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Building a realistic model of Earth's magnetic field using constrained dynamics

Dr Phil Livermore (SEE), Dr Jitse Niesen (SoM)

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A proper understanding of how the geomagnetic field is generated in Earth’s liquid core, by the so-called geodynamo, remains one of the greatest outstanding problems in Earth science. The principal difficulty is that the core is far too remote to be probed directly; scientific knowledge has advanced through exploiting the limited set of observations and computer simulations of the Earth’s core. Of significant importance is that observations from Earth’s surface can only constrain the structure of the magnetic field at the edge of the source region: the structure of the field within the core is unknown.


Figure 1: Observations of the Earth’s internal field taken from the surface and above constrain the structure of the field at the edge of the source region (the core-mantle boundary), shown by selected field lines. The sign of the outwards pointing magnetic field on the core-mantle boundary is shown as red/blue. The structure of the magnetic field inside the core is unknown, and is the focus of this project.

From a mathematical standpoint, on medium to long time scales, the core can be realistically modelled as a constrained dynamical system. Such an idea may be more familiar in mechanical systems such as (industrial) robots, which are often modelled as constrained dynamical systems: the parts move under the force exerted by motors according to the laws of mechanics under the constraint that the rods and other elements do not extend or compress. The rods in such systems can also be considered as springs in the limit that the stiffness constant goes to infinity. Specialized numerical methods are required when simulating such systems. 

The focus of this project is to consider the Earth's core as evolving under the control of a system of constraints, called the Taylor constraints (Taylor, 1963), which stem from the dominance of the rotational forces inside the core. In addition, recent evidence from seismology alongside studies of the material conditions thought to prevail within the core, further suggest the outermost part of the core is stratified (Davies et al, 2015). This leads to a further set of Malkus constraints, that add to the Taylor constraints, which the internal magnetic field must satisfy (Malkus, 1979).  The goal of this project is to construct both static and dynamical models of the Earth’s magnetic field that satisfy this large set of constraints.

Realistic modelling of the large-scale background structure of the internal magnetic field may shed light on fundamental features which are still unexplained, such as why the Earth’s field is predominantly aligned with the rotation axis, and how the magnetic field undergoes global reversals.


The goal of the project is to create realistic models of the magnetic field inside the Earth’s core, by exploiting the large family of the Taylor and Malkus constraints taken together. This project will involve both theoretical and computational aspects in modelling the Earth’s geodynamo as a constrained dynamical system. The project will be undertaken in two phases: observational snapshots models, and dynamical modelling.

  1. In the first phase, the student will reproduce existing work on the mathematical structure of the Taylor constraints within a discretised model. Coupled with observations of the magnetic field, these can be used to infer the structure of the internal field inside the core (Livermore et al, 2011). A new analysis of the mathematical structure of the Malkus constraints will then allow the whole problem to be formulated. The main task will then be to find solutions of these nonlinear constraints which are also compatible with magnetic observations. This will allow us to image the magnetic field structure inside the core assuming stratification. The student will then compare such images to those deduced from other means (e.g. data assimilation). 
  2. The student will investigate magnetic field evolution within the stratified layer by constructing models that evolve subject to the Malkus constraints. These models will be compared to observation-derived models, as well as the state-of-the-art supercomputer models of Earth’s core.


Year 1:  Familiarisation with geomagnetic data and theory of the dynamics within Earth’s core. Formulation of the Taylor constraints within a discretised model. Benchmarking against existing results.

Year 2: Formulation of the new Malkus constraints. Using these alongside the Taylor constraints to image the magnetic field inside Earth’s core. Comparison to other models.

Year 3: Dynamical modelling and comparison to observations and simulations.

Potential for high impact outcome

Explaining important features in the Earth’s magnetic field is an international endeavour and of wide interest. Studies of models of the dynamics of Earth’s core date back to the 1950's and have been published in high impact journals such as Nature and Science. Imaging inside the Earth’s core would be paradigm changing.


The student will learn both the theory and computational techniques required to model the Earth’s core, and will have access to a broad spectrum of training workshops put on by the Faculty that include an extensive range of workshops in numerical modelling, through to managing your degree, to preparing for your viva (

The student will be a part of the deep Earth research group, a vibrant part of the Institute of Geophysics and Tectonics, comprising staff members, postdocs and PhD students. The deep Earth group has a strong portfolio of international collaborators which the student will benefit from.

Although the project will be based at Leeds, there will be opportunities to attend international conferences (UK, Europe, US and elsewhere), and potentially collaborative visits within Europe.


We seek a highly motivated candidate with a strong background in mathematics, physics, computation, geophysics or another highly numerate discipline.  Knowledge of geomagnetism is not required, and training will be given in all aspects of the PhD.
For further information please contact Phil Livermore ( or Jitse Niesen (

Other opportunities

The Deep Earth Research Group in Leeds ( is one of the largest groups of scientists studying the structure and dynamics of Earth’s core and mantle in the world. Research topics include the dynamics and structure of the Earth’s magnetic field and convection in the outer core, material properties under high pressure and temperature and Global Seismology. The Group collaborates closely with the Department of Applied Mathematics in Leeds and Deep Earth research groups worldwide. Dr Livermore is interested in the dynamics of the core and geomagnetism. Please contact him ( to discuss further PhD opportunities.


Davies, C., Pozzo, M., GUBBINS, D., and Alfè, D. (2015). Constraints from material properties on the dynamics and evolution of Earth’s core. Nature Geoscience, 8(9), 678–685.

Hulot, G., Sabaka, T. J., Olsen, N. and Fournier, A. (2015) The Present and Future Geomagnetic Field. Treatise on Geophysics, Vol 5.02, Elsevier.

Livermore, P. W, Ierley, G. and  Jackson, A. (2009). The construction of exact Taylor states. I: The full sphere. Geophysical Journal International, 179(2), 923–928.

Malkus, W. (1979). Dynamo macrodynamics in rotating stratified fluids. Physics of the Earth and Planetary Interiors, 20(2-4), 181–184.

Stern, D. A Millennium of Geomagnetism, online material:

Taylor, J. (1963). The Magneto-Hydrodynamics of a Rotating Fluid and the Earth's Dynamo Problem. Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, 274, 274–283.

Related undergraduate subjects:

  • Geophysics
  • Mathematics
  • Physics