The effect of mechanical stirring on buoyancy-driven circulations

Abstract

The theoretical analysis of the energetics of mechanically-stirred horizontal convection for a Boussinesq fluid yields the formula: G(APE) = \gamma_{mixing} G(KE) + (1+\gamma_{mixing}) W_{r,laminar} where G(APE) and G(KE) are the work rate done by the buoyancy and mechanical forcing respectively, \gamma_{mixing} is the mixing efficiency, and W_{r,laminar} is the background rate of increase in gravitational potential energy due to molecular diffusion. The formula shows that mechanical stirring can easily induce a very strong buoyancy-driven overturning cell (meaning a large G(APE)) even for a relatively low mixing efficiency, whereas this is only possible in absence of mechanical stirring if \gamma_{mixing} >> 1. Moreover, the buoyancy-driven overturning becomes mechanically controlled when $\gamma_{mixing} G(KE) >> (1+\gamma_{mixing}) W_{r,laminar}$. This result explains why the buoyancy-driven overturning cell in the laboratory experiments by \cite{Whitehead2008} is amplified by the lateral motions of a stirring rod. The formula implies that the thermodynamic efficiency of the ocean heat engine, far from being negligibly small as is commonly claimed, might in fact be as large as can be thanks to the stirring done by the wind and tides. These ideas are further illustrated by means of idealised numerical experiments. A non-Boussinesq extension of the above formula is also given.