Understanding mixing efficiency in the oceans: Do the nonlinearities of the equation of state matter?

Abstract

There exist two central measures of turbulent mixing in turbulent stratified fluids, both caused by molecular diffusion: 1) the dissipation rate D(APE) of available potential energy (APE); 2) the turbulent rate of change Wr,turbulent of background potential energy GPEr. So far, these two quantities have often been regarded as the same energy conversion, namely the irreversible conversion of APE into GPEr, owing to D(APE)=Wr,turbulent holding exactly for a Boussinesq fluid with a linear equation of state. It was recently pointed out, however, that this equality no longer holds for a thermally-stratified compressible fluid, the ratio \xi=Wr,turbulent/D(APE) being then lower than unity and sometimes even negative for water/seawater. In this paper, the behavior of the ratio \xi is examined for different stratifications having the same buoyancy frequency N(z), but different vertical profiles of the parameter \Upsilon = \alpha P/(\rho C_p), where \alpha is the thermal expansion, P the hydrostatic pressure, \rho the density, and C_p the isobaric specific heat capacity, the equation of state considered being that for seawater for different particular constant values of salinity. It is found that \xi and Wr,turbulent depend critically on the sign and magnitude of d\Upsilon/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.