Reevaluating the boundary conditions of the perturbation pressure Poisson equation and an iterative solution on a non-uniform grid

Convective scale numerical models have been used by many researches over the years to simulate supercellular storms and explore their features. One common technique is to relate these features to the structure of the pressure field, which is the solution to an elliptic partial differential equation known as a Poisson equation and often solved for numerically. The solution to such an equation is dependent on the boundary conditions prescribed to the pressure field. The goals of this thesis are two-fold. The first is to evaluate whether the condition applied to the bottom boundary should be modified by a “normal force,” similar to that acting on solid objects striking the surface. This development is aided by simulations of “cold bubbles,” regions of negative potential temperature perturbations, in George Bryan’s Cloud Model 1 (CM1). Secondly, an iterative method known as Jacobi’s Method is applied to the Poisson equation on an arbitrary grid which allows for stretching of grid points and higher model resolutions at select locations, such as near the surface. The method is tested on the well known problem of two infinite parallel plates and shown to converge to the exact analytical solution in a manner predicted by theory.