Competition between Rayleigh–Bénard and horizontal convection

Abstract

We investigate the dynamics of a fluid layer subject to a bottom heat flux and a top monotonically increasing temperature profile driving horizontal convection (HC). We use direct numerical simulations and consider a large range of flux-based Rayleigh numbers $10^6 \leq Ra_F \leq 10^9$ and imposed top horizontal to bottom vertical heat flux ratios $0 \leq \varLambda \leq 1$ . The fluid domain is a closed two-dimensional box with aspect ratio $4\leq \varGamma \leq 16$ and we consider no-slip boundaries and adiabatic side walls. We demonstrate a regime transition from Rayleigh–Bénard (RB) convection to HC at $\varLambda \approx 10^{-2}$ , which is independent of $Ra_F$ and $\varGamma$ . At small $\varLambda$ , the flow is organised in multiple overturning cells with approximately unit aspect ratio, whereas at large $\varLambda$ a single cell is obtained. The RB-relevant Nusselt number scaling with $Ra_F$ and the HC-relevant Nusselt number scaling with the horizontal Rayleigh number $Ra_L=Ra_F\varLambda \varGamma ^4$ are in good agreement with previous results from classical RB convection and HC studies in the limit $\varLambda \ll 10^{-2}$ and $\varLambda \gg 10^{-2}$ , respectively. We demonstrate that the system is multi-stable near the transition $\varLambda \approx 10^{-2}$ , i.e. the exact number of cells not only depends on $\varLambda$ but also on the system's history. Our results suggest that subglacial lakes, which motivated this study, are likely to be dominated by RB convection, unless the slope of the ice–water interface, which controls the horizontal temperature gradient via the pressure-dependence of the freezing point, is greater than unity.