The feasibility of a 3D time-dependent model for predicting the area of possible sprite inception in the mesosphere based on an analytical solution to Poisson's equation

We present a formulation for calculating the electric field from the ground to the lower boundary of the ionosphere that is based on an analytical solution to Poisson's equation in three dimensions, combined with a numerical solution to the equation of charge continuity/conservation. This formulation allows one to follow the spatial and temporal evolution of the distribution of free charges and the electric field in the atmosphere and thus predict areas of possible sprite inception in the mesosphere in a quasi-electrostatic fashion for any thundercloud charge distribution. The main advantages of this formulation are: (1) for a given spatial resolution, the analytical solution to Poisson's equation is more numerically stable and accurate than a numerical solution to Poisson's equation that includes finite differencing in space; (2) unlike a numerical solution to Poisson's equation that includes finite differencing in space, the numerical stability is insensitive to the choice of spatial resolution; (3) no artificial side boundary condition need be applied; (4) no symmetry, cylindrical or otherwise, of the charge distributions nor the electric field distributions need be prescribed; and (5) the computation is readily parallelizable on multiple processors. The main limitation of the present formulation is that the electric field based on the electrostatic potential with upper and lower boundary conditions needs to be calculated for each charge in the domain, such that the larger the number of charges, the slower the computation. We explore the sensitivity of the formulation to its parameters, in order to elucidate its feasibility for performing simulations for a variety of thunderstorm charge distributions in 3D, and we demonstrate its utility in investigating the evolution of the area of possible sprite inception in cases of consecutive lightning discharges that are separated from one another in space as well as in time. In all of the simulations in the present study, the ambient electrical conductivity varies in space but is independent of time. However, as we discuss, our formulation can also accommodate a time-dependent conductivity profile.