Suppression of Rayleigh-Bénard convection and restratification by horizontal convection

Abstract

We investigate the competition between horizontal convection (HC) and Rayleigh-Bénard convection (RBC) in a fluid layer subject to a uniform destabilizing buoyancy flux at the bottom and a horizontally varying buoyancy distribution at the top. The RBC forcing imposes negative horizontal mean vertical buoyancy gradients at the top and bottom of the fluid layer. But if the HC forcing is sufficiently strong then the volume averaged vertical buoyancy gradient, $\langle b_z \rangle$, is positive i.e.~opposite in sign to destabilizing RBC buoyancy gradients at the boundaries. If $\langle b_z \rangle>0$ we say that the layer has been ''restratified''. Using scaling analysis based on power integrals together with two-dimensional direct numerical simulations at Rayleigh numbers up to $10^{10}$, we identify two cases: a neutral stratification state, in which HC first offsets RBC so that $\langle b_z \rangle = 0$, and a strong stratification regime, in which HC dominates and $\langle b_z \rangle$ is opposite in sign, and greater in magnitude, than the prescribed destabilizing vertical buoyancy gradient at the layer boundaries. For the range of parameters explored in this study, we derive scaling laws for the onset of these regimes in terms of the horizontal and vertical flux Rayleigh numbers, $\RaH$ and $\RaV$, finding $\RaHN \sim \RaV^{4/5}$ for the neutral state and $\RaHstrg \sim \RaV$ for the onset of strong stratification. The results highlight the controlling role of the top boundary layer in setting the mean stratification and clarify the conditions under which HC suppresses RBC. These findings are relevant to geophysical environments such as subglacial lakes, and the oceans of Snowball Earth and icy moons, where bottom heating and horizontal buoyancy variations jointly shape ocean stratification.

Figures (26)

• Figure 1: Horizontally- and time-averaged vertical buoyancy gradient $\bar{b}_z(z)$ in three cases. The inset shows a snapshots of the buoyancy with overlaid streamlines. The HC Rayleigh number, $\mathrm{Ra_H}$, and the RBC flux Rayleigh number, $\mathrm{Ra_V}$, are defined in \ref{['RaZ']}; other parameters are $\Gamma =8$ and $\hbox{Pr}=1$. For the joint HC and RBC case, the time and volume-averaged vertical buoyancy gradient is $\langle b_z \rangle/\beta=0.38$, where $\beta>0$ is defined in \ref{['FBC']}.
• Figure 2: Sketch of the 2D fluid layer illustrating the boundary conditions. There is a uniform buoyancy flux, $F=\kappa \beta$, through the bottom, non-uniform buoyancy at the top and no buoyancy flux through the sidewalls. All boundaries satisfy no-slip velocity conditions.
• Figure 3: Two snapshots of the buoyancy field separated by a short time interval around $0.02\,h^2/\kappa$ for $(a)$ the neutral stratification state ($\mathrm{Ra_H} = 3.35\times10^6$) and $(b)$ the onset of the strong stratification regime ($\mathrm{Ra_H} = 2.03\times10^7$). Streamlines are overlaid in both cases. Parameters are $\mathrm{Ra_V} = 10^7$, $\Gamma = 8$, and $Pr = 1$. The buoyancy unit is $\nu\kappa/h^3$.
• Figure 4: Vertical profile of the horizontal averaged buoyancy for several $\mathrm{Ra_V}$, normalized by $b_{\star}$ in the neutral stratification regime in $(a)$ and by $\beta h$ in the strong stratification regime in $(b)$. The boundary layer thickness $\delta$ is computed from the vertical profile of $\bar{b}_z(z)$ by determining the depth at which $95\%$ of the maximum value of $\bar{b}_z$ is reached. The parameters are $\Gamma = 8$ and $Pr=1$.
• Figure 5: $(a)$ Neutral horizontal Rayleigh number $\mathrm{Ra}^{\mathrm{N}}_\mathrm{H}$ (defined as the value of $\mathrm{Ra_H}$ for which $\langle b_z \rangle = 0$ at a given $\mathrm{Ra_V}$) plotted as a function of $\mathrm{Ra_V}$ for several aspect ratios. The inset shows the same data compensated by the scaling law \ref{['rahincp']}, For the two last $\mathrm{Ra_V}$ values ($10^9$, $10^{10}$), data are shown only for $\Gamma=8$. $(b)$ Strong horizontal Rayleigh number $\mathrm{Ra}^{\mathrm{S}}_\mathrm{H}$ (defined as the value of $\mathrm{Ra_H}$ for which $\langle b_z \rangle = \beta$ at a given $\mathrm{Ra_V}$) plotted as a function of $\mathrm{Ra_V}$ for several aspect ratios. The inset shows the same data compensated by the scaling law \ref{['rahstrong']}. For the last $\mathrm{Ra_V}$ value ($10^9$), data are shown only for $\Gamma=8$. Empty/full symbols indicate respectively filtered/DNS simulations.