Available potential energy and buoyancy variance in horizontal convection

Abstract

We consider the mechanical energy budget for horizontal Boussinesq convection and show that there are two distinct energy pathways connecting the mechanical energy (i.e. kinetic, available potential and background potential energies) to the internal energy reservoir and the external energy source. To obtain bounds on the magnitudes of the energy transfer rates around each cycle, we first show that the volume-averaged dissipation rate of buoyancy variance χ ≡ κ 〈|∇b|2〉, where b is the buoyancy, is bounded from above by 4.57h−1κ2/3ν−1/3b7/3max. Here h is the depth of the container, κ the molecular diffusion, ν the kinematic viscosity and bmax the maximum buoyancy difference that exists on the surface. The bound on χ is used to estimate the generation rate of available potential energy Ea and the rate at which Ea is irreversibly converted to background potential energy via diapycnal fluxes, both of which are shown to vanish at least as fast as κ1/3 in the limit κ → 0 at fixed Prandtl number Pr = ν/κ. As a thought experiment, consider a hypothetical ocean insulated at all boundaries except at the upper surface, where the buoyancy is prescribed. The bounds on the energy transfer rates in the mechanical energy budget imply that buoyancy forcing alone is insufficient by at least three orders of magnitude to maintain observed oceanic dissipation rates and that additional energy sources such as winds, tides and perhaps bioturbation are necessary to sustain observed levels of turbulent dissipation in the world's oceans.