Optimal kinematic dynamos in a sphere

A variational optimization approach is used to optimize kinematic dynamos in a unit sphere and locate the enstrophy-based critical magnetic Reynolds number for dynamo action. The magnetic boundary condition is chosen to be either pseudo-vacuum or perfectly conducting. Spectra of the optimal flows corresponding to these two magnetic boundary conditions are identical since theory shows that they are relatable by reversing the flow field (Favier & Proctor 2013 Phys. Rev. E 88, 031001 (doi:10.1103/physreve.88.031001)). A no-slip boundary for the flow field gives a critical magnetic Reynolds number of 62.06, while a free-slip boundary reduces this number to 57.07. Optimal solutions are found to possess certain rotation symmetries (or anti-symmetries) and optimal flows share certain common features. The flows localize in a small region near the sphere's centre and spiral upwards with very large velocity and vorticity, so that they are locally nearly Beltrami. We also derive a new lower bound on the magnetic Reynolds number for dynamo action, which, for the case of enstrophy normalization, is five times larger than the previous best bound.