Large-scale circulation reversals explained by pendulum correspondence

Abstract

We introduce a low-order dynamical system to describe thermal convection in an annular domain. The model derives systematically from a Fourier-Laurent truncation of the governing Navier-Stokes Boussinesq equations and accounts for spatial dependence of the flow and temperature fields. Comparison with fully-resolved direct numerical simulations (DNS) shows that the model captures parameter bifurcations and reversals of the large-scale circulation (LSC), including states of (i) steady circulating flow, (ii) chaotic LSC reversals, and (iii) periodic LSC reversals. Casting the system in terms of the fluid's angular momentum and center of mass (CoM) reveals equivalence to a damped pendulum with forcing that raises the CoM above the fulcrum. This formulation offers a transparent mechanism for LSC reversals, namely the inertial overshoot of a forced pendulum, and it yields an explicit formula for the frequency $f^*$ of regular LSC reversals in the high Rayleigh-number limit. This formula is shown to be in excellent agreement with DNS and produces the scaling law $f^* \sim Ra^{0.5}$.