## 1. Introduction

[2] For over a century now, oceanographers have debated about whether *W*_{buoyancy}, the power input due to surface buoyancy fluxes, is negligibly small or large and comparable to the mechanical power input due to the wind and tides? Part of the difficulty in addressing the issue is that while surface buoyancy fluxes give rise to buoyancy forces, they do not themselves exert any force on the fluid, and hence technically produce neither work nor power on their own. This difficulty is compounded by the fact that mechanical energy created by diabatic effects must ultimately result from a conversion from internal energy (*IE*) mediated by compressible effects via the work of expansion/contraction:

as discussed by *Welander* [1991] for instance, where *P* is the pressure, *υ* = 1/*ρ* is the specific volume, *ρ* is the density, and d*m* = *ρ*d*V* is the mass of an elementary fluid parcel. However, because low Mach number fluids such as seawater are generally regarded as nearly incompressible, they are nearly always tackled by means of the Boussinesq approximation, rarely if ever in the context of the fully compressible Navier-Stokes equations, so that no rigorous exact results currently exist that could give us insights into the actual value of *B* in the oceans.

[3] *Lorenz*'s [1955] available potential energy (APE) theory has long been the accepted framework to understand how diabatic effects drive motions, by suggesting that they do so by making a certain fraction of the total potential energy (i.e., the sum of gravitational potential and internal energies) available for conversion into kinetic energy, as recently discussed by *Hughes et al.* [2009] and *Tailleux* [2009]. In the APE theory, therefore, the concept of buoyancy power input *W*_{buoyancy} is naturally identified with the production rate of available potential energy *G*(*APE*), which physically represents the amount of potential energy being released per unit time that is in principle convertible into KE. In the oceans, *Oort et al.* [1994] estimated *G*(*APE*) = 1.2 ± 0.7TW, and concluded that *W*_{buoyancy} was comparable to the power input by the mechanical forcing.

[4] Over the past ten years or so, however, APE theory was challenged by *Munk and Wunsch* [1998] and others on account of its apparent conflict with a popular interpretation of *Sandström*'s [1908] “theorem” (see the work by *Kuhlbrodt* [2008] for a translation), according to which *W*_{buoyancy} can only be significant if heating occurs on average at higher pressure than cooling, whereas in the oceans, heating and cooling are applied at approximately constant pressure. *Sandström* [1908] failed to recognize, however, that molecular diffusion, possibly enhanced by turbulence (turbulence increases the area of isothermal surfaces, thereby enhancing the net diathermal molecular diffusive heat flux, as discussed by *Winters and D'Asaro* [1996]), always induces an additional internal heating and cooling mode such that the net (external+internal) heating always occurs on average at higher pressure that the net cooling regardless of the particular vertical arrangement of the external sources of heating and cooling, as illustrated by *Marchal* [2005]. More useful is *Paparella and Young*'s [2002] anti-turbulence theorem, which for the first time provided a rigorous quantitative constraint linking the net work of expansion contraction *B*_{bq} with the overall viscous dissipation *D*(*KE*) in the idealized context of a Boussinesq ocean with a linear equation of state, suggesting that *D*(*KE*) would be several orders of magnitude smaller than observed if the buoyancy forcing acted alone. Subsequently, *Wang and Huang* [2005] suggested that *B*_{bq} should be regarded as the relevant definition of *W*_{buoyancy}, which they estimated, using typical oceanic values, to be *B*_{bq} = *O*(15GW).

[5] The smallness of *B*_{bq}, in contrast to *Oort et al.*'s [1994] “too-large” value of *G*(*APE*), was arguably more in agreement with the prevailing idea at the time that *W*_{buoyancy} should be small because of Sandstrom's “effect”. This undoubtedly significantly contributed to the rapid and widespread acceptance of *B*_{bq} as the relevant definition of *W*_{buoyancy}, as indicated by the fact that the most recent reviews of ocean energetics by *Wunsch and Ferrari* [2004] and *Kuhlbrodt et al.* [2007] cite *Paparella and Young* [2002] and *Wang and Huang*'s [2005] studies, but elude any discussion of *Oort et al.*'s [1994] views on the oceanic energy cycle. However, it is hard to understand why we should disregard APE theory as the relevant framework to discuss the energetics of buoyancy forcing in the oceans, especially given that APE theory remains the dominant paradigm for understanding the energetics of the global atmospheric circulation. Why APE theory should work for the atmosphere but not for the oceans deserves clarification if we are to make progress. In this letter, our aim is to clarify the conceptual differences between APE theory and *Wang and Huang*'s [2005] approach, in order to help identifying which approach appears to be more suited to quantify *W*_{buoyancy}. To that end, we develop a number of physical arguments rooted in a first principles analysis of the fully compressible Navier-Stokes equations, while also trying to link the issue to the classical thermodynamic theory of heat engines, which has been the main approach to quantify how much power can be created by hot and cold sources in the literature.