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Let $h_c=\mathbf{u}\cdot \mathbf{B}$ be the cross-helicity density. The ideal magnetohydrodynamic induction equation implies $D\mathbf{B}/Dt=(\mathbf{B}\cdot \nabla )\mathbf{u}-\mathbf{B}\nabla \cdot \mathbf{u}$. 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 thereforeUsing $dh=Tds+dp/\rho$ for specific enthalpy $h$ and specific entropy $s$, together with Gauss's law for magnetism, converts this toThe 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 Ogilvie's astrophysical fluid dynamics notes.

In a homentropic flow, the cross-helicity conservation law givesConsequently cross-helicity density is conserved along each trajectory exactly when the right-hand side vanishes. In a steady state, mass conservation gives $\nabla \cdot \mathbf{u}=-\mathbf{u}\cdot \nabla \log \rho$, so the criterion becomesThe 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.

Let $e=c_{v}T$ be the specific internal energy of a calorically perfect gas and define its specific gas constant $\mathcal{R}=R^{\ast }/\mu$, so $p=\mathcal{R}\rho T$. A reversible adiabatic change has no heat input and obeysUsing $c_p-c_v=\mathcal{R}$ and $\gamma =c_p/c_v$ givesIntegration yields $p/{\rho }^{\gamma }=K$, or $\boxed{p=K{\rho }^{\gamma }}$, with $K$ fixed on that isentrope. Equivalently $s-s_{\mathrm{ref}}=c_v\ln [(p/{\rho }^{\gamma })/K_{\mathrm{ref}}]$, so $K$ specifies entropy as well as composition. In a smooth nondissipative flow this means $D(p{\rho }^{-\gamma })/Dt=0$ along each parcel. A single common $K$ throughout the fluid additionally requires a homentropic flow. No heat transfer alone does not forbid entropy production: different sides of an adiabatic shock generally have different $K$ values. Constant composition and heat capacities are needed for the fixed exponent used here.

Write $D/Dt={\partial }_t+\mathbf{u}\cdot \nabla$ for the material derivative and $q=\mathbf{u}\cdot \mathbf{B}$ for the cross-helicity density, so $H_c={\int }_{V}qdV$. The ideal magnetohydrodynamic induction equation of ideal magnetohydrodynamics, with $\nabla \cdot \mathbf{B}=0$, givesThe dot product of the Euler momentum equation with the magnetic field has no Lorentz force contribution, because $\mathbf{B}\cdot [(\nabla \times \mathbf{B})\times \mathbf{B}]=0$. Consequently,Here $h=e+p/\rho$ is the specific enthalpy, $T$ the temperature, and $s$ the specific entropy; $e$ is the internal energy per unit mass. The first law of thermodynamics in this convention is $de=Tds-pd(1/\rho )$, so $dh=Tds+dp/\rho$. Substitution, followed by $\nabla \cdot \mathbf{B}=0$, yields the cross-helicity conservation lawThe vector triple product identity $\mathbf{u}\times (\mathbf{u}\times \mathbf{B})=\mathbf{u}q-u^2\mathbf{B}$ puts this in the equivalent form

For the fixed volume, the divergence theorem gives the exact cross-helicity balanceThus cross-helicity is constant when the net source equals the outward flux. Simple sufficient hypotheses are $\mathbf{B}\cdot \nabla s=0$ throughout the volume and $\mathbf{u}\cdot \mathbf{n}=\mathbf{B}\cdot \mathbf{n}=0$ on its boundary. The first condition includes a homentropic flow; the second prevents both relevant boundary fluxes. Periodic boundary conditions, or sufficiently rapid decay at infinity, provide alternatives. These statements concern smooth ideal magnetohydrodynamics: material specific entropy conservation alone, $Ds/Dt=0$, does not eliminate spatial specific entropy gradients along the magnetic field lines.

Use the Euler equations for an inviscid fluid with no magnetic or gravitational force. Let $w$ denote specific enthalpy. For a homentropic flow, $dw=dp/\rho$, so the pressure acceleration is $-\nabla w$. The velocity identitytherefore givesTaking the curl, using that the curl of a gradient is zero and that differentiation commutes for smooth fields, proves barotropic vorticity transport:More generally the same proof works for any barotropic fluid, replacing $w$ by a pressure potential ${\int }^{\rho }{p}^{\prime }(s)ds/s$. Here “isentropic” must supply this barotropic closure, for example through one common specific entropy throughout the fluid. Merely imposing entropy advection equation $Ds/Dt=0$ allows spatial specific entropy gradients; in that case the vorticity equation contains the additional baroclinic vorticity generation term ${\rho }^{-2}\nabla \rho \times \nabla p$.

The calculation in part (a) leaves the cross-helicity conservation law sourceAt positive temperature, this vanishes exactly when the magnetic field is tangent to constant-specific entropy surfaces: $\mathbf{B}\cdot \nabla s=0$. A homentropic flow is a sufficient special case; uniform entropy throughout space is not necessary. Mere advection of specific entropy, $Ds/Dt=0$, constrains its variation along the velocity rather than along the magnetic field, so it does not by itself eliminate this source. To conserve total cross-helicity in a volume, the net boundary flux must also vanish.

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.

Use the material forms of the ideal magnetohydrodynamic induction equation and momentum equation:The magnetic force is perpendicular to $\mathbf{B}$. For $h_c=\mathbf{u}\cdot \mathbf{B}$, the two equations therefore giveSince $\nabla \cdot \mathbf{B}=0$, this is the cross-helicity conservation lawwithFor a homentropic flow, $\nabla s=0$, so the source vanishes.

Specific entropy is entropy per unit mass. For a perfect gas with constant specific heat capacity $c_v$ at constant volume and specific-heat ratio $\gamma$, differences obeywhere $p$ is pressure and $\rho$ is mass density. Thus a homentropic flow has constant $p/{\rho }^{\gamma }$.