Time-dependent relaxed magnetohydrodynamics: Inclusion of cross helicity constraint using phase-space action

A phase-space version of the ideal magnetohydrodynamic (MHD) Lagrangian is derived from first principles and shown to give a relabeling transformation when a cross-helicity constraint is added in Hamilton's Action Principle. A new formulation of time-dependent "relaxed" magnetohydrodynamics is derived using microscopic conservation of mass and macroscopic constraints on total magnetic helicity, cross helicity, and entropy under variations of density, pressure, fluid velocity, and magnetic vector potential. This gives Euler-Lagrange equations consistent with previous work on both ideal and relaxed MHD equilibria with flow, but generalizes the relaxation concept from statics to dynamics. The application of the new dynamical formalism is illustrated for short-wavelength linear waves, and the interface connection conditions for Multiregion Relaxed MHD (MRxMHD) are derived. The issue of whether E + u x B = 0 should be a constraint is discussed.