Horizontal convection with a non-linear equation of state: generalization of a theorem of Paparella and Young

An upper bound is derived for the total dissipation rate in an ocean forced exclusively by surface fluxes of heat and freshwater, assuming a non-linear equation of state. This generalizes the upper bound found by Paparella and Young, which is valid for a flow forced by an imposed temperature distribution at the surface and a linear equation of state. Like this previous result, the present one shows that the dissipation rate vanishes in the limit of vanishing molecular diffusivity of temperature and salinity, if the range of temperatures and salinities occurring in the fluid is regarded as given. A numerical evaluation for realistic ocean parameters shows that the upper bound is two orders of magnitude smaller than present estimates of the energy transformation involved in the deep ocean circulation. This supports the conclusion that mechanical forcing by winds and tides is necessary to sustain the deep ocean circulation.