= Material conservation of cross-helicity density
In a <homentropic flow>, the <cross-helicity conservation law> gives
$$
\frac{Dh_c}{Dt}
=\mathbf B\mathbin\cdot\nabla\left(\frac{u^2}{2}-h-\Phi\right)
-h_c\nabla\mathbin\cdot\mathbf u.
$$
Consequently <cross-helicity> density is conserved along each trajectory exactly when the right-hand side vanishes. In a <steady state>, <mass conservation> gives $\nabla\mathbin\cdot\mathbf u=-\mathbf u\mathbin\cdot\nabla\log\rho$, so the criterion becomes
$$
\mathbf B\mathbin\cdot\nabla\left(h+\Phi-\frac{u^2}{2}\right)
=(\mathbf u\mathbin\cdot\mathbf B)\mathbf u\mathbin\cdot\nabla\log\rho.
$$
The absence of an <entropy> source in a conservation law does not alone make its density an advected scalar: compression and transport along the <magnetic field> still change that density.
Back to article page
