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For uniform specific entropy, the material conservation of cross-helicity density criterion isThus the requested condition isIn the steady state, mass conservation gives $\nabla \cdot \mathbf{u}=-\mathbf{u}\cdot \nabla \log \rho$. An equivalent form, expressed entirely in the flow and thermodynamic variables, isIf “isentropic” means only constant specific entropy along individual trajectories, rather than a homentropic flow, the general criterion instead retains $T\mathbf{B}\cdot \nabla s$ on the right-hand side of the material derivative equation. The distinction matters because a source-free cross-helicity conservation law need not make $h_c$ constant along each trajectory.