Aspect ratio dependence of laminar horizontal convection

Horizontal convection (HC), the flow in a box heated and cooled from the bottom (or top), is a canonical model of buoyancy-driven transport with direct relevance to subglacial lakes, ocean circulation and climate in general. While HC is commonly characterised by the Rayleigh number

and Prandtl number

upper P r P r $Pr$

, the role of the domain aspect ratio

upper Gamma Γ $\varGamma$

(defined as horizontal extension over height) has received less attention. Here, we perform three-dimensional direct numerical simulations over a broad range of Rayleigh numbers and aspect ratios at fixed Prandtl number in order to quantify how geometry affects flow organisation and heat transport. We find that

upper Gamma Γ $\varGamma$

strongly influences both the development of the overturning circulation and the scaling of heat transfer (i.e. in dimensionless form, the Nusselt number

upper N u N u $Nu$

). As the aspect ratio increases, the emergence of an organised overturning flow is shifted to higher Rayleigh numbers. Moreover, with increasing

upper Gamma Γ $\varGamma$

the Nusselt number shows a stronger dependence on the Rayleigh number. The results clarify the gradual cross-over from a finite intermediate range compatible with the Rossby-type scaling

italic Nu tilde italic Ra Superscript 1 divided by 5 Nu ∼ Ra 1 / 5 $\textit{Nu}\sim \textit{Ra}^{1/5}$

(Rossby 1965 Deep-Sea Res. Oceanogr. Abstr. vol. 12, pp. 9–16) towards the

italic Nu tilde italic Ra Superscript 1 divided by 4 Nu ∼ Ra 1 / 4 $\textit{Nu}\sim \textit{Ra}^{1/4}$

behaviour suggested by Shishkina, Grossmann & Lohse (2016 Geophys. Res. Lett. vol. 43, pp. 1219–1225). It reflects the progressively decreasing vertical confinement of the kinetic boundary layer with increasing aspect ratio. These results show that the aspect ratio is an important control parameter in laminar HC and should be incorporated explicitly into scaling arguments for geophysical flows.