## Abstract

A time-dependent model that simulates the interaction of a thunderstorm with its electrical environment is introduced. The model solves the continuity equation of the Maxwell current density that includes conduction, displacement, and source currents. Lightning phenomena are neglected and the electric field is assumed to be curl free. Corona, convection, and precipitation currents are not considered in this initial study and their contribution to the source function is not specified explicitly. As a preliminary test of the model we assume that the storm is axially symmetric in spherical geometry, the conductivity depends only on the vertical coordinate, the ground is equipotential, and far from the thunderstorm region the horizontal electric field is zero. These assumptions are for computational efficiency only and can be relaxed in more realistic studies.

The mathematical energy method is applied to the continuity equation to determine boundary conditions that are sufficient to form a well-posed initial-boundary value problem. This ensures the existence of a physical solution that depends continuously on the initial and boundary data. Then analytic techniques are applied to study the dependence of the solution on the properties of the medium. There are two time scales of the problem that are analyzed and discussed: one determined by the background electrical conductivity and the other by the time dependence of the source function. The assumed source function, which represents a mechanism by which charge is separated inside the storm, contributes to a portion of the solution in which the ratio of the displacement current over the conduction current increases with decreasing altitude, i.e., in the lower atmospheric region the displacement current can have an important role in the electrical interaction between the storm and its environment. It is also demonstrated that the source function can induce temporal phase shifts in the solution, which are dependent on altitude.

To obtain details of the solution, which cannot be obtained by analytic techniques. a stable numerical approximation of the continuity equation is introduced and analyzed. The resulting numerical model is used to examine the evolution of the displacement and conduction currents during the charge buildup phase of a developing thunderstorm.