Optimized boundary driven flows for dynamos in a sphere

Abstract

We perform numerical optimization of the axisymmetric flows in a sphere to minimize the critical magnetic Reynolds number Rmcr required for dynamo onset. The optimization is done for the class of laminar incompressible flows of von Kármán type satisfying the steady-state Navier-Stokes equation. Such flows are determined by equatorially antisymmetric profiles of driving azimuthal (toroidal) velocity specified at the spherical boundary. The model is relevant to the Madison plasma dynamo experiment, whose spherical boundary is capable of differential driving of plasma in the azimuthal direction. We show that the dynamo onset in this system depends strongly on details of the driving velocity profile and the fluid Reynolds number Re. It is found that the overall lowest Rmcr≈200 is achieved at Re≈240 for the flow, which is hydrodynamically marginally stable. We also show that the optimized flows can sustain dynamos only in the range Rmcr<Rm<Rmcr2, where Rmcr2 is the second critical magnetic Reynolds number, above which the dynamo is quenched. Samples of the optimized flows and the corresponding dynamo fields are presented.