## 1. Introduction

[2] The *F* region ionosphere can become unstable if a density perturbation becomes electrically polarized by external forces from electric fields, neutral winds, and gravitational acceleration. Near the geomagnetic equator, gravity can act on the plasma attached to the nearly horizontal magnetic field lines to produce unstable conditions. After sunset when the layer is lifted by ambient electric fields, the bottomside steepens and plasma bubbles are formed. These bubbles rise through the layer in response to a Rayleigh-Taylor type instability. Also, winds or electric fields induce electric fields in both density cavities and enhancements that cause distortions in the density structures. These distorted plasma structures are responsible for degradation of radio propagation that lead to navigation errors and outages, communications systems failures and radar clutter. The modeling of ionospheric bubbles or density enhancements uses computer simulations the calculate the effects of self generated electric fields (**E**) that are driven by gravity, neutral winds, and external electric fields. Numerical computation of the electric potentials requires the most time and effort in the bubble modeling process. The electric fields for these simulations can be found numerically using direct or iterative solvers of the nonseparable potential equations that describe the self-generated electric fields [e.g., *Bernhardt and Brackbill*, 1984]. The computational time for solving for the disturbed ionosphere is often prohibitive so analytic solutions to both the transport and potential equations are useful. Exact analytic solutions for the electric potential can be used to test the numerical algorithms and to determine errors produced by boundary conditions and numerical roundoff. The analytic solutions also yield insight into the conditions for production of ionospheric bubbles.

[3] The computational and analytical techniques for simulations of equatorial bubbles are compared in Figure 1. Typically, numerical models of equatorial bubbles follow the procedure illustrated by the block diagram given in Figure 1a. A stratified model of the *F* layer is perturbed by a small density disturbance. Gravity is allowed to setup an electric potential in this plasma. The electric potential is obtained with a numerical solution of a nonseparable elliptic equation using a direct solver such as described in Appendix A. Once the electric potential is obtained, the plasma transported in response to the electric fields for a small time step. A nondissipative flux corrected transport algorithm [e.g., *Zalesak et al.*, 1982] is then used for incrementally move the plasma disturbance. The process is repeated with the generation of a revised electric potential followed by more incremental plasma transport. All of these processes are numerically intensive and can require several hours of computation.

[4] A new approach for the quasi-analytic model of the equatorial bubble is proposed using the three steps in Figure 1b. First, the electric potential is defined by an analytic function that gives self-consistent expressions for electron density (or Pedersen conductivity) structures that are obtained in the presence of background electric fields, neutral winds and gravity. This procedure is described in this paper. Second, the plasma transport is determined using incompressible motion from the induced electric fields. The plasma transport is derived with the analytic electric potential distorting the coordinates with out changing the density in each coordinate cell. The third step is to adjust the parameters in the analytic models to that the analytic solution given in step 1 matches the quasi-analytic results from step 2. The application of the transport and normalization processes will be described in a future paper.