A. Venaille, L. Gostiaux and J. Sommeria
Journal of Fluid Mechanics, 810, pp. 554–583
Predicting how much mixing occurs when a given amount of energy is injected into a Boussinesq fluid is a long-standing problem in stratified turbulence. The huge number of degrees of freedom involved in these processes renders extremely difficult a deterministic approach to the problem. Here we present a statistical mechanics approach yielding a prediction for a cumulative, global mixing efficiency as a function of a global Richardson number and the background buoyancy profile.
Assuming random evolution through turbulent stirring, the theory predicts that the inviscid, adiabatic dynamics is attracted irreversibly towards an equilibrium state characterised by a smooth, stable buoyancy profile at a coarse-grained level, upon which are fine-scale fluctuations of velocity and buoyancy. The convergence towards a coarse-grained buoyancy profile different from the initial one corresponds to an irreversible increase of potential energy, and the efficiency of mixing is quantified as the ratio of this potential energy increase to the total energy injected into the system. The remaining part of the energy is lost into small-scale fluctuations. We show that for sufficiently large Richardson number, there is equipartition between potential and kinetic energy, provided that the background buoyancy profile is strictly monotonic. This yields a mixing efficiency of 0.25, which provides statistical mechanics support for previous predictions based on phenomenological kinematics arguments. In the general case, the cumulative, global mixing efficiency predicted by the equilibrium theory can be computed using an algorithm based on a maximum entropy production principle. It is shown in particular that the variation of mixing efficiency with the Richardson number strongly depends on the background buoyancy profile. This approach could be useful to the understanding of mixing in stratified turbulence in the limit of large Reynolds and Péclet numbers.