Numerical Analysis Seminars: Geomagnetic applications of the Lagrange multiplier optimization method
11 March 2016 14:00 in CM105
The Earth’s magnetic field is generated and sustained by the complex motion of a conducting fluid in the liquid outer core. This phenomenon can be understood in the framework of dynamo theory, which mathematically describes the interaction between the flow and the magnetic field. An outstanding question is which kind of flow can amplify a seed magnetic field. The growth rate of the magnetic field is determined by the competition between magnetic advection and magnetic diffusion. The ratio between the two effects is given by a dimensionless parameter called the magnetic Reynolds number (Rm). A seed magnetic field may grow at a sufficiently high Rm, but the precise threshold for a dynamo driven by a general type of flow is unknown. Given a conducting fluid confined in a domain, what is the lowest Rm to generate a dynamo? We use a Lagrange multiplier method inspired by Willis (2012) to search for the most efficient dynamo solution. This method allows us to maximize the growth rate of the magnetic field over a time window T while imposing other constraints using Lagrange multipliers. We simultaneously look for the optimal steady flow field U and the optimal seed magnetic field B0. We choose two common geometries used in kinematic dynamo models: a cube and a sphere. In this talk, I will present the general principle and numerical methods to solve the optimization problem. In addition, I will present the most efficient dynamo solutions we have found.
Contact David.Bourne@durham.ac.uk for more information