Meridional circulation revisited

The investigation of time-dependent meridional circulation and differential rotation in radiative zones remains a central and challenging topic in stellar evolution theory. To address this, we apply the 'downward control principle' from atmospheric science under a geostrophic f-plane approximation. We confirm the known physics result that steady-state meridional circulation decays with a length scale of N/2Ω × √Pr, assuming molecular viscosity as the dominant drag mechanism. Prior to steady-state, the circulation and zonal wind (differential rotation) spread jointly via radiative diffusion, adhering to thermal wind balance. The corresponding hyper-diffusion process is reasonably well approximated by regular diffusion on scales comparable to the pressure scale-height. We derive an inhomogeneous diffusion equation for the zonal flow, offering closed-form time-dependent solutions within a finite depth domain, facilitating rapid prototyping of meridional circulation patterns. In the weak drag limit, the time to reach rotational steady-state may exceed the Eddington-Sweet time, being governed instead by the longer drag time. We also conclude that the current Sun is only one-tenth as young as its rotational steady-state and will exhaust all its fuel before reaching that stage. Our streamlined meridional circulation solutions provide leading-order internal rotation profiles, enabling the study of fluid/MHD instabilities (or waves) in angular momentum redistribution within stellar radiative zones. Despite geometric limitations and simplifying assumptions, our thin-layer geostrophic approach is expected to yield qualitatively useful insights for understanding deep meridional circulation in stars.