Meridional circulation in a star

Slow circulation in planes containing the rotation axis can transport entropy through a rotating star while the leading mechanical balance remains nearly hydrostatic equilibrium. Its steady mass flux satisfies the continuity equation. It is distinct from assuming an exactly motionless radiative interior, and from turbulent convection.

The thermal imbalance of a slowly rotating radiative star can drive stellar meridional circulation. Its fractional centrifugal distortion is ${ϵ}_{\Omega }\sim {\Omega }^2{R}^3/(GM)$, so balancing an order-${ϵ}_{\Omega }$ fraction of the luminosity by thermal advection estimates $t_{\mathrm{ES}}\sim t_{\mathrm{KH}}/{ϵ}_{\Omega }$, where $t_{\mathrm{KH}}\sim G{M}^2/(RL)$. Stable composition gradients can further impede the circulation. This radiative thermal estimate is different from rapid torque-driven circulation in a convective envelope.

For steady stellar meridional circulation, $T\nabla s=\nabla (h+\varphi )$ and $\nabla \cdot (\rho \mathbf{v})=0$. Thus $\rho \mathbf{v}\cdot T\nabla s=\nabla \cdot [\rho \mathbf{v}(h+\varphi )]$. In the leading uniformly rotating stellar hydrostatic equilibrium, $h+\varphi$ 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.