A statistical theory of mixing in stratified fluids

A statistical mechanics theory for mixing in stratified fluids Antoine Venaille 1 , Louis Gostiaux 2 and Joel Sommeria 3 Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France antoine.venaille@ens-lyon.fr Univ Lyon, CNRS, Ecole Centrale Lyon, LMFA, 69007, LYON, France Univ Grenoble-Alpes, CNRS, LEGI, 38000, GRENOBLE, France Abstract Predicting how much mixing occurs when a given amount of energy is injected into a Boussinesq fluid is a longstanding problem in stratified turbulence. Here we address this problem with the point of view of equilibrium statistical mechanics. Assuming random evolution through turbulent stirring, the theory predicts that the unforced, inviscid, adi- abatic dynamics is attracted irreversibly towards a state characterised by wild small scale velocity fluctuations carrying kinetic energy, and by a smooth buoyancy profile superim- posed with wild small scale buoyancy fluctuations. It is then possible to compute how much of the injected energy has been irreversibly lost into small scale kinetic energy, the remaining part being used to irreversibly raise the potential energy of the system. This yields to quantitative predictions for a global, cumulative mixing efficiency in freely evolv- ing configurations. We argue that this approach may be useful to the understanding of mixing in stratified turbulence in the limit of large Reynolds and P´eclet numbers. Introduction The large-scale stratification and dynamics of the oceans depend crucially on localised turbulent mixing events (Wunsch and Ferrari, 2004). These mixing processes occur on temporal and spatial scales much smaller than the current resolutions of general circulation models and must therefore be parameterised (Large et al., 1994). It is essential for that purpose to know how much mixing occurs when stratification is stirred by a turbulent flow (Hopfinger, 1987; Peltier and Caulfield, 2003; Ivey et al., 2008). More precisely, which fraction of the injected energy is lost through a direct turbulent kinetic energy cascade, which fraction contributes to modifying the background stratification, and what is the resulting vertical buoyancy profile? Building upon previous work by Tabak and Tal (2004), we propose here to use equilibrium statistical mechanics as a guideline to answer those questions (Venaille et al., 2016) The traditional approach to estimate the efficiency of mixing in stratified turbulence involves direct analyses of the diffusive destruction of small scale buoyancy variance, which in turn requires a separation of the influence of stirring from that of irreversible mixing through application of the Lorenz concept of available potential energy that can be converted into kinetic energy and a base-state potential energy which can not (Winters et al., 1995). The diffusive destruction of small scale buoyancy variance may be repre- sented by the time derivative M of base-state potential energy plus a small correction due to the action of molecular diffusion on the initial stratification, a correction that become negligible in the limit of high Reynolds number. The time dependent efficiency of turbu- lent mixing may then be computed from direct numerical or laboratory experiments as η inst = M/(M + ) where is the viscous dissipation of kinetic energy in the flow; see e.g. (Peltier and Caulfield, 2003). This definition of mixing efficiency is global in space since VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016