Stability transitions and energy pathways in horizontal convection at large Rayleigh numbers

We report three-dimensional convective circulation forced by a temperature gradient along the surface of a rectangular channel, using direct and large eddy simulations over a wide range of Rayleigh numbers, $Ra\sim 10^8-10^{15}$. The solutions are allowed to reach thermal equilibrium in which there is no net heat input. A sequence of several stability transitions lead to a change from laminar to fully-developed turbulent flow. At the smallest $Ra$ convection is maintained by a balance of viscous and buoyancy forces inside the thermal boundary layer, whereas at the largest $Ra$ inertia dominates over viscous stresses. This results in an enhancement of the overall heat transfer at $Ra\ge10^{10}$, while both dynamical balances give $Nu\sim Ra^{1/5}$. Our main focus is to analyze the mechanical energy budget. Below the transition the small scales of motion are driven predominately by thermal convection, whereas at $Ra > 10^{13}$ shear plays a dominant role in sustaining the small-scale turbulence.