On the inverse cascade and flow speed scaling behaviour in rapidly rotating Rayleigh–Bénard convection. (3rd March 2021) - Maffei, S.; Krouss, M.J.; Julien, K.; Calkins, M.A.
Abstract: Abstract : Rotating Rayleigh–Bénard convection is investigated numerically with the use of an asymptotic model that captures the rapidly rotating, small Ekman number limit, $Ek → 0$ . The Prandtl number ( $Pr$ ) and the asymptotically scaled Rayleigh number ( $\widetilde {Ra} = Ra Ek^{4/3}$, where $Ra$ is the typical Rayleigh number) are varied systematically. For sufficiently vigorous convection, an inverse kinetic energy cascade leads to the formation of a pair of large-scale vortices of opposite polarity, in agreement with previous studies of rapidly rotating convection. With respect to the kinetic energy, we find a transition from convection dominated states to a state dominated by large-scale vortices at an asymptotically reduced (small-scale) Reynolds number of $\widetilde {Re} ≈ 6$ ( $\widetilde {Re} = Re Ek^{1/3}$, where $Re$ is the Reynolds number associated with vertical flows) for all investigated values of $Pr$ . The ratio of the depth-averaged kinetic energy to the kinetic energy of the convection reaches a maximum at $\widetilde {Re} ≈ 24$, then decreases as $\widetilde {Ra}$ is increased. This decrease in the relative kinetic energy of the large-scale vortices is associated with a decrease in the convective correlations with increasing Rayleigh number. The scaling behaviour of the convective flow speeds is studied; although a linear scaling of the form $\widetilde {Re} \sim \widetilde {Ra}/Pr$ is observed over a limited range in Rayleigh number and Prandtl number, a clear departure from this scaling is observed at the highest accessible values of $\widetilde {Ra}$ . Calculation of the forces present in the governing equations shows that the ratio of the viscous force to the buoyancy force is an increasing function of $\widetilde {Ra}$, that approaches unity over the investigated range of parameters.