Understanding mixing efficiency in the oceans: do the nonlinearities of the equation of state for seawater matter?

Abstract. There exist two central measures of turbulent mixing in turbulent stratified fluids that are both caused by molecular diffusion: 1) the dissipation rate D(APE) of available potential energy APE; 2) the turbulent rate of change W_{r, turbulent} of background gravitational potential energy GPE_r. So far, these two quantities have often been regarded as the same energy conversion, namely the irreversible conversion of APE into GPE_r, owing to the well known exact equality D(APE)=W_{r, turbulent} for a Boussinesq fluid with a linear equation of state. Recently, however, Tailleux (2009) pointed out that the above equality no longer holds for a thermally-stratified compressible, with the ratio ξ=W_{r, turbulent}/D(APE) being generally lower than unity and sometimes even negative for water or seawater, and argued that D(APE) and W_{r, turbulent} actually represent two distinct types of energy conversion, respectively the dissipation of APE into one particular subcomponent of internal energy called the "dead" internal energy IE₀, and the conversion between GPE_r and a different subcomponent of internal energy called "exergy" IE_exergy. In this paper, the behaviour of the ratio ξ is examined for different stratifications having all the same buoyancy frequency N vertical profile, but different vertical profiles of the parameter Υ=α P/(ρC_p), where α is the thermal expansion coefficient, P the hydrostatic pressure, ρ the density, and C_p the specific heat capacity at constant pressure, the equation of state being that for seawater for different particular constant values of salinity. It is found that ξ and W_{r, turbulent} depend critically on the sign and magnitude of dΥ/dz, in contrast with D(APE), which appears largely unaffected by the latter. These results have important consequences for how the mixing efficiency should be defined and measured in practice, which are discussed.