Analysis of waves, instabilities and mean flows in MHD systems

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Magnetic fields are ubiquitous in the universe: for our own Sun, fields are generated in the deep interior and as they emerge from the visible surface they create spectacular phenomena such as sunspots and solar flares. The Earth's magnetic field has been a source of mystery for centuries, in particular the knowledge that the North and South poles move in time, and sometimes reverse completely. The field is now known to be generated from deep within the Earth by the motion of liquid metal, and the changes in magnetic field direction are not due to any solid body moving within the Earth, but arise from a reorganisation of the electrical currents associated with the magnetic field. Related to this are attempts to confine a fusion plasma using magnetic fields in the laboratory, for example in the ITER tokomak. Behind all these processes are the interactions of magnetic field and flow of a conducting fluid, liquid iron in the core of the Earth and ionised hydrogen plasma in the Sun. On these geophysical and astrophysical scales (much bigger than available in the laboratory), magnetic fields taken on a life of their own. Magnetic field lines have elastic properties and waves - Alfven waves - can travel, transferring energy and momentum, create instabilities, and may generate or suppress large-scale fluid, jet-like motions. The research proposed is to understand the fundamental mathematics behind these processes: rather than numerically modelling the Earth or Sun by including all possible relevant phenomena, the goal in theoretical fluid mechanics is to take problems apart into their simplest components, so as to understand basic processes, parameter regimes and how to model them simply when studying more complex phenomena. We propose a range of projects that involve studying first, magnetic field instabilities - can magnetic fields give rise to growing waves in a fluid system? Secondly, how do instabilities and waves generate mean flows, for example jet-like structures (these are famously seen as the coloured bands on Jupiter, though magnetic fields are not thought to be a key component of the physics in this instance). Third, we plan to understand how one can use mathematics to describe a whole sea of magnetic waves and their effects, using methods from differential geometry (which lies at the base of the theory of general relativity). Finally, all these theoretical questions cannot exist in a vacuum and we propose numerical simulations of carefully designed systems - numerical experiments - to motivate and validate theory, and to extend our understanding to regimes that are not easily accessible to theory. In this way, we propose to advance the theory, methods of modelling and numerical simulation, for a wide range of geophysical and astrophysical systems.