The Naval Research Laboratory (NRL) has developed a new three-dimensional code to study equatorial spread F (ESF). The code is based on the comprehensive NRL 3D ionosphere model SAMI3 and includes a potential equation to self-consistently solve for the electric field. The model assumes equipotential field lines so a 2D electrodynamic problem is considered. In this study a narrow wedge of the post-sunset ionosphere is simulated. It is found that (1) bubbles can rise to ∼1600 km, (2) extremely steep ion density gradients can develop in both longitude and latitude, (3) upward plasma velocities approach 1 km/s, and (4) the growth time of the instability is ≃15 min. These results are shown to be consistent with radar and satellite observations.
 Equatorial spread F (ESF) [Haerendel, 1974; Ossakow, 1981; Hysell, 2000] is a post-sunset phenomenon in which the equatorial F-region ionosphere becomes unstable: large-scale (10's km) electron density ‘bubbles’ can develop and rise to high altitudes (≳1000 km at times). ESF is an important space weather concern because it scintillates radio signals that can degrade and disrupt communications and navigation systems. To understand the complex and dynamic evolution of equatorial spread F, numerical simulation models are required. The majority of simulation studies have been based on a two-dimensional electrodynamic model (i.e., the geomagnetic field lines are assumed to be equipotentials) [Scannapieco and Ossakow, 1976; Zalesak and Ossakow, 1980; Zalesak et al., 1982; Huang et al., 1993; Chou and Kuo, 1996; Sekar, 2003; Huba and Joyce, 2006]. Recently, Keskinen et al.  reported results using a three-dimensional electrodynamic model. However, all of these models used a prescribed, initial background ionosphere model that did not evolve in time along the magnetic field.
 In this letter we report the results of a fully 3D spatial model of equatorial spread F that describes the motion of ions along and transverse to the geomagnetic field in a narrow longitudinal wedge of the post-sunset ionosphere. We note that similar wedge-type models of ionospheric phenomena have been developed by Retterer  and Besse et al. .
2. Numerical Model
 A modified version of the NRL 3D global ionosphere code SAMI3 is used in this analysis. SAMI3 is based on the 2D ionosphere model SAMI2 [Huba et al., 2000]. SAMI3 models the plasma and chemical evolution of seven ion species (H+, He+, N+, O+, N2+, NO2+ and O2+). The complete ion temperature equation is solved for three ion species (H+, He+ and O+) as well as the electron temperature equation. The plasma equations solved are as follows:
 Ion inertia is included in the ion momentum equation for motion along the geomagnetic field. The neutral species are specified using the empirical NRLMSISE00 model [Picone et al., 2002]. For this study the magnetic field is modeled as a dipole field aligned with the earth's spin axis and the neutral wind is set to zero (i.e., Vn = 0).
 The potential equation is derived from current conservation (∇ · J = 0) in dipole coordinates. The equation used in this study is
where Σpp = ∫(pΔ/bs)σPds, Σpϕ = ∫(1/pbsΔ)σPds, Fϕg = ∫(rEsin3θ/Δ)(B0/c)(1/Ωi)σHigpds, σP is the Pedersen conductivity, σHi is the ion component of the Hall conductivity, Δ = (1 + 3cos2θ)1/2, bs = (rE3/r3)Δ, and rE is the radius of the earth. The field-line integration is along the entire flux-tube (i.e., s direction) with the base of the field lines at 85 km. Equation (5) is similar to that derived by Haerendel et al. . The perpendicular electric field E⊥ = −∇Φ is used self-consistently in SAMI3 to calculate the E × B drifts.
 The 3D simulation model is initialized using results from the two-dimensional SAMI2 code. SAMI2 is run for 48 hrs using the following geophysical conditions: F10.7 = 170, F10.7A = 170, Ap = 4, geographic longitude is 0° so universal time and local time are the same, and day-of-year 263 (e.g., 20 Sept 2002). The plasma is modeled from hemisphere to hemisphere up to ±26° magnetic latitude; the peak altitude at the magnetic equator is ∼1600 km. The E × B drift in SAMI2 is prescribed by the Fejer/Scherliess model [Fejer and Scherliess, 1995]. The plasma parameters (density, temperature, and velocity) at time 1920 UT (of the second day) are used to initialize the 3D model at each magnetic longitudinal plane.
 The 3D model uses a grid with magnetic apex heights from 200 km to 1600 km, and a longitudinal width of 4° (e.g., ≃460 km) centered at 2°. The grid is (nz, nf, nl) = (101, 130, 96) where nz is the number grid points along the magnetic field, nf the number in ‘altitude,’ and nl the number in longitude. This grid has a resolution of ∼10 km × 5 km in altitude and longitude in the magnetic equatorial plane. The grid is periodic in longitude. In essence we are simulating a narrow ‘wedge’ of the ionosphere in the post-sunset period. Finally, a Gaussian-like perturbation in the ion density is imposed at t = 0: peak ion density perturbation of 15% centered at 2° longitude with a half-width of 0.5°, and at an altitude z = 400 km with a half-width of 60 km. A simulation study was also performed with a 5% perturbation and similar results were obtained.
Figure 1 shows the temporal and spatial evolution of the ionospheric plasma bubble from the 3D simulation model. In this figure, electron density contours are shown as a function of longitude and altitude (left panel, linear scale), and latitude and altitude (right panel, logarithmic scale). Three times are shown: 2011 UT (bottom), 2103 UT (middle), and 2137 UT (top). The first time shown is roughly 50 min after the start of the simulation.
 The rise of the plasma bubble in the magnetic equatorial plane is shown in the left panel. This is the ‘standard view’ of ESF bubble simulation results (the geomagnetic field is into the page). The bubble rises from 400 km to 1000 km in roughly one hour, and then to almost 1600 km in ∼1/2 hour. The bubble forms a narrow channel as it penetrates the thickest portion of the F-layer, but then mushrooms once it reaches the lower-density, topside F-region. The maximum width of the bubble is ∼70 km. Radar observations using the Julia radar have detected ESF bubbles near 1600 km for geophysical conditions similar to those used in this simulation study [Valladares et al., 2004]. Moreover, there is evidence that ESF bubbles can reach apex heights of over 2000 km [Mendillo et al., 2005; Martinis and Mendillo, 2007].
Figure 1 (right) shows the evolution of the ionosphere in the latitudinal plane at the center of the bubble (longitude of 2°). Figure 1 (bottom) (2011 UT) the Appleton anomaly is evident with ionization crests at ±16°. At time 2103 UT there is a very large reduction in electron density on flux tubes with a magnetic apex <1000 km; in fact, the Appleton anomaly crests are being dramatically ‘pushed’ aside. Finally, at time 2137 UT the electron density in the entire latitudinal plane has been severely reduced. Interestingly, there is evidence of a weak, secondary Appleton anomaly forming at ±20° associated with the strong uplifting of the plasma bubble. The reduction in the electron density as a function of latitude also leads to a reduction in the total electron content (TEC) as the bubble rises in altitude in this longitude sector. This result is consistent with the differential TEC analysis of Valladares et al.  that shows a temporal reduction in TEC within ±(10°–15°) of the magnetic equator when scintillations are observed.
 To better illustrate the three-dimensional nature of this phenomenon we present Figure 2. An animation of this figure with 4 min resolution is available as auxiliary material. We show electron density isosurfaces and contour maps at time 2112 UT. The ‘closed’ red isosurface is for a density 3.2 × 106 cm−3 and shows the location of the ionization crests, and the blue isosurface is for a density 6.3 × 104 cm−3. The 2D contour maps are in the plane of the magnetic equator (as in Figure 1) and orthogonal to the plane of the magnetic equator (i.e., showing the latitudinal distribution of plasma density near the base of a flux tube but not at a constant height). In this figure it becomes clearer that low density plasma (e.g., lower blue isosurface) is being lifted from low altitudes to higher altitudes all along the flux tube resulting in the reduction of plasma within the entire flux tube extending to both high altitude (up to 1600 km) and to midlatitudes (up to 25 degrees). Additionally, it is evident that the high density ionization crests are being primarily ‘pushed’ aside (in longitude) by the rising low density plasma.
 The results shown in Figures 1 and 2 are quantified in Figures 3 and 4. Figure 3 is a plot the electron density as a function of altitude (at the apex of the geomagnetic field line) at times 2011 UT, 2103 UT, and 2137 UT. The dashed line is the initial electron density at 1920 UT. At time 2103 UT there is a reduction in the peak electron density of an order of magnitude (at an altitude z ∼ 500 km); additionally, the bubble minimum density is ∼3 × 104 cm−3 at z ≃ 1000 km. Finally, at time 2137 UT the minimum bubble density has risen to z ≃ 1400 km. At altitudes below ≃1000 km there is a refilling of the flux tubes; the electron density is larger at time 2137 UT than at 2103 UT in this altitude range.
 In Figure 4 we plot the electron density as a function of latitude at an altitude z = 367 km at times 2011 UT, 2103 UT, and 2137 UT. Again, the dashed line is the initial density profile at 1920 UT. At time 2011 UT there is a reduction in the electron density in the latitude range −10° to 10° as the bubble is developing. At time 2103 UT there is a severe reduction of over three orders of magnitude in the electron density in the range −10° to 10°; moreover, there is an extremely sharp density gradient at the bubble/quiescent ionosphere density boundary (∼±16°). Finally, at time 2137 UT the bubble boundary has extended to ∼±23° and the interior region is beginning to refill.
 The results shown in Figures 3 and 4 are consistent with satellite observations made by Ogo 6 [Hanson and Sanatani, 1973]. Hanson and Sanatani [1973, Figure 5] shows an altitude profile of the ion concentration with an order-of-magnitude reduction in the peak density that is similar to Figure 3 (albeit with more vertical structure). Hanson and Sanatani [1973, Figures 2, 3, and 8] show very strong latitudinal plasma density gradients; the density drops by up to three orders-of-magnitude within 5°. Additionally, Valladares et al.  has shown a bubble extending to ≃1600 km (detected by the Julia radar) and corresponding scintillations up to a geographic latitude 11°N. This corresponds to a magnetic latitude ≃22° which is consistent with the latitude extent of the bubble shown in Figure 4.
 The E × B drift velocity reaches a maximum value ≃900 m/s roughly 1.5 hrs after the start of the simulation (at ≃2050 UT) and then slowly decreases. This result is consistent with an analytical model for the E × B bubble velocity developed by Ossakow and Chaturvedi . For a large fractional depletion of the plasma bubble they estimate the velocity to be ∼103 m/s at 500 km. Lastly, we point out that the observed bubble rise (e.g., top boundary of the bubble) is less than the maximum E × B velocity. The top of the bubble rises from 400 km to almost 1600 km in ∼1.5 hrs; this gives a bubble rise velocity of ∼222 m/s consistent with observations. Also, we find that the growth period for the mode is τg ≃ 15 min. This result is consistent with the flux-tube integrated Rayleigh-Taylor instability growth time calculated by Sultan  based on linear theory.
 We have reported new results from a fully 3D spatial model of equatorial spread F that describes the motion of ions along and transverse to the geomagnetic field. We find that (1) bubbles can rise to ∼1600 km, (2) extremely steep ion density gradients can develop in both longitude and latitude, and (3) upward plasma velocities can approach 1 km/s, and (4) the growth time of the instability is τg ≃ 15 min. These results are shown to be consistent with radar and satellite observations.
 In this simulation study, the plasma bubble eventually rises through the upper boundary at 1600 km, i.e., the upper boundary does not impede bubble rise. This begs the question ‘what physical mechanism(s) stops bubble rise?’ One answer to this question is based on buoyancy considerations. Early papers on this subject [Ott, 1978; Ossakow and Chaturvedi, 1978] suggested that bubble rise stops when the density of the plasma bubble equals the density of the background plasma. Mendillo et al.  argued in favor of a balance between flux-tube integrated densities to better explain the observation of high altitude bubbles. The simulation results presented in this Letter cannot distinguish between these two cases: the bubble is still rising at the end of the run and neither the local plasma density nor flux-tube integrated plasma density in the bubble is in balance with neighboring, undisturbed flux tubes. We intend to pursue this very important issue in a future study to determine the primary mechanism that inhibits bubble rise.
 The simulation study in this paper has made a number of simplifying assumptions: non-tilted geomagnetic dipole, no zonal or meridional neutral winds, and equipotential field lines. In the near future we intend to relax the first two assumptions and investigate their impact on ESF. Specifically, we will use an IGRF-like magnetic field model and will incorporate the effects of both a zonal wind and a meridional wind. The effects of a zonal wind are to transport plasma bubbles in the wind direction (usually eastward) and to cause a ‘tilting’ of the plumes. Preliminary simulations have been carried out with a zonal wind that show a ‘tilting’ of the ESF bubbles in longitude similar to the results of Zalesak et al. . A key issue we will also address is the impact of meridional winds on ESF. We can self-consistently model the impact of meridional winds on ESF with our 3D model. One effect of a meridional wind is to redistribute the plasma between the hemispheres. It has been suggested that the resulting asymmetry can impact the growth rate of the Rayleigh-Taylor instability [Maruyama, 1988; Zalesak and Huba, 1991]. However, an observational study by Mendillo et al.  did not find any convincing evidence for the wind suppression mechanism; nevertheless, this is an important issue that requires further investigation to better understand day-to-day variability of ESF.
 We thank S. L. Ossakow, S. Basu, and D. Hysell for helpful discussions. This research has been supported by NASA and ONR.