= Equipotential luminosity conservation with stellar circulation
{title2=$\int_S\mathbf F\cdot d\mathbf S=\int_V\rho\epsilon\,dV$}
For steady <stellar meridional circulation>, $T\nabla s=\nabla(h+\phi)$ and $\nabla\cdot(\rho\mathbf v)=0$. Thus $\rho\mathbf v\cdot T\nabla s=\nabla\cdot[\rho\mathbf v(h+\phi)]$. In the leading <uniformly rotating stellar hydrostatic equilibrium>, $h+\phi$ is constant on an <equipotential surface>; its total advected flux vanishes because the net mass flux vanishes. Integrating the energy equation then identifies the outward <luminosity> with enclosed nuclear energy generation, even though local heat advection is nonzero.
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