Prandtl--Batchelor and flux-expulsion selection for steady MHD flows in a disk

Abstract

We study the simultaneous vanishing-viscosity and vanishing-resistivity limit of steady incompressible MHD flows in a disk. The boundary velocity is a small nonaxisymmetric perturbation of a rigid rotation with mean angular speed \(α\), while the prescribed tangential magnetic trace has mean \(β\). Assuming \(α\neq0\) and the non-Alfvénic condition \(|α|\neq|β|\), we construct solutions converging on compact interior subdisks to a rigidly rotating ideal MHD core with constant vorticity and out-of-plane current density. A new MHD--Wood law determines the two core rotations from the boundary data: the velocity core is selected by a coupled kinetic--magnetic balance, whereas the magnetic core is fixed by the imposed mean circulation. Consequently, zero circulation gives complete interior magnetic expulsion, while nonzero circulation leaves a uniform magnetic rotation after the nonaxisymmetric modes are confined to a thin boundary layer. This provides a fully coupled realization of Prandtl--Batchelor selection and flux expulsion; unlike classical kinematic models, the magnetic field actively changes the flow and need not be weak. The proof combines a non-Alfvénic coercive theory for a periodic MHD boundary layer, global matching of the two fields, and a coupled stability estimate adapted to the magnetic boundary condition. A separate conditional rigidity argument, using exact viscous identities and local convergence but no interior asymptotic expansion, explains the same core structure for a broader single-eddy family.