Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-09T21:46:37.588Z Has data issue: false hasContentIssue false
  • English
  • Français

Vertical dispersion by stratified turbulence

Published online by Cambridge University Press:  16 October 2008

E. LINDBORG  and
G. BRETHOUWER
E. LINDBORG
Affiliation:
Linné Flow Centre, Department of Mechanics, KTH, SE-100 44 Stockholm, Sweden
G. BRETHOUWER
Affiliation:
Linné Flow Centre, Department of Mechanics, KTH, SE-100 44 Stockholm, Sweden
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive a relation for the growth of the mean square of vertical displacements, δz, of fluid particles of stratified turbulence. In the case of freely decaying turbulence, we find that for large times 〈δz2〉 goes to a constant value 2(EP(0) + aE(0))/N2, where EP(0) and E(0) are the initial mean potential and total turbulent energy per unit mass, respectively, a < 1 and N is the Brunt–Väisälä frequency. In the case of stationary turbulence, we find that 〈δz2〉 = 〈δb2〉/N2 + 2εPt/N2, where εP is the mean dissipation of turbulent potential energy per unit mass and 〈δb2〉 is the Lagrangian structure function of normalized buoyancy fluctuations. The first term is the same as that obtained in the case of adiabatic fluid particle dispersion. This term goes to the finite limit 4EP/N2 as t → ∞. Assuming that the second term represents irreversible mixing, we show that the Osborn & Cox model for vertical diffusion is retained. In the case where the motion is dominated by a turbulent cascade with an eddy turnover time T ≫ N−1, rather than linear gravity waves, we suggest that there is a range of time scales, t, between N−1 and T, where 〈δb2〉 = 2πCPLεPt, where CPL is a constant of the order of unity. This means that for such motion the ratio between the adiabatic and the diabatic mean-square displacement is universal and equal to πCPL in this range. Comparing this result with observations, we make the estimate CPL ≈ 3.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 614 , 10 November 2008 , pp. 303 - 314
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Batchelor, G. K. 1949 Diffusion in a field of homogeneous turbulence. Aust. J. Sci. Res. A 2, 437–450.Google Scholar
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and numerical simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343–368.CrossRefGoogle Scholar
Britter, R. E., Hunt, J. C. R., Marsh, G. L. & Snyder, W. H. 1983 The effects of stable stratification on turbulent diffusion and the decay of grid turbulance. J. Fluid Mech. 127, 27–44.CrossRefGoogle Scholar
Csanady, G. T. 1964 Turbulent diffusion in a stratified fluid. J. Atmos. Sci. 21, 439–447.2.0.CO;2>CrossRefGoogle Scholar
D'Asaro, E. A. & Lien, R.-C. 2000 Lagrangian measurements of waves and turbulence in stratified flows. J. Phys. Oceanogr. 30, 641–655.2.0.CO;2>CrossRefGoogle Scholar
Einstein, A. 1905 On the movement of small particles suspended in stationary in a stationary liquid demanded by the molecular-kinetic theory of heat. Reprinted in Investigations of the Theory of the Brownian Movement (ed. Fürth, F.). Dover (1956).Google Scholar
Kaneda, Y. & Ishida, T. 2000 Suppression of vertical diffusion in strongly stratified turbulence. J. Fluid Mech. 402, 311–327.CrossRefGoogle Scholar
Kennard, E. H. 1938 The Kinetic Theory of Gases. McGraw-Hill.Google Scholar
Kimura, Y. & Herring, J. R. 1996 Diffusion in stably stratified turbulence. J. Fluid Mech. 328, 253–269.CrossRefGoogle Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl-Akad Nauk SSSR, 30, 301–305; English transl. reprinted in Turbulence and stochastic processes: Kolmogorov's ideas 50 years on. Proc. R. Soc. Lond. A 434, 9–13.Google Scholar
Lien, R.-C., d'Asaro, E. & Dairiki, G. T. 1998 Lagrangian frequency spectra of vertical velocity and vorticity in high-Reynolds-number oceanic turbulence. J. Fluid Mech. 362, 177–198.CrossRefGoogle Scholar
Lien, R.-C. & D'Asaro, E. A. 2004 Lagrangian spectra and diapycnal mixing in stratified flow. J. Phys. Oceanogr. 34, 978–984.2.0.CO;2>CrossRefGoogle 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. 586, 83–108.CrossRefGoogle Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics: Mechanics of Turbulence Vol. 2, The MIT Press.Google Scholar
Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 83–89.2.0.CO;2>CrossRefGoogle Scholar
Osborn, T. R. & Cox, C. S. 1972 Oceanic fine structure. Geophys. Fluid Dyn. 3, 321–345.CrossRefGoogle Scholar
Pardyjak, E. R., Monti, P. & Fernando, H. J. S. 2002 Flux Richardson number measurements in stable atmospheric shear flows. J. Fluid Mech. 459, 307–316.CrossRefGoogle Scholar
Pearson, H. J., Puttock, J. S. & Hunt, J. C. 1983 A statistical model of fluid-element motions and vertical diffusion in a homogeneous stratified turbulent flow. J. Fluid Mech. 129, 219–249.CrossRefGoogle Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.CrossRefGoogle Scholar
Riley, J. J. & deBruynKops, S. T. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15, 2047–2059.CrossRefGoogle Scholar
Riley, J. J. & Lindborg, E. 2008 Stratified turbulence: a possible interpretation of some geophysical turbulence measurements. J. Atmos. Sci. 65, 2416–2424.CrossRefGoogle Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. Lond. Math. Soc. 2, 20, 196–212.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. The MIT Press.CrossRefGoogle Scholar
Venayagomoorthy, S. K. & Stretch, D. D. 2006 Lagrangian mixing in decaying stably stratified turbulence. J. Fluid Mech. 564, 197–226.CrossRefGoogle Scholar
Winters, K. B. & D'Asaro, E. A. 1996 Diapycnal fluxes in density stratified flows. J. Fluid Mech. 317, 179–193.CrossRefGoogle Scholar