Relaxed magnetohydrodynamics with cross-field flow

The phase-space Lagrangian model of Dewar et al. (Phys. Plasmas 27, 062507, 2020) provides a framework for incorporating cross-field flow into relaxed equilibria while retaining ideal magnetohydrodynamics force balance. Here, we characterize the steady-state solution space and identify a solvability condition that couples the prescribed constrained flow to the geometry through the metric tensor. Using this condition, we construct equilibria in slab, cylindrical, and toroidal geometries. In toroidal geometry, the cross-field flow strongly correlates with magnetic-island structure: varying the rotation frequency modifies the dominant Fourier harmonic of the radial component of the magnetic field and can drive a transition from a primary (m = 1) island to secondary (m = 2) islands. In slab and cylindrical geometries, flow parameters weakly affect island width but strongly modify equilibrium profiles.