The steady incompressible ideal free-boundary flows of a hydromagnetic star

This self-contained theoretical study treats incompressible, free-boundary flows in a gravitating, ideal hydromagnetic star abutting vacuum, centered on the steady field-aligned flows of Chandrasekhar, Prendergast and Tsinganos, together with a novel family of steady cross-field flows, all as solutions of the axisymmetric Tsinganos equation. In the absence of compressive waves and shocks, an incompressible fluid evolves by its frozen-in magnetic field propagating as transverse Alfv\`{e}n waves along the field lines, with pressure reacting instantly in place. The origin of the steady flows rests on the Parker theory that everywhere-continuous flows are the exception rather than the rule because of a basic propensity for tangential field/flow discontinuities. Astrophysical viscosity and electrical resistivity are not zero but are significant only over scales much smaller than macroscopic scales. Such near-ideal fluids have the same propensity for tangential discontinuities but the near-discontinuities readily dissipate by small-scale, viscous-resistive magnetic reconnections. The study treats the strictly ideal fluid separately in its own right, to construct a conceptual understanding of the turbulent creation of a steady flow in a self-organizing near-ideal fluid via irrepressible energy loss and field-topology changes as episodic reconnections run out of free energy. The study suggests that metastable storage of steady vortices and twisted fields is a natural product of the solar internal dynamo, to explain a recent, multi-instrument observation of solar-coronal eruptions persisting coherently in preferred longitudinal locations over solar-rotational timescales.