The dynamical evolution of fluids appears in many areas of science. Theoretical understanding and reliable computational description of complex and turbulent flows are one of the grand challenges of science. Although much has been accomplished recently, there is a significant amount of work remaining to get an accurate and effective description of fluid dynamics. This thesis uses two different approaches on different topics in geophysical fluid dynamics.
First, rigorous theoretical bounds on the transport of heat are discovered for convection driven both by an internal heat source, and convection driven by an enforced temperature gradient. For stress-free vertical boundaries it is shown that at arbitrary Prandtl number in two dimensions (or at infinite Prandtl number in three dimension) the enhanced heat transport due to convection is bounded as Nu < Ra^(5/12) where Ra is a measure of the strength of the driving force. For these same type of boundaries (and under the identical assumptions on the Prandtl number and dimension) with internal heating, the spatially and temporally averaged temperature is bounded from below by H^(12/17) where H is the strength of the internal heating. For no-slip boundaries at infinite Prandtl number the temperature is bounded by H^(3/4) log(H)^(-1/4)$.
Second, methods from numerical analysis and physical intuition are used to test the numerical models intended to describe the evolution of the earth's climate and weather. A stability analysis is carried out to test the numerical stability of divergence damping (a form of numerical dissipation meant to model unresolved sub-grid processes) applied on a latitude-longitude grid. The analysis yields sharp stability constraints, and highlights some of the issues inherent to the choice of grid. A test is also proposed to consider the consistency between the integration of the primitive equations, and the advection of passive tracers in a atmospheric dynamical core. Potential voriticity is used to examine the level of inconsistency between dynamics and tracers in the four dynamical cores present in the National Center for Atmospheric Research's Community Atmosphere Model (CAM5.0).Ph.D.Applied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91464/1/jaredwh_1.pd
