We are actively engaged in research in a wide-range of areas of astrophysical and geophysical fluid dynamics from planetary dynamics (the geodynamo and planetary dynamos) through solar, stellar and galactic dynamics to highly compressible and relativistic dynamics on the largest scales. We have a large number of research students and post-doctoral research associates and are actively seeking students who are interested in pursuing research to gain a PhD in mathematics.
Atmosphere and ocean dynamics
The Earth's atmosphere and ocean are continuously in motion. Familiar examples are winds at the Earth's surface, and the rise and fall of the ocean surface (perhaps as wind-generated water waves with a time-scale of a few seconds, or as tides with a time-scale of 12 hours or more). However, all sorts of other motions occur throughout the depths of the atmosphere and ocean, some of which can be observed from satellites or with other modern technology. These motions interact with one another in a complicated way, helping to determine the climate of the atmosphere-ocean system. Our research involves modelling such motions using the equations of fluid dynamics.
Present areas of research include:
- The stability of shear flows (linear analysis and nonlinear equilibration)
- The generation and propagation of internal waves in the ocean
- Global modelling of ocean tides
The Life of black holes and neutron stars
When massive stars exhaust their nuclear fuel they die in the most spectacular fashion — their central core collapses either to a superdense ball of only few kilometres in size, neutron stars, or to a black hole. The gravitational energy released during or soon after the collapse drives huge explosions known as Supernovae and Hypernovae. The life of neutron stars and black holes produced in this way remains most spectacular even after these explosions. In spite of decades of observational and theoretical research many key aspects of black hole/neutron star physics remains unclear and are waiting for new generations of researchers to take on them. One of the most noticeable advances of recent years has been the development of powerful computational tools that allow invaluable insights to be gained into the phenomena of relativistic astrophysics, via numerical simulations.
Past and current areas of research include:
- Theory of relativistic magnetised plasma
- Numerical methods
- Dynamics of relativistic jets
- Magnetospheres of black holes
- Magnetospheres of neutron stars and pulsar wind nebulae
Cosmic gas dynamics
Cosmic gas dynamics is concerned with the dynamics of the diffuse gas between the stars in galaxies (the Interstellar Medium) and with even more rarefied gas between galaxies in cluster of galaxies (the Intracluster Gas). Although the density of this material can be extremely small (less than a million atoms per cubic meter), for most purposes it can be regarded as a compressible fluid. Despite its low density, it occupies such a large volume that it constitutes a significant fraction of the mass of a galaxy (about 30% in our galaxy). Most of the Interstellar Medium is not at rest, but is in supersonic motion under the action of supernova explosions, stellar winds and radiation fields and the time dependent gravitational field due to spiral density waves in the stellar disc of the galaxy. Since supersonic motions almost always involve shock waves, these play a crucial role. The galaxy also contains a dynamically significant magnetic field which means that the dynamics are governed by the equations of compressible magnetohydrodynamics. One of the most interesting problems is that of star formation. It is known that stars form out of the Interstellar Medium and that this mostly occurs in Giant Molecular Clouds such as the Rosette Nebula.
Solar dynamo theory
Solar Dynamo theory is concerned with the origin of the solar magnetic field. It is an exciting time for research into the Solar Dynamo — the problem of the origin of the sun's magnetic activity. The beginning of the new Millennium has seen great increases in both theory (made possible by the breath-taking advances in computational power) and high resolution observations (driven by the recent and planned space missions). Taken together, these advances have significantly increased our understanding of the magnetic activity and variability of our nearest star.
The magnetic field of the Earth is generated in its molten iron core. As the Earth gradually cools down, slowly moving convection currents keep stirring the core, and this motion is sufficient to generate the geomagnetic field by dynamo action. This is possible because the iron core of the Earth is electrically conducting. In the last ten years, it has become possible to model this process by solving the fundamental equations that govern the motion of fluids and the generation of magnetic fields. These simulations have been remarkably successful in reproducing many of the observed features of the geomagnetic field. At Leeds, we are running dynamo codes and comparing their output with the important features of the geomagnetic field. Because the field depends on the rate of cooling of the Earth, we can use these codes to study processes going on in the deep interior of the Earth.
Solar active regions are the surface manifestations of a deep-seated, predominantly toroidal magnetic field. Although it is generally agreed that the solar magnetic field is maintained by some sort of hydromagnetic dynamo, the mechanism by which this is effected is far from understood. Consequently the strength, structure and location of the interior magnetic field are similarly unknown. However, most recent solar dynamo models, despite their significant differences in other respects, postulate that the tachocline, the thin layer of strong velocity shear located at the base of the convection zone, plays an important role in the generation of toroidal field by the shearing of a weaker poloidal component. Inspection of active regions on the solar surface suggests indeed the emergence through the photosphere of a buckled, toroidal magnetic field. However, very little is known about the structure, strength and orientation of the sub-surface field. It is therefore important to understand the nature of the initial escape of the magnetic field from the tachocline, its subsequent ascent through the convection zone, and its eventual emergence at the photosphere. A vertically stratified, horizontal magnetic field can be unstable to magnetic buoyancy instability provided that the field decreases sufficiently rapidly with height.
It is well-known from the field of thermonuclear fusion that a plasma confined by strong toroidal magnetic fields is generically unstable; in this context, MHD instabilities are able to destroy the plasma configuration on very short time-scales. MHD instabilities include pressure-driven ones, whose linear growth is due to the destabilising interplay of the equilibrium pressure gradient and of the magnetic field line curvature. The excited pressure-driven modes include the so-called sausage mode, and are usually divided into interchange modes and ballooning modes; they share some common features with the Rayleigh-Taylor and magnetic buoyancy, or Parker, instabilities. In astrophysics and geophysics, pressure-driven instabilities have been considered in the context of the Sun's corona to explain solar flares and in the context of the near-Earth magnetotail where they may cause geomagnetic substorms. Turbulence due to pressure-driven instabilities may also provide the source of heat responsible for the high-energy emission which takes place in the interior of astrophysical jets, confined on large scales by strong toroidal magnetic fields.
Most research in astrophysical and geophysical fluids is either observational or theoretical. There are a number of groups though, in America, France, Germany, Russia, and other countries, that have taken on the challenge of trying to reproduce some of the relevant physics in laboratory experiments. Working with liquid sodium or gallium obviously involves considerable technical difficulties, not only in handling them safely, but also in measuring the resulting fluid flows. It is the only way though to study magnetohydrodynamic phenomena in the lab. In the astrophysical and geophysical fluids group at Leeds we do not do experiments ourselves (we are mathematicians, after all, not physicists). Instead, we use theoretical/computational modelling to try to understand some of the existing results, and also suggest new experimental possibilities.
Massive astrophysical bodies such as forming stars or compact objects exert gravitational attraction onto the gas at rest in their surroundings. However, owing to the conservation of its angular momentum, the cosmic material falling in the gravitational well undergoes rapid rotation and, after the loss of the potential energy released in this process through radiation, tends to form a disc. In discs formed of sufficiently cool gas, the gravitational attraction of the central body is at first approximation balanced by centrifugal forces — as in planetary systems. Such discs are found in a variety of astrophysical environments: around young stars where they are the sites of planetary formation, around super-massive black holes in the centre of active galaxies and around the compact stars that disrupt their companion in interacting binary systems.
Fluids heated from below often undergo convection. As particles of fluid become hotter, they become less dense than the surrounding fluid and they rise due to their buoyancy. When they reach the surface regions they release their heat, become denser, and sink back. Convecting fluids are thus in a perpetual state of motion. The Sun, the Earth, and many stars and planets transport heat outwards by convection. The resulting fluid motion drives the solar magnetic field, the Earth's magnetic field, plate tectonics and many other key phenomena in astrophysical and geophysical fluid dynamics. In consequence, the study of convection is fundamental to many aspects of astrophysics and geophysics.
Naturally, there has been much work devoted to understanding the fundamentals of convection theory, but much still remains to be done. If a fluid layer of uniform depth is heated from below (the Rayleigh-Benard convection problem), and the temperature at the lower boundary is gradually increased while the upper boundary is held at constant temperature, there is a critical temperature difference at which convection onsets. The behaviour close to this critical value has been studied intensively, to see which patterns of fluid motion set in: hexagonal cells, square cells and two-dimensional rolls are all possible. Convection can occur as steady convection, oscillatory convection or as travelling waves.
All upcoming seminars can be found in our events section.
We have opportunities for prospective PhD students. Potential projects can be found in our postgraduate research opportunities directory.