The optimal kinematic dynamo driven by steady flows in a sphere

Chen, L.; Herreman, W.; Li, K.; Livermore, P.W.; Luo, J.W.; Jackson, A.

Abstract

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.

Citation

Chen, L., Herreman, W., Li, K., Livermore, P., Luo, J., & Jackson, A. (2018). The optimal kinematic dynamo driven by steady flows in a sphere. Journal of Fluid Mechanics, 839, 1-32. https://doi.org/10.1017/jfm.2017.924

• Journal Article Type: Article
• Acceptance Date: Dec 19, 2017
• Online Publication Date: Jan 25, 2018
• Publication Date: Mar 25, 2018
• Deposit Date: Apr 23, 2018
• Publicly Available Date: Jul 25, 2018
• Journal: Journal of Fluid Mechanics
• Print ISSN: 0022-1120
• Electronic ISSN: 1469-7645
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 839
• Pages: 1-32
• DOI: https://doi.org/10.1017/jfm.2017.924
• Public URL: https://durham-repository.worktribe.com/output/1328699