## The Role of Large-Scale Shear Flows in the Geodynamo

Prof David Hughes (SoM), Prof S.M. Tobias (SoM)

**Contact email:**d.w.hughes@leeds.ac.uk

One of the most fascinating problems in geophysics is that of understanding the generation of the Earth’s magnetic field. It is universally accepted that the magnetic field is maintained by what is known as a hydromagnetic dynamo in the liquid metal outer core, in which the fluid motions, driven by convection, can maintain the magnetic field by induction against its otherwise natural tendency to decay. That said, the details of this process are still far from being fully understood. Indeed, the dynamo problem in the wider context of other planets, the Sun and other stars remains one of the great challenges in cosmical fluid dynamics. Reviews of dynamo theory in the earth, planets and the Sun can be found in references [1-3].

One particularly fascinating feature of the Earth's magnetic field is that, every so often, it reverses its polarity. The field maintains a certain polarity for long periods of time (up to 10 million years), and then (randomly it seems) changes sign in quite a short timescale (of the order of 10,000 years). To account for this tendency for reversals, it has been suggested that the dynamo operates close to the boundary in parameter space separating dipolar from multi-polar solutions. The aim of this project is to explore this idea in some detail. In particular it will investigate the competition between a dynamo produced by small-scale flows, for which the magnetic field may be small scale (what we might think of as multi-polar) or large-scale (what we might think of as a dipolar field), and the tendency of a large scale shear flow to stretch out a large-scale magnetic field.

The dynamo problem is governed by the equations of MagnetoHydroDynamics (MHD); these are the standard (Navier-Stokes) partial differential equations of fluid dynamics, augmented by an additional magnetic (Lorentz) force term in the momentum equation and a new equation (the induction equation) for the magnetic field. Solving them at the parameter values applicable for the Earth is computationally impossible, and will remain so for the foreseeable future. It is therefore important to construct simpler models that isolate certain crucial aspects of the physics.

This project, described in more detail in the Objectives section, will make certain simplifications, particularly in terms of the geometry of the problem, but will then analyse the model mathematically, using both analytical and numerical (computational) approaches, in order to gain insight into the underlying physics. It will provide the student with a training not only in the dynamo problem, but in widely applicable skills in fluid dynamics and applied mathematics more generally.

### Objectives

The project will first look at the role of a large-scale shear flow on small-scale dynamo action resulting from an idealized forced cellular flow of the type found in rapidly rotating environments such as the Earth's outer core, seeking to determine the boundary between small- and large-scale dynamo action. It will consider, for examples, dynamos studied in papers [4-5]. Although some of the work will necessarily be computational, this will be made relatively straightforward through the choice of specific flows. The student will then be able to construct his or hew own computational code to solve the governing equations. This part of the project will provide new insight into the competition between small- and large-scale dynamos.

The model will then be extended to consider strongly rotating convective flows more realistically, via cellular flows with a well-defined (long) correlation time that can be set as a variable parameter, and will incorporate shear flows of the type found in the Earth's interior. This will then get to the heart of one of the key problems in understanding the geodynamo.

### Potential for high impact outcome

Understanding the means by which the Earth’s magnetic field is maintained – the geodynamo problem – remains one of the classical unsolved problems in magnetohydrodynamics. One of the difficulties is that a direct computational solution at the correct parameter values for the Earth is currently impossible, even with the most powerful computers, and will remain so for the foreseeable future. It is therefore important to explore other ways into the problem, such as that proposed here. The project thus has the potential to make inroads into this important problem.

### Training

The student will work under the supervision of Profs David Hughes and Steve Tobias within the School of Mathematics. The student will thus be a member of one of the UK’s strongest research groups in astrophysical and geophysical fluid dynamics and magnetohydrodynamics. The group has weekly seminars in fluid dynamics and MHD, in addition to the Applied Mathematics seminars, and holds a weekly discussion meeting in which scientific papers of interest are discussed. A wide variety of courses are on offer to PhD students – ranging from level 5 (Masters level) courses, to specialised computational training, through to training in more general aspects of the PhD. Through the project, the student will be trained not only not only in the geodynamo, fluid dynamics and magnetohydrodyanmics, but will also acquire valuable analytical and computational skills.

### Student profile

The student should have a strong background in either maths or physics, certainly with some knowledge of, and interest in, fluid dynamics. Exposure to magnetohydrodynamics (MHD) would be useful, but is not essential.

### References

- Roberts, P.H., Glatzmaier, G.A. 2000 Geodynamo theory and simulations, Rev. Mod. Phys. 72, 1081.
- Jones, C.A. 2011 Planetary magnetic field and fluid dynamos, Ann. Rev. Fluid Mechanics 45, 583.
- Charbonneau, P.A. 2014 Solar dynamo theory, Ann. Rev. Astron. Astrophys. 52, 251.
- Galloway, D.J., Proctor, M.R.E. 1992 Numerical calculations of fast dynamos in smooth velocity fields with realistic diffusion, Nature, 356, 692.
- Otani, N.F. 1993 A fast kinematic dynamo in two-dimensional time-dependent flows, J. Fluid Mech. 253, 327.

### Related undergraduate subjects:

- Applied mathematics
- Mathematics
- Physics