Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-27T05:23:21.669Z Has data issue: false hasContentIssue false
  • English
  • Français

Shear instability of internal solitary waves in Euler fluids with thin pycnoclines

Published online by Cambridge University Press:  29 August 2012

A. Almgren ,
R. Camassa  and
R. Tiron
A. Almgren
Affiliation:
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
R. Camassa
Affiliation:
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA
R. Tiron*
Affiliation:
UCD School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
*
†Email address for correspondence: roxana.tiron@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability with respect to initial condition perturbations of solitary travelling-wave solutions of the Euler equations for continuously, stably stratified, near two-layer fluids is examined numerically and analytically for a set of parameters of relevance for laboratory experiments. Numerical travelling-wave solutions of the Dubreil–Jacotin–Long equation are first obtained with a variant of Turkington, Eyland and Wang’s iterative code by testing convergence on the equation’s residual. In this way, stationary solutions with very thin pycnoclines (and small Richardson numbers) approaching the near two-layer configurations used in experiments can be obtained, allowing for a stability study free of non-stationary effects, introduced by lack of numerical resolution, which develop when these solutions are used as initial conditions in a time-dependent evolution code. The thin pycnoclines in this study permit analytical results to be derived from strongly nonlinear models and their predictions compared with carefully controlled numerical simulations. This brings forth shortcomings of simple criteria for shear instability manifestations based on parallel shear approximations due to subtle higher-order effects. In particular, evidence is provided that the fore–aft asymmetric growth observed in all simulations requires non-parallel shear analysis. Collectively, the results of this study reveal that while the wave-induced shear can locally reach unstable configurations and give rise to local convective instability, the global wave/self-generated shear system is in fact stable, even for extreme cases of thin pycnoclines and near-maximum-amplitude waves.

JFM classification

Instability: Absolute/convective instability Geophysical and Geological Flows: Internal waves Wakes/Jets: Shear layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 710 , 10 November 2012 , pp. 324 - 361
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. Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H. & Welcome, M. L. 1998 A conservative adaptive projection method for the variable density incompressible Navier–Stokes equations. J. Comput. Phys. 142, 1–46.CrossRefGoogle Scholar
2. Arnold, V. I. 1992 Ordinary Differential Equations. Springer.Google Scholar
3. Barad, M. F. & Fringer, O. B. 2010 Simulations of shear instabilities in interfacial gravity waves. J. Fluid Mech. 644, 61–95.CrossRefGoogle Scholar
4. Benjamin, T. B. 1966 Internal waves of finite amplitude and permanent form. J. Fluid Mech. 25, 241–270.CrossRefGoogle Scholar
5. Benjamin, T. B. 1972 The stability of solitary waves. Proc. R. Soc. A 328, 153–182.Google Scholar
6. Benjamin, T. B. 1986 On the Boussinesq model for two-dimensional wave motions in heterogeneous fluids. J. Fluid Mech. 165, 445–474.CrossRefGoogle Scholar
7. Bona, J. 1975 On the stability theory of solitary waves. Proc. R. Soc. Ser. A 344, 363–374.Google Scholar
8. Briggs, R. J. 1964 Electron-stream Interaction with Plasmas. MIT Press.CrossRefGoogle Scholar
9. Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816.CrossRefGoogle Scholar
10. Camassa, R., Choi, W., Michallet, H., Rusås, P.-O. & Sveen, J. K. 2006 On the realm of validity of strongly nonlinear asymptotic approximations for internal waves. J. Fluid Mech. 549, 1–23.CrossRefGoogle Scholar
11. Camassa, R. & Tiron, R. 2011 Optimal two-layer approximation for continuous density stratification. J. Fluid Mech. 669, 32–54.CrossRefGoogle Scholar
12. Camassa, R. & Viotti, C. 2012 On the response to upstream disturbances of large-amplitude internal waves. J. Fluid Mech. 702, 59–88.CrossRefGoogle Scholar
13. Carr, M., King, S. E. & Dritschel, D. G. 2011 Numerical simulation of shear-induced instabilities in internal solitary waves. J. Fluid Mech. 683, 263–288.CrossRefGoogle Scholar
14. Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 1–36.CrossRefGoogle Scholar
15. Duda, T. F., Lynch, J. F., Irish, J. D., Beardsley, R. C., Ramp, S. R., Chiu, C.-S., Tang, T. Y. & Yang, Y. J. 2004 Internal tide and nonlinear internal wave behaviour at the continental slope in the northern South China Sea. IEEE J. Ocean. Engng 29, 1105–1130.CrossRefGoogle Scholar
16. Friedlander, S. 2001 On nonlinear instability and stability for stratified shear flow. J. Math. Fluid Mech. 3, 82–97.CrossRefGoogle Scholar
17. Fructus, D., Carr, M., Grue, J., Jensen, A. & Davies, P. 2009 Shear-induced breaking of large internal solitary waves. J. Fluid Mech. 620, 1–29.CrossRefGoogle Scholar
18. Grue, J., Jensen, A., Rusås, P.-O. & Sveen, J. K. 1999 Properties of large-amplitude internal waves. J. Fluid Mech. 380, 257–278.CrossRefGoogle Scholar
19. Hazel, P. 1972 Numerical studies of the stability of stratified shear flows. Annu. Rev. Fluid Mech. 38, 395–425.Google Scholar
20. Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.CrossRefGoogle Scholar
21. Holyer, J. Y. 1979 Large amplitude progressive interfacial waves. J. Fluid Mech. 93, 433–448.CrossRefGoogle Scholar
22. Howard, L. N 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.CrossRefGoogle Scholar
23. Huerre, P. 2007 Open shear flow instabilities. In Perspectives in Fluid Dynamics: A Collective Introduction to Current Research, sixth edition, pp. 159–229. Cambridge University Press.Google Scholar
24. Huerre, P. & Monkewitz, P. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.CrossRefGoogle Scholar
25. Huerre, P. & Monkewitz, P. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
26. Kataoka, T. 2006 The stability of finite-amplitude interfacial solitary waves. Fluid Dyn. Res. 38, 831–867.CrossRefGoogle Scholar
27. Lamb, K. G. 2002 A numerical investigation of solitary internal waves with trapped cores formed via shoaling. J. Fluid Mech. 451, 109–144.CrossRefGoogle Scholar
28. Lamb, K. G. & Farmer, D. 2011 Instabilities in an internal solitary-like wave in the Oregon Shelf. J. Phys. Oceanogr. 41, 67–87.CrossRefGoogle Scholar
29. Long, R. 1965 On the Boussinesq approximation and its role in the theory of internal waves. Tellus 15, 46–52.CrossRefGoogle Scholar
30. Maslowe, S. A. & Thompson, J. M. 1971 Stability of a stratified free shear layer. Phys. Fluids 14, 453–458.CrossRefGoogle Scholar
31. Miles, J. W. 1963 On the stability of heterogeneous shear flows (part 2). J. Fluid Mech. 16, 209–227.CrossRefGoogle Scholar
32. Moum, J. N., Farmer, D. M., Smyth, W. D., Armi, L. & Vagle, L. 2003 Structure and generation of turbulence at interfaces strained by internal solitary waves propagating shoreward over the continental shelf. J. Phys. Oceanogr. 33, 2093–2112.2.0.CO;2>CrossRefGoogle Scholar
33. Puckett, E. G., Almgren, A. S., Bell, J. B., Marcus, D. L. & Rider, W. G. 1997 A higher-order projection method for tracking fluid interfaces in variable density incompressible flows. J. Comput. Phys. 130, 269–282.CrossRefGoogle Scholar
34. Tiron, R. 2009 Strongly nonlinear internal waves in near two-layer stratifications: generation, propagation and self-induced shear instabilities. PhD thesis, Mathematics Department, University of North Carolina at Chapel Hill.Google Scholar
35. Troy, D. & Koseff, J. R. 2005 The instability and breaking of long internal waves. J. Fluid Mech. 543, 107–136.CrossRefGoogle Scholar
36. Turkington, B., Eydeland, A. & Wang, S. 1991 A computational method for solitary internal waves in a continuously stratified fluid. Stud. Appl. Math. 85, 93–127.CrossRefGoogle Scholar
37. Yih, C. 1980 Stratified Flows. Academic.Google Scholar