A linear dual-field finite element method for incompressible MHD, part I: cross-helicity and Alfv\'enic alignment in periodic domains

Abstract

Cross helicity, together with total energy, determines the Els\"asser energies and thus quantifies departures from Alfv\'enic alignment in incompressible MHD. Building on the dual-field method in [Mao and Xi, \emph{J. Comput. Phys.}, 2025], we construct a linear finite element method for periodic domains that reproduces the cross-helicity balance. We connect the discrete energy and cross-helicity conservation laws to Alfv\'enic alignment through a nonnegative Els\"asser-energy representation, establishing preservation of the corresponding exact relation when either Els\"asser energy is initially zero in the unforced ideal case. We reassign the two discrete approximations of the magnetic field to the integer and half-integer time levels and redesign the Lorentz and induction couplings within the Mao-Xi formulation. The resulting scheme uses linear staggered updates and retains the energy balances, strong and weak divergence-free properties, and magnetic- and fluid-helicity balances. Numerical experiments illustrate the role of the cross-helicity balance in preserving an aligned state in the ideal case and in alignment dynamics under weak dissipation.