On the inverse cascade and flow speed scaling behaviour in rapidly rotating Rayleigh–Bénard convection

• Published in: Journal of fluid mechanics 2021-03, Vol.913, Article A18
• Main Authors: Maffei, S., Krouss, M.J., Julien, K., Calkins, M.A.
• Format: Article
• Language: English
• Subjects: Asymptotic properties; Cascade flow; Computational fluid dynamics; Convection; Convective flow; Fluid flow; Fluids; JFM Papers; Kinetic energy; Numerical analysis; Polarity; Prandtl number; Rayleigh number; Rayleigh-Benard convection; Reynolds number; Rotation; Scaling; Simulation; Vortices
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2020.1058

Rotating Rayleigh–Bénard convection is investigated numerically with the use of an asymptotic model that captures the rapidly rotating, small Ekman number limit, $Ek \rightarrow 0$. 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 $\widetilde {Re} \approx 6$ ($\widetilde {Re} = Re Ek^{1/3}$, 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 $\widetilde {Re} \approx 24$, 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.