A convective fluid pendulum revealing states of order and chaos

We examine thermal convection in a two-dimensional annulus using fully resolved direct numerical simulation (DNS) in conjunction with a low-dimensional model deriving from Galerkin truncation of the governing Navier-Stokes Boussinesq (NSB) equations. The DNS is based on fast and accurate pseudo-spectral discretization of the full NSB system with implicit-explicit time-stepping. Measurements of the frequency power spectrum indicate that the DNS accurately resolves turbulent fluctuations present at high Rayleigh number. The reduced model is based on a Fourier-Laurent truncation of the NSB system and can generalize to any degree of accuracy. We demonstrate that the lowest-order model capable of satisfying all boundary conditions on the annulus successfully captures reversals of the large-scale circulation (LSC) in certain regimes. Based on both the DNS and stability analysis of the reduced model, we identify a sequence of transitions between (i) a motionless conductive state, (ii) a state of steady circulation, (iii) non-periodic dynamics and chaotic reversals of the LSC, (iv) a high Rayleigh-number state in which LSC reversals are periodic despite turbulent fluctuations at the small scale. Measurements of the fractal dimension and Lyapunov exponent characterize the chaotic state and show the return to orderly dynamics at very high Rayleigh number. The reduced model reveals a link to a damped pendulum system with a particular form of external forcing. Calculations based on an energy balance of this system yield accurate predictions for the frequency of regular LSC reversals in the high Rayleigh-number regime.