M. Sc. and D. Phil. projects

• River network formation* (Alun James 1992, Henry Winstanley 2001)
  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 ice sheet/basal hydrology model

• 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 Drobnjak 2001)
  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.