Part 1: We address the problem of meridional circulation and differential rotation in stellar radiative zones using the “downward control principle” of atmospheric science under a geostrophic f-plane approximation. Before steady-state, meridional circulation and zonal wind spread together via radiative diffusion under thermal wind balance. We derive an advection-diffusion equation for the zonal flow starting from primitive equations, yielding time-dependent analytical solutions that enable rapid prototyping of differential rotation profiles. In the weak drag limit, stars may never reach rotational steady state within their main-sequence lifetimes.

Part 2: Tidal locking causes a planet’s atmospheric circulation to have a fixed hotspot, while asynchronously rotating planets feature moving hotspots. We show that such a planet can mimic a tidally locked circulation if its diurnal period resonates with time-varying instellation, as in pulsating or multiple star systems. Slight period mismatches cause the hotspot to drift slowly East-West or West-East, governed by the difference between diurnal and instellation variation rates—analogous to beat patterns in wave theory. We term this phenomenon "beating." Dynamical constraints on beating arise from radiative, rotational, wave propagation, and drag timescales. Applying this to Kepler and TESS circumbinary planets, we identify two candidates with climatic deviations from the no-variation case. Hotter, faster-spinning planets are more prone to these effects, which may also extend habitable zones.