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Scaling analysis and simulation of strongly stratified turbulent flows

Published online by Cambridge University Press:  07 August 2007

G. BRETHOUWER ,
P. BILLANT ,
E. LINDBORG  and
J.-M. CHOMAZ
G. BRETHOUWER
Affiliation:
Linné Flow Centre, KTH Mechanics, KTH, SE-100 44 Stockholm, Swedengeert@mech.kth.se
P. BILLANT
Affiliation:
LadHyX, Ecole Polytechnique, F-91128 Palaiseau Cedex, France
E. LINDBORG
Affiliation:
Linné Flow Centre, KTH Mechanics, KTH, SE-100 44 Stockholm, Swedengeert@mech.kth.se
J.-M. CHOMAZ
Affiliation:
LadHyX, Ecole Polytechnique, F-91128 Palaiseau Cedex, France
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Abstract

Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re ≫ 1 and horizontal Froude number Fh ≪ 1 are presented. The results are interpreted on the basis of a scaling analysis of the governing equations. The analysis suggests that there are two different strongly stratified regimes according to the parameter . When , viscous forces are unimportant and lv scales as lv ∼ U/N (U is a characteristic horizontal velocity and N is the Brunt–Väisälä frequency) so that the dynamics of the flow is inherently three-dimensional but strongly anisotropic. When , vertical viscous shearing is important so that (lh is a characteristic horizontal length scale). The parameter is further shown to be related to the buoyancy Reynolds number and proportional to (lO/η)4/3, where lO is the Ozmidov length scale and η the Kolmogorov length scale. This implies that there are simultaneously two distinct ranges in strongly stratified turbulence when : the scales larger than lO are strongly influenced by the stratification while those between lO and η are weakly affected by stratification. The direct numerical simulations with forced large-scale horizontal two-dimensional motions and uniform stratification cover a wide Re and Fh range and support the main parameter controlling strongly stratified turbulence being . The numerical results are in good agreement with the scaling laws for the vertical length scale. Thin horizontal layers are observed independently of the value of but they tend to be smooth for < 1, while for > 1 small-scale three-dimensional turbulent disturbances are increasingly superimposed. The dissipation of kinetic energy is mostly due to vertical shearing for < 1 but tends to isotropy as increases above unity. When < 1, the horizontal and vertical energy spectra are very steep while, when > 1, the horizontal spectra of kinetic and potential energy exhibit an approximate k−5/3h-power-law range and a clear forward energy cascade is observed.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 585 , 25 August 2007 , pp. 343 - 368
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Alvelius, K. 1999 Random forcing of three-dimensional homogeneous turbulence. Phys. Fluids 11, 1880–1889.CrossRefGoogle Scholar
Basak, S. & Sarkar, S. 2006 Dynamics of a stratified shear layer with horizontal shear. J. Fluid Mech. 568, 19–54.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2000 a Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 418, 167–188.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2000 b Theoretical analysis of the zigzag instability of a vertical columnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 419, 29–63.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 1645–1651.CrossRefGoogle Scholar
de Bruyn Kops, S. M. & Riley, J. J. 1998 Direct numerical simulation of laboratory experiments in isotropic turbulence. Phys. Fluids 10, 2125–2127.CrossRefGoogle Scholar
Carnevale, G. F., Briscolini, M. & Orlandi, P. 2001 Buoyancy- to inertial-range transition in forced stratified turbulence. J. Fluid Mech. 427, 205–239.CrossRefGoogle Scholar
Cho, J. Y. N. & Lindborg, E. 2001 Horizontal velocity structure functions in the upper troposphere and lower stratosphere 1. Observations. J. Geophys. Res. 106 (D10), 10223–10232.CrossRefGoogle Scholar
Cot, C. 2001 Equatorial mesoscale wind and temperature fluctuations in the lower atmosphere. J. Geophys. Res 106 D2, 1523–1532.CrossRefGoogle Scholar
Dewan, E. M. 1979 Stratospheric spectra resembling turbulence. Science 204, 832–835.CrossRefGoogle ScholarPubMed
Dewan, E. M. 1997 Saturated-cascade similitude theory of gravity wave spectra. J. Geophys. Res 102 D25, 29, 799-817.CrossRefGoogle Scholar
Dewan, E. M. & Good, R. E. 1986 Saturation and the ‘universal’ spectrum for vertical profiles of horizontal scalar winds in the atmosphere. J. Geophys. Res 91, 2742–2748.CrossRefGoogle Scholar
Gage, K. S. 1979 Evidence for a k−5/3 law inertial range in mesoscale two-dimensional turbulence. J. Atmos. Sci. 36, 1950–1954.2.0.CO;2>CrossRefGoogle Scholar
Gargett, A. E., Osborn, T. R. & Nasmyth, P. W. 1984 Local isotropy and the decay of turbulence in a stratified fluid. J. Fluid Mech. 144, 231–280.CrossRefGoogle Scholar
Godeferd, F. S. & Cambon, C. 1994 Detailed investigation of energy transfers in homogeneous stratified turbulence. Phys. Fluids 6, 2084–2100.CrossRefGoogle Scholar
Godeferd, F. S. & Staquet, C. 2003 Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. Part 2. Large-scale and small-scale anisotropy. J. Fluid Mech. 486, 115–159.CrossRefGoogle Scholar
Godoy-Diana, R., Chomaz, J.-M. & Billant, P. 2004 Vertical length scale selection for pancake vortices in strongly stratified viscous fluids. J. Fluid Mech. 504, 229–238.CrossRefGoogle Scholar
Hebert, D. A. & de Bruyn Kops, S. M. 2006 Relationship between vertical shear rate and kinetic energy dissipation rate in stably stratified flows. Geophys. Res. Lett. 33, L06602.CrossRefGoogle Scholar
Herring, J. R. & Métais, O. 1989 Numerical experiments in forced stably stratified turbulence. J. Fluid Mech. 202, 97–115.CrossRefGoogle Scholar
Hines, C. O. 1991 The saturation of gravity waves in the middle atmosphere. Part I: Critique of linear-instability theory. J. Atmos. Sci. 48, 1348–1359.2.0.CO;2>CrossRefGoogle Scholar
Holford, J. M. & Linden, P.F. 1999 Turbulent mixing in stratified fluid. Dyn. Atmos. Oceans 30, 173–198.CrossRefGoogle Scholar
Holloway, G. 1988 The buoyancy flux from internal gravity wave breaking. Dyn. Atmos. Oceans 12, 107–125.CrossRefGoogle Scholar
Kitamura, Y. & Matsuda, Y. 2006 The k−3h and k−5/3h energy spectra in stratified turbulence. Geophys. Res. Lett. 33, L05809.CrossRefGoogle Scholar
Laval, J.-P., McWilliams, J. C. & Dubrulle, B. Forced stratified turbulence: Successive transitions with Reynolds number. Phys. Rev. E 68, 036308.Google Scholar
Lienhard, J. H. & Van Atta, C. W. 1990 The decay of turbulence in thermally stratified flow. J. Fluid Mech. 210, 57–112.CrossRefGoogle Scholar
Lilly, D. K. 1983 Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci. 40, 749–761.2.0.CO;2>CrossRefGoogle Scholar
Lilly, D. K., Bassett, G., Droegemeier, K. & Bartello, P. 1998 Stratified turbulence in the atmospheric mesoscales. Theor. Comput. Fluid Dyn. 11, 139–153.CrossRefGoogle Scholar
Lindborg, E. 2002 Strongly stratified turbulence: a special kind of motion. Advances in Turbulence IX, Proc. Ninth European Turbulence Conference, Southampton.Google Scholar
Lindborg, E. 2005 The effect of rotation on the mesoscale energy cascade in the free atmosphere. Geophys. Res. Lett. 32, L01809.CrossRefGoogle Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.CrossRefGoogle Scholar
Lindborg, E. & Brethouwer, G. 2007 Stratified turbulence forced in rotational and divergent modes. J. Fluid Mech. (in press).CrossRefGoogle Scholar
Lindborg, E. & Cho, J. Y. N. 2001 Horizontal velocity structure functions in the upper troposphere and lower stratosphere 2. Theoretical considerations. J. Geophys. Res. 106 (D10), 10223–10232.CrossRefGoogle Scholar
Lumley, J. L. 1964 The spectrum of nearly inertial turbulence in a stably stratified fluid. J. Atmos. Sci. 21, 99–102.2.0.CO;2>CrossRefGoogle Scholar
Majda, A. J. & Shefter, M. G. 1998 Elementary stratified flows with instability at large Richardson number. J. Fluid Mech. 376, 319–350.CrossRefGoogle Scholar
Métais, O., Bartello, P., Garnier, E., Riley, J. J. & Lesieur, M. 1996 Inverse cascade in stably stratified rotating turbulence. Dyn. Atmos. Oceans 23, 193–203.CrossRefGoogle Scholar
Moum, J. 1996 Energy-containing scales of turbulence in the ocean thermocline. J. Geophys. Res. 101, 14095.CrossRefGoogle Scholar
Nastrom, G. D. & Gage, K. S. 1985 A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci. 42, 950–960.2.0.CO;2>CrossRefGoogle Scholar
Park, Y.-G., Whitehead, J. A. & Gnanadeskian, A. Turbulent mixing in stratified fluids: layer formation and energetics. J. Fluid Mech. 279, 279–311.CrossRefGoogle Scholar
Praud, O., Fincham, A. M. & Sommeria, J. 2005 Decaying grid turbulence in a strongly stratified fluid. J. Fluid Mech. 522, 1–33.CrossRefGoogle Scholar
Riley, J. J. & deBruynKops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15, 2047–2059.CrossRefGoogle Scholar
Riley, J. J. & Lelong, M.-P. 2000 Fluid motion in the presence of strong stable stratification. Annu. Rev. Fluid Mech. 32, 613–657.CrossRefGoogle Scholar
Riley, J. J., Metcalfe, R. W. & Weismann, M. A. 1981 Direct numerical simulations of homogeneous turbulence in density stratified fluids. in Proc. AIP Conf. on Nonlinear Properties of Internal Waves, pp. 79–112. AIP, Woodbury.Google Scholar
Shih, L. H., Koseff, J. R., Ivey, G. N. & Ferziger, J. H. 2005 Parameterization of turbulent fluxes and scales using homogeneous sheared stably stratified turbulence simulations. J. Fluid Mech. 525, 193–214.CrossRefGoogle Scholar
Smith, L. M. & Waleffe, F. 2002 Generation of slow large scales in forced rotating stratified turbulence. J. Fluid Mech. 451, 145–168.CrossRefGoogle Scholar
Smyth, W. D. & Moum, J. N. 2000 Anisotropy of turbulence in stably stratified mixing layers. Phys. Fluids 12, 1343–1362.CrossRefGoogle Scholar
Staquet, C. & Godeferd, F. S. 1998 Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. Part 1. Flow energetics.. J. Fluid Mech. 360, 295–340CrossRefGoogle Scholar
Taylor, G. I. 1935 Statistical theory of turbulence: Parts I-II. Proc. R. Soc. Lond. A 151, 421–64.Google Scholar
Waite, M. L. & Bartello, P. 2004 Stratified turbulence dominated by vortical motion. J. Fluid Mech. 517, 281–308.CrossRefGoogle Scholar
Waite, M. L. & Bartello, P. 2006 Stratified turbulence generated by internal gravity waves. J. Fluid Mech. 546, 313–339.CrossRefGoogle Scholar