Editorial Type:
Article
Article Type:
Research Article

Stably Stratified Flows: A Model with No Ri(cr)

V. M. Canuto NASA Goddard Institute for Space Studies, New York, and Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York

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Y. Cheng NASA Goddard Institute for Space Studies, New York, and Center for Climate Systems Research, Columbia University, New York, New York

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A. M. Howard NASA Goddard Institute for Space Studies, New York, New York, and Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts

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I. N. Esau Nansen Environmental and Remote Sensing Center, Bjerknes Centre for Climate Research, Bergen, Norway

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Print Publication:
01 Jul 2008
DOI:
https://doi.org/10.1175/2007JAS2470.1
Page(s):
2437–2447
Restricted access

Abstract

A large set of laboratory, direct numerical simulation (DNS), and large eddy simulation (LES) data indicates that in stably stratified flows turbulent mixing exists up to Ri ∼ O(100), meaning that there is practically no Ri(cr). On the other hand, traditional local second-order closure (SOC) models entail a critical Ri(cr) ∼ O(1) above which turbulence ceases to exist and are therefore unable to explain the above data. The authors suggest how to modify the recent SOC model of Cheng et al. to reproduce the above data for arbitrary Ri.

Corresponding author address: V. M. Canuto, NASA GISS, 2880 Broadway, New York, NY 10025. Email: vcanuto@giss.nasa.gov

Abstract

A large set of laboratory, direct numerical simulation (DNS), and large eddy simulation (LES) data indicates that in stably stratified flows turbulent mixing exists up to Ri ∼ O(100), meaning that there is practically no Ri(cr). On the other hand, traditional local second-order closure (SOC) models entail a critical Ri(cr) ∼ O(1) above which turbulence ceases to exist and are therefore unable to explain the above data. The authors suggest how to modify the recent SOC model of Cheng et al. to reproduce the above data for arbitrary Ri.

Corresponding author address: V. M. Canuto, NASA GISS, 2880 Broadway, New York, NY 10025. Email: vcanuto@giss.nasa.gov

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  • Abarbanel, H. D., D. Holm, J. E. Marsden, and T. Ratiu, 1984: Richardson number criterion for the nonlinear stability of 3D stratified flow. Phys. Rev. Lett., 52 , 2352–2355.

    • Search Google Scholar
    • Export Citation

  • André, J. C., G. DeMoor, P. Lacarrére, and R. du Vachat, 1978: Modeling the 24-hour evolution of the mean and turbulent structures of the planetary boundary layer. J. Atmos. Sci., 35 , 1861–1883.

    • Search Google Scholar
    • Export Citation

  • Banta, R. M., R. K. Newsom, J. K. Lundquist, Y. L. Pichugina, R. L. Coulter, and L. Mahrt, 2002: Nocturnal low-level jet characteristics over Kansas during CASES-99. Bound.-Layer Meteor., 105 , 221–252.

    • Search Google Scholar
    • Export Citation

  • Bertin, F., J. Barat, and R. Wilson, 1997: Energy dissipation rates, eddy diffusivity, and the Prandtl number: An in situ experimental approach and its consequences on radar estimate of turbulent parameters. Radio Sci., 32 , 791–804.

    • Search Google Scholar
    • Export Citation

  • Canuto, V. M., 1992: Turbulent convection with overshooting: Reynolds stress approach. Astrophys. J., 392 , 218–232.

  • Canuto, V. M., and M. S. Dubovikov, 1996a: A dynamical model for turbulence. I: General formalism. Phys. Fluids, 8 , 571–586.

  • Canuto, V. M., and M. S. Dubovikov, 1996b: A dynamical model for turbulence. II: Shear-driven flows. Phys. Fluids, 8 , 587–598.

  • Canuto, V. M., and M. S. Dubovikov, 1997: A dynamical model for turbulence. IV: Buoyancy driven flows. Phys. Fluids, 9 , 2118–2131.

  • Canuto, V. M., and M. S. Dubovikov, 1998: Stellar turbulent convection. I: Theory. Astrophys. J., 493 , 834–847.

  • Canuto, V. M., F. Minotti, C. Ronchi, R. M. Ypma, and O. Zeman, 1994: Second-order closure PBL model with new third-order moments: Comparison with LES data. J. Atmos. Sci., 51 , 1605–1618.

    • Search Google Scholar
    • Export Citation

  • Canuto, V. M., Y. Cheng, and A. M. Howard, 2001: New third-order moments for the convective boundary layer. J. Atmos. Sci., 58 , 1169–1172.

    • Search Google Scholar
    • Export Citation

  • Canuto, V. M., Y. Cheng, and A. M. Howard, 2007: Non-local ocean mixing model and a new plume model for deep convection. Ocean Modell., 16 , 28–46.

    • Search Google Scholar
    • Export Citation

  • Cheng, Y., and V. M. Canuto, 1994: Stably stratified shear turbulence: A new model for the energy dissipation length scale. J. Atmos. Sci., 51 , 2384–2396.

    • Search Google Scholar
    • Export Citation

  • Cheng, Y., V. M. Canuto, and A. M. Howard, 2002: An improved model for the turbulent PBL. J. Atmos. Sci., 59 , 1550–1565.

  • Cheng, Y., V. M. Canuto, and A. M. Howard, 2005: Nonlocal convective PBL model based on new third- and fourth-order moments. J. Atmos. Sci., 62 , 2189–2204.

    • Search Google Scholar
    • Export Citation

  • Esau, I. N., and A. A. Grachev, cited. 2007: Turbulent Prandtl number in stably stratified atmospheric boundary layer: Intercomparison between LES and SHEBA data. [Available online at http://ejournal.windeng.net/16/01/Esau_Grachev_manuscript_published.pdf.].

  • Fedorovich, E., R. Conzemius, and D. Mironov, 2004: Convective entrainment into a shear-free, linearly stratified atmosphere: Bulk models reevaluated through large eddy simulations. J. Atmos. Sci., 61 , 281–295.

    • Search Google Scholar
    • Export Citation

  • Galperin, B., S. Sukoriansky, and P. S. Anderson, 2007: On the critical Richardson number in stably stratified turbulence. Atmos. Sci. Lett., 8 , 65–69.

    • Search Google Scholar
    • Export Citation

  • Gerz, T., U. Schumann, and S. E. Elghobashi, 1989: Direct numerical simulation of stratified homogeneous turbulent shear flows. J. Fluid Mech., 200 , 563–594.

    • Search Google Scholar
    • Export Citation

  • Grachev, A. A., C. W. Fairall, P. O. G. Persson, E. L. Andreas, and P. S. Guest, 2005: Stable boundary-layer scaling regimes: The Sheba data. Bound.-Layer Meteor., 116 , 201–235.

    • Search Google Scholar
    • Export Citation

  • Kondo, J., O. Kanechika, and N. Yasuda, 1978: Heat and momentum transfers under strong stability in the atmospheric surface layer. J. Atmos. Sci., 35 , 1012–1021.

    • Search Google Scholar
    • Export Citation

  • Launder, B. E., G. Reece, and W. Rodi, 1975: Progress in the development of a Reynolds-stress turbulent closure. J. Fluid Mech., 68 , 537–566.

    • Search Google Scholar
    • Export Citation

  • Lumley, J. L., 1978: Computational modeling of turbulent flows. Adv. Appl. Mech., 18 , 123–176.

  • Mack, S. A., and H. C. Schoeberlein, 2004: Richardson number and ocean mixing: Towed chain observations. J. Phys. Oceanogr., 34 , 736–754.

    • Search Google Scholar
    • Export Citation

  • Mahrt, L., and D. Vickers, 2005: Boundary layer adjustment over small-scale changes of surface heat flux. Bound.-Layer Meteor., 116 , 313–330.

    • Search Google Scholar
    • Export Citation

  • Martin, P. J., 1985: Simulation of the mixed layer at OWS November and Papa with several models. J. Geophys. Res., 90 , 903–916.

  • Mellor, G. L., and T. Yamada, 1974: A hierarchy of turbulence closure models for planetary boundary layers. J. Atmos. Sci., 31 , 1791–1806.

    • Search Google Scholar
    • Export Citation

  • Mellor, G. L., and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. Space Phys., 20 , 851–875.

    • Search Google Scholar
    • Export Citation

  • Ohya, Y., 2001: Wind-tunnel study of atmospheric stable boundary layers over a rough surface. Bound.-Layer Meteor., 98 , 57–82.

  • Poulos, G. S., and Coauthors, 2002: CASES-99: A comprehensive investigation of the stable nocturnal boundary layer. Bull. Amer. Meteor. Soc., 83 , 555–581.

    • Search Google Scholar
    • Export Citation

  • Rehmann, C. R., and J. R. Koseff, 2004: Mean potential energy change in stratified grid turbulence. Dyn. Atmos. Oceans, 37 , 271–294.

  • Rotta, J. C., 1951: Statistische Theorie nichthomogener Turbulenz. Z. Phys., 129 , 547–572.

  • Schumann, U., 1991: Subgrid length-scales for large-eddy simulation of stratified turbulence. Theor. Comput. Fluid Dyn., 2 , 279–290.

  • Shih, T-H., and A. Shabbir, 1992: Advances in modeling the pressure correlation terms in the second moment equations. Studies in Turbulence, T. B. Gatski, S. Sarkar, and C. G. Speziale, Eds., Springer-Verlag, 91–128.

    • Search Google Scholar
    • Export Citation

  • Strang, E. J., and H. J. S. Fernando, 2001: Vertical mixing and transports through a stratified shear layer. J. Phys. Oceanogr., 31 , 2026–2048.

    • Search Google Scholar
    • Export Citation

  • Stretch, D. D., J. W. Rottman, K. K. Nomura, and S. K. Venayagamoorthy, 2001: Transient mixing events in stably stratified turbulence. Proc. 14th Australasian Fluid Mechanics Conf., Adelaide, Australia, 612–628 pp.

  • Sukoriansky, S., B. Galperin, and I. Staroselsky, 2005: A quasinormal scale elimination model of turbulent flow with stable stratification. Phys. Fluids, 17 , 085107. 1–28.

    • Search Google Scholar
    • Export Citation

  • Uttal, T., and Coauthors, 2002: Surface heat budget of the Arctic Ocean. Bull. Amer. Meteor. Soc., 83 , 255–275.

  • Weinstock, J., 1978: On the theory of turbulence in the buoyancy subrange of stably stratified flows. J. Atmos. Sci., 35 , 634–649.

  • Zeman, O., and J. L. Lumley, 1979: Buoyancy effects in entraining turbulent boundary layers: A second-order closure study. Turbulent Shear Flows, F. Durst et al., Eds, Vol. 1, Turbulent Shear Flows, Springer-Verlag, 295–306.

    • Search Google Scholar
    • Export Citation

  • Zilitinkevich, S., and I. N. Esau, 2007: Similarity theory and calculation of turbulent fluxes at the surface for the stably stratified atmospheric boundary layers. Bound.-Layer Meteor., 125 , 193–206.

    • Search Google Scholar
    • Export Citation

  • Zilitinkevich, S. S., T. Elperin, N. Kleeorin, and I. Rogachevskii, 2007: Energy- and flux-budget (EFB) turbulence closure model for the stably stratified flows. Part I: Steady-state, homogeneous regimes. Bound.-Layer Meteor., 125 , 167–192.

    • Search Google Scholar
    • Export Citation

  • Zilitinkevich, S. S., T. Elperin, N. Kleeorin, I. Rogachevskii, I. N. Esau, T. Mauritsen, and M. W. Miles, 2008: Turbulence energetics in stably stratified geophysical flows: Strong and weak mixing regimes. Quart. J. Roy. Meteor. Soc., in press.

    • Search Google Scholar
    • Export Citation
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