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).

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

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
(`κ``t`)^{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.

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.