## 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.