Polarization of elliptic E region plasma irregularities and implications for coherent radar backscatter from Farley-Buneman waves



[1] The problem of two-dimensional, homogeneous, elliptical irregularities in an otherwise homogeneous plasma with anisotropic conductivity is considered. We find an analytic solution for the potential inside and outside the irregularities. In the special case of circular irregularities, the internal electric field is reduced from the background field in both depletions and enhancements. The internal field is rotated in different directions for depletion and enhancements, however. When the irregularity is elongated, the electric field inside can be larger or smaller than the background field in both depletions and enhancements, depending on the attack angle of the background field. The effects of ion inertia can further suppress the internal electric field in small-scale circular irregularities. These electrodynamics considerations may help explain some aspects of radar observations of irregularities excited by Farley-Buneman waves and instabilities in the electrojets, in particular, their tendency to exhibit Doppler shifts significantly smaller than the line-of-sight background electron convection speed and proportional to the cosine of the flow angle. The analysis generalizes that of St.-Maurice and Hamza (2001), who introduced this avenue of investigation.

1. Introduction

[2] It is well established that Farley-Buneman waves in the equatorial and auroral E region ionospheres propagate at phase speeds significantly less than the background electron convection speed and roughly equal to the ion acoustic speed [e.g., Farley, 1985; Haldoupis, 1989; Sahr and Fejer, 1996]. This is inconsistent with the prediction of linear theory based on plane wave analysis, although linear theory does suggest that waves propagating at this speed will be marginally stable [e.g., Fejer et al., 1984]. Evidence that nonlinear effects are responsible for limiting the propagation speed of the waves to the ion acoustic speed is compelling [Otani and Oppenheim, 1998, 2006]. The main nonlinear process at work seems to be the generation of secondary Farley-Buneman waves that propagate at oblique angles to the primary waves [Oppenheim, 1997]. The superposition of the primary and secondary waves produces a checkerboard pattern of discrete enhancements and depletions that propagate en masse in (nearly) the direction of the background flow but at significantly slower speeds and with opposing transverse drifts in the direction normal to it. This behavior is seen clearly in numerical simulations of Farley-Buneman instabilities driven well above threshold [Oppenheim et al., 1996; Otani and Oppenheim, 1998]. Figure 3 of Otani and Oppenheim [1998] along with Figures 7–9 of Otani and Oppenheim [2006] in particular show the configurations and flow patterns to which we refer.

[3] Recent experimental evidence indicates that the Doppler shifts of radar echoes from Farley-Buneman waves can obey a cosine dependence on flow angle [Woodman and Chau, 2002; Bahcivan et al., 2005]. The same cosine dependence has also been found in the numerical simulations using advanced diagnostics [Oppenheim et al., 2005]. This suggests that the Doppler shifts may be indicative of the line-of-sight projection of the proper motion of the irregularity patches in the simulations, each one behaving like a hard target (see also J. Drexler and J.-P. St.-Maurice, Nonlocal waves in vertical density gradients in the high-latitude E region, submitted to Annales Geophysicae, 2006). We are therefore interested in investigating the dynamics of these irregularities, which can be modeled as elliptical regions of enhanced or depleted plasma. The purpose of this paper is to examine how such irregularities drift apart from explicit consideration of any instability physics. Note that there is no contradiction between the cosine dependence alluded to above and the preponderance of echoes with Doppler shifts close to the ion acoustic speed observed with the STARE and Millstone Hill radars [Haldoupis, 1989] so-called “type 1” echoes with this property dominate coherent scatter from Farley-Buneman waves at frequencies above 50 MHz [Balsley and Farley, 1971] and are associated with scatter from small flow angles. At longer wavelengths, echoes from all flow angles can be more readily observed and exhibit Doppler shifts bounded by the ion acoustic speed.

[4] The problem of elliptical irregularities in two dimensional, homogeneous, anisotropic conductors was solved formally by Jones [1945] but not in the space physics context of interest here. If it is found that the irregularities move much more slowly than and turn from the background plasma because of purely electrodynamic considerations, it may be possible to explain the failure of linear theory on the basis of plane wave analysis to predict the irregularity drift speeds and the associated radar Doppler shifts. This was the reasoning that led St.-Maurice and Hamza [2001] to undertake the first study of the electrodynamics of discrete, small-scale E region plasma enhancements and depletions (blobs and holes) undergoing Farley-Buneman turbulence. They showed that discrete, small-scale density irregularities both slow and turn from the background flow in a predictable way. The present study aims to generalize their findings somewhat utilizing a different mathematical approach (conformal mapping) that permits the explicit assessment of the effects of elliptical irregularity geometry and orientation.

[5] We limit our analysis to two-dimensional irregularities and therefore neglect the effects of finite aspect sensitivity, which are outside the scope of this study. The analysis considers irregularities with jump discontinuity boundaries and therefore must also exclude the effects of diffusion and diamagnetic drift. We consider irregularities of sufficiently small scale to avoid issues pertaining to electrostatic potential mapping along geomagnetic field lines, although the results could be generalized by replacing local conductivities with flux tube-integrated conductivities. Although we focus on Farley-Buneman waves, many of the results could be applicable to other waves and flows.

[6] This paper is organized as follows. We begin by finding an analytic solution for the electrostatic potential in the vicinity of two-dimensional elliptical irregularities in an E region plasma in a background electric field. We highlight the special case of circular irregularities but consider also how elongation affects their dynamics. The effects of ion inertia are also considered. Finally, we compare our results to those from other related studies and assess the implications for Farley-Buneman waves.

2. Electric Field Solutions

[7] This will be a two-dimensional treatment of plasma density irregularities and the associated electric fields in the plane perpendicular to the background magnetic field. The plasma has uniform Pedersen and Hall conductivities, and the irregularity is a uniform, elliptical enhancement or depletion with a step boundary. We seek solutions for steady flow around the irregularity boundary at the moment its center coincides with the origin of a polar coordinate system. As the electrons are magnetized, equipotential contours are streamlines of the electron flow around and through the irregularity. Speeding, slowing, and deflection of the irregularity will be signaled by an internal electric field differing from the background field.

[8] We regard the plasma as a dielectric and assert that the electrostatic potential obeys Laplace's equation inside and outside the irregularity boundary. The boundary conditions are that the potential and the normal component of the current density be continuous across the irregularity wall. In addition, the electric field outside the irregularity should transition to a uniform background field at large distances. In an isotropic dielectric, the electric field inside is shielded by induced dipole moments and polarization surface charge at the wall that emerges to maintain the boundary conditions. In an anisotropic conductor (i.e., with second rank conductivity), the field inside is also rotated.

[9] Following Smythe [1939], we solve this problem by solving two related ones. The construction involved is counterintuitive but mathematically expedient and accurate. Consider a disk of radius r1 surrounded by a concentric ring of enhanced or depleted plasma of outer radius r2, itself surrounded by homogeneous background plasma. This situation is depicted in the left plot of Figure 1. A uniform background electric field is applied. We seek solutions to Laplace's equation everywhere outside the disk for two distinct configurations where Dirichlet and Neumann boundary conditions are applied, respectively, at the surface of the disk. Results from the two configurations will contribute to the complete solution of the elliptical irregularity problem.

Figure 1.

Figures illustrating the electrostatic potential in the vicinity of an elliptical, anisotropic conductor in a background electric field. The problem is scale-invariant, so the scaling and labeling are arbitrary. We consider a 20% elliptical depletion with an aspect ratio of 5:3. The ratio of the Hall to Pedersen conductivity is 15. (left) Equipotentials outside a disk of radius r1, surrounded by a concentric ring-shaped irregularity with outer radius r2, itself surrounded by homogeneous plasma. Solid (dashed) lines result from applying Dirichlet (Neumann) boundary conditions at the surface of the disk. (middle) Same as the left plot, only transformed into the w plane. Here the solid and dashed lines represent equipotentials aligned with the major and minor axes of the ellipse with foci at u = ±r1. (right) Superposition of equipotentials from the middle plot. Equipotentials in the vicinity of an elliptical irregularity in an arbitrary background electric field are thus constructed (see text).

[10] Given the configuration with Dirichlet boundary conditions, the solutions for the potential in the regions within and outside the ring of plasma must have the forms

equation image
equation image

where E is the amplitude of the background electric field and θ is the polar angle measured counterclockwise from the background field direction. The coefficients are set by applying boundary conditions. One of these is that the potential should be continuous across the r = r2 boundary. The other is that the component of the current density J normal to the boundary should be continuous across it. (This is the integral form of the plasma quasineutrality condition.) For the moment, we regard the current density to comprise Pedersen and Hall terms, with radial components given by

equation image

where σP and σH are the Pedersen and Hall conductivities, respectively, which we take as

equation image

The equations to be solved are consequently

equation image
equation image

with (3) imposed by the continuity of the potential and (4) by the continuity of the current density across the irregularity boundary. Here, the i and o subscripts denote quantities inside and outside the irregularity, respectively. We also introduce the notation r± = r2 ± r12/r2. Sine and cosine terms must equate separately. Solving the system gives

equation image

Substituting these expressions into (1) and (2) yields the potential everywhere outside the disk, which behaves like a conductor in this configuration. Equipotential lines for this solution are shown in the left plot of Figure 1 as solid lines. Note that the solution has been rotated so that the zero equipotential inside the plasma ring is coincident with the horizontal axis. In view of (1), the necessary rotation angle is given by θ = tan−1(b/a).

[11] Given the configuration with Neumann boundary conditions at r = r1, the correct form of the potential becomes

equation image
equation image

The primed coefficients can be solved by applying the same boundary conditions at r = r2. Doing so produces expressions for a′, b′, A′ and B′ of precisely the same form as listed above except with the r+ and r terms exchanging roles. (That is, wherever r+ appears, replace it with r, and vice versa.) It can be shown that (2) is conjugate to (1), meaning that the equipotential curves they give rise to fall at mutual right angles. The equipotentials for one configuration are the lines of force for the other. The same relationship does not hold for (6) and (2), however.

[12] Equipotential contours of ϕ′ are plotted in the left plot of Figure 1 using dashed lines. This time, the solution has been rotated so that the zero equipotential inside the plasma ring is coincident with the vertical axis. The angle of rotation this time is given by θ′ = tan−1(b′/a′), where the primed coefficients are calculated according to the prescription in the preceding paragraph. While both the primed and unprimed potential solutions denote uniform electric fields at large distances from the origin, those electric fields are not orthogonal to one another when the solutions inside the plasma ring are orthogonal, as θ ≠ θ′ in general.

[13] The horizontal and vertical axes of the left plot of Figure 1 may be regarded as the real and imaginary axes of the complex plane z = x + iy. The next objective is to map this plane into another one, w = u + iv, where the boundaries transform into the desired ellipse but where the form of Laplace's equation and the boundary conditions are maintained. The necessary conformal transformation is given by

equation image

which maps the circle of radius r1 in the z plane to a line segment on the real axis between u = ±r1 in the w plane (see middle and right plots of Figure 1). Concentric circles with radii r2 > r1 in the z plane map to ellipses with foci at the endpoints of the line segment in the w plane. The major and minor axes of the ellipse are given by r±r2 ± r12/r2, respectively. At large radial distances, the z and w planes become the same except for a constant factor of two. This means that the boundary conditions at infinity are also the same and that the background electric field that dominates at large distances in both is the same field (within a constant). Note that the w plane plots in Figure 1 have been scaled by this factor of two for easier viewing; correct proportions are otherwise maintained.

[14] The potential solutions for the two configurations illustrated in the left plot of Figure 1 have been transformed according to (7) and plotted as solid and dashed lines, respectively, in the middle plot. In the configuration with Dirichlet boundary conditions, the circle of radius r1 in the z plane was an equipotential, making the line segment in the w plane between the foci of the ellipse an equipotential. Lines parallel to that inside the ellipse are also equipotentials. In the Neumann boundary condition configuration, the circle of radius r1 was a line of force. Consequently, the associated equipotentials in the w plane are normal to the line segment between the foci (vertical).

[15] The potential in the vicinity of the ellipse arising from an arbitrary background field can be formed from the superposition of the results from the two configuration. This is illustrated in the right plot of Figure 1, which gives the result of adding equal parts of the solutions in the middle plot. Far from the irregularity center, the potential is that of a uniform background electric field. Inside the irregularity, the electric field is uniform, but its direction and magnitude differ from the background field. The rotation angle is between θ and θ′, depending on the weighting. Fringing fields distort the flow just outside the irregularity boundary.

2.1. Circular Irregularities

[16] The formulas derived above become considerably less cumbersome in the illustrative case of a circular irregularity, for which r1 = 0 and r± = r2r. The potential is given by (1) and (2) without further transformation or construction. Furthermore, the coefficients on the harmonics assume relatively simple forms:

equation image

where use has been made of the following abbreviations:

equation image

and where the subscripts i and o continue to imply quantities inside (r < r) and outside (r > r) the circular irregularity. Here, Δ > 0 (Δ < 0) denotes and enhancement (depletion).

[17] The solution describes electron flow around the circular irregularity that is nearly tangential to the boundary except in the ram and wake. Inside the irregularity, the electric field is uniform. The magnitude of the interior field is

equation image

While it is possible for (8) to predict electric fields stronger than E in a depleted irregularity given small enough values of R, values of R greater than equation image imply attenuation of the field inside the irregularity for either sign of Δ.

[18] The angle the interior field makes with the background field is

equation image

which can have either sign depending on the sign of Δ. The effects of anisotropic conductivity (R > 0) are made clear by (8) and (9).

[19] Potential solutions for a circular density depletion and an enhancement are shown in Figures 2 and 3, respectively. The interior number density is 20% below and above background in the two cases. We take R = 15 in both examples. Equipotential contours are plotted. The background electric field points toward the right, and the angle θ is measured counterclockwise from the 3 o'clock position. Taking the magnetic field direction to be into the paper, the streamlines of the flow go from bottom to top.

Figure 2.

Potential contours for a 20% circular depletion.

Figure 3.

Potential contours for a 20% circular enhancement.

[20] Equation (8) predicts internal electric fields 57% and 54% of the background field for the depletion and enhancement, respectively. Equation (9) predicts rotations of −59° and 54° for these two cases. The rotation has opposite senses for depletions and enhancements, but the internal electric field is reduced (shielded) in both cases (as first pointed out by St.-Maurice and Hamza [2001]). Note also that depletions and enhancements behave asymmetrically, the former giving rise to stronger electric field perturbations than the latter. The asymmetry arises from the fact that it is the difference between the inner and outer conductivities that causes polarization but the sum of the inner and outer conductivities that arrests it and maintains quasineutrality. Symmetry is approximately restored when the density perturbations are small.

2.2. Effect of Irregularity Elongation

[21] The effects of irregularity elongation can be assessed from two extreme examples. Figure 4 shows the equipotentials in the vicinity of an elliptical irregularity oriented such that the background electric field is parallel to its minor axis. The irregularity is a 20% depletion with an aspect ratio of 6:1. The equipotentials outside the irregularity are nearly parallel to its surface, and the flow around it is almost uninterrupted. The interior electric field is slightly smaller than what (8) predicts, but the rotation angle is much less than (9). This is a case where elongation counteracts rotation.

Figure 4.

Elliptical depletion irregularity with an aspect ratio of 6:1. The minor axis is aligned with the background electric field.

[22] In contrast, Figure 5 shows the same irregularity only oriented with its major axis parallel to the background electric field. This time, flow around the irregularity is drastically interrupted. The rotation angle of the electric field is somewhat greater this time than what (9) predicts. Notably, the electric field inside the irregularity is larger than the background field. This is a case where elongation counteracts the suppression of the internal field. Enhanced irregularities behave similarly, producing internal fields both weaker and stronger than the background field depending on the attack angle of the latter. The angle of rotation is reversed from that of depletions.

Figure 5.

Elliptical depletion irregularity with an aspect ratio of 6:1. The major axis is aligned with the background electric field.

[23] The effects of elongation are summarized by Figure 6, which plots the magnitude and direction of the electric field inside a 20% depleted irregularity relative to the external field. The abscissa is the angle that the background field makes with the major axis of the ellipse. Three ellipses, with aspect ratios of 5:3, 3:1, and 6:1, are considered. Figure 6 illustrates that elongation invalidates the results from the circular irregularity analysis in the manner exemplified above. The rotation and attenuation of the internal electric field is counteracted when the irregularity is elongated perpendicular to and parallel to the background electric field, respectively. Sufficiently elongated depletions can have internal electric fields stronger than the background field and rotation angles greater than 90°. Similar remarks apply to Figure 7, which shows the results for 20% enhanced elliptical irregularities.

Figure 6.

Amplitude and rotation angle of electric field within elliptical depletion irregularities as a function of the angle between the major axis and the background electric field. Solid lines depict the modulus of the electric field as a fraction of the background field. Dashed lines give the angle between the internal and background fields. Three sets of curves represent irregularities with aspect ratios of 5:3, 3:1, and 6:1. Larger aspect ratios give more eccentric curves.

Figure 7.

Same as Figure 6 but for 20% elliptical enhancement irregularities.

[24] The tendency for the interior field in circular E region irregularities to be reduced for either sign of Δ arises from the dissimilar role of the Hall conductivity inside and outside the irregularity. Outside, the equipotential lines are nearly tangent to the irregularity, the electric field is nearly normal to the boundary, and the Hall current does not contribute significantly to the radial current flowing through the boundary. This is not true in the interior, where the equipotentials are oblique to the irregularity wall and the Hall current contributes substantially to the radial current balance. In that regard and given sufficiently large ratios R, density depletions and enhancements tend to behave like conductivity enhancements alike. Only for small values of R can the interior electric field in circular depletions exceed the background field. When the irregularity is elongated and oriented as it is in Figure 5, however, the interruption in the flow causes the equipotentials outside the irregularity to approach it normally instead of tangentially. In that event, Hall currents flowing outside the irregularity can contribute to the boundary current, and the electric field inside the depletion can exceed the background electric field.

2.3. Effects of Ion Inertia

[25] Thus far, only Pedersen and Hall currents have been included in the boundary conditions of the potential problem. For small-scale irregularities, it is also necessary to include polarization currents associated with ion inertia. The “irregularity” in question here is the boundary between the background and the depleted or enhanced plasma. Since the electrons are incompressible, the boundary must move with the velocity of the electrons inside. The component of the electron drift velocity normal to the boundary is continuous across the boundary, which is just another statement of incompressibility. The tangential component is not continuous, but tangential flow is immaterial to the boundary motion.

[26] Ions generally move much more slowly than the electrons. Quasineutrality is nevertheless maintained as ions converge or diverge in front of and behind the irregularity so as to preserve the integrity of the boundary. To the extent they are unable to change configuration rapidly, the electric field inside the irregularity and the irregularity velocity are also limited. Such limiting depends on the shape of the irregularity and on the inner and outer conductivities, in the manner shown above.

[27] Ion inertia causes an ion current to flow when the electric field structure associated with the irregularity boundary drifts past. The smaller the scale of the structure r and the faster the drift speed v, the stronger the current. Polarization currents should compete with Pedersen currents when the time it takes for an irregularity to drift past is comparable to a collision time, or when r/v ∼ νi−1. Extending the analysis to include polarization currents introduces a scale size dependence to the problem.

[28] We consider the effects of polarization currents on the dynamics of circular irregularities only. The polarization current density is carried by the ions and has the form (in the collisional limit)

equation image

which is obtained by solving the ion momentum equation, retaining the Lorentz force, ion neutral collision, and inertia terms. The strategy for evaluating the time derivative is to consider steady flow around and through the circular irregularity. The ions have negligible velocity in the lab frame of reference considered to this point in the analysis. In the frame of reference fixed to the irregularity and moving with velocity v in the lab frame, the total time derivative above can be replaced with the convective derivative, −(v · ∇), where the irregularity velocity is again the E × B drift velocity associated with the internal electric field. Note that ∇E is invariant and can be evaluated in any inertial frame of reference; the remainder of the calculations take place in the lab frame.

[29] Polarization current flows only outside the irregularity where the electric field is nonuniform. Only the radial component of the current density enters into the boundary conditions for the problem. In polar coordinates, we evaluate this current density at the boundary using (1) and (2):

equation image

where vr and vθ are the radial and azimuthal components of the irregularity drift velocity at the irregularity boundary. A general solution to the potential problem incorporating the current density implied by (10) in the boundary conditions does not exist, and it appears that the flow is not steady but rather evolves when the effects of ion inertia are included in the analysis. The boundary will consequently distort in a manner this analysis cannot predict. However, we can arrive at an approximate solution by assuming steady flow at the leading edge of the irregularity, near the ram, and enforcing the boundary conditions there. Near the ram, vrv and vθ ∼ 0 by definition, and the surviving terms in (10) can be added to the right side of (4) and combined there with the terms representing the Pedersen current flowing outside the irregularity, which have the same form. In effect, ion inertia reduces or counteracts the Pedersen current driven by polarization charge accumulated at the irregularity boundary. These Pedersen currents tend to diminish the polarization charge and its inherent shielding effect, but inertia helps to restore the shielding. Since the background electric field ultimately responsible for the charge accumulation is uniform, it gives rise to no polarization current to offset the Pedersen current it drives.

[30] The prescription for including the effects of ion inertia is then to define an effective Pedersen conductivity for the region outside the irregularity:

equation image

and to substitute this term into (4), which now reads

equation image

Notice that σpo still modifies the E term, which is unaffected. Propagating the changes through the preceding formalism leads to the following results:

equation image
equation image

Finally, (13) must be solved self consistently since the v in (11) is equation image/B. The result depends on the irregularity radius r through (11) as well as on the collision frequency and irregularity amplitude.

3. Comparison With Other Studies

[31] There have been a number of studies evaluating the electric field inside F region plasma density irregularities associated with barium cloud releases and equatorial spread F “bubbles.” Ott [1978] calculated the ascent rate of circular bubbles in the collision- and inertia-dominated flow regimes. His collisional regime calculation was general, but his inertial regime calculation was limited to the case of 100% plasma depletions. Ossakow and Chaturvedi [1978] undertook a related study, considering partial enhancements and depletions and allowing for ellipsoidal geometries but neglecting inertial effects. Most recently, McDaniel and Hysell [1997] considered the case of partial, circular enhancements and depletions, including inertial effects. Taken together, the studies found that F region depletions can develop interior electric fields much larger than the background field but that inertia limits the field amplitude, particularly when the irregularity is small. Enhancements, meanwhile have reduced interior electric fields. The interior field does not rotate. Depletions give rise to relatively larger electric field perturbations than enhancements for the same reason discussed above. Our results (taking R = 0) agree with the aforementioned studies in the collisional regime. As the effects of ion inertia are different in E and F region plasmas, we do not expect agreement in the inertial regime.

[32] The present results can also be compared to those obtained computationally by Hysell et al. [2002]. In their simulations, which followed on the analysis of Shalimov et al. [1998], large electric fields many times background were produced inside some E region plasma density enhancements. However, those results were obtained with a three-dimensional model that included electrical coupling to the F region. Coupling was scale size–dependent, and electric field enhancements were only obtained for elongated E region structures with major and minor axes perpendicular to B longer and shorter than about 1 km, respectively. The results had a narrow range of applicability and do not apply to the submeter-scale structures of primary interest here.

[33] Finally, this study is obviously most closely related to the pioneering one by St.-Maurice and Hamza [2001], who calculated the electric field inside elongated E region density irregularities using a limiting approach and assuming a balance between inertia and diffusion. Their approach did not permit the precise specification of the irregularity shape and neglected the fringing fields that form outside the irregularity. Given the differences in approaches, it is interesting to note that their equation (19) predicts internal field rotations in agreement with our (9). The amplitude of the internal field they predicted is consistent with (8) except with the Δ term in the numerator in our expression deleted. Their results evidently describe the behavior of circular irregularities with small density perturbations (but large values of the ratio R).

[34] In a plane wave treatment of Farley-Buneman waves, currents driven by ion inertia and diffusion balance in the case of marginal stability. While the effects of diffusion have not been addressed in the present analysis, we might expect inertia and diffusion to balance in a global sense but not in a local sense, the dependence of the two mechanisms on angle being different. Our treatment of inertia should therefore hold, at least approximately, in the ram and the wake of the irregularity where polarization currents are largest and, presumably, uncompensated by diffusion. This hypothesis can be verified through numerical simulation.

4. Discussion

[35] For illustrative purposes, we have considered irregularities with 20% density depletions and enhancements in the examples shown throughout the paper. Density fluctuations in Farley-Buneman turbulence are meanwhile limited to about 5% RMS [Pfaff, 1991], implying much weaker polarization effects. In such cases, (8) and (9) predict electric field rotations of about ∼±20° and modest field reductions proportional to the amplitude perturbation. When the irregularity is elongated to a 6:1 aspect ratio, however, the internal electric field can be reduced or enhanced over background by as much as 20%, depending on the attack angle of the background field. Larger aspect ratios imply even larger perturbations and rotations. We can model a primary Farley-Buneman wave packets as series of elongated, parallel plasma irregularities, oriented to the background field as shown in Figure 5 and alternating between enhancements and depletions, the crests and troughs of the primary waves. The rotation angles within the irregularities would alternate signs accordingly. This picture is consistent with the wave turning described by Oppenheim et al. [1996], who pointed out the asymmetry between enhancements and depletions that we have accounted for on purely electrodynamic considerations.

[36] Because of the rotations, convection electric fields with components transverse to the background field would be available to drive secondary Farley-Buneman waves at right angles to the primary wave. The superposition of the secondary waves and the primary would then result in a patchwork of roughly circular enhancements and depletions. In simulation, these irregularities predominate and have submeter scales [Oppenheim et al., 1996].

[37] Ion inertia and the resulting polarization currents would tend to arrest the motion of the small-scale irregularities. For example, setting the parameter 2v/rνi in (11) to unity, considering a circular irregularity with a 5% density perturbation, and propagating the results through to (13) predicts an internal electric field reduced to 4/5 the value of the background field. Taking E/B to be 1200 m/s then implies an internal drift velocity of E/B = 960 m/s. The rotation angles predicted by (14) in this case are about ∼±40° and is slightly different for enhancements and depletions. The component of the irregularity drift velocity in the direction of propagation of the primary wave is therefore about 740 m/s, which is something like the propagation speed of the waves observed in simulation.

[38] Given an ion neutral collision frequency of 1000 s−1 then requires, in consideration of (11), an irregularity radius of 0.8 m for self consistency. This is even larger than the size of the irregularities emerging in the numerical simulations of Oppenheim et al. [1996] and consistent with wave speed saturation well below the background electron convection speed. In view of their opposing and slightly asymmetric rotations, irregularities in opposing phases of the primary wave would tend to drift almost at right angles with respect to each other, with the average direction being slightly different than the background convection direction.

[39] Secondary wave pumping and the subsequent amplification of the secondary waves continues until the primary wave speed is reduced to the ion acoustic speed, at which time the primary wave becomes marginally stable. Assuming the Doppler shift of radar backscatter from the irregularities is indicative of their proper motion, we would expect a mean Doppler shift of ∼Cscos θ and a Doppler width of ∼Cssin θ, where θ is the angle between the background electron convection and the radar line of sight. Experiments conducted recently support this picture [Woodman and Chau, 2002; Bahcivan et al., 2005].


[40] D.L.H. is thankful for helpful discussions with J. P. St.-Maurice. This work was supported by the National Science Foundation through NSF grant ATM-0436114 to Cornell University. The Jicamarca Radio Observatory is a facility of the Instituto Geofísico del Perú and is operated with support from NSF Cooperative Agreement ATM-0432565 through Cornell University. The help of the staff was much appreciated.