A quasi-electrodynamic model for examining the effects of induction and causality in simulating regions of possible sprite inception in the mesosphere

Previous studies have demonstrated that electrodynamic effects can sometimes be important in simulations of elves, sprite halos, and sprite initiation. To examine the extent to which such effects contribute to the evolution of regions of possible sprite inception, we extend our fully three-dimensional quasi-electrostatic (QES) model of the electric field above thunderstorms to include dynamic effects. The original QES model employed the method of images for every charge in the domain at every time step, eliminating the need for spatial finite differencing of the electric potential or electric field and yielding a numerically stable and accurate solution of the QES equations (see, e.g., Haspel and Yair, 2025; doi:10.1016/j.asr.2025.01.013). In the present implementation, we add the electric induction (“velocity”) term to the Coulomb term in the expression for the electric field produced by each charge in the domain. In addition, we replace instantaneous time with retarded time, such that the model is also fully causal; a change in charge density at point A does not manifest in a change in the electric field at point B until that “signal” has time to propagate from A to B. The resulting Coulomb and induction contributions are structurally equivalent to the corresponding terms in Jefimenko’s formulation. This approach lies between traditional QES models and full-wave electromagnetic models and may be described as quasi-electrodynamic rather than quasi-electrostatic. It allows induction and causality effects to be included throughout the entire domain without spatial finite differencing and without an explicit representation of the lightning channel as used in transmission-line or EMP models. We find that the inclusion of causality delays the formation of regions of possible sprite inception and, together with the induction term, produces regions that persist longer than in traditional QES simulations with otherwise identical simulation parameters. Initial results from this extended model will be presented and discussed.