M. Sc. and D. Phil. projects
I don't necessarily distinguish between M. Sc. and
D. Phil. projects. The aim is conceptually the same. Typically, an
M. Sc. will try and do one problem, whereas a D. Phil. will survey a
whole research area.
Projects below with an asterisk have been `done' already, although
there is inevitably scope for following on projects.
- River network formation* (Alun James 1992, Henry Winstanley
Continuous models (pdes) for channel formation in erodible substrates
exist: rills form as an instability analogous to cave formation in
limestone, or vent formation in thermochemical groundwater systems
(geysers, black smokers). The issue is whether such continuous models
can be used to predict the formation of (fractal) arterial networks in rivers
(and other systems), and under what circumstances.
- Drumlin formation* (Christian Schoof 2002)
Drumlins are small elongate hills which occur in swarms, notably in
Northern Ireland (County Down), and are formed by the action of ice
sheet flow over a subglacial layer of unconsolidated sediments. The
problem of why they form may be analogous to the formation of dunes
and ripples in river beds and deserts, and this study will involve a
coupled model of ice sheet flow, till deformation and subglacial hydrology.
- Chaos in delay equations* (Jonathan Wattis 1990)
Delay-recruitment equations are important in population biology and
physiology (and lasers and...), and reduce in the singular limit of
large delay to a difference equation. The relationship between chaos
in the differential equation and in the difference equation can be
partly but incompletely understood by methods of singular perturbation
theory, and this project seeks to extend that understanding.
- Chaos and turbulence in partial differential equations
The origin of chaos in ordinary differential equations through the
occurrence of homoclinic bifurcations is well understood. This project
will extend that understanding to partial differential equations; a
related project will seek to apply the idea to the origin of shear
flow turbulence, initially through simplified model equations, such as
those describing the evolution of a line vortex in a shear flow.
- Exponential asymptotics in differential equations
Much of the interest generated by Berry's rediscovery of Dingle's work
on asymptotics `beyond all orders' lies in its application to certain
integrals. An open question is whether one can derive exponentially
accurate approximations directly from differential equations in a step
by step procedure, and this
project seeks to further understanding in this direction.
- Ice streams in ice sheets
Large ice sheets have a propensity to form laterally inhomogeneous
patterns, manifested by the formation of fast moving `ice streams'
within the ice; notable examples occur in the Ross and Filchner-Ronne
ice shelves in Antarctica. Such behaviour can occur because of a
positive feedback between ice velocity and lubricated basal friction,
and this project seeks to predict this (numerically) with a combined
- Double porosity models for segregated flows
The typical example is for fractured rock, where permeability occurs
in two ways: through the rock, and through fissures. Homogenisation
theory can be used derive averaged models in a standard way. This
project aims to extend this type of theory to more exotic situations,
e.g. subglacial drainage and, particularly, magma transport, where it
is likely to be applicable.
- Strongly variable viscosity flows in mantle convection
Mantle convection (the formative mechanism for continental drift)
occurs through the solid state creep of crystalline rocks to a depth
of 3,000 km in the Earth. The effective viscosity depends strongly on
temperature (and probably pressure), and the rheological behaviour is
viscoelastic for sufficiently short time scales. Fundamental problems
(the depth of convection, the initiation of subduction) remain
unresolved, and this project develops asymptotic methods to solve the
governing equations in a sequence of increasingly complicated models.
- The formation of layered igneous intrusions
A number of basaltic magma chambers form layered structures as they
crystallise. The best known example is probably the Skaergaard
intrusion in East Greenland; both chemical and mineralogical layering
occurs, with a typical layer width of order metres. Processes involved
during solidification include double diffusive convection, and nucleation
and crystal growth in a developing porous `mushy' crystal pile; a
satisfactory explanation for the formation of layers remains to be found.
- Subglacial sediment transport and channel hydrology
In the last ten years it has become clear that large scale oscillatory
behaviour of glaciers and ice sheets (the latter associated with
Heinrich events) is often fundamentally controlled by the interaction
between ice flow and the hydraulically controlled deformation of
subglacially sediments. This project studies the hydraulics of
subglacial channel flows, and their interaction with deforming till layers.
- Surging glaciers
A successful theory to describe surging glaciers was enunciated in
1987, based on subglacial sliding laws which depend on the degree of
cavitation. Glaciers such as Trapridge in Alaska or Bakaninbreen in
Svalbard are thermally controlled, and a coherent theory for their
surging behaviour does not exist. This project will provide that theory.
- Waves in two phase flows
Bubbly flows break down to form slug flows in one-dimensional two phase
flows through an instability which promotes the formation of voidage
waves. The same instability causes the waves which can be seen in Guinness
when first formed. Similar waves form in stratified sub-horizontal
gas-liquid flows. The theory which describes these flows is mathematically
similar to that which describes the onset of roll waves in steep free
surface (river) flows. This project will formulate a complete theoretical
description of such flows.
- Nonlinear dynamical modelling in clinical time series
Clinical monitoring provides time series of such variables as heart
rate, blood pressure, ventilation, as well as ECGs, EEGs and other
such diagnostic measures. Whereas the technology to obtain such data
is sophisticated, theoretical techniques to assess their significance
are primitive, and this project will develop techniques based on
nonlinear dynamics to provide useful statistics whereby effective
diagnoses can be made.
- Oscillations in the respiratory control system* (Giri
Kalamangalam 1991, 1995; Chris Ham 2002)
Various kinds of breathing irregularities involve periodic breathing, a
slow waxing and waning of the amplitude of breathing. It is commonly
observed at high altitude, and in infants, where it is one mooted
cause of sudden infant death syndrome (SIDS, or `cot death'). In
extreme forms, periodic cessation of breath occurs in the
Cheyne-Stokes breathing experienced by patients with heart failure or
stroke. We aim to provide a useful, predictive model for this
phenomenon which can be clinically tested.
- Oscillations in blood cell diseases* (Brent Neiman 2000, Ivana
Various kinds of leukaemia and other blood diseases are characterised
by oscillations in the blood cell count. Models of this typically
involve feedback control systems with maturation delays. In this
project, such models will be studied with a view to understanding,
both analytically and numerically, the causes and nature of the
oscillations. The models are like age-related population models with
delays and nonlinear feedback terms.
- Oscillations in the cardio-respiratory system
Heart rate and respiration are two of the simplest non-invasive
diagnostics which it is possible to measure. They are coupled, and
through the mechanism of a variety of feedback systems, cause various
oscillations, such as respiratory sinus arrythmia and Mayer waves. It
is clinically important to understand the
mechanisms responsible for such oscillations, and this project seeks
to further that understanding.
- Mathematical modelling of leukaemias and other blood diseases
Chronic myelogenous leukaemia is a genetically caused disease of
haematopoietic stem cells in which a dormant (pre-clinical) phase is
followed by an apparently stable
chronic phase lasting several years, and a subsequent
acute phase of about six months, which inevitably leads to death. The
cause of this sequence (which is very similar to AIDS) is unknown, but
may be caused (as in AIDS) by a slow attrition of the cellular immune
system. The project will study models of the system, incorporating
time delays in stem cell control, maturation of different cell lines,
and their interaction with the immune system, to build a viable
portrait of the progression of this disease. This model may then act
as a blueprint for other such diseases.
- Grounding line migration in ice sheets
Ice sheets are essentially large, shallow viscous droplets, dominated
by shear stress; ice shelves are their oceanward extension, floating
shelves of ice which are essentially elongated threads, and dominated
by longitudinal stresses. Transition between the two limiting states
involves asymptotic analysis in the intermediate régime, and
computation of the location of the grounding line. Several
research groups have failed to achieve a satisfactory numerical
solution. An immediate
application is to the proposed break-up of the West Antarctic Ice
Sheet (with consequent sea-level rise).
This project is suitable either as a numerical or modelling project.
- Bioremediation of polluted soil
An inevitable by-product of any kind of human industrial activity is
soil contamination. This applies as much to agriculture, where the
contaminants may be pesticides or fertilisers, as to factories, power
plants, airports, garages, and so on, where heavy metals or
hydrocarbons may be the agent of pollution. Bioremediation is the
process whereby naturally occurring or artificially added microbial
agents clean the soil by consuming the contaminant. This project will
study differential equation models which describe the interaction
between substrate (contaminant), biomass (microbes) and nutrient
(oxygen) in situations of practical concern. Particular application
will be to industrial sites under study by the
Groundwater Protection and
Restoration Group at the University of Sheffield.
- Ice sheets on Mars* (Rachel Zammett, 2004)
Mars has two polar ice caps. The northern cap is about three
kilometres thick and a thousand kilometres in extent. Its surface is
furrowed by kilometre deep spiral valleys completely unlike the surfaces of
the two major Terran ice sheets. We will pose and study a model for
this extraordinary morphology.
- Braiding in rivers
Rivers come in diverse forms. Braided rivers occur when large gravel
bars split the stream into a number of different threads, like an
unravelling rope. This project studies a mechanism for braiding, based
on river flow coupled with bed erosion and transport.
- Plastic crustal deformation
The Earth's crust is often taken to deform as an elastic medium,
despite plentiful evidence of faulting and creep. This project will
attempt to understand mechanisms of crustal deformation using a model
of plastic deformation. This work will be done in conjunction with
Marta Perez-Gusinye (Earth Sciences).
- Melt bands
The magma which is erupted in volcanoes and at mid-ocean ridges is
formed when rock melts by pressure release melting, as it ascends as
part of the convective circulation of the Earth's mantle. This occurs
at depths of perhaps 100 kilometres. Quite how this molten magma then
ascends through the relatively cold overlying lithosphere remains
mysterious, despite thirty years of intensive research. This project
will seek a theory to explain the formation of 'melt bands', which
form when a molten rock is sheared in the laboratory, and which may be
instrumental in developing channelised drainage networks which allow
magma to ascend to the Earth's surface. This project is in
collaboration with Ben Holtzman at Lamont-Doherty Earth Observatory.
- Tidal bores
The most notable examples are the Severn bore and the Trent Aegir, but
a number of other examples are known worldwide; the literature on the
phenomenon is, however, particularly sparse. This (M. Sc.) project will produce
a viable theory of bores based on modern analytic and numerical techniques.
- Roll waves
Roll waves form in an open channel flow when the Froude number is
larger than 2, and can be seen on steeply sloping tarmac during a
downpour. They form through an instability in the uniform state, and
can be modelled as a periodic series of shock waves. This project
reviews the instability and its nonlinear development, and describes
the rôle of eddy viscosity in providing a shock
structure. Applications of the theory to other waves, such as waves in
guinness, will be made. The theory will be compared with high speed
digital photography of settling guinness
- Two phase flow in volcanic eruptions
Villarrica volcano in Chile is a Strombolian volcano which erupts
bubbly magma in a time-periodic fashion, apparently oscillating
between slug flow (which causes explosions in the vent) and bubbly
flow. This oscillation may be a manifestation of the density wave
instability which occurs in other heated two phase flows. This project
will provide and analyse a model this oscillatory style of
eruption. The project will be done in collaboration with the
volcanologist Eliza Calder
at the Open University.
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