The Naval Research Laboratory three-dimensional simulation code SAMI3/ESF is used to study the response of the postsunset ionosphere to circular gravity waves. We model the coupling of both circular (local) and plane wave (nonlocal) gravity waves to the bottomside F layer as a mechanism for triggering equatorial plasma bubbles. Results support the hypothesis that nonplane gravity waves can more strongly couple to the F layer than plane gravity waves. Results also show that the coupling of the seed wave to the F layer depends on the (nonlocal) growth rate and the local electron density at the position of the seed wave.
 The mechanisms that trigger the equatorial spread F (ESF) [Haerendel, 1974; Ossakow, 1981; Hysell, 2000; Makela, 2006] instability, also known as the generalized Rayleigh-Taylor instability, remain a topic of intense interest in the ionospheric research community. ESF is a postsunset phenomenon in which the equatorial F-region ionosphere becomes unstable: large-scale (~ 10 km) electron density “bubbles” can develop and rise to high altitudes (~ 1000 km). These quickly evolve to produce steep plasma density gradients and small-scale irregularities that can scatter radar signals [Booker and Wells, 1938] and disrupt radio waves [Kintner and Ledvina, 2005]. If the seed mechanisms of ESF could be determined, it might be possible to better predict or even control the formation of the resulting equatorial plasma bubbles (EPBs) and reduce their deleterious effects on electromagnetic signals.
 In previous modeling studies [e.g., Huba et al., 2008; Huba and Joyce, 2010; Krall et al., 2011], we have shown that ionospheric disturbances couple to the F layer through the electric potential: a local perturbation of the electron density on a geomagnetic field line can excite an E field that acts everywhere along that field line. Because this coupling occurs along field lines and because field lines can be reasonably treated as equipotentials, we have already shown that a disturbance without a finite zonal wave number will not trigger EPBs [Krall et al., 2011]. Based on this general approach, Tsunoda [2010a] argues that circular gravity waves represent a potentially effective seed mechanism for EPBs. Tsunoda [2010a, Figure 1] argues that a plane gravity wave propagating at an oblique angle to the geomagnetic field will produce alternating contributions to the electric potential that tend to cancel each other out and that, in contrast, circular wavefronts would more effectively seed the ESF instability. In this paper, we use the Naval Research Laboratory SAMI3/ESF code [Huba et al., 2008] to test this idea.
 We note that gravity waves [see also Huang and Kelley, 1996; Fritts et al., 2009] are only one of many possible seed mechanisms for ESF. Others include sheared plasma drift velocities [Huang and Kelley, 1996], a combination of velocity shear and a zonal E field [Sekar and Kelley, 1998], a collisional-shear instability [Hysell and Kudeki, 2004], a sporadic E-layer instability [Tsunoda, 2007], mesoscale traveling ionospheric disturbances [Miller et al., 2009], and large-scale wave structure in the bottomside ionosphere [Tsunoda, 2005]. Some of these have been observed in association with ESF [Saito and Maruyama, 2007; Fritts et al., 2009; Miller et al., 2009].
2 SAMI3/ESF Modeling of EPBs
 The Naval Research Laboratory SAMI3/ESF code [Huba et al., 2008], which is based on the SAMI2 (Sami2 is Another Model of the Ionosphere) [Huba et al., 2000] and SAMI3 [Huba and Joyce, 2010] ionosphere codes, has been used for numerous studies of ESF [e.g., Huba et al., 2009a and 2009b; Krall et al., 2009]. The two-dimensional potential equation in SAMI3/ESF is based on current conservation (∇ ⋅ J = 0). This version of SAMI3/ESF includes the vertical component of the neutral wind in the potential and momentum equations.
 As configured for this study, SAMI3/ESF is limited to 8° in longitude with periodic boundary conditions. For simplicity SAMI3/ESF uses a nontilted dipole field, so magnetic latitude and geographic latitude are the same, and the geographic longitude is centered on 0°, so universal time and local time are the same. In all cases the geophysical parameters are F10.7 = 150, F10.7A = 150, Ap = 4 and day-of-year 80. As in previous studies, the initial state of the SAMI3/ESF ionosphere is computed using SAMI2 [Huba et al., 2000]. These same parameters have been used in previous SAMI3/ESF simulations [e.g., Krall et al., 2011], but with different seed perturbations.
 For these runs the gravity wave is modeled as an imposed horizontally-propagating wind perturbation with horizontal wavelength λ = –250 km, vertical wavelength λz = –50 km, and period 20 min. The amplitude of the horizontal component of the wind perturbation is UGW = 40 ms–1. These simple parameters are consistent with observed gravity wave periods and wavelengths [Vadas et al., 2009] and are chosen to obtain good coupling between the gravity waves and the ionosphere.
 The zonal (δux), meridional (δuy) and vertical (δuz) components of these winds are
where x is a zonal coordinate that has its origin at the gravity wave source location, y the corresponding meridional coordinate, z is the vertical coordinate, k = 2π/λ, and m = 2π/λz. In these simulations we have set the background winds to zero such that the only winds in the system are these perturbations, which propagate outward from the source with circular phase fronts. In the case where the source location is far from the region of interest, the phase fronts are approximately planar.
 We control the effect of our model gravity wave field by localizing its effects to altitudes 150 km < z < 450 km, longitudes between –3° and +3° (avoiding the periodic boundaries at ±4°) and in latitude to between 5° and 15°. Because the potential field calculation treats geomagnetic field lines as equipotentials, the resulting disturbance in the electric potential extends along the field.
 Before considering the question of circular waves (local, with the source inside the simulated gravity wave field) versus plane gravity waves (nonlocal; source location far from the simulated gravity wave field), we considered the optimum location of the gravity wave field itself. In particular, past simulations [Krall et al., 2011] have shown that a disturbance can couple effectively to the F layer if that disturbance occurs in a region of the lower ionosphere that “maps” along the geomagnetic field to the bottom side of the F layer at the equator. With our gravity wave “seed” field centered at height 300 km and longitude 0, we varied its location in latitude and found that it was most effective in generating EPBs when centered at latitude ±10°. This result can be explained as follows.
 The growth rate for the ESF instability can be written in terms of field-line integrated quantities [Sultan, 1996]. In the absence of winds,
s is a coordinate along the field line, and p is a coordinate perpendicular to the zonal direction and (approximately) to the field line [see also Krall et al., 2009, Appendix]. Gravity being directed downwards, gp < 0 and γ is always positive (unstable) in the bottomside F-layer, where Ln > 0. In a previous study [Krall et al., 2010] we showed that the approximate condition for ESF growth to cease is for the vertical gradient in the field-line averaged density <ρ > to fall to zero, ∂ < ρ >/∂ p = 0.
 For example, consider Figure 1 (top panel), which shows the electron density in the F layer at 20:00, before the development of EPBs (this simulation will be referred to below as the ∆ = 2.8° case). Also shown are representative field lines (white curves), with a white dashed curve indicating the field line where the theoretical growth rate is maximum. Because γ and < ρ > are functions of field line, a common way to visualize them is to plot them versus height at the equator (i.e., along line A in Figure 1), as is done in Figure 1 (lower left). We see that the growth rate peaks in the bottomside F layer, falling to zero near the peak of the < ρ > profile. Of more interest for this problem, however, is the plot of the growth rate versus latitude at constant altitude (line B), at the altitude where the seed perturbation will be applied. This is shown in Figure 1 (lower right). At this altitude, the peak growth rate corresponds to latitude 7°. However, the local electron density at this point (solid curve) is more than 2 orders of magnitude lower than the peak density. That we found the strongest coupling with the gravity waves centered at 10° instead of 7° indicates that the coupling of a perturbation to the F layer within a period of interest (i.e., during the night) requires sufficiently large values for both the growth rate and the local electron density. In fact, the effect of the imposed gravity wave can be seen in the modulations of the local electron density in Figure 1 (lower right). A yellow dashed curve in Figure 1 (top panel) marks the field line of optimum coupling between the seed perturbation and the ESF instability, as determined by our simulations. Additional simulations (not shown) indicate that shifting the gravity wave field to higher altitudes moves the optimum position closer to the equator, as expected [Tsunoda, 2010b].
 Having established the optimum location for our gravity wave field, we will now consider simulations in which the location of the gravity wave field is fixed, while the location of the gravity wave source is varied. In this way we will see the transition from circular to plane wavefronts in the gravity-wave perturbation region.
 We performed several runs of the SAMI3/ESF code, each beginning at 19:20 UT, after the F layer has been lifted by the prereversal enhancement. In each case there was no initial perturbation of the ionospheric density, but the gravity-wave wind field was imposed as described above. The effect of the gravity wave winds on the ions is both direct, with ions being pushed along the magnetic field, and indirect, with ion density perturbations being “blown” by zonal winds via E × B drifts.
 For example, Figure 2 shows log10(ne) − log10(ne(t = 0)) at constant height 290 km plotted versus longitude and latitude in the northern hemisphere, in the region affected by the gravity-wave wind field. By differencing the log of the electron density in this way, the differences in the initial gravity wave perturbations are clear, as well as the development of EPBs. Shown are three simulations, each represented by an early-time plot on the left (20:00, before EPBs develop) and a later-time plot on the right (22:00, during the growth of EPBs). Note that contour levels are different in the right-hand and left-hand panels. The top row of Figure 2 shows a simulation in which the gravity wave source location is at longitude 4° and latitude 10°, at the edge of the simulation region and just outside the right edge of the plot. The middle row shows a simulation in which the gravity wave source has been displaced ∆ = 2.8° northeast (2° north and 2° east). In the lower row, ∆ = 5.7°. In the earlier plots (20:00 UT, left-hand column) the effect of the imposed winds on the ionosphere is evident. In the top row, the perturbation shows circular phase fronts. In the middle row, the phase fronts are curved, but generally with a larger radius of curvature than in the top row. In the bottom row the gravity wave source is further outside of the simulation region and the phase fronts are more plane-wave-like. Plots in the right-hand column show the growth of EPBs, with lower density values at higher latitudes indicating faster growth of the instability. The lowest-density regions are indicated by the dark blue regions that correspond to the EPB voids. Clearly, cases where the imposed wave phase-fronts have a smaller radius of curvature correspond to faster growth of the ESF instability. As always, the simulated ESF-related airglow depletions move outward in latitude as the corresponding EPBs move upward in altitude [Makela, 2004].
 We analyzed ESF growth in each case, using the peak vertical E × B drift as a proxy for the ESF amplitude. In Figure 3 shows this proxy plotted versus time for each of three simulations (∆ = 0, 2.8°, 5.7°). Relative to the circular-phase-front case (∆ = 0), the slowest growing case takes over 40 min to reach the same given level of ESF growth.
 As suggested by Tsunoda [2010a], we find the fastest growth when the model gravity-wave phase fronts are most strongly curved (see Figure 2, upper left). As the gravity wave source point is displaced north and east, the waves within the simulation region become more plane-like with phase fronts propagating at an angle of 45° to the magnetic equator (see Figure 2, lower left). These two cases are intended to be similar to the two panels of Figure 1 of Tsunoda [2010a]. However, whereas Tsunoda [2010a] describes an idealized situation in which the coupling of the wave to the F layer has little spatial variation with the result that a plane-wave perturbation does not excite an E field, we instead find weak coupling. Specifically, the time to reach a given level of ESF growth increases by over 40 min between the most unstable and least unstable cases (see Figure 3). A key difference between the model ionosphere and Tsunoda's idealized case can be seen in Figure 1 above, where both the growth rate and the local electron density (lower right panel) vary rapidly across the region where the model gravity waves were imposed (outlined by a white box in the upper panel). Nevertheless we conclude that his concept is essentially correct.
 In addition, we have analyzed the result, first published in Krall et al. , that for an imposed wave centered at altitude 300 km, magnetic latitude 10° is the optimal location to excite the ESF instability for these conditions. Specifically, we find that the coupling of a seed wave to the F layer not only requires that the seed wave map to the bottomside F layer, where the ESF growth rate is largest (see Figure 1), it also requires a sufficiently large values local electron density. The latter is needed to support the E field that drives the instability. If the local value of ne was not important the optimum position for the seed wave in our example would be latitude 7°, where the growth rate at altitude 300 km is maximum, instead of 10°, where ne is much larger. Of key importance is the altitude of the imposed seeding: higher altitudes correspond to optimum locations closer to the magnetic equator.
 This work was supported by the Naval Research Laboratory Base Funds and NASA.