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Scaling laws for mixing and dissipation in unforced rotating stratified turbulence

Published online by Cambridge University Press:  06 April 2018

A. Pouquet Open the ORCID record for A. Pouquet [Opens in a new window] ,
D. Rosenberg ,
R. Marino  and
C. Herbert
A. Pouquet*
Affiliation:
National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307, USA Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80309, USA
D. Rosenberg
Affiliation:
1401 Bradley Drive, Boulder, CO 80305, USA
R. Marino
Affiliation:
Laboratoire de Mécanique des Fluides et d’Acoustique, CNRS, École Centrale de Lyon, Université de Lyon, INSA de Lyon Écully, 69134, France
C. Herbert
Affiliation:
Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
*
†Email address for correspondence: [email protected]
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Abstract

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.

JFM classification

Turbulent Flows: Stratified turbulence Turbulent Flows: Turbulent Flows Mixing: Turbulent mixing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 844 , 10 June 2018 , pp. 519 - 545
Copyright
© 2018 Cambridge University Press 

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References

Barry, M., Ivey, G., Winters, K. & Imberger, J. 2001 Measurements of diapycnal diffusivities in stratified fluids. J. Fluid Mech. 442, 267–291.10.1017/S0022112001005080Google Scholar
Bartello, P. 1995 Geostrophic adjustment and inverse cascade in rotating stratified turbulence. J. Atmos. Sci. 52, 4410–4428.10.1175/1520-0469(1995)052<4410:GAAICI>2.0.CO;22.0.CO;2>Google Scholar
Billant, P. & Chomaz, J. M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 1645–1651.10.1063/1.1369125Google Scholar
Bluteau, C. E., Jones, N. L. & Ivey, G. N. 2013 Turbulent mixing efficiency at an energetic ocean site. J. Geophys. Res. 118, 1–11.10.1002/jgrc.20292Google Scholar
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, 096601.10.1063/1.3197876Google Scholar
Bouffard, D. & Boegman, L. 2013 A diapycnal diffusivity model for stratified environmental flows. Dyn. Atmos. Oceans 61–62, 14–34.10.1016/j.dynatmoce.2013.02.002Google Scholar
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J. M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343–368.10.1017/S0022112007006854Google Scholar
de Bruyn Kops, S. M. 2015 Classical scaling and intermittency in strongly stratifed Boussinesq turbulence. J. Fluid Mech. 775, 436–463.10.1017/jfm.2015.274Google Scholar
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, 231–255.10.1017/S0022112003007055Google Scholar
Cambon, C. & Jacquin, L. 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295–317.10.1017/S0022112089001199Google Scholar
D’Asaro, E., Lee, C., Rainville, L., Harcourt, R. & Thomas, L. 2011 Enhanced turbulence and energy dissipation at ocean fronts. Science 332, 318–322.10.1126/science.1201515Google Scholar
Davidson, P. A., Staplehurst, P. J. & Dalziel, S. B. 2006 On the evolution of eddies in a rapidly rotating system. J. Fluid Mech. 557, 135–144.10.1017/S0022112006009827Google Scholar
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, 2223–2241.10.1175/2011JPO4344.1Google Scholar
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, 663–681.10.1175/JPO-D-14-0259.1Google Scholar
Dillon, T. M. 1982 Vertical overturns: A comparison of Thorpe and Ozmidov length scales. J. Geophys. Res. 87, 9601–9613.10.1029/JC087iC12p09601Google Scholar
Dimotakis, P. E. 2005 Turbulent mixing. Annu. Rev. Fluid Mech. 37, 329–356.10.1146/annurev.fluid.36.050802.122015Google Scholar
Dritschel, D. G. & McKiver, W. J. 2015 Effect of Prandtl’s ratio in geophysical turbulence. J. Fluid Mech. 777, 569–590.10.1017/jfm.2015.348Google Scholar
Ferrari, R., Mashayek, A., McDougall, T. J., Nikurashin, M. & Campin, J. M. 2016 Turning ocean mixing upside down. J. Phys. Oceanogr. 46, 2239–2261.10.1175/JPO-D-15-0244.1Google Scholar
Finnigan, J. 1999 A note on wave-turbulence interactions and the possibility of scaling the very stable planetary boundary layer. Boundary-Layer Meteorol. 90, 529–539.10.1023/A:1001756912935Google Scholar
Fleury, M. & Lueck, R. G. 1994 Direct heat flux estimates using a towed vehicle. J. Phys. Oceangr. 24, 801–818.10.1175/1520-0485(1994)024<0801:DHFEUA>2.0.CO;22.0.CO;2>Google Scholar
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, 1–7.10.1002/2016GL068032Google Scholar
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, 165–204.10.1017/jfm.2016.581Google Scholar
Herring, J. R. 1980 Statistical theory of quasi-geostrophic turbulence. J. Atmos. Sci. 37, 969–977.10.1175/1520-0469(1980)037<0969:RDOWTS>2.0.CO;22.0.CO;2>Google Scholar
Ishihara, T., Gotoh, T. & Kaneda, Y. 2009 Study of high Reynolds number isotropic turbulence by direct numerical simulation. Annu. Rev. Fluid Mech. 41, 165–180.10.1146/annurev.fluid.010908.165203Google Scholar
Ivey, G., Winters, K. & Koseff, J. 2008 Density stratification, turbulence but how much mixing? Annu. Rev. Fluid Mech. 40, 169–184.10.1146/annurev.fluid.39.050905.110314Google Scholar
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).Google Scholar
Karimpour, F. & Venayagamoorthy, S. K. 2015 On turbulent mixing in stably stratified wall-bounded flows. Phys. Fluids 27, 046603.10.1063/1.4918533Google Scholar
Kimura, Y. & Herring, J. R. 1996 Diffusion in stably stratified turbulence. J. Fluid Mech. 328, 253–269.10.1017/S0022112096008713Google Scholar
Klymak, J. M., Pinkel, R. & Rainville, L. 2008 Direct breaking of the internal tide near topography: Kaena Ridge, Hawaii. J. Phys. Oceanogr. 38, 380–399.10.1175/2007JPO3728.1Google Scholar
Kurien, S. & Smith, L. M. 2014 Effect of rotation and domain aspect-ratio on layer formation in strongly stratified Boussinesq flows. J. Turbul. 15, 241–271.10.1080/14685248.2014.895832Google Scholar
Laval, J.-P., McWilliams, J. C. & Dubrulle, B. 2003 Forced stratified turbulence: Successive transitions with reynolds number. Phys. Rev. E 68, 036308.Google Scholar
Lelong, M.-P. & Riley, J. J. 1991 Internal wave-vortical mode interactions in strongly stratified flows. J. Fluid Mech. 232, 1–19.10.1017/S0022112091003609Google Scholar
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, 2352–2367.10.1175/JPO2835.1Google Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.10.1017/S0022112005008128Google Scholar
Lindborg, E. & Brethouwer, G. 2008 Vertical dispersion by stratified turbulence. J. Fluid Mech. 614, 303–314.10.1017/S0022112008003595Google Scholar
Liu, H. L., Yudin, V. & Roble, R. 2013 Day-to-day ionospheric variability due to lower atmosphere perturbations. Geophys. Res. Lett. 40, 665–670.10.1002/grl.50125Google Scholar
Lozovatsky, I. D. & Fernando, H. J. S. 2013 Mixing efficiency in natural flows. Phil. Trans. R. Soc. Lond. A 371, 20120213.10.1098/rsta.2012.0213Google Scholar
Luketina, D. & Imberger, J. 1989 Turbulence and entrainment in a buoyant surface plume. J. Geophys. Res. 94, 12619–12636.10.1029/JC094iC09p12619Google Scholar
Maffioli, A., Brethouwer, G. & Lindborg, E. 2016 Mixing efficiency in stratified turbulence. J. Fluid Mech. 794, R3.10.1017/jfm.2016.206Google Scholar
Maffioli, A. & Davidson, P. A. 2016 Dynamics of stratified turbulence decaying from a high buoyancy Reynolds number. J. Fluid Mech. 786, 210–233.10.1017/jfm.2015.667Google Scholar
Marino, R., Pouquet, A. & Rosenberg, D. 2015a Resolving the paradox of oceanic large-scale balance and small-scale mixing. Phys. Rev. Lett. 114, 114504.10.1103/PhysRevLett.114.114504Google Scholar
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, 49001.10.1209/0295-5075/112/49001Google Scholar
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, 216–261.10.1017/jfm.2013.176Google Scholar
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, 6296–6306.10.1002/2016GL072452Google Scholar
Mater, B. D., Schaad, S. M. & Venayagamoorthy, S. K. 2013 Relevance of the Thorpe length scale in stably stratified turbulence. Phys. Fluids 25, 076604.10.1063/1.4813809Google Scholar
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, 4646–4653.10.1002/2014GL060571Google Scholar
McWilliams, J. 2016 Submesoscale currents in the ocean. Proc. R. Soc. Lond. A 472, 2016.0117.10.1098/rspa.2016.0117Google Scholar
Métais, O. & Herring, J. 1989 Numerical simulations of freely evolving turbulence in stably stratified fluids. J. Fluid Mech. 202, 117–148.10.1017/S0022112089001126Google Scholar
Mininni, P. D., Rosenberg, D. & Pouquet, A. 2012 Isotropization at small scale of rotating helically driven turbulence. J. Fluid Mech. 699, 263–279.10.1017/jfm.2012.99Google Scholar
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, 316–326.10.1016/j.parco.2011.05.004Google Scholar
Monin, A. S. & Yaglom, A. M. 1979 Statistical Fluid Mechanics. MIT Press, Cambridge.Google Scholar
Oks, D., Mininni, P. D. & Pouquet, A.2018 Generation of turbulence through frontogenesis in sheared stratified flows. Phys. Fluids (submitted) arXiv:1706.10287v2.Google Scholar
Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 83–89.10.1175/1520-0485(1980)010<0083:EOTLRO>2.0.CO;22.0.CO;2>Google Scholar
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, 5037–5055.10.5194/acp-14-5037-2014Google Scholar
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, L15608.10.1029/2006GL026949Google Scholar
Peltier, W. & Caulfield, C. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.10.1146/annurev.fluid.35.101101.161144Google Scholar
Phillips, O. M. 1972 Turbulence in a strongly stratified fluid: Is it unstable? Deep-Sea Res. 19, 79–81.Google Scholar
Pouquet, A. & Marino, R. 2013 Geophysical turbulence and the duality of the energy flow across scales. Phys. Rev. Lett. 111, 234501.10.1103/PhysRevLett.111.234501Google Scholar
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, 111108.10.1063/1.5000730Google Scholar
Praud, O., Sommeria, J. & Fincham, A. 2006 Decaying grid turbulence in a rotating stratified fluid. J. Fluid Mech. 547, 389–412.10.1017/S0022112005007068Google Scholar
Pumir, A., Xu, H. & Siggia, E. D. 2016 Small-scale anisotropy in turbulent boundary layers. J. Fluid Mech. 804, 5–23.10.1017/jfm.2016.529Google Scholar
Riley, J. J. & deBruynKops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15, 2047–2059.10.1063/1.1578077Google Scholar
Rorai, C., Mininni, P. D. & Pouquet, A. 2014 Turbulence comes in bursts in stably stratified flows. Phys. Rev. E 89, 043002.Google Scholar
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.Google Scholar
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.Google Scholar
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, 055105.10.1063/1.4921076Google Scholar
Rubinstein, R., Clark, T. T. & Kurien, S. 2017 Leith diffusion model for homogeneous anisotropic turbulence. Comput. Fluids 151, 108–114.10.1016/j.compfluid.2016.07.009Google Scholar
Salehipour, H. & Peltier, W. R. 2015 Diapycnal diffusivity, turbulent Prandtl number and mixing efficiency in Boussinesq stratified turbulence. J. Fluid Mech. 775, 464–500.10.1017/jfm.2015.305Google Scholar
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, 193–214.10.1017/S0022112004002587Google Scholar
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, 1969–1992.10.1175/1520-0485(2001)031<1969:TEOMIT>2.0.CO;22.0.CO;2>Google Scholar
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, 035112.10.1063/1.4915074Google Scholar
Stacey, M., Monismith, S. & Burau, J. 1999 Observations of turbulence in a partially stratified estuary. J. Phys. Oceanogr. 29, 1950–1970.10.1175/1520-0485(1999)029<1950:OOTIAP>2.0.CO;22.0.CO;2>Google Scholar
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: From instabilities to turbulence. Annu. Rev. Fluid Mech. 34, 559–593.10.1146/annurev.fluid.34.090601.130953Google Scholar
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, 91–122.10.1017/S0022112083001251Google Scholar
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, 25–36.10.1016/j.dynatmoce.2008.11.002Google Scholar
Sukoriansky, S., Galperin, B. & Staroselsky, I. 2005 A quasinormal scale elimination model of turbulent flows with stable stratification. Phys. Fluids 17, 085107.10.1063/1.2009010Google Scholar
Thorpe, S. A. 1987 Transitional phenomena and the development of turbulence in stratified fluids: a review. J. Geophys. Res. 92, 5231–5248.10.1029/JC092iC05p05231Google Scholar
Venayagamoorthy, S. K. & Koseff, J. R. 2016 On the flux Richardson number in stably stratified turbulence. J. Fluid Mech. 798, R1–R10.10.1017/jfm.2016.340Google Scholar
Waite, M. & Bartello, P. 2006 The transition from geostrophic to stratified turbulence. J. Fluid Mech. 568, 89–108.10.1017/S0022112006002060Google Scholar
Wells, M., Cenedese, C. & Caulfield, C. P. 2010 The relationship between flux coefficient and entrainment ratio in density currents. J. Phys. Oceanogr. 40, 2713–2727.10.1175/2010JPO4225.1Google Scholar
Zakharov, V. E., L’vov, V. S. & Falkovich, G. 1992 Kolmogorov spectra of turbulence: Wave turbulence. In Non-Linear Dynamics, Springer.Google Scholar
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, 341–373.10.1007/s10546-012-9768-8Google Scholar
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, 793–799.10.1002/qj.264Google Scholar