Dynamo theory

Planetary dynamos involve phenomena spanning an enormous range of different length and timescales. For instance, the Earth's liquid outer core is about 3000 km deep, while the viscosity of liquid iron is so small that the usual order of magnitude calculations imply a viscous boundary layer thickness of about a metre on the inner core boundary. The state of the art in direct numerical simulations is limited to the equivalent of 64³ in spherical geometry for two reasons. Firstly, there is no fast Legendre transform for spherical harmonics analogous to the fast Fourier transform, and secondly the integration times have to be extremely long (millions of years) compared with the inertial oscillation period (one day).

For these reasons, we (like many others) have adopted a two-pronged approach. On the one hand, we study large scale phenomena, particularly magnetic field reversals, using low-dimensional models based on mechanical analogues that capture some of the basic mechanisms. On the other hand we study the small scale hydrodynamics of simple model problems with the aim of devising an effective turbulence parametrisation for use in numerical simulations.

Hydromagnetic geodynamo theory

Characterisation of small-scale turbulence in parameter regimes relevant to the Earth's liquid inner core (rapidly rotating, small viscosity, large magnetic diffusivity). Unlike planetary atmospheres and oceans, the Earth's fluid outer core is not shallow (the aspect ratio is order one), and so not hydrostatic. In general the rotation and ambient magnetic field vectors are not aligned so the problem is genuinelly three dimensional. Inertia and viscosity are both relatively unimportant in the balance between rotation, magnetic, and buoyancy forces. While in principle the governing equations are all known (fluid dynamics, thermodynamics, and Maxwell's equations) they are so far from being directly tractable that there is still a lot of scope for modelling.

Investigation of instabilities in small-scale coherent structures. The main tool is a reformulation as a spatially varying linear stability problem that is tackled by a combination of breeding and other iterative methods to determine growing modes.

Construction of turbulence models that might be relevant to such parameter regimes, in particular where magnetic diffusivity is much larger than viscosity.

Investigation of improved numerical algorithms for integrating PDE systems with fast linear waves, notably rapid rotation and consequent fast inertial waves.

Related models for liquid metal magnetohydrodynamics on a laboratory or industrial scale, for instance in processing of liquid metals, or electrolytically reducing bauxite to make aluminium.

Dynamo theory

Hide (1997) has proposed a hierarchy of related self-exciting coupled Faraday-disk dynamos incorporating electric motors as additional electromechanical elements and driven by steady mechanical couples has been proposed. Each system comprises N interacting units which can be arranged in a ring or a lattice. Within each unit are electric motors, driven into motion by the dynamo and connected either in series or in parallel with the coil. Nonlinearity enters solely through the coupling between components. By introducing additional terms into the equations, it is possible to include the effects of biasing from impressed electromotive forces due to thermoelectric or chemical processes, as well as from the presence of ambient magnetic fields. Dissipation is introduced into the models via ohmic heating and mechanical friction in the disk and motors, with the latter playing a crucial role in the generation of chaos (see e.g. Hide et al (1996)).

We are also interested in the comparison between numerical simulations of low order systems of nonlinear differential equations (digital) with experimental realisations of such systems as electronic circuits (analogue) with a view to understanding similarities and differences between models and systems, especially in the problem of extraction of unstable periodic ordbits. This work, which will use low-dimensional models of self-exciting dynamos as the systems, builds on an MSc summer project on the Moore-Spiegel equation and is expected to involve collaboration with colleagues in the Physics Department at GMU, Virginia, Oxford and Drexel University, Philadelphia.

People working in this area within OCIAM are

• Dr Irene Moroz

For detailed information see

• R. Hide, A.C. Skeldon and D.J. Acheson (1996), `A study of two novel self-exciting single-disk homopolar dynamos: theory'. Proc. R. Soc. Lond. A vol. 452, 1369-95

• R. Hide (1997), 'The nonlinear Differential equations governing a hierarchy of self-exciting coupled Faraday-disk Homopolar dynamos'. Phys. Earth Plan. Int. vol 103 281-291

• I.M. Moroz (2001), 'Self-exciting Faraday disk homopolar dynamos'. I.J.B.C. vol 11 2961-75.