Date of Award

Spring 1-1-2011

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Keith Julien

Second Advisor

Bengt Fornberg

Third Advisor

Baylor Fox-Kemper

Abstract

This thesis documents three investigations carried out in pursuance of a doctoral degree in applied mathematics at the University of Colorado (Boulder). The first investigation concerns the properties of rotating Rayleigh-Benard convection -- thermal convection in a rotating infinite plane layer between two constant-temperature boundaries. It is noted that in certain parameter regimes convective Taylor columns appear which dominate the dynamics, and a semi-analytical model of these is presented. Investigation of the columns and of various other properties of the flow is ongoing. The second investigation concerns the interactions between planetary-scale and mesoscale dynamics in the oceans. Using multiple-scale asymptotics the possible connections between planetary geostrophic and quasigeostrophic dynamics are investigated, and three different systems of coupled equations are derived. Possible use of these equations in conjunction with the method of superparameterization, and extension of the asymptotic methods to the interactions between mesoscale and submesoscale dynamics is ongoing. The third investigation concerns the linear stability properties of semi-implicit methods for the numerical integration of ordinary differential equations, focusing in particular on the linear stability of IMEX (Implicit-Explicit) methods and exponential integrators applied to systems of ordinary differential equations arising in the numerical solution of spatially discretized nonlinear partial differential equations containing both dispersive and dissipative linear terms. While these investigations may seem unrelated at first glance, some reflection shows that they are in fact closely linked. The investigation of rotating convection makes use of single-space, multiple-time-scale asymptotics to deal with dynamics strongly constrained by rotation. Although the context of thermal convection in an infinite layer seems somewhat removed from large-scale ocean dynamics, the asymptotic methods generalize directly to the second investigation which simply adds large spatial scales -- the transition from convectively unstable to convectively stable dynamics does not change the mathematical framework. The rotating Navier-Stokes equations in the Boussinesq approximation and the equations derived from them asymptotically in the investigation of rotating convection include dispersive and dissipative linear terms that are stiff, i.e. that hinder numerical solution by explicit methods. A variety of methods which purport to alleviate this difficulty have been derived, and have been tested on and applied largely to problems with purely dissipative linear terms. But it was heretofore unfortunately quite difficult to judge and compare how effectively these methods achieve their goal when the stiff linear term is both dissipative and dispersive. The third investigation therefore introduces a visual, analytical method for comparing the linear stability properties of the various methods (the linear stability properties being a proxy for their ability to alleviate stiffness) and supports the results of this analysis by comprehensive.

Share

COinS