Radiative-equilibrium obstruction in a rotating barotropic star ($\nabla \cdot (f\nabla \varphi )=f^{\prime }|\nabla \varphi {|}^2+f{\nabla }^2\varphi$)

On a regular equipotential surface, $f,f^{\prime },\rho$ and ${\nabla }^2\varphi$ are constant, but $|\nabla \varphi {|}^2$ generally varies. Consequently a flux $\mathbf{F}=f(\varphi )\nabla \varphi$ cannot ordinarily satisfy local radiative equilibrium $\nabla \cdot \mathbf{F}=\rho ϵ$ at all latitudes. Special cases such as $f^{\prime }=0$ or spherical symmetry evade this obstruction. Stellar meridional circulation supplies heat advection in the leading hydrostatic treatment.