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Scaling laws for mixing and dissipation in unforced rotating stratified turbulence

  • A. Pouquet (a1) (a2), D. Rosenberg (a3), R. Marino (a4) and C. Herbert (a5)

We present a model for the scaling of mixing in weakly rotating stratified flows characterized by their Rossby, Froude and Reynolds numbers $Ro,Fr$ , $Re$ . This model is based on quasi-equipartition between kinetic and potential modes, sub-dominant vertical velocity, $w$ , and lessening of the energy transfer to small scales as measured by a dissipation efficiency $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D716}_{V}/\unicode[STIX]{x1D716}_{D}$ , with $\unicode[STIX]{x1D716}_{V}$ the kinetic energy dissipation and $\unicode[STIX]{x1D716}_{D}=u_{rms}^{3}/L_{int}$ its dimensional expression, with $w,u_{rms}$ the vertical and root mean square velocities, and $L_{int}$ the integral scale. We determine the domains of validity of such laws for a large numerical study of the unforced Boussinesq equations mostly on grids of $1024^{3}$ points, with $Ro/Fr\geqslant 2.5$ , and with $1600\leqslant Re\approx 5.4\times 10^{4}$ ; the Prandtl number is one, initial conditions are either isotropic and at large scale for the velocity and zero for the temperature $\unicode[STIX]{x1D703}$ , or in geostrophic balance. Three regimes in Froude number, as for stratified flows, are observed: dominant waves, eddy–wave interactions and strong turbulence. A wave–turbulence balance for the transfer time $\unicode[STIX]{x1D70F}_{tr}=N\unicode[STIX]{x1D70F}_{NL}^{2}$ , with $\unicode[STIX]{x1D70F}_{NL}=L_{int}/u_{rms}$ the turnover time and $N$ the Brunt–Väisälä frequency, leads to $\unicode[STIX]{x1D6FD}$ growing linearly with $Fr$ in the intermediate regime, with a saturation at $\unicode[STIX]{x1D6FD}\approx 0.3$ or more, depending on initial conditions for larger Froude numbers. The Ellison scale is also found to scale linearly with $Fr$ . The flux Richardson number $R_{f}=B_{f}/[B_{f}+\unicode[STIX]{x1D716}_{V}]$ , with $B_{f}=N\langle w\unicode[STIX]{x1D703}\rangle$ the buoyancy flux, transitions for approximately the same parameter values as for $\unicode[STIX]{x1D6FD}$ . These regimes for the present study are delimited by ${\mathcal{R}}_{B}=ReFr^{2}\approx 2$ and $R_{B}\approx 200$ . With $\unicode[STIX]{x1D6E4}_{f}=R_{f}/[1-R_{f}]$ the mixing efficiency, putting together the three relationships of the model allows for the prediction of the scaling $\unicode[STIX]{x1D6E4}_{f}\sim Fr^{-2}\sim {\mathcal{R}}_{B}^{-1}$ in the low and intermediate regimes for high $Re$ , whereas for higher Froude numbers, $\unicode[STIX]{x1D6E4}_{f}\sim {\mathcal{R}}_{B}^{-1/2}$ , a scaling already found in observations: as turbulence strengthens, $\unicode[STIX]{x1D6FD}\sim 1$ , $w\approx u_{rms}$ , and smaller buoyancy fluxes together correspond to a decoupling of velocity and temperature fluctuations, the latter becoming passive.

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Barry, M., Ivey, G., Winters, K. & Imberger, J. 2001 Measurements of diapycnal diffusivities in stratified fluids. J. Fluid Mech. 442, 267291.
Bartello, P. 1995 Geostrophic adjustment and inverse cascade in rotating stratified turbulence. J. Atmos. Sci. 52, 44104428.
Billant, P. & Chomaz, J. M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 16451651.
Bluteau, C. E., Jones, N. L. & Ivey, G. N. 2013 Turbulent mixing efficiency at an energetic ocean site. J. Geophys. Res. 118, 111.
van Bokhoven, L. J. A., Clercx, H. J. H., van Heijst, G. J. F. & Trieling, R. R. 2009 Experiments on rapidly rotating turbulent flows. Phys. Fluids 21, 096601.
Bouffard, D. & Boegman, L. 2013 A diapycnal diffusivity model for stratified environmental flows. Dyn. Atmos. Oceans 61–62, 1434.
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J. M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343368.
de Bruyn Kops, S. M. 2015 Classical scaling and intermittency in strongly stratifed Boussinesq turbulence. J. Fluid Mech. 775, 436463.
Cambon, C., Godeferd, F. S., Nicolleau, F. & Vassilicos, J. C. 2004 Turbulent diffusion in rapidly rotating flows with and without stable stratification. J. Fluid Mech. 499, 231255.
Cambon, C. & Jacquin, L. 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295317.
D’Asaro, E., Lee, C., Rainville, L., Harcourt, R. & Thomas, L. 2011 Enhanced turbulence and energy dissipation at ocean fronts. Science 332, 318322.
Davidson, P. A., Staplehurst, P. J. & Dalziel, S. B. 2006 On the evolution of eddies in a rapidly rotating system. J. Fluid Mech. 557, 135144.
Davis, K. A. & Monismith, S. G. 2011 The modification of bottom boundary layer turbulence and mixing by internal waves shoaling on a barrier reef. J. Phys. Oceanogr. 41, 22232241.
de Lavergne, C., Madec, G., Le Sommier, J., Nurser, A. J. G. & Garabato, A. C. Naveira 2016 The impact of a variable mixing efficiency on the abyssal overturning. J. Phys. Ocean. 46, 663681.
Dillon, T. M. 1982 Vertical overturns: A comparison of Thorpe and Ozmidov length scales. J. Geophys. Res. 87, 96019613.
Dimotakis, P. E. 2005 Turbulent mixing. Annu. Rev. Fluid Mech. 37, 329356.
Dritschel, D. G. & McKiver, W. J. 2015 Effect of Prandtl’s ratio in geophysical turbulence. J. Fluid Mech. 777, 569590.
Ferrari, R., Mashayek, A., McDougall, T. J., Nikurashin, M. & Campin, J. M. 2016 Turning ocean mixing upside down. J. Phys. Oceanogr. 46, 22392261.
Finnigan, J. 1999 A note on wave-turbulence interactions and the possibility of scaling the very stable planetary boundary layer. Boundary-Layer Meteorol. 90, 529539.
Fleury, M. & Lueck, R. G. 1994 Direct heat flux estimates using a towed vehicle. J. Phys. Oceangr. 24, 801818.
van Haren, H., Cimatoribus, A. A., Cyr, F. & Gostiaux, L. 2016 Insights from a 3-D temperature sensors mooring on stratified ocean turbulence. Geophys. Res. Lett. 43, 17.
Herbert, C., Marino, R., Pouquet, A. & Rosenberg, D. 2016 Waves and vortices in the inverse cascade regime of rotating stratified turbulence with or without rotation. J. Fluid Mech. 806, 165204.
Herring, J. R. 1980 Statistical theory of quasi-geostrophic turbulence. J. Atmos. Sci. 37, 969977.
Ishihara, T., Gotoh, T. & Kaneda, Y. 2009 Study of high Reynolds number isotropic turbulence by direct numerical simulation. Annu. Rev. Fluid Mech. 41, 165180.
Ivey, G., Winters, K. & Koseff, J. 2008 Density stratification, turbulence but how much mixing? Annu. Rev. Fluid Mech. 40, 169184.
Iyer, K. P., Sreenivasan, K. R. & Yeung, P. K. 2017 Reynolds number scaling of velocity increments in isotropic turbulence. Phys. Rev. E 95, 021101(R).
Karimpour, F. & Venayagamoorthy, S. K. 2015 On turbulent mixing in stably stratified wall-bounded flows. Phys. Fluids 27, 046603.
Kimura, Y. & Herring, J. R. 1996 Diffusion in stably stratified turbulence. J. Fluid Mech. 328, 253269.
Klymak, J. M., Pinkel, R. & Rainville, L. 2008 Direct breaking of the internal tide near topography: Kaena Ridge, Hawaii. J. Phys. Oceanogr. 38, 380399.
Kurien, S. & Smith, L. M. 2014 Effect of rotation and domain aspect-ratio on layer formation in strongly stratified Boussinesq flows. J. Turbul. 15, 241271.
Laval, J.-P., McWilliams, J. C. & Dubrulle, B. 2003 Forced stratified turbulence: Successive transitions with reynolds number. Phys. Rev. E 68, 036308.
Lelong, M.-P. & Riley, J. J. 1991 Internal wave-vortical mode interactions in strongly stratified flows. J. Fluid Mech. 232, 119.
Lelong, M.-P. & Sundermeyer, M. 2005 Geostrophic adjustment of an isolated diapycnal mixing event and its implications for small-scale lateral dispersion. J. Phys. Oceanogr. 35, 23522367.
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207242.
Lindborg, E. & Brethouwer, G. 2008 Vertical dispersion by stratified turbulence. J. Fluid Mech. 614, 303314.
Liu, H. L., Yudin, V. & Roble, R. 2013 Day-to-day ionospheric variability due to lower atmosphere perturbations. Geophys. Res. Lett. 40, 665670.
Lozovatsky, I. D. & Fernando, H. J. S. 2013 Mixing efficiency in natural flows. Phil. Trans. R. Soc. Lond. A 371, 20120213.
Luketina, D. & Imberger, J. 1989 Turbulence and entrainment in a buoyant surface plume. J. Geophys. Res. 94, 1261912636.
Maffioli, A., Brethouwer, G. & Lindborg, E. 2016 Mixing efficiency in stratified turbulence. J. Fluid Mech. 794, R3.
Maffioli, A. & Davidson, P. A. 2016 Dynamics of stratified turbulence decaying from a high buoyancy Reynolds number. J. Fluid Mech. 786, 210233.
Marino, R., Pouquet, A. & Rosenberg, D. 2015a Resolving the paradox of oceanic large-scale balance and small-scale mixing. Phys. Rev. Lett. 114, 114504.
Marino, R., Rosenberg, D., Herbert, C. & Pouquet, A. 2015b Interplay of waves and eddies in rotating stratified turbulence and the link with kinetic-potential energy partition. Eur. Phys. Lett. 112, 49001.
Mashayek, A. & Peltier, W. R. 2013 Shear-induced mixing in geophysical flows: does the route to turbulence matter to its efficiency? J. Fluid Mech. 725, 216261.
Mashayek, A., Salehipour, H., Bouffard, D., Caulfield, C. P., Ferrari, R., Nikurashin, M., Peltier, W. R. & Smyth, W. D. 2017 Efficiency of turbulent mixing in the abyssal ocean circulation. Geophys. Res. Lett. 44, 62966306.
Mater, B. D., Schaad, S. M. & Venayagamoorthy, S. K. 2013 Relevance of the Thorpe length scale in stably stratified turbulence. Phys. Fluids 25, 076604.
Mater, B. D. & Venayagamoorthy, S. K. 2014 The quest for an unambiguous parameterization of mixing efficiency in stably stratified geophysical flows. Geophys. Res. Lett. 41, 46464653.
McWilliams, J. 2016 Submesoscale currents in the ocean. Proc. R. Soc. Lond. A 472, 2016.0117.
Métais, O. & Herring, J. 1989 Numerical simulations of freely evolving turbulence in stably stratified fluids. J. Fluid Mech. 202, 117148.
Mininni, P. D., Rosenberg, D. & Pouquet, A. 2012 Isotropization at small scale of rotating helically driven turbulence. J. Fluid Mech. 699, 263279.
Mininni, P. D., Rosenberg, D., Reddy, R. & Pouquet, A. 2011 A hybrid MPI-OpenMP scheme for scalable parallel pseudospectral computations for fluid turbulence. Parallel Comput. 37, 316326.
Monin, A. S. & Yaglom, A. M. 1979 Statistical Fluid Mechanics. MIT Press, Cambridge.
Oks, D., Mininni, P. D. & Pouquet, A.2018 Generation of turbulence through frontogenesis in sheared stratified flows. Phys. Fluids (submitted) arXiv:1706.10287v2.
Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 8389.
Paoli, R., Thouron, O., Escobar, J., Picot, J. & Cariolle, D. 2014 High-resolution large-eddy simulations of stably stratified flows: application to subkilometer-scale turbulence in the upper troposphere–lower stratosphere. Atm. Chem. Phys. 14, 50375055.
Patterson, M. D., Caulfield, C. P., McElwaine, J. N. & Dalziel, S. B. 2006 Time-dependent mixing in stratified Kelvin–Helmholtz billows: Experimental observations. Geophys. Res. Lett. 33, L15608.
Peltier, W. & Caulfield, C. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135167.
Phillips, O. M. 1972 Turbulence in a strongly stratified fluid: Is it unstable? Deep-Sea Res. 19, 7981.
Pouquet, A. & Marino, R. 2013 Geophysical turbulence and the duality of the energy flow across scales. Phys. Rev. Lett. 111, 234501.
Pouquet, A., Marino, R., Mininni, P. D. & Rosenberg, D. 2017 Dual constant-flux energy cascades to both large scales and small scales. Phys. Fluids 29, 111108.
Praud, O., Sommeria, J. & Fincham, A. 2006 Decaying grid turbulence in a rotating stratified fluid. J. Fluid Mech. 547, 389412.
Pumir, A., Xu, H. & Siggia, E. D. 2016 Small-scale anisotropy in turbulent boundary layers. J. Fluid Mech. 804, 523.
Riley, J. J. & deBruynKops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15, 20472059.
Rorai, C., Mininni, P. D. & Pouquet, A. 2014 Turbulence comes in bursts in stably stratified flows. Phys. Rev. E 89, 043002.
Rosenberg, D., Marino, R., Herbert, C. & Pouquet, A. 2016 Variations of characteristic time-scales in rotating stratified turbulence using a large parametric numerical study. Eur. Phys. J. E 39, 8.
Rosenberg, D., Marino, R., Herbert, C. & Pouquet, A. 2017 Correction to: Variations of characteristic time scales in rotating stratified turbulence using a large parametric numerical study. Eur. Phys. J. E 40, 87.
Rosenberg, D., Pouquet, A., Marino, R. & Mininni, P. D. 2015 Evidence for Bolgiano–Obukhov scaling in rotating stratified turbulence using high-resolution direct numerical simulations. Phys. Fluids 27, 055105.
Rubinstein, R., Clark, T. T. & Kurien, S. 2017 Leith diffusion model for homogeneous anisotropic turbulence. Comput. Fluids 151, 108114.
Salehipour, H. & Peltier, W. R. 2015 Diapycnal diffusivity, turbulent Prandtl number and mixing efficiency in Boussinesq stratified turbulence. J. Fluid Mech. 775, 464500.
Shih, L., Koseff, J., Ivey, G. & Ferziger, J. 2005 Parameterization of turbulent fluxes and scales using homogeneous sheared stably stratified turbulence simulations. J. Fluid Mech. 525, 193214.
Smyth, W. D., Moum, J. N. & Caldwell, D. R. 2001 The efficiency of mixing in turbulent patches: Inferences from direct simulations and microstructure observations. J. Phys. Oceanogr. 31, 19691992.
Sozza, A., Boffetta, G., Muratore-Ginanneschi, P. & Musacchio, S. 2015 Dimensional transition of energy cascades in stably stratified forced thin fluid layers. Phys. Fluids 27, 035112.
Stacey, M., Monismith, S. & Burau, J. 1999 Observations of turbulence in a partially stratified estuary. J. Phys. Oceanogr. 29, 19501970.
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: From instabilities to turbulence. Annu. Rev. Fluid Mech. 34, 559593.
Stillinger, D., Helland, K. & van Atta, C. 1983 Experiments on the transition of homogeneous turbulence to internal waves in a stratified fluid. J. Fluid Mech. 131, 91122.
Stretch, D. D., Rottman, J., Venayagamoorthy, S. K., Nomura, K. & Rehmann, C. R. 2010 Mixing efficiency in decaying stably stratified turbulence. Dyn. Atmos. Oceans 49, 2536.
Sukoriansky, S., Galperin, B. & Staroselsky, I. 2005 A quasinormal scale elimination model of turbulent flows with stable stratification. Phys. Fluids 17, 085107.
Thorpe, S. A. 1987 Transitional phenomena and the development of turbulence in stratified fluids: a review. J. Geophys. Res. 92, 52315248.
Venayagamoorthy, S. K. & Koseff, J. R. 2016 On the flux Richardson number in stably stratified turbulence. J. Fluid Mech. 798, R1R10.
Waite, M. & Bartello, P. 2006 The transition from geostrophic to stratified turbulence. J. Fluid Mech. 568, 89108.
Wells, M., Cenedese, C. & Caulfield, C. P. 2010 The relationship between flux coefficient and entrainment ratio in density currents. J. Phys. Oceanogr. 40, 27132727.
Zakharov, V. E., L’vov, V. S. & Falkovich, G. 1992 Kolmogorov spectra of turbulence: Wave turbulence. In Non-Linear Dynamics, Springer.
Zilitinkevich, S. S., Elperin, T., Kleeorin, N., Rogachevskii, I. & Esau, I. 2013 A hierarchy of energy- and flux-budget (EFB) turbulence closure models for stably-stratified geophysical flows. Boundary-Layer Meteorol. 146, 341373.
Zilitinkevich, S. S., Elperin, T., Kleeorin, N., Rogachevskii, I., Esau, I., Mauritsen, T. & Miles, M. W. 2008 Turbulence energetics in stably stratified geophysical flows: Strong and weak mixing regimes. Q. J. R. Meteorol. Soc. 134, 793799.
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