## 1. Introduction

[2] Equatorial spread *F* (ESF) refers collectively to a family of plasma irregularities that form in the equatorial *F*region 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 in*Woodman* [2009]). 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* [1976], *Zalesak and Ossakow* [1980], *Zalesak et al.* [1982], *Zargham and Seyler* [1989], *Huang and Kelley* [1996], and *Chou and Kuo* [1996]) or quasi-three-dimensional models incorporating magnetic flux tube integrated quantities (as, e.g.,*Huba et al.* [2008] and *Retterer* [2010]). *Keskinen et al.* [2003]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.* [2011]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 at*F* 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.

[3] 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.