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 Three different approaches to the evaluation of the electrostatic potential in the ionosphere under equatorial spread F(ESF) conditions are considered. First, we calculate the potential using an analytical approach, applying force balance laws to a simplified ionosphere. Second, we compute the potential around a cylinder-like plasma depletion in an idealized ionosphere using both the equipotential field line (EFL) approach and the full 3-D solution to the electrostatic potential problem. Our third approach involves an initial boundary value simulation in a realistic ionosphere using both EFL and 3-D potential solutions. The results show that the equipotential field line assumption does not fully capture the 3-D structure of the ionospheric current system and leads to an underestimation of the growth rate of ESF irregularities in numerical simulations.
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 Equatorial spread F (ESF) refers collectively to a family of plasma irregularities that form in the equatorial Fregion ionosphere after sunset. ESF has been extensively studied with support of coherent scatter radars, ionosondes, airglow imagers, ground-based scintillation receivers, and instruments onboard rockets and satellites (see details inWoodman ). Numerical simulations of ESF has been used as a tool for the understanding of the mechanisms behind it. Most of those simulation studies were either based on two-dimensional models (as, e.g.,Scannapieco and Ossakow , Zalesak and Ossakow , Zalesak et al. , Zargham and Seyler , Huang and Kelley , and Chou and Kuo ) or quasi-three-dimensional models incorporating magnetic flux tube integrated quantities (as, e.g.,Huba et al.  and Retterer ). Keskinen et al. were the first to present ESF simulations incorporating 3-D solutions for the electrostatic potential. Using three-dimensional simulations of ESF under realistic conditions,Aveiro et al. showed that simulated magnetic field perturbations show good qualitative agreement with CHAMP satellite measurements, arguing against an Alfvénic interpretation of all of the CHAMP magnetic field observations in ESF. The field-aligned currents (FAC) showed large amplitudes associated with the divergence of the (mainly gravity-driven) zonal current atF region altitudes in the presence of ESF. FAC flows poleward (equatorward) on the external edges of the western (eastern) walls of the plasma depletions, i.e., strong FAC are associated with transverse currents closing around deep density depletions.
 Here, we evaluate the role of potential variations along the magnetic field and the parallel currents in ESF development. To evaluate how the current loops close in the presence of plasma depletions, we use three different approaches: analytical theory for a simplified ionosphere, numerical computation of the electrostatic potential for an idealized ionosphere, and initial boundary value simulations of ESF under realistic ionospheric conditions. The numerical studies are performed for both the equipotential field line approximation and the three-dimensional computation of the electrostatic potential. The main issue we address here is the degree to which it is possible to describe the ionospheric current circulation with the EFL approximation and how the approximation affects ESF simulations.
2. Analytical Approach
 The three-dimensional equation for the electrostatic potential (Φ) of a warm, single-ion plasma irregularity in a uniform magnetic field ( ) and background neutral wind ( ) is given by [Drake et al., 1985]:
where νin and Ωi are the ion collision frequency and gyrofrequency, respectively, and νe is the electron collision frequency. We assumed thermodynamical equilibrium (Te = Ti = T). The terms U and B represent the wind velocity and magnetic field. Equation (1)neglects background electric fields, Hall terms, gravity, and ion parallel and electron perpendicular diffusion. The time scales for ion parallel and electron perpendicular diffusion are much longer than the process under evaluation. Gravity driven-currents and background electric fields become important near theF peak. Hall terms are negligible in the F region, but become important at E region altitudes. In this coordinate system, ‘x’ points eastward, ‘y’ points upward, and ‘z’ is parallel to the magnetic field.
 We can define a plasma irregularity length along the magnetic field Lz such that Lz = rc(ΩeΩi/νeνin)1/2, where rc is the radius of the circular irregularity in the perpendicular plane. The parallel diffusion is scaled to Γ = 2T/eBUrc and we obtain a residual potential of the form ψ = [Φ + Tln(n)/e]/BrcU. The remaining dimensionless variables are rc∇⊥ → ∇⊥, Lz∂/∂z → ∂/∂z, and n/nb → n, where nb represents the background density. With this new set of dimensionless variables, equation (1) can be rewritten as [Drake and Huba, 1987]
We examine the potential solution of a waterbag plasma irregularity which is a sphere of unity radius in the dimensionless units, i.e., n(r) = 1 + MH(1 − r), where H is the Heaviside step function and the constant M represents the ratio n(r)/nb inside the sphere.
 The potential is linearly separated into two parts (ψ = ψa + ψw): (a) the polarization potential (ψw) of the plasma due to the neutral winds and (b) the ambipolar potential (ψa) due to gradients in the plasma pressure.
 The solution of the polarization potential in the drifting frame of reference is given by
The solution of the ambipolar potential gives
where P2is the second-order Legendre polynomial and the coefficients are given by
A similar set of solutions was derived by Drake and Huba  for ionospheric plasma clouds and applied to plasma cloud stability. Here, we analyze the potential around a plasma irregularity elongated along the magnetic field direction. We assume a 90% depletion to the background density (M = −0.9) and compute the ambipolar and polarization potential.
Figure 1(top left) shows the ambipolar potential in the y-z (vertical-parallel) plane andFigure 1(top right) shows the polarization potential in the x-y (zonal-vertical) plane for a spherical (in normalized coordinates) plasma irregularity embedded in a homogeneous plasma. The ambipolar potential shows a quadrupole structure, with larger gradients along the magnetic field (z-direction). The polarization potential shows a dipolar structure with null potential inside the sphere in the frame of reference that moves with the plasma irregularity.
Figure 1(bottom) depicts the combination of the two potentials into the residual potential in the x-y plane at three zonal cuts: z = 0.0, 0.5, and 1.0. Variation in the residual potential inside and in the vicinity of the plasma irregularity along the magnetic field is evident. In a realistic ionosphere atF-region altitudes, the parallel conductivity is 104–105 times larger than the Pedersen conductivity. Thus, even for small electric fields along B, parallel currents can still be larger than Pedersen currents. Figure 1 (top right) shows the parallel currents estimated from the residual potential. Since parallel currents are proportional to the gradient of the residual potential, large (parallel) currents are flowing in the regions where the gradients in the potential are larger, i.e., at the boundaries of the sphere. Also, small parallel currents are observed flowing both inside and outside the sphere opposite to the main current.
3. Numerical Approach
 Below, we solve the electrostatic potential using two different approaches: (a) equipotential field line assumption (EFL) and (b) full three-dimensional (3-D) solution. The model was constructed using tilted magnetic dipole coordinates (p, q, ϕ), where the tilt is matched to the magnetic declination in the longitude of interest. In our terminology, p represents the McIlwain parameter (L), qis the magnetic co-latitude, andϕ is longitude [see, e.g., Hysell et al., 2004]. The background current density is given by
where the terms in the RHS represent the ohmic, diffusion, and gravity-driven currents, respectively ( , , and represent the conductivity, diffusion, and gravity tensors, respectively).
 The EFL approach consist in using the fact that parallel conductivities are much larger than transverse terms and assume no variations in the electrostatic potential along B. The partial differential equation that describes the instantaneous behavior of the electrostatic potential based on the solenoidal current density (∇ · J = 0) for the EFL approach is given by
The terms and are unit vectors, and σP and σH are the Pedersen and Hall conductivities, respectively. The hi coefficients are the scale factors, where the index i refers to the direction.
 The partial differential equation obtained for the computation of the electrostatic potential in 3-D (also from ∇ · J = 0) can be written in a compact form as
Both equations (7) and (10) are solved using the BiConjugate Gradient Stabilized (BiCGStab) method [e.g., van der Vorst, 1992] using the algorithms described by Saad . The resultant current density is the combination of the background (J0), the polarization ( ), and the (non-divergent) diamagnetic (Jd) current densities (i.e., ).
 To compute the potential, we derive plasma density and composition from the International Reference Ionosphere (IRI-007) Model [Bilitza and Reinisch, 2008], and neutral composition and temperature estimates from the Mass Spectrometer and Incoherent Scatter (NRL-MSISE00) model [Picone et al., 2002]. Zonal neutral winds are obtained from the Horizontal Wind Model (HWM-07) [Drob et al., 2008]. We take the ionosphere and neutral atmosphere to be in thermodynamic equilibrium after sunset (Tn = Te = Ti). Expressions for the ion-neutral and electron-neutral collision frequencies used to compute conductivities can be taken fromRichmond . Expressions for the Pedersen, Hall, and parallel mobilities and diffusivities themselves are found in Kelley , for example. The background electric field is specified and partly controls the forcing. The ion composition includes O+, NO+, and O2+.
 The simulation is cast on a rectangular grid 139 × 133 × 189 points wide in (p, q, ϕ) space constructed using a tilted magnetic dipole coordinate system. A cut through the equatorial plane spans altitudes between 90–510 km and longitudes between ±6°. The flux tubes covered by the parallel coordinate all reach to the lower E region. The computations are performed for moderate solar activity.
3.1. Solution for a Cylindrical Plasma Depletion
 Using a symmetrical ionosphere in relation to the magnetic equator, we compute the background ionospheric conditions. Figure 2 shows the electrostatic potential results for the EFL (Figure 2, middle) and full 3-D solution (Figure 2, bottom). Transverse currents are indicated in Figure 2 (left), and meridional currents are displayed in Figure 2 (right). Figure 2 (top) depicts electron density in transverse (Figure 2, left) and meridional (Figure 2, right) cuts.
 The electrostatic potential obtained using the equipotential field line approach showed vertical currents on the bottomside of the Fregion. Above the bottomside, gravity-driven and Pedersen currents flow eastward. Residual field-aligned currents due to gradients in the electron density along the magnetic field flow equatorward at the topside and poleward near theF peak.
Figure 2(bottom) shows the electrostatic potential computed using the full 3-D solution. Again, gravity-driven and Pedersen currents exist above the bottomside, but large vertical currents flowing from the bottomside to theF peak dominate the scenario. The meridional current panel shows that the vertical currents are partially supplied by equatorward currents connecting the Fregion bottomside to the off-equatorialE region. In order to close the loop, poleward currents connect the topside to the low latitude ionosphere.
 Using the same initially symmetric ionosphere, we added a cylindrical plasma irregularity aligned with the magnetic field and computed the electrostatic potential (Figure 3). In the transverse plane for both the EFL and 3-D solutions, the zonal electric field is enhanced inside the irregularity. Diamagnetic currents flowing around the plasma depletion are evident. The main difference is in the meridional plane, where the 3-D solution shows a complicated current system flowing poleward (equatorward) at the bottom (upper) edge of the plasma depletion. At the meridional borders of the irregularity, a competition between ambipolar currents and parallel currents are evident. In the EFL scenario, only localized ambipolar currents are detected at the meridional edges of the irregularities.
3.2. Initial Boundary-Value Simulation
 Next, we use consider an initial boundary-value simulation applied to a realistic ionosphere. The runs shown below are centered on the dip equator near Kwajalein (5.5°N latitude, 166.5°E longitude, 7.4° declination) with background conditions modeled for August 11, 2004. To seed the simulation run, we added a independent Gaussian white noise to the initial number density with a 20% relative amplitude. We solve a discretized version of the continuity equation for each ion species using a monotone upwind scheme for conservation laws (MUSCL) (a review can be found inTrac and Pen ) directly applicable to the ion continuity problem [see Aveiro et al., 2012]. The characteristic of the neutral atmosphere (densities, temperature, and wind velocity) are updated in time on the basis of inputs from climatological models.
Figure 4shows the results for the (left) EFL approximation and (right) the 3-D solution. The time evolution of the simulation (for t = 0h30, 1h15, and 2h00 after an initialt0 = 0.25 UT) is organized from top to bottom.
Figure 4 (top) depicts the evolution of ionospheric irregularities due to the collisional shear instability (CSI) [Hysell and Kudeki, 2004; Kudeki et al., 2007]. The spatial scales of the irregularities forming at the base of the bottomside range between 30–50 km. These irregularities cannot undergo vertical development and remain confined to altitudes where the plasma flow is retrograde.
 By the time of Figure 4 (middle) (t0+ 1 h), the transient phase of the CSI was ending, and the asymptotic phase was underway. In the EFL approximation, the plasma flows nearly at the neutral speed and the irregularities are damped by diffusion. In the 3-D solution, the 30–50 km waves at the base of the bottomside coalesced into large-scale waves with scale sizes between 100–200 km. Under the action of the background zonal electric field, the bottomside irregularities start to exhibit vertical development.
 In the final panel of the 2-D solution, damping of the irregularities by diffusion is still observed. In the 3-D solution, the irregularities penetrate to the topside where gravity-driven currents can contribute further to growth. Some of the original intermediate scale structure from the first panel in the simulation survives, contributing to the overall fine structure. Bifurcation is evident, as well as secondary instabilities growing mainly on the western walls of the primaries. Some descending, wedge-shaped plasma enhancements accompany the ascending depletions.
 Three different approaches have been used to evaluate the role of the parallel currents in equatorial spread F.Using an idealized ionosphere, analytical results show that parallel electric fields (and currents) are an inherent feature of ionosphere flows. In the second approach, we computed the background current circulation in a simplified ionosphere both with and without irregularity. One of the main results is the inability of the equipotential field line approximation to describe the three-dimensional structure of the currents. The EFL approach underestimates the vertical current flowing through the equatorialF region. Our final approach used an initial boundary value simulations in a realistic postsunset equatorial F region ionosphere. The result show that the suppression of the vertical current flowing throughout the F region and associated differential speed between ions and neutrals seemed to play a major role in the underestimation of the growth rate of equatorial spread F irregularities in the numerical simulations based on the EFL approximation.
 This work was supported by award NNX11A069G from the NASA LWS to Cornell University.
 The Editor thanks Joseph Huba and an anonymous reviewer for assisting in the evaluation of this paper.