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Durham University

Department of Mathematical Sciences


Publication details for Dr Long Chen

Chen, L., Herreman, W., Li, K., Livermore, P. W., Luo, J. W. & Jackson, A. (2018). The optimal kinematic dynamo driven by steady flows in a sphere. Journal of Fluid Mechanics 839: 1-32.

Author(s) from Durham


We present a variational optimization method that can identify the most efficient
kinematic dynamo in a sphere, where efficiency is based on the value of a magnetic
Reynolds number that uses enstrophy to characterize the inductive effects of the
fluid flow. In this large-scale optimization, we restrict the flow to be steady and
incompressible, and the boundary of the sphere to be no-slip and electrically
insulating. We impose these boundary conditions using a Galerkin method in terms of
specifically designed vector field bases. We solve iteratively for the flow field and the
accompanying magnetic eigenfunction in order to find the minimal critical magnetic
Reynolds number Rmc,min for the onset of a dynamo. Although nonlinear, this iteration
procedure converges to a single solution and there is no evidence that this is not
a global optimum. We find that Rmc,min = 64.45 is at least three times lower than
that of any published example of a spherical kinematic dynamo generated by steady
flows, and our optimal dynamo clearly operates above the theoretical lower bounds
for dynamo action. The corresponding optimal flow has a spatially localized helical
structure in the centre of the sphere, and the dominant components are invariant
under rotation by π.