Flux expulsion with dynamics

In the process of flux expulsion, a magnetic field is expelled from a region of closed streamlines on a TR1/3 m time scale, for magnetic Reynolds number Rm ≫ 1 (T being the turnover time of the flow). This classic result applies in the kinematic regime where the flow field is specified independently of the magnetic field. A weak magnetic ‘core’ is left at the centre of a closed region of streamlines, and this decays exponentially on the TR1/2 m time scale. The present paper extends these results to the dynamical regime, where there is competition between the process of flux expulsion and the Lorentz force, which suppresses the differential rotation. This competition is studied using a quasi-linear model in which the flow is constrained to be axisymmetric. The magnetic Prandtl number Rm/Re is taken to be small, Rm large, and a range of initial field strengths b0 is considered. Two scaling laws are proposed and confirmed numerically. For initial magnetic fields below the threshold bcore = O(UR−1/3 m ), flux expulsion operates despite the Lorentz force, cutting through field lines to result in the formation of a central core of magnetic field. Here U is a velocity scale of the flow and magnetic fields are measured in Alfv´en units. For larger initial fields the Lorentz force is dominant and the flow creates Alfv´en waves that propagate away. The second threshold is bdynam = O(UR−3/4 m ), below which the field follows the kinematic evolution and decays rapidly. Between these two thresholds the magnetic field is strong enough to suppress differential rotation leaving a magnetically controlled core spinning in solid body motion, which then decays slowly on a time scale of order TRm.