Horizontal convection in shallow enclosures scales with height, not length, at low Rayleigh numbers

Horizontal convection is a distinct type of natural convection, where the flow is driven by non-uniform buoyancy supplied along a horizontal boundary. Horizontal convection has been studied in enclosures with aspect ratios (ratio of enclosure height H to length L) down to O(10⁻¹), though the flow dynamics and heat transfer in very shallow enclosures remain unexplored. This study employs a high-order spectral-element method and the Boussinesq buoyancy approximation to simulate horizontal convection in rectangular enclosures for aspect ratios 10⁻³ ≤ A = H/L ≤ 0.16, over Rayleigh numbers 10 ≤ Ra ≤ 10¹⁶ and a fixed Prandtl number Pr = 6.14 representative of water. The flow is driven by imposing a linear temperature variation along the bottom boundary of the enclosure, and insulating temperature conditions on the remaining boundaries. This work, for the first time, explores small aspect ratios towards the shallow-enclosure limit up to 100 times shallower than those studied previously to elucidate the effect of ocean-relevant confinement on horizontal convection. The Rayleigh number delineating the transition between the diffusion-dominated and convection-influenced regime is identified. The aspect ratio dependence of these Rayleigh numbers and their corresponding Nusselt numbers are found to follow power-law scalings Nu ~ A and Ra ~ A⁻⁴. These scalings illuminate a modified Nusselt number and Rayleigh number that govern horizontal convection at lower Rayleigh numbers, and reveal that height, rather than the horizontal length, governs the flow dynamics. Away from the side-walls the velocity and temperature profiles in these regimes exhibit self-similarity features and are well-described by the analytical solution for a one-dimensional horizontal channel flow driven by horizontal temperature gradient. A previously unseen behaviour is discovered whereby at aspect ratios A ≲ 0.1, an increasingly broad horizontally uniform zone with no heat transfer into the base occupies the middle of the enclosure, thereby localising the heating and cooling to within a distance of approximately 4H of the sidewalls.