Hostname: page-component-7c8c6479df-nwzlb Total loading time: 0 Render date: 2024-03-28T03:15:32.969Z Has data issue: false hasContentIssue false

Instability and hydraulics of turbulent stratified shear flows

Published online by Cambridge University Press:  20 February 2012

Zhiyu Liu*
Affiliation:
State Key Laboratory of Marine Environmental Science, Xiamen University, Xiamen 361005, China
S. A. Thorpe
Affiliation:
School of Ocean Sciences, Bangor University, Gwynedd LL59 5EY, UK
W. D. Smyth
Affiliation:
College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331, USA
*
Email address for correspondence: zyliu@xmu.edu.cn

Abstract

The Taylor–Goldstein (T–G) equation is extended to include the effects of small-scale turbulence represented by non-uniform vertical and horizontal eddy viscosity and diffusion coefficients. The vertical coefficients of viscosity and diffusion, and , respectively, are assumed to be equal and are expressed in terms of the buoyancy frequency of the flow, , and the dissipation rate of turbulent kinetic energy per unit mass, , quantities that can be measured in the sea. The horizontal eddy coefficients, and , are taken to be proportional to the dimensionally correct form, , found appropriate in the description of horizontal dispersion of a field of passive markers of scale . The extended T–G equation is applied to examine the stability and greatest growth rates in a turbulent shear flow in stratified waters near a sill, that at the entrance to the Clyde Sea in the west of Scotland. Here the main effect of turbulence is a tendency towards stabilizing the flow; the greatest growth rates of small unstable disturbances decrease, and in some cases flows that are unstable in the absence of turbulence are stabilized when its effects are included. It is conjectured that stabilization of a flow by turbulence may lead to a repeating cycle in which a flow with low levels of turbulence becomes unstable, increasing the turbulent dissipation rate and so stabilizing the flow. The collapse of turbulence then leads to a condition in which the flow may again become unstable, the cycle repeating. Two parameters are used to describe the ‘marginality’ of the observed flows. One is based on the proximity of the minimum flow Richardson number to the critical Richardson number, the other on the change in dissipation rate required to stabilize or destabilize an observed flow. The latter is related to the change needed in the flow Reynolds number to achieve zero growth rate. The unstable flows, typical of the Clyde Sea site, are relatively further from neutral stability in Reynolds number than in Richardson number. The effects of turbulence on the hydraulic state of the flow are assessed by examining the speed and propagation direction of long waves in the Clyde Sea. Results are compared to those obtained using the T–G equation without turbulent viscosity or diffusivity. Turbulence may change the state of a flow from subcritical to supercritical.

Type
Papers
Copyright
Copyright © Cambridge University Press 2012

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

1. Ashford, O. M. 1985 Prophet or Professor? The Life and Work of Lewis Fry Richardson. Adam Hilger.Google Scholar
2. Baines, P. G. 1995 Topographic Effects in Stratified Flows. Cambridge University Press.Google Scholar
3. Baines, P. G. & Mitsudera, H. 1994 On the mechanism of shear flow instabilities. J. Fluid Mech. 276, 327342.CrossRefGoogle Scholar
4. Bell, T. H. 1974 Effects of shear on the properties of internal gravity waves. Deutsche Hydrograph. Z. 27, 5762.CrossRefGoogle Scholar
5. Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
6. Gage, K. S. 1971 The effect of stable thermal stratification on the stability of viscous parallel flows. J. Fluid Mech. 47, 120.CrossRefGoogle Scholar
7. Galperin, B., Sukoriansky, S. & Anderson, P. S. 2007 On the critical Richardson number in stably stratified turbulence. Atmos. Sci. Lett. 8, 6569.CrossRefGoogle Scholar
8. 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, 231280.CrossRefGoogle Scholar
9. Gregg, M. C. & Pratt, L. J. 2010 Flow and hydraulics near the sill of Hood Canal, a strongly sheared, continuously stratified fjord. J. Phys. Oceanogr. 40, 10871105.CrossRefGoogle Scholar
10. Hogg, A. M., Winters, K. B. & Ivey, G. N. 2001 Linear internal waves and the control of stratified exchange flows. J. Fluid Mech. 447, 357375.CrossRefGoogle Scholar
11. Howard, L. N. 1961 Note on a paper by John W. Miles. J. Fluid Mech. 10, 509512.CrossRefGoogle Scholar
12. Kantha, L. H. & Clayson, C. A. 2000 Numerical Models of Oceans and Oceanic Processes. Academic.Google Scholar
13. Koppel, D. 1964 On the stability of flow of a thermally stratified fluid under the action of gravity. J. Math. Phys. 5, 963982.CrossRefGoogle Scholar
14. Ledwell, J. R., Watson, A. J. & Law, C. S. 1998 Mixing of a tracer in the pycnocline. J. Geophys. Res. 103 (21), 499–21, 529.Google Scholar
15. Liu, Z. 2010 Instability of baroclinic tidal flow in a stratified fjord. J. Phys. Oceanogr. 40, 139154.CrossRefGoogle Scholar
16. Maslowe, S. A. & Thompson, J. M. 1971 Stability of a stratified free shear layer. Phys. Fluids 14, 453458.Google Scholar
17. Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496508.Google Scholar
18. Mittendorf, G. H. 1961 The instability of stratified flow. MSc thesis. State University of Iowa.Google Scholar
19. Okubo, A. 1971 Oceanic diffusion diagrams. Deep-Sea Res. 18, 789802.Google Scholar
20. Ollitrault, M., Gabillet, C. & De Verdiere, A. C. 2005 Open ocean regimes of relative dispersion. J. Fluid Mech. 533, 381407.CrossRefGoogle Scholar
21. Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 8389.2.0.CO;2>CrossRefGoogle Scholar
22. Pratt, L. J., Deese, H. E., Murray, S. P. & Johns, W. 2000 Continuous dynamical modes in straits having arbitrary cross sections. J. Phys. Oceanogr. 30, 25152534.2.0.CO;2>CrossRefGoogle Scholar
23. Richardson, L. F. 1952 Transforms for the eddy-diffusion of clusters. Proc. R. Soc. Lond. A 214, 120.Google Scholar
24. 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, 193214.CrossRefGoogle Scholar
25. Smyth, W. D., Moum, J. N. & Caldwell, D. R. 2001 The efficiency of mixing in turbulent patches: inferences from direct simulations and microstructure observations. J. Phys. Oceanogr. 31, 19691992.Google Scholar
26. Smyth, W. D., Moum, J. N. & Nash, J. D. 2011 Narrowband, high-frequency oscillations at the equator. Part II. Properties of shear instabilities. J. Phys. Oceanogr. 41, 412428.Google Scholar
27. Sun, C., Smyth, W. & Moum, J. 1998 Dynamic instability of stratified shear flow in the upper equatorial Pacific. J. Geophys. Res. 103, 1032310337.Google Scholar
28. Sundermeyer, M. A. & Ledwell, J. R. 2001 Lateral dispersion over the continental shelf: analysis of dye release experiments. J. Geophys. Res. 106, 96039621.CrossRefGoogle Scholar
29. Sundermeyer, M. A., Ledwell, J. R., Oakey, N. S. & Greenan, B. J. W. 2005 Stirring by small-scale vortices caused by patchy mixing. J. Phys. Oceanogr. 35, 12451262.Google Scholar
30. Taylor, G. I. 1935 Statistical theory of turbulence. Proc. R. Soc. Lond. A 151, 421478.Google Scholar
31. Thorpe, S. A. 1969 Neutral eigensolutions of the stability equations for stratified shear flow. J. Fluid Mech. 36, 673683.Google Scholar
32. Thorpe, S. A. 1971 Experiments on the instability of stratified shear flows: miscible fluids. J. Fluid Mech. 46, 299319.CrossRefGoogle Scholar
33. Thorpe, S. A. 2005 The Turbulent Ocean. Cambridge University Press.CrossRefGoogle Scholar
34. Thorpe, S. A. 2010 Turbulent hydraulic jumps in a stratified shear flow. J. Fluid Mech. 654, 305350.CrossRefGoogle Scholar
35. Thorpe, S. A. & Hall, A. J. 1977 Mixing in upper layer of a lake during heating cycle. Nature 265, 719722.Google Scholar
36. Thorpe, S. A. & Liu, Z. 2009 Marginal stability? J. Phys. Oceanogr. 39, 23732381.CrossRefGoogle Scholar
37. Thorpe, S. A. & Ozen, B. 2007 Are cascading flows stable? J. Fluid Mech. 589, 411432, (and Corrigendum 631, 2009, 441–442).CrossRefGoogle Scholar
38. Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar
39. Yih, C.-S. 1955 Stability of two-dimensional parallel flows for three-dimensional disturbances. Q. Appl. Maths 12, 434435.Google Scholar
40. Zika, J. 2008 The stability of boundary layer flow. 2007 Program of Study: Boundary Layers, WHOI GFD Summer School 2007 Tech. Rep. WHOI-2008-05, pp. 143–170.Google Scholar
41. Zilitinkevich, S. S., Elperin, T., Kleeorin, T., Rogachevskii, I., Esqu, I., Mauritsen, T. & Miles, M. W. 2008 Turbulence energetics in stably stratified geophysical flows: strong and weak mixing regimes. Q. J. R. Meteorol. Soc. 134, 793799.CrossRefGoogle Scholar