Optimal velocity fields for instantaneous dynamo action

We consider a variant of the kinematic dynamo problem. Rather prescribing the velocity field and solving an eigenvalue problem to determine the magnetic field that exhibits maximal growth rate, we treat the seed magnetic-field structure as given and ask which velocity field maximally enhances its growth. We show this second problem has an elegant formulation in terms of variational calculus. Constraining a weighted sum of kinetic energy and enstrophy yields a forced Helmholtz equation with the Lorentz force acting as the inhomogeneity. For the special case of fixed kinetic energy, the problem can be solved exactly and the optimal velocity field everywhere opposes the divergence-free projection of the Lorentz force. Under more general constraints, the optimal velocity profile can differ from the Lorentz force field and can be found by solving the forced Helmholtz equation numerically. We demonstrate these results through 2.5-dimensional periodic examples.