A direct link appears to exist between gravity wave (GW) activity in the troposphere and plasma structure in the nighttime equatorial F layer during the solstices. This finding strongly favors a role being played by a neutral-ion coupling process, which involves (1) a spatially-varying dynamo current, generated in situ by a GW, and (2) perturbation transfer to the plasma by a polarization electric field. We show, for the first time, how two more of the puzzling features in observed plasma structure, referred to as equatorial spread F (ESF), can finally be explained, if we allow GWs to have circular, instead of plane, phase fronts. These features include (1) early appearance of large-scale wave structure (LSWS) when the F layer is low, and (2) common appearances of a monochromatic LSWS with a long zonal wavelength.
 There is mounting evidence that the development of plasma structure in the nighttime equatorial F layer, known as equatorial spread F (ESF), is not solely controlled by the post-sunset rise (PSSR) of the F layer, at least not during the solstices; seeding by atmospheric gravity waves (GWs) is beginning to emerge as an important contributor. As case in point, there are features in ESF morphology (i.e., longitudinal and seasonal behavior) that are not easily accountable only in terms of a PSSR [see McClure et al., 1998, and references therein]. Most apparent are occurrences of ESF during solstices, when PSSR is weak. These occurrences have now been shown to be associated with longitudes, where the inter-tropical convergence zone (ITCZ) aligns with the magnetic dip equator [Tsunoda, 2010]. This finding provides the first direct link between GW source regions and ESF.
 Hence, we now have a sound basis to believe that wavelike, spatial modulation of isodensity contours in the bottomside equatorial F layer [Tsunoda and White, 1981], referred to as large-scale wave structure (LSWS), is the seed for ESF [Tsunoda, 2005]. All indications are that LSWS always precedes the appearance of ESF [Tsunoda, 2005, 2008, 2009; Thampi et al., 2009; Tsunoda et al., 2010]. Most conclusive, thus far, is the finding that LSWS followed by ESF still developed, despite an absence of the PSSR, on a night of very low solar activity [Tsunoda et al., 2010]. This finding clearly shows that both LSWS and ESF are not strictly dependent on the presence of a PSSR, hence, an interchange instability. These findings, taken together, make it clear that seeding participates actively in the development of ESF.
 An important next step is to better understand the relationship between GWs and LSWS. Tsunoda  suggested that neutral-ion coupling likely involves a spatially-varying dynamo current associated with a GW, and the appearance of a polarization electric field (p), when phase fronts of the GW align with the geomagnetic field () [e.g., Klostermeyer, 1978; Huang and Kelley, 1996; Keskinen and Vadas, 2009]. This requirement for alignment provides a reason why it may be necessary for the ITCZ, where the most intense sources of GWs reside [e.g., Waliser and Gautier, 1993], to be in proximity of the magnetic dip equator. Given the likelihood that this is the process that links GWs and LSWS, we take a closer look at some of its properties in this paper.
1.2. Problem Statement
 Thus far, all theoretical treatments of this coupling process have assumed GWs to be plane waves (referred to hereafter as PGWs) [Klostermeyer, 1978; Huang and Kelley, 1996; Keskinen and Vadas, 2009]. But, there is evidence, both experimental and theoretical, that GWs actually have circular phase fronts (referred to hereafter as CGWs). For example, all-sky images of nightglow in the mesosphere, have shown that a thunderstorm cell launches CGWs with phase fronts that emanate from its center [e.g., Taylor and Hapgood, 1988; Sentman et al., 2003]. Computer simulations have also demonstrated that the appearance of circular wave fronts is likely [Vadas and Fritts, 2004]. For the GWs of interest, the horizontal distance of propagation before dissipation appears to be in the range, from 250 to 2000 km [e.g., Waldock and Jones, 1987; Vadas, 2007]. At these distances, the phase fronts are still circular. Hence, we have reason to consider the response of p to CGWs, and to compare the differences in response to PGWs.
2.1. Plane Waves
 The basic process involves a dynamo-driven Pedersen current (p), driven by neutral-wind perturbations () associated with a PGW; is in the vertical plane and directed transverse to its wave vector (). For simplicity, we assume that the background neutral wind () is zero. Ion velocity in the F region is given by
where ρi = νin /Ωi is ion-neutral collision frequency normalized by the angular ion gyrofrequency. The electric field () could include background and GW-related components. Without , we have i = ρi ( × )/B. Hence, p oscillates spatially as p ≈ Neρi ( × ) = σP( × ), in the direction of , where σP is the Pedersen conductivity. Note that the spatial variation in p is produced by , and not by gradients in plasma density (N) or νin [e.g., Klostermeyer, 1978], as is usually the case. For simplicity, we use Cartesian coordinates, whose origin is on the magnetic dip equator. We simply assume is horizontal and directed eastward along the x axis, and is along the z axis (positive upward). Thus, p is aligned with , in the zonal direction, and given by
 The net Pedersen current of a PGW wave train on any given line must be zero to be divergence-free. Integrating (2) along (y axis) and equating it to a polarization current, we have,
where ΣP is field-line-integrated σP (with superscripts to indicate E or F layer), and Epx arises to maintain current continuity. (Strictly speaking, E layers would not be on horizontal lines in the F region, as they would be on dipolar lines. We include ΣPE just for discussion purposes.) Solving (3) for Epx, we have
where z is the net uz after integration. For the ‘open circuit’ case (i.e., ΣPE = 0), we have Epx = zB. Hence, p is aligned with and plasma transport (i.e., p × /B) is identical with z. An upward (downward) z sets up an eastward (westward) p, and z = uz when phase fronts align with .
 According to (4), the dynamo and polarization process occurs in the F layer with electrical loading by the E layer. The finding that LSWS has been detected most often around E-region sunset [Tsunoda and White, 1981; Singh et al., 1997; Tsunoda, 2009] is consistent with this description. The finding of p fluctuations [Eccles, 2004] and LSWS [Thampi et al., 2009] prior to E-region sunset is not inconsistent, but it may imply a role being played in the E region.
 The polarization response to GW presence, described by (3) and (4), is illustrated in Figure 1. Red and blue lines indicate phase fronts where uz has maximum amplitude but opposite signs. The dotted lines indicate where uz is zero. When plane phase fronts are aligned with (vertical lines with arrowheads), the behavior is as described above. When they are not aligned, the geometry becomes that in Figure 1a. Notice what happens on a given line, such as along the bold green line. A red (blue) dot indicates where uz > 0 (uz < 0), and where the largest local p would appear. The net p on that line is determined by integrating along . The mapping distance is proportional to (where σ0 is direct conductivity) and to the transverse wavelength of p [Farley, 1959]. With a transverse wavelength (λ) of p (hence, GW) on the order of 100 km, mapping would occur without any attenuation. Because oppositely directed p cancel, the net p is negligible; hence, a modulated p pattern would not appear.
2.2. Circular Waves
 Polarization response in presence of a CGW is sketched in Figure 1b. We assume that a localized, mesoscale convective cell (MCC), located at the origin, launches a CGW. All of the circular phase fronts will intersect each line. Hence, some cancellation of p occurs, but, there is a region (near the origin), where cancellation does not occur. We call it the first Fresnel zone (from an analogy in optics). To show that the residual p is still substantial, we assume, for any given instant in time, that uz varies sinusoidally, as follows,
where, in Figure 1b, d is the zonal distance from MCC to the line of interest (bold green line), λ is the distance between two adjacent blue (or red) curves, r is the radius of curvature (i.e., distance from MCC to any point on the line of interest), and is the angle between d and r. According to (4), we can use (5), integrated along to represent the behavior of z, and hence, Epx. Notice that Epx depends on uz0, which is the peak amplitude of uz, and the ratio d/λ.
 As an example, uz/uz0, which is equivalent to a normalized Epx (in the open-circuit case), and obtained by using (5) for d = 400 km and λ = 400 km, is plotted in Figure 2a. Notice how broad the positive-valued region is, compared to areas under the remaining regions of alternating sign. Hence, most of Epx from the smaller regions cancel, while leaving a large contribution from the broad positive-valued region; this is shown by the upper curve in Figure 2b, where we have plotted (5) after integration along up to the value of shown along the abscissas. (Full integration along a straight line would be out to = ±90°.) We see that the net uz/uz0 approaches a constant value, when integration is taken beyond, say θ ≈ 60°. Hence, the strength of the net Epx (on a given line) depends on d/λ, as seen by comparing upper and lower curves in Figure 2b. Reducing λ to 40 km, d/λ increases by a factor of 10, and net uzB is reduced by a factor of four. As distance from the MCC increases, the phase fronts begin to approach those of a PGW, Epx approaches the value for the aligned PGW case. The modulation of net Epx with d (or x) is sketched at the bottom of Figure 1b.
3. Discussion and Further Results
 In the following, we discuss how the neutral-ion coupling process (involving CGWs) appears capable of explaining two puzzling features that are observed in ESF development: (1) low-altitude seeding, and (2) long, monochromatic nature of LSWS.
3.1. Low-Altitude Seeding
 Observations indicate that ESF can develop, when the PSSR is weak (e.g., during solstices) [Tsunoda, 2010], or even absent (e.g., during very low solar activity) [Tsunoda et al., 2010]. In other words, seeding of LSWS must be efficient, when the F layer is low. According to (4), other than E-region loading (which is represented by ΣPE/ΣPF), the strength of Epx depends strictly on z. Given that we must be concerned with dissipation of GWs at higher altitudes [e.g., Vadas, 2007], it may turn out that uz is larger at lower altitudes. It is well known that the Rayleigh-Taylor (RT) instability is weak and not a likely contributor at low altitudes.
3.2. Long Wavelengths
 We have already demonstrated that the polarization response favors longer λ; that is, Epx would be larger for d/λ values less than unity. For example, if the base of the F layer is at 250 km altitude, and CGW propagation occurs at an elevation angle of, say 40°, d would be about 400 km. Then, d/λ= 1 for λ = 400 km, which is the example in Figure 2a. For experimental estimates of d, we referred to Waldock and Jones , assuming GW propagation is similar at low and middle latitudes. Through inverse ray tracing, they found that the source region was located between 250 and 1500 km in distance from the observation location. This means λ would have to be larger, if d increases, in order to keep the ratio near unity. Hence, according to our model, the observed λ should have values similar to d, between 250 and 1500 km.
 In fact, this range of λ values appears to match observations. Given that ESF occurs in patches, which are collocated with upwellings in LSWS [Tsunoda and White, 1981], we can use the longitudinal spacing of ESF patches as a measure of λ. Röttger  found that patch spacing can vary from less than 100 km to more than 1500 km, with a median of 380 km. The zonal λ of LSWS, measured with an incoherent-scatter radar, was 400 km [Tsunoda and White, 1981]. In situ satellite measurements indicate ranges of λ, as follows: (1) 350 to 630 km [Oya et al., 1982], (2) 150 to 800 km with an extreme value of 3000 km [Singh et al., 1997], and (3) 300 to 800 km [Eccles, 2004]. More recently, zonal λ of 300 to 600 km were found in LSWS detected with all-sky images of 630 nm nightglow [Makela and Miller, 2008]. These values are also comparable with theoretical estimates of λ for GWs that can reach the thermosphere [e.g., Vadas and Fritts, 2004].
3.3. Interaction Time
 Given that p is proportional to z, which means that p should exist whenever CGWs are present. According to Eccles , p appears ubiquitous; hence, CGWs must be ubiquitous. But, he also found that detection of perturbations in p was not always followed by the development of LSWS and plasma bubbles. He concluded that the RT instability must assist in the seeding process. But, in the absence of a PSSR, hence, a low F layer, it is not clear how large LSWS amplitude can become from neutral-ion coupling alone. It is not uncommon for LSWS to have a peak-to-trough amplitude of 60 km [e.g., Tsunoda and White, 1981]. Amplitude growth rate, however, cannot be faster than z. If uz is 100 m/s (which seems unrealistically large), the time required would be only 10 min; but if uz is 10 m/s (which is more reasonable), the time required would have to be 100 min. In comparison, the lifetimes of GWs could be as much as 4 hr after excitation [Vadas and Fritts, 2004]. Hence, in the absence of a boost from an interchange instability, LSWS amplitude likely depends on time available for neutral-ion coupling to take place.
 To prolong interaction time, the GW and F-region plasma must travel together with minimum relative motion. We already know that LSWS grows in amplitude without significant zonal transport [Tsunoda and White, 1981; Tsunoda, 2005; Thampi et al., 2009]. If there is any noticeable transport, it appears to be slowly westward [Tsunoda, 2007]. This behavior of F-region plasma is consistent with drift, which occurs below the vertical shear in zonal plasma motion. This means that GW phase velocity must be nearly zero in the Earth-fixed frame. Given that the mean zonal is typically eastward around sunset, the GWs of interest must be those that propagate westward (in the frame of the neutral gas). This scenario seems reasonable because westward-propagating GWs appear more capable of penetrating into the thermosphere, in the presence of a mean eastward , than eastward-propagating GWs [e.g., Fritts and Vadas, 2008; Fritts et al., 2008]. Further, we note that GWs usually have a downward component of phase velocity, seeding must be insignificant during the PSSR.
3.4. Monochromatic Waves
 An interesting possibility arises, if we recognize that long interaction time (“spatial resonance”) is basically a narrow-band process, whereas GWs are dispersive. That is, the phase velocity of a GW is a function of its λ. This means that only GWs within a narrow band of λ will contribute to LSWS amplification. In fact, observations seem to indicate that the appearance of LSWS is usually wavelike and monochromatic [e.g., Tsunoda and White, 1981; Tsunoda et al., 2010]. Until now, this appearance of LSWS had remained unexplained.
3.5. Scenario for LSWS Development
 Given the above findings, we envision the following. Seeding should be possible, starting around sunset, when E-region loading is reduced [Tsunoda and White, 1981; Singh et al., 1997; Tsunoda, 2009]. Seeding occurs in westward-drifting plasma, which suggests that E-region loading is likely still important. Amplification by spatial resonance occurs under relatively stable conditions, which allows selection of λ, and occurs over a time interval long enough for the monochromatic LSWS to grow substantially relative to those associated with other λ. In the absence of a PSSR [e.g., Tsunoda et al., 2010], these conditions may persist longer, which would allow the needed amplification. At times after sunset, polarization response should improve with further decay of the E layer. The growth of LSWS would likely continue, but its waveform would likely be distorted by contributions from GWs with other values of λ, which should occur when there are small changes in zonal transport of plasma, or in . This notion is consistent with the fact that ESF tends to develop well after the expected time of the PSSR [e.g., Tsunoda et al., 2010], and perhaps even after local midnight.
 If a PSSR does occur, the seeding process is likely disrupted by the vertical bulk-plasma transport and the acceleration of the zonal ; the latter is a consequence of reduced ion drag, which occurs because of the vertical plasma transport. Seeding may stop at this point, but amplification of existing LSWS likely continues, via an interchange instability. This scenario is consistent with observations of LSWS, PSSR, and ESF [Tsunoda and White, 1981; Tsunoda, 2008, 2009; Thampi et al., 2009; Tsunoda et al., 2010].
4. Summary and Conclusions
 The findings reported herein are promising. They appear to support the neutral-ion coupling process by which uz perturbations are transferred via p to the plasma in the bottomside F layer. The process is appealing because it favors seeding at the base of the F layer, when it is low in altitude, and this property is consistent with the findings that (1) seeding appears to be important during solstices, when PSSR is weak [Tsunoda, 2010], and (2) LSWS and ESF can occur in the absence of a PSSR [Tsunoda et al., 2010]. We also seem to have explanations for why LSWS tends to have a long zonal λ (e.g., 400 km or more), and to appear monochromatic. The reason is not simply because it is the λ of the GW that reaches the thermosphere or that selected by velocity-shear effects in the plasma [e.g., Hysell and Kudeki, 2004]. Instead, the appearance of LSWS with long λ appears to be related to the Fresnel zone of CGWs, and its appearance in the form of monochromatic waves appears to be related to the spatial resonance process.
 This research was supported by the National Science Foundation under grant ATM-0720396, and by the Air Force Office of Scientific Research under contract FA9550-10-C-0004.