this boundary condition at an extreme value of z, integrate equation (10) numerically across the shear region, and then find the eigenvalue that makes the solution vanish at an extreme value of z on the other side. At this point, the complex eigenvalue and eigenfunction are known.
 Profiles for background number density n∘, collision frequency, νin, and velocity v∘ have to be specified so as to support an equilibrium initial state. The vertical current density flowing in steady state is proportional to the product n∘(z)νin(z)(u − (z)), which must therefore be constant if the equilibrium current density is to be solenoidal. The idealized profile shapes considered here are then:
where ν∘ ≥ ν1 and where u > V∘ is required to avoid singularities. In the nondimensional analysis which follows, we scale all of the parameters to V∘ and L and control the relative steepness of the number density and collision frequency profiles with respect to the velocity profile through adjustments to u, ν∘ and ν1.
Keskinen et al.  investigated the emergence of transverse electrostatic Kelvin-Helmholtz instabilities in the auroral ionosphere where the magnetic field lines are nearly vertical and so were not concerned with the effects of transverse variations in the collision frequency. They therefore neglected transverse gradients in νin in their analysis of equations (6) and (7) but allowed for transverse plasma density gradients associated with patches and other inhomogeneities. They found that the Kelvin-Helmholtz instability was stabilized by collisions, in agreement with other studies [Satyanarayana et al., 1987a; Willig et al., 1997]. We can reproduce their results here by taking u = 2, ν1 = 0, and varying ν∘. The results of such an analysis are presented in Figure 2. It is evident that the flow is stabilized by normalized collision frequencies (normalized to V∘/L) approaching unity. In the bottomside equatorial ionosphere, where we may take V∘ ≲ 100 m/s and L ≳ 20 km, stabilization occurs for νin ≳ 0.005 s−1. Classic Kelvin-Helmholtz instability is associated with the inertial flow regime and is therefore not normally expected to function near or below the F peak at high or low latitudes.
 What is not evident in Figure 2, however, is that the introduction of the density gradient offsets the stabilizing effect of collisions which would be even more severe in a homogeneous plasma. Shortening the density gradient scale length increases the growth rate to a value that can actually exceed that of the classic, collisionless Kelvin-Helmholtz instability, even in the collisional domain. A distinct instability functions in the collisional domain which does not depend on the inertial terms in the model equations and which has a growth rate that is essentially independent of collision frequency. An analogy can be drawn between the inertial and collisional shear instabilities and inertial and collisional interchange instabilities in the ionosphere, which behave similarly.
Figure 3 presents the results of nonlocal growth rate calculations for collisional regime (νin ≫ ωr) instability excited in the vicinity of a steep density gradient. Here, normalized growth rates are plotted as a function of normalized horizontal wave number for different values of ν1 given ν∘ = 2 and u = 2. The growth rate is relatively insensitive to the collision frequency profile itself, and the main effect of changing ν1 is to change the steepness of the equilibrium background density profile. As n∘(z) is made steeper by increasing ν1, the growth rate increases to the point that it exceeds that of the classic Kelvin Helmholtz instability for the same velocity profile. Given V∘ = 100 m/s and L = 20 km, a normalized growth rate of γ = 0.25 corresponds to a physical e-folding time of 800 s, a figure comparable to the growth time of interchange instabilities in the equatorial F region. To the extent that the parameters of this calculation are geophysically representative and that the boundary value analysis is applicable, shear instabilities may even be expected to compete with interchange instabilities in the equatorial ionosphere.
Figure 4 shows the amplitude of the fastest growing eigenmode ∣ϕ(z)∣ for the ν1 = 1 case. The mode shape is inherently asymmetrical about the shear node at z = 0, exhibiting two distinct peaks above and below the node maximizing at altitudes where the velocity profile is nearly flat. The amplitude of the lower peak is greater than the upper, and the solution is confined mainly within a span of altitudes about 10–15 L wide. Figure 5 furthermore shows that the upper and lower peaks are not quite 180° out of phase. The asymmetry of the solution, which is representative of a broad range of instability parameters, is crucial to the underlying instability mechanism.
 The normalized linear growth rate and frequency of the instability, along with the most unstable wave number k, increase with u and ν1, albeit in a complicated and coupled way that depends strongly on the shapes of the density, velocity, and collisionality profiles. The instability appears to be inherently nonlocal, and although a local analysis of equation (10) including a vertical wave number component reveals the existence of a growing root in the collisional domain, the local growth rate thus calculated is misleading and does not agree closely with or scale like the results presented above. Indeed, the interdependence of the controlling parameters makes isolating the importance of any one of them difficult. In order to evaluate whether instability can occur under conditions present in the equatorial ionosphere, we will turn to numerical simulation of the complete, nonlinear system of equations. The simulation should reproduce both the transient response in the early stages of the instability and late-stage saturation, providing a complete picture of all of the processes involved and yielding a physically accurate prediction of the most unstable mode wavelength and growth rate.
4. Numerical Simulation
 In order to evaluate whether collisional shear instabilities can play a role in the dynamics of the twilight equatorial F region ionosphere, realistic number density, collision frequency, and velocity profiles must be considered in the analysis. By utilizing a time-dependent approach to this problem, we can also assess the validity of the linear eigen analysis and gain insights into the physical mechanism underlying the instability. Moreover, it should be possible to allow shear flow to arise self-consistently in a time-dependent simulation rather than through imposition. Shear flow can be realized in studies of the equatorial ionosphere by incorporating dynamo theory in the model equations.
 Numerical simulations of the collisional shear instability have been conducted utilizing the simple, two-dimensional fluid model of the equatorial F region ionosphere first described by Zargham and Seyler . The model can be forced by zonal neutral winds. One of the model equations is ion continuity
where Da is the ambipolar diffusion coefficient and vi is the ion drift velocity, which mainly represents the E × B drift velocity with a small diamagnetic drift correction and which can be expressed in terms of a stream function ψ as
where ψ is nearly proportional to the electrostatic potential and ρ is the component of the curl of the plasma flux or the generalized vorticity. A dynamical equation for ρ can be derived from the conservation of electron and ion momentum. Adding the conservation equations for fluid electrons and ions and neglecting terms involving the electron mass produces
where the J × B force has been absorbed into a generalized pressure . Taking the curl of equation (14) then yields the desired dynamical equation for ρ:
where an ad hoc viscosity term has been added. The vorticity equation resembles the driven Navier Stokes equation for two-dimensional neutral fluid flows, only for inhomogeneous fluids in this case and with an additional, cubicly nonlinear term representing density convection. Zargham  and Hysell and Shume  determined that this new term has a minute effect in simulations of collisional regime flow, and so it will be neglected here. Although we do not explicitly neglect the convective derivative term remaining on the left side of equation (15) in this derivation or in our simulation runs, we note that it too by definition has only a small effect on the evolution of the collisional regime flow.
 This simple model can be generalized to include three important phenomena: the zonal neutral wind, altitude variations in the collision frequency, and electrostatic coupling along magnetic field lines to the conjugate E region ionosphere. Following the recipe used by Zargham , we take
in the ion momentum equation, where u is the zonal neutral wind speed and F is a parameter meant to represent the integrated E region Pedersen conductivity. Such a treatment neglects the scale-size dependence on the coupling between the E and F regions and is appropriate for studying intermediate- and large-scale (λ ≳ 1 km)waves only.
 The dynamical equation for the generalized vorticity may now be written
This model describes collisional and inertial interchange instabilities as well as shear instabilities. In order to suppress interchange instabilities, we set the term representing gravity to zero and include no background zonal electric field. Forcing therefore comes entirely from the zonal wind term u. Shear is produced self-consistently by the variation in dynamo efficiency with altitude. The model does not contain an E region dynamo or other influences that might produce retrograde (westward) motion given an eastward neutral wind, as is generally found in nature around twilight. Nevertheless, the components necessary to excite shear instability have been included.
 The model has been solved computationally as an initial value problem using a simulation code which evolves the density and generalized vorticity in time using a leapfrog method (the stream function is a derived quantity which need not be evolved). The hybrid code uses spectral methods in the horizontal direction and finite differencing in the vertical direction and enforces Neumann/Dirichlet boundary conditions at the top and bottom, respectively. The simulation is initialized with a density profile with a positive vertical gradient in the lower half space. The initial profile is accompanied by broadband seed noise. Simulation parameters are listed in Table 1. These parameters were chosen so as to be physically reasonable with the caveats discussed below, although compromises were made in the interest of numerical stability. The simulation space is a 62.8 km square region centered on the F peak.
The ion-neutral collision frequency is 0.2 s−1 at the base and decreases exponentially with altitude with a scale height of 20 km. Note that the simulation length and timescales are unrelated to the dimensional scales L and V∘ discussed previously in the nonlocal analysis.
2π × 10 km
7.0 × 10−5 l2/t
5.0 × 10−5 l2/t
Figure 6 shows the results of the numerical simulation of the instability. The initial density profile follows a hyperbolic tangent law below the peak and an exponential decay law above it. The stream function initially exhibits only the vertical gradient necessary to sustain the equilibrium zonal plasma flow, which is given by
This profile takes on a maximum value of about 70 m/s at the bottom of the simulation and again near the F peak and is represented by the dotted lines in the lower frames of the figure. Shear arises mainly in the bottomside region, where the density is increasing rapidly with altitude, resulting in a bottomside drift velocity profile that also resembles a hyperbolic tangent with a length scale L of about 5–10 km. The height of the shear node is very close to the height of the maximum background density gradient. Note that v∘(z) only differs by about 10 m/s between its maximum and minimum in the bottomside. This is modest shear by geophysical standards.
 The perturbed stream function plotted in Figure 6 is the difference between the current stream function and the stream function at the initial time step. By time step 16, evidence of shear instability can clearly be seen in the perturbed stream function plot. Here, we find periodic islands or cells of low and high stream function, ψ, straddling the altitude of the shear node. The magnitude of ψ taken through the vertical cut shown bears close resemblance to what was predicted in Figure 4 for the shear instability, exhibiting a null at the shear node and peaks that maximize at altitudes where the horizontal drift velocity profile becomes flat. A dominant horizontal wavelength is also clearly evident by time step 16. The wavelength is precisely half the horizontal size of the simulation or about 31.4 km. This figure corresponds to kL > 0.5 and so represents a wavelength shorter than what was predicted by the boundary value analyses. Over time, the dominant horizontal wavelength increases, so that the kL ∼ 0.5 condition is ultimately satisfied. (That an integer number of waveforms always exist within the simulation is a consequence of the horizontally periodic boundary conditions.) Notice that the regions of low and high ψ are not quite vertically aligned. This signifies that the solutions above and below the shear node are not precisely 180° out of phase, as predicted by Figure 5. Lines of constant ψ are streamlines of the flow, and the plasma circulates clockwise and counterclockwise around the cells of low and high ψ, respectively.
 By time step 64, the instability has grown to the point of being detectable in the plasma density plot. Here, we find elongated regions of depleted and enhanced plasma penetrating above and below the shear node, respectively. A rotational pattern is suggested by the morphology of the enhancements and depletions, and the circulation cells visible at time step 16 have merged into one or two main cells by time step 64. By monitoring the amplitude of the perturbed stream function throughout the early stages of the simulation, it is possible to estimate the e-folding time of the instability to be about τ = 30 time steps or about 50 min.
 By time step 90, the unstable flow pattern exhibits a distinct “cat's eye” surrounding the most prominent circulation cell. The primary density irregularity is unstable to secondary, wind-driven gradient instabilities, accounting for the intermediate-scale structuring in the cat's eye walls. Also by time step 90, the background flow profile is departing visibly from the initial profile given by equation (17). The amplitude of the density and stream function perturbations grow to large fractions of the background values before the instability saturates. However, irregularities never penetrate significantly into the topside. Instability ceases as the flow dynamics eradicate the steep vertical density and velocity profiles that set it in motion and the associated free energy is released.
 The simulation reveals how the collisional shear instability functions. In a classic electrostatic Kelvin Helmholtz instability, shear flow implies rotational plasma velocity, which in turn implies a divergence in the polarization current at the center of the rotation via the convective derivative term in equation (2). The resulting accumulation of space charge at the center of the vortex gives rise to enhanced radial electric fields and faster rotational E × B drift. Collisions allow the dissipation of the space charge through Pedersen currents that decrease the growth rate of the instability. However, instability returns in the collisional regime if the background plasma density gradient is sufficiently steep.
 It was noted that the product n(z)νin(z)(u − v(z)), which is proportional to the vertical current density in a magnetized plasma forced by the wind, is constant in an equilibrium configuration. However, if the height of the layer is displaced vertically by an amount δz, the resulting current density is proportional to n(z − δz)νin(z)(u − v(z)) and no longer constant if n is nonuniform. Expanding n in a Taylor series and taking the divergence results in the expression −δz[n′(z)νin(z)(u − v(z))]′ which is proportional to the divergence of the Pedersen current. For typical parameter profiles, this function passes through zero near the density gradient peak and has oppositely signed maxima above and below it. Polarization electric fields will arise to arrest the development of space charge, and the net electrostatic potential will resemble what was predicted in Figures 4 and 5. Note that high-potential regions above low potential regions result from downward layer displacements, and vice versa. This process is clearly evident in the simulation by time step 16.
 For instability to occur, the convection driven by the polarization electric fields must deform the plasma such that the initial upward and downward perturbations in layer height are amplified. The near symmetry of the eight convection cells shown in the bottom left panel of Figure 6 might suggest that no such amplification should occur. However, the symmetry is broken by the offset of the upper and lower rows of convection cells. The effect is to produce what look like plane waves propagating upward and eastward (toward the right). If we join together the diagonal regions of high and low potential and consider that the circulation is clockwise (counter clockwise) in the low (high) potential islands, it becomes evident that the initial altitude perturbations that created the potential perturbations will be amplified by the convection that results. This is the mechanism responsible for instability. It is robust, but the growth rate depends strongly on the actual shapes of the number density, collision, and velocity profiles in a complicated way that evolves over time with the profiles.
 Initial value and boundary value analyses point to the existence of a collisional shear instability that can operate under conditions found in the equatorial F region ionosphere around twilight. Wave growth is due to a density gradient driven fluid instability driven by a transverse wind, where sheared flow is necessary to preserve quasineutrality. The dominant wavelength of the waves initially produced by a time-dependent numerical simulation was consistent with the 30 km figure inferred from radar images of irregularities within bottom-type scattering layers. Over time, the dominant wavelength increased until satisfying the approximate relationship kL ∼ 0.5, as predicted by linear boundary value analysis. That a shorter wavelength was observed early in the simulation suggests that a transient response dominated initially.
 The growth time of the simulated instability in the linear regime was about 50 min, which is several times longer than the growth time of the collisional interchange (generalized Rayleigh Taylor) instability in the ionosphere. Stronger shear flow in the simulation would increase the growth rate, and it may be possible for the shear instability to compete with the interchange instability when conditions are particularly conducive of the former. However, with few exceptions, the morphology of the large-scale waves observed with coherent scatter radars under spread F conditions is consistent with interchange instability [Hysell and Burcham, 1998], and this paper does not purport to downgrade the role of interchange instabilities in producing spread F.
 However, the coherent scatter radar database also shows that it is common for fully developed spread F plumes penetrating well into the topside to appear within an hour (within 4–5 e-folding times) of sunset, implying the existence of large-scale seed waves at twilight from which they can rapidly develop. Such waves may also be inferred from radar imagery, which reveals periodic distributions of bottom-type layer irregularities [Hysell and Chau, 2004]. While gravity waves are often thought to be a source of seed irregularities for spread F [Kelley et al., 1981], the evening vortex and associated dynamical effects are another likely source that must be considered [Kudeki and Bhattacharyya, 1999].
 Shear flow develops in the equatorial ionosphere beginning around 14 LT each day and intensifies at twilight as the E and valley regions recombine. Despite having a modest growth rate, shear instabilities may therefore undergo several e-folds before interchange instabilities activate and thereby create the seed irregularities necessary for full-blown spread F to occur. Determining whether this actually happens in nature will require examination of the incoherent scatter drifts database and correlative study of the occurrence or nonoccurrence of spread F plumes after sunset.
 D.L.H. wishes to thank Charles Seyler at Cornell University for his insightful comments. This work was supported by the National Science Foundation through cooperative agreement ATM-9911209 to Cornell University and by NSF grant ATM-0225686 to Cornell University. Additional support was received from the Air Force Research Laboratory through award 03C0067. The Jicamarca Radio Observatory is operated by the Instituto Geofísico del Perú, Ministry of Education, with support from the NSF cooperative agreements just mentioned. The help of the staff was much appreciated.
 Arthur Richmond thanks Bamandas Basu and M. J. Keskinen for their assistance in evaluating this paper.