A statistical mechanics approach to mixing in stratified fluids

Article, Preprint English OPEN
Venaille , Antoine ; Gostiaux , Louis ; Sommeria , Joël (2016)
  • Publisher: HAL CCSD
  • Related identifiers: doi: 10.1017/jfm.2016.721
  • Subject: stratified flows | Mixing efficiency | Physics - Fluid Dynamics | Stratified Fluids | [ SDU.OCEAN ] Sciences of the Universe [physics]/Ocean, Atmosphere | [ PHYS.MECA.MEFL ] Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph] | turbulence modelling | turbulent mixing | Turbulence | Statistical Mechanics
    arxiv: Physics::Fluid Dynamics

Accepted for the Journal of Fluid Mechanics; Predicting how much mixing occurs when a given amount of energy is injected into a Boussinesq fluid is a longstanding 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 Richard-son 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 literally lost into small-scale fluctuations. We show that for sufficiently large Richardson number, there is equiparti-tion 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.
  • References (90)
    90 references, page 1 of 9

    BARTELLO, P. 1995 Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atmos. Sci. 52 (24), 4410-4428.

    BOUCHER, C., ELLIS, R. S. & TURKINGTON, B. 2000 Derivation of maximum entropy principles in two-dimensional turbulence via large deviations. J. Stat. Phys. 98 (5-6), 1235-1278.

    BOUCHET, F. & CORVELLEC, M. 2010 Invariant measures of the 2d Euler and Vlasov equations. J. Stat. Mech. 2010 (08), P08021.

    BOUCHET, F. & SIMONNET, E. 2009 Random changes of flow topology in two-dimensional and geophysical turbulence. Phys. Rev. Lett. 102 (9), 094504.

    BOUCHET, F. & SOMMERIA, J. 2002 Emergence of intense jets and jupiter's great red spot as maximum-entropy structures. J. Fluid Mech. 464, 165-207.

    BOUCHET, F. & VENAILLE, A. 2012 Statistical mechanics of two-dimensional and geophysical flows. Phys. Rep. 515 (5), 227-295.

    BOUFFARD, D. & BOEGMAN, L. 2013 A diapycnal diffusivity model for stratified environmental flows. Dyn. Atmos. Oceans 61, 14-34.

    CAULFIELD, C. P. & KERSWELL, R. R. 2001 Maximal mixing rate in turbulent stably stratified couette flow. Phys. Fluids 13 (4), 894-900.

    CHAVANIS, P.-H. 2002 Statistical mechanics of two-dimensional vortices and stellar systems. In Dynamics and Thermodynamics of Systems with Long-range Interactions, pp. 208-289. Springer.

    CORRSIN, S. 1951 On the spectrum of isotropic temperature fluctuations in an isotropic turbulence. J. Appl. Phys. 22 (4), 469-473.

  • Metrics
    2
    views in OpenAIRE
    0
    views in local repository
    0
    downloads in local repository
Share - Bookmark