Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-29T14:43:10.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of a stratified shear layer above a region of uniform stratification

Published online by Cambridge University Press:  10 July 2009

HIEU T. PHAM ,
SUTANU SARKAR  and
KYLE A. BRUCKER
HIEU T. PHAM
Affiliation:
Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093, USA
SUTANU SARKAR*
Affiliation:
Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093, USA
KYLE A. BRUCKER
Affiliation:
Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Direct numerical simulations (DNS) are performed to investigate the behaviour of a weakly stratified shear layer in the presence of a strongly stratified region beneath it. Both, coherent Kelvin–Helmholtz (KH) rollers and small-scale turbulence, are observed during the evolution of the shear layer. The deep stratification measured by the Richardson number Jd is varied to study its effect on the dynamics. In all cases, a pycnocline is found to develop at the edges of the shear layer. The region of maximum shear shifts downward with increasing time. Internal waves are excited, initially by KH rollers, and later by small-scale turbulence. The wave field generated by the KH rollers is narrowband and of stronger amplitude than the broadband wave field generated by turbulence. Linear theory based on Doppler-shifted frequency of the KH mode is able to predict the angle of the internal wave phase lines during the direct generation of internal waves by KH rollers. Waves generated by turbulence are relatively weaker with a broader range of excitation angles which, in the deep region, tend towards a narrower band. The linear theory that works for the internal waves excited by KH rollers does not work for the turbulence generated waves. The momentum transported by the internal waves into the interior can be large, about 10% of the initial momentum in the shear layer, when Jd ≃ 0.25. Integration of the turbulent kinetic energy budget in time and over the shear layer thickness shows that the energy flux can be up to 17% of the turbulent production, 33% of the turbulent dissipation rate and 75% of the buoyancy flux. These numbers quantify the dynamical importance of internal waves. In contrast to linear theory where the effect of deep stratification on the shear layer instabilities has been found to be weak, the present nonlinear simulations show that the evolution of the shear layer is significantly altered because of the significant momentum and energy carried away by the internal waves.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 630 , 10 July 2009 , pp. 191 - 223
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

Andrews, D. G. & McIntyre, M. E. 1978 An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.CrossRefGoogle Scholar
Basak, S. & Sarkar, S. 2006 Dynamics of a stratified shear layer with horizontal shear. J. Fluid Mech. 568, 19–54.CrossRefGoogle Scholar
Brucker, K. A. & Sarkar, S. 2007 Evolution of an initially turbulent stratified shear layer. Phys. Fluids 19, 101105.CrossRefGoogle Scholar
Caulfield, C. P. & Peltier, W. R. 1994 Three dimensionalization of the stratified mixing layer. Phys. Fluids 6, 3803–3805.CrossRefGoogle Scholar
Caulfield, C. P. & Peltier, W. R. 2000 The anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid Mech. 413, 1–47.CrossRefGoogle Scholar
D'Asaro, E. A., Eriksen, C. C., Levine, M. D., Niiler, P. A., Paulson, C. & Meurs, P. V. 1995 Upper-ocean inertial currents forced by a strong storm. Part I. Data and comparisons with linear theory. J. Phys. Oceanogr. 25, 2909–2936.2.0.CO;2>CrossRefGoogle Scholar
de loncle, A., Chomaz, J. & Billant, P. 2007 Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid. J. Fluid Mech. 570, 297–305.CrossRefGoogle Scholar
Diamessis, P. J. & Nomura, K. K. 2004 The structure and dynamics of overturns in stably stratified homogeneous turbulence. J. Fluid Mech. 499, 197–229.CrossRefGoogle Scholar
Dohan, K. & Sutherland, B. R. 2003 Internal waves generated from a turbulent mixed region. Phys. Fluids 15 (2), 488–498.CrossRefGoogle Scholar
Drazin, P. G., Zaturska, M. B. & Banks, W. H. H. 1979 On the normal modes of parallel flow of inviscid stratified fluid. Part 2. Unbounded flow with propagation at infinity. J. Fluid Mech. 95, 681–705.CrossRefGoogle Scholar
Xuequan, E. & Hopfinger, E. J. 1986 On mixing across an interface in stably stratified fluid. J. Fluid Mech. 166, 227–244.CrossRefGoogle Scholar
Einaudi, F., Lalas, D. P. & Perona, G. E. 1978 The role of gravity waves in tropospheric processes. Pure Appl. Geophys. 117, 627–663.CrossRefGoogle Scholar
Eliassen, A. & Palm, E. 1960 On the transfer of energy in stationary mountain waves. Geofys. Publ. (Geophys. Norv.) 22, 1–23.Google Scholar
Eriksen, C. 1982 Geostrophic equatorial deep jets. J. Mar. Res. 40, 143–157.Google Scholar
Firing, E. 1987 Deep zonal currents in the central equatorial Pacific. J. Mar. Res. 45, 791–812.CrossRefGoogle Scholar
Fritts, D. 1982 Shear excitation of atmospheric gravity waves. J. Atmos. Sci. 39, 1936–1952.2.0.CO;2>CrossRefGoogle Scholar
Gerz, T., Schumann, U. & Elghobashi, S. E. 1989 Direct numerical simulation of stratified homogeneous turbulent shear flows. J. Fluid Mech. 200, 563–594.CrossRefGoogle Scholar
Hazel, P. 1972 Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech. 51, 39–61.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
Holton, J. R., Haynes, P. H., McIntyre, M. E., Douglass, A. R., Rood, R. B. & Pfister, L. 1995 Stratosphere–troposphere exchange. Rev. Geophys. 33, 403–439.CrossRefGoogle Scholar
Jacobitz, F. G. & Sarkar, S. 1999 a A direct numerical study of transport and anisotropy in a stably stratified turbulent flow with uniform horizontal shear. Flow Turbul. Combust. 63, 343–360.CrossRefGoogle Scholar
Jacobitz, F. G. & Sarkar, S. 1999 b On the shear number effect in stratified shear flow. Theor. Comput. Fluid Dyn. 13, 171–188.CrossRefGoogle Scholar
Jacobitz, F. G., Sarkar, S. & VanAtta, C. W. 1997 Direct numerical simulations of the turbulence evolution in a uniformly sheared and stably stratified flow. J. Fluid Mech. 342, 231–261.CrossRefGoogle Scholar
Johnston, T. & Rudnick, D. 2009 Observations of the transition layer. J. Phys. Oceanogr. in press.CrossRefGoogle Scholar
Kaltenbach, H.-J., Gerz, T. & Schumann, U. 1994 Large-eddy simulation of homogeneous turbulence and diffusion in stably stratified shear flow. J. Fluid Mech. 280, 1–40.CrossRefGoogle Scholar
Koop, C. G. & Browand, F. K. 1979 Instability and turbulence in a stratified fluid with shear. J. Fluid Mech. 93, 135–159.CrossRefGoogle Scholar
Linden, P. F. 1975 The deepening of a mixed layer in a stratified fluid. J. Fluid Mech. 71, 385–405.CrossRefGoogle Scholar
Luyten, J. & Swallow, J. 1976 Equatorial undercurrents. Deep Sea Res. 23, 499–524.Google Scholar
Monkewitz, P. A. & Huerre, P. 1982 Influence of the velocity ratio on the spatial instability of mixing layers. Phys. Fluids 25, 1137–1143.CrossRefGoogle Scholar
Moum, J. N., Hebert, D., Paulson, C. A. & Caldwell, D. R. 1992 Turbulence and internal waves at the equator. Part I. Statistics from towed thermistors and a microstructure profiler. J. Phys. Oceanogr. 22, 1330–1345.2.0.CO;2>CrossRefGoogle Scholar
Oakey, N. S. 1985 Statistics of mixing parameters in the upper ocean during JASIN phase 2. J. Phys. Oceanogr. 15, 1662–1675.2.0.CO;2>CrossRefGoogle Scholar
Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 80–89.2.0.CO;2>CrossRefGoogle Scholar
Piccirillo, P. & VanAtta, C. W. 1997 The evolution of a uniformly sheared thermally stratified turbulent flow. J. Fluid Mech. 334, 61–86.CrossRefGoogle Scholar
Rohr, J. J., Itsweire, E. C., Helland, K. N. & VanAtta, C. W. 1988 Growth and decay of turbulence in a stably stratified shear flow. J. Fluid Mech. 195, 77–111.CrossRefGoogle Scholar
Rosenlof, K. H. 1996 Summer hemisphere differences in temperature and transport in the lower stratosphere. J. Geophys. Res. 101, 19129–19136.CrossRefGoogle Scholar
Skyllingstad, E. D. & denbo, D. W. 1994 The role of internal gravity waves in the equatorial current system. J. Phys. Oceanogr. 24, 2093–2110.2.0.CO;2>CrossRefGoogle Scholar
Smyth, W. D. & Moum, J. N. 2000 a Anisotropy of turbulence in stably stratified mixing layers. Phys. Fluids 12 (6), 1343–1362.CrossRefGoogle Scholar
Smyth, W. D. & Moum, J. N. 2000 b Length scales of turbulence in stably stratified mixing layers. Phys. Fluids 12 (6), 1327–1342.CrossRefGoogle Scholar
Smyth, W. D. & Moum, J. N. 2002 Shear instability and gravity wave saturation in an asymmetrically stratified jet. Dyn. Atmos. Oceans 35, 265–294.CrossRefGoogle Scholar
Smyth, W. D., Moum, J. N. & Caldwell, D. R. 2001 The efficiency of mixing in turbulent patches: Inferences from direct simulations and microstrucure observations. J. Phys. Oceanogr. 31, 1969–1992.2.0.CO;2>CrossRefGoogle Scholar
Staquet, C. 2000 Mixing in a stably stratified shear layer: two- and three-dimensional numerical experiments. Fluid Dyn. Res. 27, 367–404.CrossRefGoogle Scholar
Staquet, C. & Riley, J. J. 1989 A numerical study of a stably-stratified mixing layer. In Turbulent Shear Flows 6, pp. 381–397. Springer-Verlag.CrossRefGoogle Scholar
Strang, E. J. & Fernando, H. J. S. 2001 Entrainment and mixing in stratified shear flows. J. Fluid Mech. 428, 349–386.CrossRefGoogle Scholar
Sun, C., Smyth, W. D. & Moum, J. N. 1998 Dynamic instability of stratified shear flow in the upper equatorial Pacific. J. Geophys. Res. 103, 10323–10337.CrossRefGoogle Scholar
Sutherland, B. R. 1996 Dynamic excitation of internal gravity waves in the equatorial oceans. J. Phys. Oceanogr. 26, 2398–2419.2.0.CO;2>CrossRefGoogle Scholar
Sutherland, B. R. 2006 Rayleigh wave-internal wave coupling and internal wave generation above a model jet stream. J. Atmos. Sci. 63, 1042–1055.CrossRefGoogle Scholar
Sutherland, B. R. & Linden, P. F. 1998 Internal wave excitation from stratified flow over a thin barrier. J. Fluid Mech. 377, 223–252.CrossRefGoogle Scholar
Taylor, J. R. & Sarkar, S. 2007 Internal gravity waves generated by a turbulent bottom Ekman layer. J. Fluid Mech. 590, 331–354.CrossRefGoogle Scholar
Thorpe, S. A. 1973 Experiments on instability and turbulence in a stratified shear flow. J. Fluid Mech. 61, 731–751.CrossRefGoogle Scholar
Tse, K. L., Mahalov, A., Nicolaenko, B. & Fernando, H. J. S. 2003 Quasi-equilibrium dynamics of shear-stratified turbulence in a model tropospheric jet. J. Fluid Mech. 496, 73–103.CrossRefGoogle Scholar
Weller, R. A. & Plueddemann, A. J. 1996 Observations of the vertical structure of the oceanic boundary layer. J. Geophys. Res. 101, 8789–8806.CrossRefGoogle Scholar