Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-13T01:42:01.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Vortex pairing and resonant wave interactions in a stratified free shear layer

Published online by Cambridge University Press:  21 April 2006

D. A. Collins  and
S. A. Maslowe
D. A. Collins
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, PQ, H3A 2K6, Canada Present address: Computer Modelling Group, Calgary, Alberta T2L 2A6, Canada.
S. A. Maslowe
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, PQ, H3A 2K6, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

In a previous study using finite-amplitude techniques (Maslowe 1977), a strong instability mechanism was discovered that takes effect at the Richardson numbers consistent with turbulence observations in the atmosphere and oceans. The mechanism involves second harmonic resonance of two neutral or nearly neutral modes at a Richardson number of roughly 0.22. In the present investigation, the nonlinear Boussinesq equations have been solved numerically to further explore this instability and to assess the limits of validity of the theory. Qualitative agreement between the theory and numerical simulations was satisfactory as the most significant numerical results were predicted by the theory. In particular, the wave interaction leads to impressive instabilities at Richardson numbers large enough that a single linearly unstable wave would amplify only weakly. At a Richardson number of 0.14, for example, the saturation amplitude of the long wave in the two-wave interacting case was 15 times as large as the amplitude of the linearly most unstable wave (evolving by itself) at the same Richardson number.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 191 , June 1988 , pp. 465 - 480
Copyright
© 1988 Cambridge University Press

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

Brown, S. N., Rosen, A. S. & Maslowe, S. A. 1981 The evolution of a quasi-steady critical layer in a stratified viscous shear layer. Proc. R. Soc. Lond. A 375, 271–293.Google Scholar
Browning, K. A. 1971 Structure of the atmosphere in the vicinity of large-amplitude Kelvin-Helmoholtz billows. Q. J. R. Met. Soc. 97, 283–299.Google Scholar
Churilov, S. M. & Shukhman, I. G. 1987 Nonlinear stability of a stratified shear flow: a viscous critical layer. J. Fluid Mech. 180, 1–20.Google Scholar
Collins, D. A. 1982 A numerical study of the stability of a stratified mixing layer. PhD thesis, McGill University.
Corcos, G. M. & Sherman, F. S. 1984 The mixing layer: deterministic models of a turbulent flow. Part 1. Introduction and the two-dimensional flow. J. Fluid Mech. 139, 29–65.Google Scholar
Craik, A. D. D. 1985 Wave Interactions in Fluid Flows. Cambridge University Press.
Curry, J. H., Herring, J. R., Loncaric, J. & Orszag, S. A. 1984 Order and disorder in two- and three-dimensional Bénard convection. J. Fluid Mech. 147, 1–38.Google Scholar
Davey, R. F. & Roshko, A. 1972 The effect of a density difference on shear-layer instability. J. Fluid Mech. 53, 523–543.Google Scholar
Davis, P. A. & Peltier, W. R. 1979 Some characteristics of the Kelvin-Helmholtz and resonant over-reflection modes of shear flow instability and of their interaction through vortex pairing. J. Atmos. Sci. 36, 2394–2412.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Fritts, D. C. 1984 Shear excitation of atmospheric gravity waves: Part II. J. Atmos. Sci. 41, 524–537.Google Scholar
Gottlieb, D. & Orszag, S. A. 1977 Numerical analysis of spectral methods: theory and applications. NSF-CBMS Monograph no. 26, Soc. Ind. App. Math.
Greene, G. E. & Hooke, W. H. 1979 Scales of gravity waves generated by instability in tropospheric shear flows. J. Geophys. Res. 84, 6362–6364.Google Scholar
Hardy, K. R., Reed, R. J. & Mather, G. K. 1973 Observation of Kelvin-Helmholtz billows and their mesoscale environment by radar, instrumented aircraft, and a dense radiosonde network. Q. J. R. Met. Soc. 99, 279–293.Google Scholar
Ho, C.-M. & Huerre, P. 1984 Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365–424.Google Scholar
Kelly, R. E. 1967 On the stability of an inviscid shear layer which is periodic in space and time. J. Fluid Mech. 27, 657–689.Google Scholar
Kelly, R. E. 1968 On the resonant interaction of neutral disturbances in two inviscid shear flows. J. Fluid Mech. 31, 789–799.Google Scholar
Laroche, H. 1987 Simulation numérique directe de couches de mélange en présence de rotation différentielle. Mémoire de DEA, Institut de Mécanique de Grenoble.
McGoldrick, L. F. 1970 An experiment on second-order capillary gravity resonant wave interactions. J. Fluid Mech. 40, 251–271.Google Scholar
Maslowe, S. A. 1977 Weakly nonlinear stability theory of stratified shear flows. Q. J. R. Met. Soc. 103, 769–783.Google Scholar
Maslowe, S. A. 1986 Critical layers in shear flows. Ann. Rev. Fluid Mech. 18, 405–432.Google Scholar
Maslowe, S. A. & Thompson, J. M. 1971 Stability of a stratified free shear layer. Phys. Fluids 14, 453–458.Google Scholar
Metcalf, J. I. & Atlas, D. 1973 Microscale ordered motions and atmospheric structure associated with thin echo layers in stably stratified zones. Boundary Layer Met. 4, 7–36.Google Scholar
Miles, J. W. & Howard, L. N. 1964 Note on a heterogeneous shear flow. J. Fluid Mech. 20, 331–336.Google Scholar
Nayfeh, A. H. 1973 Second-harmonic resonance in the interaction of an air stream with capillary-gravity waves. J. Fluid Mech. 59, 803–816.Google Scholar
Patnaik, P. C., Sherman, F. S. & Corcos, G. M. 1976 A numerical simulation of Kelvin-Helmholtz waves of finite amplitude. J. Fluid Mech. 73, 215–240.Google Scholar
Riley, J. J. & Metcalfe, R. W. 1980 Direct numerical simulation of a perturbed, turbulent mixing layer. AIAA paper 80-0274.
Scotti, R. S. & Corcos, G. M. 1972 An experiment on the stability of small disturbances in a stratified free shear layer. J. Fluid Mech. 52, 499–528.Google Scholar
Stuart, J. T. 1971 Nonlinear stability theory. Ann. Rev. Fluid Mech. 3, 347–370.Google Scholar
Usher, J. R. & Craik, A. D. D. 1975 (with an Appendix by F. Hendriks) Nonlinear wave interactions in shear flows. Part 2. Third-order theory. J. Fluid Mech. 70, 437–461.Google Scholar
Woods, J. D. 1968 Wave induced shear instability in the summer thermocline. J. Fluid Mech. 32, 791–800.Google Scholar