A trio of simple optimized axisymmetric kinematic dynamos in a sphere

Holdenried-Chernoff, D.; Chen, L.; Jackson, A.

Abstract

Planetary magnetic fields are generated by the motion of conductive fluid in the planet's interior. Complex flows are not required for dynamo action; simple flows have been shown to act as efficient kinematic dynamos, whose physical characteristics are more straightforward to study. Recently, Chen et al. (2018, J. Fluid Mech.839, 1–32. (doi:10.1017/jfm.2017.924)) found the optimal, unconstrained kinematic dynamo in a sphere, which, despite being of theoretical importance, is of limited practical use. We extend their work by restricting the optimization to three simple two-mode axisymmetric flows based on the kinematic dynamos of Dudley & James (1989, Proc. R. Soc. Lond. A425, 407–429. (doi:10.1098/rspa.1989.0112)). Using a Lagrangian optimization, we find the smallest critical magnetic Reynolds number for each flow type, measured using an enstrophy-based norm. A Galerkin method is used, in which the spectral coefficients of the fluid flow and magnetic field are updated in order to maximize the final magnetic energy. We consider the t01s01, t01s02 and t02s02 flows and find enstrophy-based critical magnetic Reynolds numbers of 107.7, 142.4 and 125.5 (13.7, 19.6 and 16.4, respectively, with the energy-based definition). These are up to four times smaller than the original flows. These simple and efficient flows may be used as benchmarks in future studies.

Citation

Holdenried-Chernoff, D., Chen, L., & Jackson, A. (2019). A trio of simple optimized axisymmetric kinematic dynamos in a sphere. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 475(2229), Article 20190308. https://doi.org/10.1098/rspa.2019.0308

• Journal Article Type: Article
• Acceptance Date: Aug 15, 2019
• Online Publication Date: Sep 18, 2019
• Publication Date: Sep 4, 2019
• Deposit Date: Oct 7, 2019
• Publicly Available Date: Nov 1, 2019
• Journal: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
• Print ISSN: 1364-5021
• Electronic ISSN: 1471-2946
• Publisher: The Royal Society
• Peer Reviewed: Peer Reviewed
• Volume: 475
• Issue: 2229
• Article Number: 20190308
• DOI: https://doi.org/10.1098/rspa.2019.0308
• Public URL: https://durham-repository.worktribe.com/output/1289680