Nonlinear hydrodynamic and hydromagnetic spin-up driven by Ekman-Hartmann boundary layers

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Abstract
Finite amplitude, impulsively started spin-up and spin-down is analysed for axially symmetric flow of a viscous, incompressible, electrically conducting fluid confined between infinite, flat, parallel, insulating boundaries. A uniform axial magnetic field is present in the initial state, but is subsequently distorted by fluid motions. The method of matched asymptotic expansions reduces the problem to a first-order, ordinary, nonlinear, integro-differential equation for the transient development of the interior angular velocity on the time scale of spin- up, as driven by quasi-steady nonlinear Ekman-Hartmann boundary layers. This two-parameter equation is solved analytically in certain limits and numeric-ally in general. The solutions show that nonlinear non-magnetic spin-up and spin-down take longer than for linearized flow, spin-down occurring more rapidly in the early stages but requiring more time for completion than spin-up. A magnetic field promotes both spin-up and spin-down, but a weak field is relatively ineffective for spin-down yet very effective for spin-up. A strong magnetic field dominates nonlinear processes and gives identical spin-up and spin-down times, which coincide with that found from linear hydromagnetic theory.