Theoretical Solutions for Convective Flows in Geophysically Motivated Regimes

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A dissertation submitted to the University of Cambridge for the degree of Doctor of Philosophy.

Author: Robert J. Whittaker, supervised by John R. Lister.

Submitted: September 2006; and with minor corrections: January 2007.

Approved: February 2007.

Full text download: thesis.pdf (PDF file, 2.7 MB).

Table of Contents

  1. Introduction
  2. Steady Axisymmetric Creeping Plumes above a Planar Boundary. Part I: A Point Source
  3. Steady Axisymmetric Creeping Plumes above a Planar Boundary. Part II: A Distributed Source
  4. On Modelling Steady Sheared Plumes in Viscous Fluid
  5. The Self-Similar Rise of a Buoyant Thermal in Stokes Flow
  6. Free Convection Beneath a Heated Horizontal Plate in a Rapidly Rotating System
  7. A Note on Circulation in Differentially Heated Rotating Stratified Fluid and Other Elliptic Problems
  8. Epilogue

Summary

This dissertation is concerned with theoretical solutions to problems in convection motivated by flows in geophysical systems. Six problems are presented. Four involve infinite-Prandtl-number convection in a non-rotating system (applicable to conditions in the Earth's mantle), and the other two are at finite Prandtl number and also include strong rotational effects (more applicable to convection in oceans and the outer core).

In Chapters 2 and 3, two asymptotic solutions are obtained for the rise of an axisymmetric plume from a source at the base of a half-space filled with very viscous fluid. Solutions are obtained first for a point source with a prescribed buoyancy flux B, and secondly for a heated disk with a prescribed temperature difference ΔT. The internal structure of the plume is found using stretched coordinates, and this is matched to a slender-body expansion for the external Stokes flow. For the disk, the boundary layer which forms above it is analysed to determine the total heat flux.

In Chapter 4, a simple model is developed to describe steady sheared plumes in very viscous fluid that are subject to deflection by a background flow. Key parameters and regimes are identified, and results are compared with previous models and experimental studies.

In Chapter 5, a similarity solution is obtained numerically for the rise of a buoyant thermal in Stokes flow, in which both the thermal's linear extent and the height risen scale like (κt)1/2. For weak thermals there are only slight deformations to a spherically symmetric Gaussian temperature distribution. For strong thermals, the temperature distribution becomes elongated vertically, with a long wake left behind the head. A simple analytic model for strong thermals is obtained using slender-body theory.

In Chapter 6, the flow beneath a finite heated horizontal plate in a rapidly rotating system is considered, for the case where the Ekman layer is confined within a much deeper thermal boundary layer. Solutions are derived in both planar and axisymmetric geometries. In particular, the relationships between the Nusselt and Rayleigh numbers are determined for the case of uniform plate temperature.

In Chapter 7, An alternative method of solution for a previously studied problem of circulation in differentially heated rotating stratified fluid is presented. An annular volume of fluid with a background thermal stratification maintained from the top and bottom boundaries is subjected to a weak perturbing heat flux from the outer boundary. The internal flow and temperature field are found using a Fourier–Bessel expansion and effective asymptotic boundary conditions.

Colophon

The thesis was typeset on a Linux computer system using LaTeX 2ε, with a custom thesis class I wrote myself (see my LaTeX packages page). Figures and graphs were mainly produced using XFig and Gnuplot, with the occasional use of Matlab and POV-Ray. The PDF version was produced with using the dvips and ps2pdf utilities.