Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-09T22:05:54.932Z Has data issue: false hasContentIssue false
  • English
  • Français

Theory for differential transport of scalars in sheared stratified turbulence

Published online by Cambridge University Press:  12 February 2009

P. RYAN JACKSON  and
CHRIS R. REHMANN
P. RYAN JACKSON*
Affiliation:
Applied Ocean Physics and Engineering Department, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, [email protected]
CHRIS R. REHMANN
Affiliation:
374 Town Engineering Building, Department of Civil, Construction, and Environmental Engineering, Iowa State University, Ames, IA 50014, [email protected]
*
†Present address and email address for correspondence: US Geological Survey, Illinois Water Science Center, 1201 W. University Ave, Urbana, IL 61801, USA. [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Scalars with different molecular diffusivities can be transported at different rates in a strongly stratified, weakly turbulent flow. Rapid distortion theory (RDT) is used to examine the mechanisms responsible for differential diffusion of scalars in a sheared stratified flow. The theory, which applies when the flow is strongly stratified, predicts upgradient flux and its wavenumber dependence, which previous direct numerical simulations have shown to be important in differential diffusion. The net effect of shear on differential diffusion depends on the Grashof number, or the relative importance of buoyancy and viscous effects. RDT also allows the effects of the density ratio, Schmidt number, Lewis number, scalar activity and mean shear to be examined without the high computational cost of direct numerical simulation. RDT predicts that differential diffusion will increase with increasing density ratio, but only at low Grashof number. When the Lewis number is fixed, the Grashof number below which differential diffusion occurs decreases with increasing Schmidt number, and when one of the Schmidt numbers is fixed, differential diffusion decreases with increasing Lewis number. Also, differential transport of passive scalars increases when the Schmidt number of the scalar stratifying the flow increases.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 621 , 25 February 2009 , pp. 1 - 21
Copyright
Copyright © Cambridge University Press 2009

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

Altman, D. B. & Gargett, A. E. 1990 Differential property transport due to incomplete mixing in a stratified fluid. In Proc. 3rd Intl Symp. on Stratified Flows (ed. List, E. J. & Jirka, G. H.), pp. 454–460. American Society of Civil Engineers, New York.Google Scholar
Baker, M. A. & Gibson, C. H. 1987 Sampling turbulence in the stratified ocean: statistical consequences of strong intermittency. J. Phys. Oceanogr. 17, 1817–1836.2.0.CO;2>CrossRefGoogle Scholar
Bilger, R. W. 1989 Turbulent diffusion flames. Annu. Rev. Fluid Mech. 21, 101–135.CrossRefGoogle Scholar
Canuto, V. M., Howard, A., Cheng, Y. & Dubovikov, M. S. 2002 Ocean turbulence. Part II: Vertical diffusivities of momentum, heat, salt, mass, and passive scalars. J. Phys. Oceanogr. 32, 240–264.2.0.CO;2>CrossRefGoogle Scholar
De Silva, I. P. D., & Fernando, H. J. S. 1992 Some aspects of mixing in a stratified turbulent patch. J. Fluid Mech. 240, 601–625.CrossRefGoogle Scholar
Gargett, A. E. 2003 Differential diffusion: an oceanographic primer. Prog. Oceanogr. 56, 559–570.CrossRefGoogle Scholar
Gargett, A. E., Merryfield, W. J. & Holloway, G. 2003 Direct numerical simulation of differential scalar diffusion in three-dimensional stratified turbulence. J. Phys. Oceanogr. 33, 1758–1782.CrossRefGoogle Scholar
Gerz, T. & Schumann, U. 1991 Direct simulation of homogeneous turbulence and gravity waves in sheared and unsheared stratified flows. Turbulent Shear Flows 7, pp. 27–45. Springer.CrossRefGoogle Scholar
Gerz, T. & Yamazaki, H. 1993 Direct numerical simulation of buoyancy-driven turbulence in stably stratified fluid. J. Fluid Mech. 249, 415–440.CrossRefGoogle Scholar
Gregg, M. C. 1987 Diapycnal mixing in the thermocline: a review. J. Geophys. Res. 92, 5249–5286.Google Scholar
Hanazaki, H. 2003 Effects of initial conditions on the passive and active scalar fluxes in unsteady stably stratified turbulence. Phys. Fluids 15, 841–848.CrossRefGoogle Scholar
Hanazaki, H. & Hunt, J. C. R. 1996 Linear processes in unsteady stably stratified turbulence. J. Fluid Mech. 318, 303–337.CrossRefGoogle Scholar
Hanazaki, H. & Hunt, J. C. R. 2004 Structure of unsteady stably stratified turbulence with mean shear. J. Fluid Mech. 507, 1–42.CrossRefGoogle Scholar
Hebert, D. 1999 Intrusions: What drives them? J. Phys. Oceanogr. 29, 1382–1391.2.0.CO;2>CrossRefGoogle Scholar
Hebert, D. & Ruddick, B. R. 2003 Differential mixing by breaking internal waves. Geophys. Res. Lett. 30, 1042–1045.CrossRefGoogle Scholar
Hindmarsh, A. C. 1983 ODEPACK, A systematized collection of ODE solvers. In Scientific Computing (ed. Stepleman, R. S.) pp. 55–64. North-Holland, Amsterdam.Google Scholar
Holloway, G. 2006 Statistically stationary differential diffusion in a large-scale internal waves-vortical modes environment. Deep Sea Res. II 53, 116–127.CrossRefGoogle Scholar
Holt, S. E., Koseff, J. R. & Ferziger, J. H. 1992 A numerical study of the evolution and structure of homogeneous stably stratified sheared turbulence. J. Fluid Mech. 237, 499–539.CrossRefGoogle Scholar
Hunt, J. C. R., Stretch, D. D. & Britter, R. E. 1988 Length scales in stably stratified turbulent flows and their use in turbulence models. In Stably Stratified Flow and Dense Gas Dispersion (ed. Puttock, J. S.), pp. 285–321. Clarendon.Google Scholar
Hwang, J. H., Yamazaki, H. & Rehmann, C. R. 2006 Buoyancy generated turbulence in stably stratified flow with shear. Phys. Fluids 18, 045104.CrossRefGoogle Scholar
Ivey, G. N., Winters, K. B. & De Silva, I. P. D. 2000 Turbulent mixing in a sloping benthic boundary layer energized by internal waves. J. Fluid Mech. 418, 59–76.CrossRefGoogle Scholar
Jackson, P. R. 2006 Differential diffusion of scalars in sheared, stratified turbulence. PhD thesis, University of Illinois at Urbana-Champaign.Google Scholar
Jackson, P. R. & Rehmann, C. R. 2003 a Kinematic effects of differential transport on mixing efficiency in a diffusively stable, turbulent flow. J. Phys. Oceanogr. 33, 299–304.2.0.CO;2>CrossRefGoogle Scholar
Jackson, P. R. & Rehmann, C. R. 2003 b Laboratory measurements of differential diffusion in a diffusively stable, turbulent flow. J. Phys. Oceanogr. 33, 1592–1603.CrossRefGoogle Scholar
Jackson, P. R., Rehmann, C. R., Saenz, J. A. & Hanazaki, H. 2005 Rapid distortion theory for differential diffusion. Geophys. Res. Lett. 32, L10601, doi:10.1029/2005GL022443.CrossRefGoogle Scholar
Joseph, B., Mahalov, A., Nicolaenko, B. & Leung Tse, K. 2004 Variability of turbulence and its outer scales in a model tropopause jet. J. Atmos. Sci. 61, 621–643.2.0.CO;2>CrossRefGoogle Scholar
King, D. B. & Saltzman, E. S. 1995 Measurement of the diffusion coefficient of sulfur hexafluoride in water. J. Geophys. Res. 100, 7083–7088.CrossRefGoogle Scholar
Komori, S. & Nagata, K. 1996 Effects of molecular diffusivities on counter-gradient scalar and momentum transfer in strongly stable stratification. J. Fluid Mech. 326, 205–237.CrossRefGoogle Scholar
Ledwell, J. R., Watson, A. J. & Law, C. S. 1993 Evidence for slow mixing across the pycnocline from an open-ocean tracer-release experiment. Nature 364, 701–703.CrossRefGoogle Scholar
Liu, H.-T. 1995 Energetics of grid turbulence in a stably stratified fluid. J. Fluid Mech. 296, 127–157.CrossRefGoogle Scholar
Liu, Y. N., Maxworthy, T. & Spedding, G. R. 1987 Collapse of a turbulent front in a stratified fluid 1. Nominally two-dimensional evolution in a narrow tank. J. Geophys. Res. 92 (C5), 5247–5433.Google Scholar
Martin, J. E. & Rehmann, C. R. 2006 Layering in a flow with diffusively stable temperature and salinity stratification. J. Phys. Oceanogr. 36, 1457–1470.CrossRefGoogle Scholar
Merryfield, W. J. 2005 Dependence of differential mixing on N and R ρ. J. Phys. Oceanogr. 35, 991–1003.CrossRefGoogle Scholar
Merryfield, W. J., Holloway, G. & Gargett, A. E. 1998 Differential vertical transport of heat and salt by weak stratified turbulence. Geophys. Res. Lett. 25, 2773–2776.CrossRefGoogle Scholar
Nash, J. D. & Moum, J. N. 2002 Microstructure estimates of turbulent salinity flux and the dissipation spectrum of salinity. J. Phys. Oceanogr. 32, 2312–2333.2.0.CO;2>CrossRefGoogle Scholar
Nazarenko, S., Kevlahan, N. K.-R. & Dubrulle, B. 1999 WKB theory for rapid distortion of inhomogeneous turbulence. J. Fluid. Mech. 390, 325–348.CrossRefGoogle Scholar
Rehmann, C. R. 1995 Effects of stratification and molecular diffusivity on the mixing efficiency of decaying grid turbulence. PhD thesis, Stanford University.Google Scholar
Rehmann, C. R. & Hwang, J. H. 2005 Small-scale structure of strongly stratified turbulence. J. Phys. Oceanogr. 35, 151–164.CrossRefGoogle Scholar
Riley, J. J. & Lelong, M. P. 2000 Fluid motions in the presence of strong stable stratification. Annu. Rev. Fluid Mech. 32, 613–657.CrossRefGoogle Scholar
Shih, L. H., Koseff, J. R., Ferziger, J. H. & Rehmann, C. R. 2000 Scaling and parameterization of stratified homogeneous turbulent shear flow. J. Fluid Mech. 412, 1–20.CrossRefGoogle 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
Smyth, W. D., Nash, J. D. & Moum, J. N. 2005 Differential diffusion in breaking Kelvin–Helmholtz billows. J. Phys. Oceanogr. 35, 1004–1022.CrossRefGoogle Scholar
Socolofsky, S. A. & Jirka, G. H. 2002 Environmental Fluid Mechanics Part I: Mass Transfer and Diffusion, 2nd edn. Institut für Hydromechanik, Universität Karlsruhe, Germany.Google Scholar
Spedding, G. R., Browand, F. K. & Fincham, A. M. 1996 The long-time evolution of the initially-turbulent wake of a sphere in a stable stratification. Dyn. Atmos. Oceans 23, 171–182.CrossRefGoogle Scholar
Strang, E. J. 1997 Entrainment and mixing in stratified shear flows. PhD thesis, Arizona State University.Google Scholar
Townsend, A. A. 1980 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Turner, J. S. 1968 The influence of molecular diffusivity on turbulent entrainment across a density interface. J. Fluid Mech. 33, 639–656.CrossRefGoogle Scholar