Relaxed Magnetohydrodynamics with Ideal Ohm's Law Constraint

Recently, a new magnetofluid dynamics, Relaxed MHD (RxMHD), was constructed using Hamilton's Principle with a phase-space Lagrangian incorporating constraints of magnetic and cross helicity. A key difference between RxMHD and Ideal Magnetohydrodynamics (IMHD) is that IMHD implicitly constrains the magnetofluid to obey the zero-resistivity "Ideal" Ohm's Law (IOL) pointwise whereas RxMHD discards the IOL constraint completely, which can violate the desideratum that all equilibrium solutions of RxMHD form a subset of all IMHD equilibria. The present paper lays the formal groundwork for rectifying this deficiency. In order to impose a weak form of the IOL constraint on RxMHD two forms of the iterative augmented Lagrangian penalty function method are proposed and discussed. It is conjectured this weak-form regularization will allow reconnection and thus avoid the formation of the singularities that plague three-dimensional IMHD equilibria. A unified dynamical formalism is developed that can treat a number of MHD versions. Euler-Lagrange equations and a gauge-invariant momentum equation in conservation form are derived, in which the IOL constraint contributes an external force and internal stress terms until convergence is achieved.