= Cross-helicity conservation law
Let $h_c=\mathbf u\mathbin\cdot\mathbf B$ be the <cross-helicity> density. The <ideal magnetohydrodynamic induction equation> implies $D\mathbf B/Dt=(\mathbf B\mathbin\cdot\nabla)\mathbf u-\mathbf B\nabla\mathbin\cdot\mathbf u$. Dotting the momentum equation with $\mathbf B$ eliminates the <Lorentz force density>, since it is perpendicular to the <magnetic field>. The <material derivative> of $h_c$ is therefore
$$
\frac{Dh_c}{Dt}
=\mathbf B\mathbin\cdot\nabla\left(\frac{u^2}{2}-\Phi\right)
-\frac{\mathbf B\mathbin\cdot\nabla p}{\rho}
-h_c\nabla\mathbin\cdot\mathbf u.
$$
Using $dh=T\,ds+dp/\rho$ for <specific enthalpy> $h$ and <specific entropy> $s$, together with <Gauss's law for magnetism>, converts this to
$$
\partial_th_c+\nabla\mathbin\cdot\left[
\mathbf u h_c+\mathbf B\left(h+\Phi-\frac{u^2}{2}\right)
\right]
=T\mathbf B\mathbin\cdot\nabla s.
$$
The source vanishes when <magnetic field lines> lie in constant-<specific entropy> surfaces, in particular in a <homentropic flow>. Integrating over a volume conserves its total <cross-helicity> only if the boundary flux also vanishes. This local law is derived in https://arxiv.org/abs/1604.03835[Ogilvie's astrophysical fluid dynamics notes].
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