Investigations of Reduced Equations for Rotating, Stratified and Non-hydrostatic Flows

Date of Award

Spring 1-1-2016

Document Type


Degree Name

Doctor of Philosophy (PhD)


Applied Mathematics

First Advisor

Keith Julien

Second Advisor

Ian Grooms

Third Advisor

Bengt Fornberg

Fourth Advisor

Mark Hoefer

Fifth Advisor

Jeffrey Weiss


This thesis is a collection of studies concerning an asymptotically reduced equation set derived from the Boussinesq approximation describing rotationally constrained geophysical flow.

The first investigation is concerned with a statistical identification of coherent and long-lived structures in rotationally constrained Rayleigh-Bénard convection. Presently, physical laboratory limitations challenge experimentalists while spatio-temporal resolution requirements challenges numericists performing direct numerical simulations of the Boussinesq equations. These challenges prevent an exhaustive analysis of the flow morphology in the rapid rotating limit. In this study the flow morphologies obtained from simulations of the reduced equations are investigated from a statistical perspective. Auto- and cross-correlations are computed from temporal and spatial signals that synthesize experimental data that may be obtained in laboratory experiments via thermistor measurements or particle image velocimetry. The statistics used can be employed in laboratory experiments to identify regime transitions in flow morphology, capture radial profiles of coherent structures, and extract transport properties belonging to these structures. These results provide a foundation for comparison and a measure for understanding the extent to which rotationally constrained regime has been accessed by laboratory experiments and direct numerical simulations.

A related study comparing the influence of fixed temperature and fixed heat flux thermal boundary conditions on rapidly rotating convection in the plane layer geometry is also investigated and briefly summarized for the case of stress-free mechanical boundary conditions. It is shown that the difference between these thermal boundary conditions on the interior geostrophically balanced convection is asymptotically weak. Through a simple rescaling of thermal variables, the leading order reduced system is shown to be equivalent for both thermal boundary conditions. These results imply that any horizontal thermal variation along the boundaries that varies on the scale of the convection has no leading order influence on the interior convection, thus providing insight into geophysical and astrophysical flows where stress-free mechanical boundary conditions are often assumed.

The final study presented here contrasts the previous investigations. It presents an investigation of rapidly rotating and stably stratified turbulence where the stratification strength is varied from weak (large Froude number) to strong (small Froude number). The investigation is set in the context of the asymptotically reduced model which efficiently retains anisotropic inertia-gravity waves with order-one frequencies and highlights a regime of wave-eddy interactions. Numerical simulations of the reduced model are performed where energy is injected by a stochastic forcing of vertical velocity. The simulations reveal two regimes: one characterized by the presence of well-formed, persistent and thin turbulent layers of locally-weakened stratification: the other characterized by the absence of layers at large Froude numbers. Both regimes are characterized by a large-scale barotropic dipole in a sea of small-scale turbulence. When the Reynolds number is not too large a direct cascade of barotropic kinetic energy is observed and leads to an equilibration of total energy. We examine net energy exchanges that occur through vortex stretching and vertical buoyancy flux and diagnose the horizontal scales active in these exchanges. We find that baroclinic motions inject energy directly to the largest scales of the barotropic mode governed by the two-dimensional vorticity equation, and implies that the large-scale barotropic dipole is not the end result of an inverse cascade within the two-dimensional barotropic mode. An additional yet brief look into the linear vortical and wave modes is considered.

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