Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 314 / 4 / b / Solution

If $\mathbf{u}$ is parallel to $\mathbf{B}$, write $\mathbf{u}=\kappa \mathbf{B}$. Thenand the flux from part a reduces towhere $C_B$ is the gravitational Bernoulli function. If it is constant along field lines, $\mathbf{B}\cdot \nabla C_B=0$, thenThe source also vanishes because the flow is homentropic. Hence ${\partial }_t{h}_c=0$, and every fixed volume satisfiesThus the cross-helicity is conserved even for this time-dependent aligned flow.