Impacts of sigma coordinates on the Euler and Navier-Stokes equations using continuous Galerkin methods

Abstract

In this thesis, the impacts of transforming the coordinate system of an existing x-z mesoscale model to x-[sigma]z are analyzed and discussed as they were observed in three test cases. The three test cases analyzed are: A rising thermal bubble, a linear hydrostatic mountain, and a linear nonhydrostatic mountain. The methods are outlined for the transformation of two sets (set 1, the non-conservative form using Exner pressure, momentum, and potential temperature; and set 2, the nonconservative form using density, momentum, and potential temperature) of the x-z Navier-Stokes equations to x-[sigma]z and their spatial (Continuous Galerkin) and temporal (Runge-Kutta 35) discretization methods are shown in detail. For all three test cases evaluated, the x-[sigma]z models performed worse than their x-z counterparts, yielding higher RMS errors, which were observed predominantly in intensity values and not in placement of steady state features. Since the models did converge to a fairly representative steady-state solution the results found by this project are promising, even though they did indicate that x-[sigma]z coordinates are not as accurate or efficient as x-z coordinates. With further fine-tuning of the model environment, these issues could be made minimal enough to warrant their utility with semi-implicit methods.