Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-29T12:50:39.722Z Has data issue: false hasContentIssue false
  • English
  • Français

Shear-induced mixing in geophysical flows: does the route to turbulence matter to its efficiency?

Published online by Cambridge University Press:  14 May 2013

A. Mashayek  and
W. R. Peltier
A. Mashayek*
Affiliation:
Department of Physics, University of Toronto, Ontario, Canada M5S 1A7
W. R. Peltier
Affiliation:
Department of Physics, University of Toronto, Ontario, Canada M5S 1A7
*
†Email address for correspondence: amashaye@atmosp.physics.utoronto.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Motivated by the importance of diapycnal mixing parameterizations in large-scale ocean general circulation models, we provide a detailed analysis of high-Reynolds-number mixing in density stratified shear flows which constitute an archetypical example of the small-scale physical processes occurring in the oceanic interior that control turbulent diffusion. Our focus is upon the issue as to whether the route to fully developed turbulence in the stratified mixing layer is in any significant way determinant of diapycnal mixing efficiency as represented by an effective turbulent diffusivity. We characterize different routes to fully developed turbulence by the nature of the secondary instabilities through which a primary Kelvin–Helmholtz billow executes the transition to this state. We then demonstrate that different mechanisms of turbulence transition characterized in these different transition mechanisms lead to considerably different values for the efficiency of diapycnal mixing and also for the effective vertical flux of buoyancy. We show that the widely employed value of 0.15–0.2 for the efficiency of mixing in shear-induced stratified turbulence based upon both laboratory measurements and similarly low-Reynolds-number numerical simulations may be too low for the high-Reynolds-number regime characteristic of geophysical flows. Our results show that the mixing efficiency tends to a value of approximately $1/ 3$ for sufficiently large Reynolds number at an intermediate value of 0.12 for the Richardson number. This is in agreement with a theoretical predictions of Caulfield, Tang and Plasting (J. Fluid Mech., vol. 498, 2004, pp. 315–332) for the asymptotic value of mixing efficiency in stratified Couette flows. In the high-Reynolds-number regime, mixing efficiency is shown to vary over a considerable range during the course of a particular shear-induced mixing event. We explain this variation on the basis of a detailed examination of the underlying dynamics. Since values in the range 0.15–0.2 for mixing efficiency have been extensively employed to infer an effective diffusivity from ocean microstructure measurements and also in energy balance analyses of the requirements of the global ocean circulation, our findings have potentially important implications for large-scale ocean modelling. We also quantify the errors introduced by employing the Osborn (J. Phys. Oceanogr., vol. 10, 1980, pp. 83–89) formula along with an efficiency of 0.15 to infer values for effective diffusivity, and explain the logical underpinnings of this conclusion. One of the more important aspects of this work from the perspective of our theoretical understanding of stratified turbulence is the demonstration that the inverse cascade of energy, which is facilitated by the vortex-merging process that is typical of laboratory experiments and of the low-Reynolds-number simulations of shear flow evolution, is strongly suppressed by increase of the Reynolds number to values typical of geophysical flows. Based on this finding, the application of results based on low-Reynolds-number (numerical or laboratory) experiments to high-Reynolds-number geophysical shear flows needs to be reconsidered.

JFM classification

Turbulent Flows: Shear layer turbulence Turbulent Flows: Stratified turbulence Mixing: Turbulent mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 725 , 25 June 2013 , pp. 216 - 261
Copyright
©2013 Cambridge University Press 

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

Aucan, J., Merrifield, M. A., Luther, D. S. & Flament, P. 2006 Tidal mixing events on the deep flanks of Kaena Ridge, Hawaii. J. Phys. Oceanogr. 36, 1202–1219.Google Scholar
Balmforth, N. J., Roy, A. & Caulfield, C. P. 2012 Dynamics of vorticity defects in stratified shear flow. J. Fluid Mech. 694, 292–331.Google Scholar
Carpenter, J. R., Balmforth, N. J. & Lawrence, G. A. 2010a Identifying unstable modes in stratified shear layers. Phys. Fluids 22, 1–13.CrossRefGoogle Scholar
Carpenter, J. R., Lawrence, G. A. & Smyth, W. D. 2007 Evolution and mixing of asymmetric Holmboe instabilities. J. Fluid Mech. 582, 103–132.Google Scholar
Carpenter, J. R., Tedford, E. W., Rahmani, M. & Lawrence, G. A. 2010b Holmboe wave fields in simulation and experiment. J. Fluid Mech. 648, 205–223.Google Scholar
Caulfield, C. P. 1994 Multiple linear instability of a layered stratified shear flow. J. Fluid Mech. 258, 155–285.Google Scholar
Caulfield, C. & Peltier, W. R. 2000 Anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid Mech. 413, 1–47.CrossRefGoogle Scholar
Caulfield, C. P., Tang, W. & Plasting, S. C. 2004 Reynolds number dependence of an upper bound for the long-time-averaged buoyancy flux in plane stratified couette flow. J. Fluid Mech. 498 (1), 315–332.Google Scholar
Caulfield, C. P., Yoshida, S. & Peltier, W. R. 1996 Secondary instability and three-dimensionalization in a laboratory accelerating shear layer. Dyn. Atmos. Oceans 23, 125–138.Google Scholar
Corcos, G. & Sherman, F. 1976 Vorticity concentration and the dynamics of unstable free shear layers. J. Fluid Mech. 73, 241–264.Google Scholar
Cortesi, A. B., Yadigaroglu, G. & Bannerjee, S. 1998 Numerical investigation of the formation of three-dimensional structures in stably stratified mixing layers. Phys. Fluids 10, 1449–1473.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Fukao, S., Luce, H., Megac, T. & Yamamoto, M. K. 2011 Extensive studies of large-amplitude Kelvin–Helmholtz billows in the lower atmosphere with VHF middle and upper atmosphere radar. Q. J. R. Meteorol. Soc. 137, 1019–1041.Google 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 (1), 231–280.Google Scholar
Gemmrich, J. R. & van Haren, H. 2001 Thermal fronts generated by internal waves propagating obliquely along the continental slope. J. Phys. Oceanogr. 31, 649–655.Google Scholar
Geyer, W. R., Lavery, A. C., Scully, M. E. & Trowbridge, J. H. 2010 Mixing by shear instability at high Reynolds number. Geophys. Res. Lett. 37, L22607.Google Scholar
Gossard, E. E. 1990 Radar research on the atmospheric boundary layer. In Radar in Meteorology (ed. Atlas, D.). pp. 477–527. American Meteorological Society.Google Scholar
Gregg, M. C. 1987 Diapycnal mixing in the thermocline: a review. J. Geophys. Res. 92 (C5), 5249–5286.Google Scholar
Griffiths, S. D. & Peltier, W. R. 2009 Modelling of polar ocean tides at the last glacial maximum: amplification, sensitivity, and climatological implications. J. Clim. 22, 2905–2924.CrossRefGoogle Scholar
Helmholtz, P. 1868 XLIII. On discontinuous movements of fluids. Philos. Mag. 36 (244), 337–346.CrossRefGoogle Scholar
Holmboe, J. 1962 On the behaviour of symmetric waves in stratified shear layers. Geofys. Publ. Oslo 24, 67–113.Google Scholar
Howard, 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.Google Scholar
Jayne, S. R. & Laurent, L. C. St. 2001 Parameterizing tidal dissipation over rough topography. Geophys. Res. Lett. 28, 811–814.CrossRefGoogle Scholar
Kelvin, Lord 1871 Hydrokinetic solutions and observations. Phil. Mag. 10, 155–168.Google Scholar
Klaassen, G. P. & Peltier, W. R. 1985a The effect of Prandtl number on the evolution and stability of Kelvin–Helmholtz billows. Geophys. Astrophys. Fluid Dyn. 32, 23–60.CrossRefGoogle Scholar
Klaassen, G. P. & Peltier, W. R. 1985b The onset of turbulence in finite amplitude Kelvin–Helmholtz billows. J. Fluid Mech. 155, 1–35.Google Scholar
Klaassen, G. P. & Peltier, W. R. 1989 The role of transverse secondary instabilities in the evolution of free shear layers. J. Fluid Mech. 202, 367–402.Google Scholar
Laurent, L. St. & Simmons, H. 2006 Estimates of power consumed by mixing in the ocean interior. J. Clim. 19, 4877–4890.Google Scholar
Laurent, L. C. St., Simmons, H. L. & Jayne, S. R. 2002 Estimates of tidally driven enhanced mixing in the deep ocean. Geophys. Res. Lett. 29 (23), 2106.Google Scholar
Ledwell, J. R., Montgomery, E. T., Polzin, K. L., Laurent, L. C. St., Schmitt, R. W. & Toole, J. M. 2000 Evidence for enhanced mixing over rough topography in the abyssal ocean. Nature 403, 182–189.Google 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
Ledwell, J. R., Watson, A. J. & Law, C. S. 1998 Mixing of a tracer in the pycnocline. J. Geophys. Res. 103, 21499–21529.CrossRefGoogle Scholar
Lee, V. & Caulfield, C. P. 2001 Nonlinear evolution of a layered stratified shear flow. Dyn. Atmos. Oceans 34, 103–124.Google Scholar
Legg, S. 2009 Improving oceanic overflow representation in climate models: the gravity current entrainment climate process team. Bull. Am. Meteorol. Soc. 90, 657–670.Google Scholar
Ley, B. E. & Peltier, W. R. 1978 Wave generation and frontal collapse. J. Atmos. Sci. 35, 3–17.Google Scholar
Luce, H., Mega, T., Yamamoto, M. K., Yamamoto, M., Hashiguchi, H., Fukao, S., Nishi, N., Tajiri, T. & Nakazato, M. 2010 Observations of Kelvin–Helmholtz instability at a cloud base with the middle and upper atmosphere (MU) and weather radars. J. Geophys. Res. 115, D19116.Google Scholar
Marshall, J. & Speer, K. 2012 Closure of the meridional overturning circulation through southern ocean upwelling. Nature Geosci. 5, 171–180.CrossRefGoogle Scholar
Mashayek, A. & Peltier, W. R. 2011a Three-dimensionalization of the stratified mixing layer at high Reynolds number. Phys. Fluids 23, 111701.CrossRefGoogle Scholar
Mashayek, A. & Peltier, W. R. 2011b Turbulence transition in stratified atmospheric and oceanic shear flows: Reynolds and Prandtl number controls upon the mechanism. Geophys. Res. Lett. 38, L16612.CrossRefGoogle Scholar
Mashayek, A. & Peltier, W. R. 2012a The ‘zoo’ of secondary instabilities precursory to stratified shear flow transition. Part 1. Shear aligned convection, pairing, and braid instabilities. J. Fluid Mech. 708, 5–44.CrossRefGoogle Scholar
Mashayek, A. & Peltier, W. R. 2012b The ‘zoo’ of secondary instabilities precursory to stratified shear flow transition. Part 2. The influence of stratification. J. Fluid Mech. 708, 45–70.Google Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.CrossRefGoogle Scholar
Moum, J. N. 1996 Energy-containing scales of turbulence in the ocean thermocline. J. Geophys. Res. 101, 14095–14109.Google Scholar
Moum, J. N., Farmer, D. M., Smith, W. D., Armi, L. & Vagle, S. 2003 Structure and generation of turbulence at interfaces strained by internal solitary waves propagating shoreward over the continental shelf. J. Phys. Oceanogr. 33, 2093–2112.Google Scholar
Munk, W. H. 1966 Abyssal recipes. Deep-Sea Res. 113, 207–230.Google Scholar
Munk, W. H. & Wunsch, C. 1998 Abyssal recipes II: energetics of tidal and wind mixing. Deep-Sea Res. 45, 1977–2010.Google Scholar
Nikurashin, M. & Legg, S. 2011 A mechanism for local dissipation of internal tides generated at rough topography. J. Phys. Oceanogr. 41, 378–395.Google Scholar
Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 83–89.Google Scholar
Peltier, W. R. & Caulfield, C. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.Google Scholar
Peltier, W. R. & Clark, T. L. 1979 The evolution and stability of finite amplitude mountain waves. Part II: surface wave drag and severe downslope windstorms. J. Atmos. Sci. 36, 1498–1529.Google Scholar
Peltier, W. R. & Scinocca, J. F. 1990 The origin of severe downslope windstorm pulsations. J. Atmos. Sci. 47, 2853–2870.Google Scholar
Pham, H. T., Brucker, S. & Sarkarand, K. 2009 Dynamics of a stratified shear layer above a region of uniform stratification. J. Fluid Mech. 630, 191–223.Google Scholar
Pham, H. T. & Sarkar, S. 2010 Transport and mixing of density in a continuously stratified shear layer. J. Turbul. 24, 1–23.Google Scholar
Polzin, K. L., Toole, J. M., Ledwell, J. R. & Schmitt, R. W. 1997 Spatial variability of turbulent mixing in the abyssal ocean. Science 276, 93–96.Google Scholar
Salehipour, H., Stuhne, G. R. & Peltier, W. R. 2013 A higher order discontinuous Galerkin, global shallow water model: global ocean tides and aquaplanet benchmarks, Ocean Model. (submitted).Google Scholar
Smyth, W. D. 2003 Secondary Kelvin–Helmholtz instability in weakly stratified shear flow. J. Fluid Mech. 497, 67–98.Google Scholar
Smyth, W. D., Carpenter, J. R. & Lawrence, G. A. 2007 Mixing in symmetric holmboe waves. J. Phys. Oceanogr. 37, 1566–1583.Google Scholar
Smyth, W. D., Klaassen, G. P. & Peltier, W. R. 1988 Finite amplitude Holmboe waves. Geophys. Astrophys. Fluid Dyn. 43 (2), 181–222.Google Scholar
Smyth, W. D. & Moum, J. N. 2000 Length scales of turbulence in stably stratified mixing layers. Phys. Fluids 12, 1327.CrossRefGoogle Scholar
Smyth, W. D., Moum, J. & Caldwell, D. 2001 The efficiency of mixing in turbulent patches: inferences from direct simulations and microstructure observations. J. Phys. Oceanogr. 31, 1969–1992.Google Scholar
Smyth, W. D. & Peltier, W. R. 1991 Instability and transition in finite amplitude Kelvin–Helmholtz and Holmboe waves. J. Fluid Mech. 228, 387–415.Google Scholar
Smyth, W. D. & Peltier, W. R. 1994 Three-dimensionalization of barotropic vortices on the f-plane. J. Fluid Mech. 265, 25–64.Google Scholar
Smyth, W. D. & Winters, K. B. 2003 Turbulence and mixing in Holmboe waves. J. Phys. Oceanogr. 33, 694–711.Google Scholar
Staquet, 2000 Mixing in a stably stratified shear layer: two- and three-dimensional numerical experiments. Fluid. Dyn. Res. 27, 367–404.CrossRefGoogle Scholar
Stillinger, D. C., Helland, K. N. & Van Atta, C. W. 1983 Experiments on the transition of homogeneous turbulence to internal waves in a stratified fluid. J. Fluid Mech. 131, 91–122.CrossRefGoogle Scholar
Sutherland, B. R., Caulfield, C. P. & Peltier, W. R. 1994 Internal gravity wave generation and hydrodynamic instability. J. Atmos. Sci. 51, 3261–3280.Google Scholar
Sutherland, B. R. & Peltier, W. R. 1994 Turbulence transition and internal wave generation in density stratified jets. Phys. Fluids A 6, 1267–1284.Google Scholar
Taylor, G. I. 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. A 132, 499–523.Google Scholar
Thorpe, S. A. 2005 The Turbulent Ocean. Cambridge University Press.Google Scholar
van Haren, H. & Gostiaux, L. 2010 A deep-ocean Kelvin–Helmholtz billow train. Geophys. Res. Lett. 37, L03605.CrossRefGoogle Scholar
van Haren, H. & Gostiaux, L. 2012 Detailed internal wave mixing above a deep-ocean slope. J. Mar. Res. 70, 173–197.Google Scholar
Winters, K., Lombard, P., Riley, J. & D’Asaro, E. A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115–128.Google Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 28–314.Google Scholar