 Solution

The logarithmic density term is the barotropic fluid energy for $p=c_s^2\rho$, rather than the thermodynamic heat content of an isolated gas. DefineFor any scalar $q$, the continuity equation implies ${\partial }_t(\rho q)+{\partial }_z(\rho wq)=\rho Dq$. Therefore $De=-c_s^2{w}^{\prime }$, $D\Phi =wg$, and dotting the momentum equations with $\rho \mathbf{v}$ givesThe MHD induction equation gives the complementary magnetic energy balanceAdding these two identities, and collecting the product derivatives, yields the barotropic magnetic energy equationHere $E=E_h+E_B$. The two stress terms in $S$ describe magnetohydrodynamic shear work: the maintained background shear flow can supply energy to, or remove energy from, the perturbation flow and magnetic field. The sign depends on the off-diagonal total stress; this is why the energy excluding the background shear flow is not generally conserved. Changing ${\rho }_0$ merely adds a constant multiple of the conserved density to $E$ and the corresponding mass flux to $F$, leaving $S$ unchanged.