On the Energetics of Turbulent Mixing in Stratified Fluids

According to classical turbulence theory, there exists two main dissipation routes for kinetic energy (KE) in a turbulent strati?ed fluid. The first one is the well-known "viscous" route, while the second one is the much less well understood "diffusive" route, which, in the traditional view, works as follows. First, the KE is converted into available potential energy (APE). Then, lateral molecular diffusion converts the APE irreversibly into background Potential Energy (PEr). When the incompressibility assumption is made, which is systematically done for turbulent fluids at low Mach number, the internal energy (IE) contribution to both APE and PEr is neglected, which leads to the apparent conclusion that the diffusive route irreversibly convert a fraction of the KE, called the "mixing effciency", into the background GPEr.
In this talk, the validity of the above scenario is re-examined in the context of the fully compressible Navier-Stokes equations (CNSE), taken here as the measure of all ulimate truth. To that end, the results of Winters et al. (1995) are ?rst extended to the fully compressible case, to provide an exact and complete description of the energetics of a turbulent strati?ed ?uids that unambiguously de?ne the pathways of all form of mechanical and thermodynamical energy in the system. The main conclusions are: 1) While the diffusive route still exists in the CNSE, its nature appears to strongly differ from that predicted by the Boussinesq approximation, in that: a) the magnitude of the conversion has a different form in the NCSE, and appears to be much smaller in magnitude in the context of freely decaying turbulence; b) the KE dissipated via this route appears to end up in IE, as if it were viscously dissipated, not in mean GPE; c) the observed increase in GPE appears to be due to the thermodynamic internal energy reservoir associated with the vertical strati?cation, as in the laminar case. 2) The IE conversion terms appear to be as important as the other energy conversion terms, in contradiction with the assumptions of the Boussinesq approximation. The present results impose important constraints on the parameterizations for turbulent mixing in numerical ocean models. In particular, we will stress the importance of ensuring that reversible (adiabatic) and irreversible (diabatic) processes only affect the available and unavailable parts of the energy respectively, or face the possibility that supposedly adiabatic processes be responsible for a large fraction of spurious diapycnal mixing.