Hostname: page-component-84b7d79bbc-lrf7s Total loading time: 0 Render date: 2024-07-28T00:17:54.306Z Has data issue: false hasContentIssue false
  • English
  • Français

On the response of large-amplitude internal waves to upstream disturbances

Published online by Cambridge University Press:  22 May 2012

Roberto Camassa  and
Claudio Viotti
Roberto Camassa
Affiliation:
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
Claudio Viotti*
Affiliation:
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
*
†Email address for correspondence: viotti@email.unc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Large-amplitude internal solitary waves generate shear flows that intensify from the wings of the waves to their maxima. Upstream perturbations of the hydrostatic equilibrium in the form of wave packets along the path of wave propagation are expected to trigger shear instability and ultimately generate Kelvin–Helmholtz roll-ups. In contrast, as shown here with accurate simulations of incompressible stratified Euler equations, large internal waves can act as suppressors of perturbations. The precise understanding of the mechanisms leading to different outcomes, including whether instability is excited, is the focus of this work. Under the action of shear flows, small-amplitude wave packets undergo stretching and filamentation, which lead to significant absorption of perturbation energy into the background shear. It is found that this typical behaviour is present in the self-induced shear by internal waves, regardless of whether the shear is stable or unstable, and can leave a quieter state in the wave’s wake for a wide range of perturbation parameters. In the unstable case, even once perturbations are selected to excite the instability, our results show that this absorption can act to reduce growth in the strong-shear region, effectively making roll-up development observable only downstream of the wave crest. Our approach is both analytical and numerical; a model valid for relatively thin pycnoclines and suitable for local spectral analysis is devised and used. Energy diagnostics on the simulations are implemented to validate the numerics and illustrate the energy exchanges between background wave flow and its shear. A link between the absorption mechanism and the clustering of local eigenvalues along the wave is proposed. This promotes an energetic coupling among neutral modes stronger than what may be expected to occur in slowly varying flows, and gives rise to multi-modal transient dynamics of the kind often referred to as non-normality effects. For those cases in which the wave-induced shear meets the conditions for local instability, it is found that the growth of disturbances is selective with respect to the sign of the mode excited upstream. Elements of this phenomenon are interpreted by asymptotic analysis for spatial growth in time-independent slowly varying media.

JFM classification

Instability: Instability Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 702 , 10 July 2012 , pp. 59 - 88
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. Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
2. 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
3. Banks, W. H. H., Drazin, P. G. & Zaturska, M. B. 1976 On the normal modes of parallel flow of inviscid stratified fluid. J. Fluid Mech. 75, 149–171.CrossRefGoogle Scholar
4. Barad, M. F. & Fringer, O. B. 2010 Simulations of shear instabilities in interfacial gravity waves. J. Fluid Mech. 644, 61–95.CrossRefGoogle Scholar
5. Belan, M. & Tordella, D. 2006 Convective instability in wake intermediate asymptotics. J. Fluid Mech. 552, 127–136.CrossRefGoogle Scholar
6. 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
7. Camassa, R., McLaughlin, R. M. & Viotti, C. 2010 Analysis of passive scalar advection in parallel shear flows: sorting of modes at intermediate time scales. Phys. Fluids 22, 12660.CrossRefGoogle Scholar
8. Camassa, R. & Tiron, R. 2011 Optimal two-layer approximation for continuous density stratification. J. Fluid Mech. 669, 32–54.CrossRefGoogle Scholar
9. Carr, M., Fructus, D., Grue, J., Jensen, A. & Davies, P. A. 2008 Convectively induced shear instability in large amplitude internal solitary waves. Phys. Fluids 20, 12660.CrossRefGoogle Scholar
10. Carr, M., King, S. E. & Dritschel, D. G. 2011 Numerical simulations of shear-induced instabilities in internal solitary waves. J Fluid. Mech. 683, 263–288.CrossRefGoogle Scholar
11. 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
12. Choi, W. & Camassa, R. 1999 Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech. 396, 1–36.CrossRefGoogle Scholar
13. Crighton, D. G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77, 397–413.CrossRefGoogle Scholar
14. Diamessis, P. J. & Redekopp, L. G. 2006 Numerical investigation of solitary internal wave-induced global instability in shallow water benthic boundary layers. J. Phys. Oceanogr. 36, 784–812.CrossRefGoogle Scholar
15. Farrel, B. F. & Ioannou, P. J. 1993 Transient development of perturbations in stratified shear flows. J. Atmos. Sci 50, 2201–2214.2.0.CO;2>CrossRefGoogle Scholar
16. Fructs, D., Carr, M., Grue, J., Jensen, A. & Davies, P. A. 2009 Shear-induced breaking of large internal solitary waves. J. Fluid Mech. 620, 1–29.CrossRefGoogle Scholar
17. Fructs, D. & Grue, J. 2004 Fully nonlinear solitary waves in a layered stratified fluid. J. Fluid Mech. 505, 323–347.CrossRefGoogle Scholar
18. Gelfgat, A. Y & Kit, E. 2006 Spatial versus temporal instabilities in a parametrically forced stratified mixing layer. J. Fluid Mech. 552, 189–227.CrossRefGoogle Scholar
19. Hazel, P. 1972 Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech. 51, 39–61.CrossRefGoogle Scholar
20. Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395–425.CrossRefGoogle Scholar
21. Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.CrossRefGoogle Scholar
22. Howard, L. N. 1963 Neutral curves and stability boundaries in stratified flow. J. Fluid Mech. 16, 333–342.CrossRefGoogle Scholar
23. Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instability in free shear layers. J. Fluid Mech. 159, 161–168.CrossRefGoogle Scholar
24. Kataoka, T. 2006 The stability of finite-amplitude interfacial solitary waves. Fluid Dyn. Res. 38, 831–867.CrossRefGoogle Scholar
25. Lamb, K. G. 2008 On the calculation of the available potential energy of an isolated perturbation in a density-stratified fluid. J. Fluid Mech. 597, 415–427.CrossRefGoogle Scholar
26. Lamb, K. G. & Farmer, D. 2011 Instability in an internal solitary-like wave on the Oregon shelf. J. Phys. Oceanogr. 41, 67–87.CrossRefGoogle Scholar
27. Lamb, K. G. & Wan, B. 1998 Conjugate flows and flat solitary waves for a continuously stratified fluid. Phys. Fluids 10, 2061–2079.CrossRefGoogle Scholar
28. Lombard, N. L. & Riley, J. J. 1996 Instability and breakdown of internal gravity waves. I. Linear stability analysis. Phys. Fluids 8, 3271–3287.CrossRefGoogle Scholar
29. Miles, J. W. 1961 On the stability of heterogeneous shear flows. Part 1. J. Fluid Mech. 10, 496–508.CrossRefGoogle Scholar
30. Miles, J. W. 1963 On the stability of heterogeneous shear flows. Part 2. J. Fluid Mech. 16, 209–227.CrossRefGoogle Scholar
31. Miles, J. W. & Howard, L. N. 1964 Note on a heterogeneous shear flow. J. Fluid Mech. 20, 331–336.CrossRefGoogle Scholar
32. Miyata, M. 1985 An internal solitay wave of large amplitude. La Mer 23, 43–48.Google Scholar
33. Monkewitz, P. A., Huerre, P. & Chomaz, J. 1993 Global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech. 251, 1–20.CrossRefGoogle Scholar
34. Moum, J. N., Farmer, D. M., Smyth, 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, 2093–2112.2.0.CO;2>CrossRefGoogle Scholar
35. Pullin, D. I. & Grimshaw, R. H. J. 1985 Stability of finite-amplitude interfacial waves. Part 2. Numerical results. J. Fluid Mech. 20, 317–336.CrossRefGoogle Scholar
36. Rhines, P. B. & Young, W. R. 1983 How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133–145.CrossRefGoogle Scholar
37. Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
38. 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, Chapel Hill.Google Scholar
39. Tricomi, F. G. 1957 Integral Equations. Interscience.Google Scholar
40. Troy, C. & Koseff, J. R. 2005 The instability and breaking of long internal waves. J. Fluid Mech. 543, 107–136.CrossRefGoogle Scholar
41. Turkington, B., Eydeland, A. & Wang, S. 1991 A computational method for solitary internal waves in a continuously stratified fluid. Stud. Appl. Maths 85, 93–127.CrossRefGoogle Scholar
42. Yih, C. 1960 Gravity waves in a stratified fluid. J. Fluid Mech. 8, 481–508.CrossRefGoogle Scholar
43. Yih, C. 1980 Stratified Flows. Academic Press.Google Scholar