Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T17:51:38.597Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of vorticity defects in stratified shear flow

Published online by Cambridge University Press:  25 January 2012

N. J. Balmforth ,
A. Roy  and
C. P. Caulfield
N. J. Balmforth
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada Department of Earth and Ocean Science, University of British Columbia, 6339 Stores Road, Vancouver, BC, V6T 1Z4, Canada
A. Roy
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore, India
C. P. Caulfield*
Affiliation:
BP Institute for Multiphase Flow, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, UK Department of Applied Mathematics & Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the linear stability and nonlinear evolution of two-dimensional shear flows that take the form of an unstratified plane Couette flow that is seeded with a localized ‘defect’ containing sharp density and vorticity variations. For such flows, matched asymptotic expansions furnish a reduced model that allows a straightforward and computationally efficient exploration of flows at sufficiently high Reynolds and Péclet numbers that sharp density and vorticity gradients persist throughout the onset, growth and saturation of instability. We are thereby able to study the linear and nonlinear dynamics of three canonical variants of stratified shear instability: Kelvin–Helmholtz instability, the Holmboe instability, and the lesser-considered Taylor instability, all of which are often interpreted in terms of the interactions of waves riding on sharp interfaces of density and vorticity. The dynamics near onset is catalogued; if the interfaces are sufficiently sharp, the onset of instability is subcritical, with a nonlinear state existing below the linear instability threshold. Beyond onset, both Holmboe and Taylor instabilities are susceptible to inherently two-dimensional secondary instabilities that lead to wave mergers and wavelength coarsening. Additional two-dimensional secondary instabilities are also found to appear for higher Prandtl numbers that take the form of parasitic Holmboe-like waves.

JFM classification

Instability: Instability Wakes/Jets: Shear layers Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 694 , 10 March 2012 , pp. 292 - 331
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. Alexakis, A. 2005 On Holmboe’s instability for smooth shear and density profiles. Phys. Fluids 17, 084103.CrossRefGoogle Scholar
2. Baines, P. G. & Mitsudera, H. 1994 On the mechanism of shear flow instabilities. J. Fluid Mech. 276, 327–342.CrossRefGoogle Scholar
3. Balmforth, N. J. 1998 Stability of vorticity defects in viscous shear. J. Fluid Mech. 357, 199–224.CrossRefGoogle Scholar
4. Balmforth, N. J., del Castillo-Negrete, D. & Young, W. R. 1997 Dynamics of vorticity defects in shear. J. Fluid Mech. 333, 197–230.CrossRefGoogle Scholar
5. Balmforth, N. J., Morrison, P. J. & Thiffeault, J.-L. 2012 Pattern formation in systems with continuous spectra and the single-wave model. Rev. Mod. Phys. (to be submitted).Google Scholar
6. Brown, S. N. & Stewartson, K. 1978 The evolution of the critical layer of a Rossby wave. Part II. Geophys. Astrophys. Fluid Dyn. 10, 1–24.CrossRefGoogle Scholar
7. Carpenter, J. R., Balmforth, N. J. & Lawrence, G. A. 2010a Identifying unstable modes in stratified shear layers. Phys. Fluids 22, 054104.CrossRefGoogle Scholar
8. 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.CrossRefGoogle Scholar
9. Caulfield, C. P. 1994 Multiple linear instability of layered stratified shear flow. J. Fluid Mech. 258, 255–285.CrossRefGoogle Scholar
10. 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
11. Caulfield, C. P., Yoshida, S., Peltier, W. R. & Ohtani, M. 1995 An experimental investigation of the instability of a shear flow with multi-layer density stratification. Phys. Fluids 7, 3028–3041.CrossRefGoogle Scholar
12. Cheng, C. Z. & Knorr, G. 1976 Integration of Vlasov equation in configuration space. J. Comput. Phys. 22, 330–351.CrossRefGoogle Scholar
13. Churilov, S. M. & Shukhman, I. G. 1987 Nonlinear stability of a stratified shear-flow – a viscous critical layer. J. Fluid Mech. 180, 1–20.CrossRefGoogle Scholar
14. Churilov, S. M. & Shukhman, I. G. 1996 The nonlinear critical layer resulting from the spatial or temporal evolution of weakly unstable disturbances in shear flows. J. Fluid Mech. 318, 189–221.CrossRefGoogle Scholar
15. Constantinou, N. C. & Ioannou, P. J. 2011 Optimal excitation of two-dimensional Holmboe instabilities. Phys. Fluids 23, 074102.CrossRefGoogle Scholar
16. Corcos, G. M. & Sherman, F. S. 1976 Vorticity concentration and the dynamics of unstable free shear layers. J. Fluid Mech. 73, 241–264.CrossRefGoogle Scholar
17. Fujiwara, T. 1981 Vlasov simulations of stellar systems: infinite homogeneous case. Publ. Astron. Soc. Japan 33, 531–540.Google Scholar
18. Gill, A. E. 1965 A mechanism for instability of plane Couette flow and of Poiseuille flow in a pipe. J. Fluid Mech. 21, 503–511.CrossRefGoogle Scholar
19. Hogg, A. M. & Ivey, G. N. 2003 The Kelvin–Helmholtz to Holmboe instability transition in stratified exchange flows. J. Fluid Mech. 477, 339–362.CrossRefGoogle Scholar
20. Holmboe, J. 1962 On the behaviour of symmetric waves in stratified shear layers. Geophys. Publ. 24, 67–113.Google Scholar
21. Ivey, G. N., Winters, K. B. & Koseff, J. R. 2008 Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech. 40, 169–184.CrossRefGoogle Scholar
22. 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.CrossRefGoogle Scholar
23. Lee, V. & Caulfield, C. P. 2001 Nonlinear evolution of a layered stratified shear flow. Dyn. Atmos. Oceans 34, 103–124.CrossRefGoogle Scholar
24. Lin, Z. 2001 Instability of periodic BGK waves. Math. Res. Lett. 8, 1–14.CrossRefGoogle Scholar
25. Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13, 2–23.CrossRefGoogle Scholar
26. Lovelace, R. V. E., Li, H., Colgate, S. A. & Nelson, A. F. 2000 Rossby wave instability of Keplerian accretion disks. II. Detailed linear theory. Astrophys. J. 533, 1023–1034.Google Scholar
27. Mashayek, A. & Peltier, W. R. 2011a Turbulence transition in stratified atmospheric and oceanic shear flows: Reynolds and Prandtl number controls upon the mechanism. Geophys. Res. Lett. 38, L16612.CrossRefGoogle Scholar
28. Mashayek, A. & Peltier, W. R. 2011b Three-dimensionalization of the stratified mixing layer at high Reynolds number. Phys. Fluids 23, 111701.CrossRefGoogle Scholar
29. Nicholson, D. R. 1983 Introduction to Plasma Theory. Wiley.Google Scholar
30. O’Neil, T. M., Winfrey, J. H. & Malmberg, J. H. 1971 Nonlinear interaction of a small cold beam and a plasma. Phys. Fluids 14, 1204–1212.CrossRefGoogle Scholar
31. Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.CrossRefGoogle Scholar
32. Rahmani, M. 2011 Kelvin–Helmholtz instability in sheared density stratified flows, PhD thesis, University of British Columbia.Google Scholar
33. Schmitt, R. W. 1994 Double diffusion in oceanography. Annu. Rev. Fluid Mech. 26, 255–285.CrossRefGoogle Scholar
34. Sellwood, J. A. & Kahn, F. D. 1991 Spiral modes driven by narrow features in angular-momentum density. Mon. Not. R. Astron. Soc. 250, 278–299.CrossRefGoogle Scholar
35. Shukhman, I. G. & Churilov, S. M. 1997 Effect of slight stratification on the nonlinear spatial evolution of a weakly unstable wave in a free shear layer. J. Fluid Mech. 343, 197–233.CrossRefGoogle Scholar
36. Smyth, W. D., Klaassen, G. P. & Peltier, W. R. 1988 Finite amplitude Holmboe waves. Geophys. Astrophys. Fluid Dyn. 43, 181–222.CrossRefGoogle Scholar
37. Smyth, W. D. & Moum, J. N. 2000 Length scales of turbulence in stably stratified mixing layers. Phys. Fluids 12, 1327–1342.CrossRefGoogle Scholar
38. Smyth, W. D. & Winters, K. B. 2003 Turbulence and mixing in Holmboe waves. J. Phys. Oceanogr. 33, 649–671.2.0.CO;2>CrossRefGoogle Scholar
39. Staquet, C. 1995 Two-dimensional secondary instabilities in a strongly stratified shear layer. J. Fluid Mech. 296, 73–126.CrossRefGoogle Scholar
40. Staquet, C. 2000 Mixing in a stably stratified shear layer: two- and three-dimensional numerical experiments. Fluid Dyn. Res. 27, 367–404.CrossRefGoogle Scholar
41. Sutherland, B. R. 2010 Internal Gravity Waves. Cambridge University Press.CrossRefGoogle Scholar
42. 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
43. Tedford, E. W., Pieters, R. & Lawrence, G. A. 2009 Symmetric Holmboe instabilities in a laboratory exchange flow. J. Fluid Mech. 636, 137–153.CrossRefGoogle Scholar
44. Thorpe, S. A. 1971 Experiments on instability of stratified shear flows: miscible fluids. J. Fluid Mech. 46, 299–319.CrossRefGoogle Scholar
45. Winant, C. D. & Browand, F. K. 1974 Vortex pairing: the mechanism of turbulent mixing-layer growth at moderate Reynolds number. J. Fluid Mech. 63, 237–255.CrossRefGoogle Scholar
46. Yoshida, S., Ohtani, M., Nishida, S. & Linden, P. F. 1998 Mixing processes in a highly stratified river. In Physical Processes in Lakes & Oceans (ed. Imberger, J. ). Coastal & Estuarine Studies , vol. 54. pp. 389–400. American Geophysical Union.CrossRefGoogle Scholar
47. Zhu, D. & Lawrence, G. A. 2001 Holmboe’s instability in exchange flows. J. Fluid Mech. 429, 391–401.CrossRefGoogle Scholar