Date of Award

Spring 1-1-2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

First Advisor

Keith Julien

Second Advisor

Michael A. Calkins

Third Advisor

Ian Grooms

Fourth Advisor

James Meiss

Fifth Advisor

Peter Hamlington

Abstract

Geophysical flows, such as the turbulent motion within natural systems, are characterized by a wide range of spatial and temporal scales. Due to this complexity, approaches that rely on solving the full Navier-Stokes equations are limited to values of parameters that are far from the extreme values characteristic of geophysical regimes. Therefore, results from these simulations must be extrapolated over orders of magnitude to apply to Earth's core. In this thesis, I'll present an alternative strategy for modeling these flows, which relies on deriving reduced models using asymptotic techniques. These models are investigated using efficient scientific computing methods and yield results that are within geophysically relevant parameter regimes.

This thesis presents asymptotically reduced models for rapidly rotating convection in a plane layer geometry and highlights a few applications of these reduced models. Specifically, the effect of different boundary conditions on key dynamics is investigated to facilitate comparison with experimental studies. The main scientific question is if these boundary conditions are passive in geophysical parameter regimes and how the rate of rotation influences the heat flux in the systems. While direct numerical simulation and laboratory experimental results can only examine this for moderate values of rotation, the asymptotic model is applied to determine an empirical scaling for the impact of these boundary layers within the rapidly rotating regime.

The dynamics of electrically conducting fluids and self-sustained magnetic fields are also investigated with the use of asymptotic models. The presence of electromagnetic fields presents added complexity in the dynamics and the use of asymptotic analysis allows us to investigate a geo/astrophysically relevant limit that is not attainable in current DNS simulations. The asymptotic reduction leads to multiple timescales that require new numerical strategies for solving and two multiscale numerical methods are tested on this new model for both single mode and multimode cases. The model is used to characterize the influence of dynamo action on convection and explore the large scale structure of the flows, especially in comparison to both non-magnetic convective results of rapidly rotating asymptotic models as well as results from dynamo simulations that are not in the geophysical parameter space. We find that dynamo action leads to fundamental changes in the dynamics, compared to non-magnetic convection, and current work is aimed at understanding the implications for natural dynamos.

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