A statistical mechanics approach to mixing in stratified fluids

Predicting how much mixing occurs when a given amount of energy is injected into a Boussinesq fluid is a longstanding problem in stratified turbulence. This study presents a statistical mechanics approach that yields a prediction for a cumulative, global mixing efficiency based on the global Richardson number and the background buoyancy profile. By assuming random evolution through turbulent stirring, the theory predicts that in inviscid, adiabatic dynamics, the system irreversibly approaches an equilibrium characterized by a stable buoyancy profile, which develops fine-scale velocity and buoyancy fluctuations. The convergence toward a new buoyancy profile corresponds to an irreversible increase in potential energy, and mixing efficiency is quantified as the ratio of potential energy increase to total energy injected. For sufficiently large Richardson numbers, we find equipartition between potential and kinetic energy under a monotonic buoyancy profile, predicting a mixing efficiency of 0.25. This provides statistical mechanics support for previous phenomenological models. Additionally, the study shows the variation of mixing efficiency with Richardson number is heavily dependent on the buoyancy profile, offering insights into mixing phenomena in stratified turbulence relevant for high Reynolds and Péclet number scenarios.