Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-18T04:03:53.316Z Has data issue: false hasContentIssue false
  • English
  • Français

Generation of surface waves by shear-flow instability

Published online by Cambridge University Press:  18 December 2013

W. R. Young  and
C. L. Wolfe
W. R. Young*
Affiliation:
Scripps Institution of Oceanography, La Jolla, CA 92093-0213, USA
C. L. Wolfe
Affiliation:
Scripps Institution of Oceanography, La Jolla, CA 92093-0213, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the linear stability of an inviscid parallel shear flow of air over water with gravity and capillarity. The velocity profile in the air is monotonically increasing upwards from the sea surface and is convex, while the velocity in the water is monotonically decreasing from the surface and is concave. An archetypical example, the ‘double-exponential’ profile, is solved analytically and studied in detail. We show that there are two types of unstable mode which can, in some cases, co-exist. The first type is the ‘Miles mode’ resulting from a resonant interaction between a surface gravity wave and a critical level in the air. The second unstable mode is an interaction between surface gravity waves and a critical level in the water, resulting in the growth of ripples. The gravity–capillary waves participating in this second resonance have negative intrinsic phase speed, but are Doppler shifted so that their actual phase speed is positive, and matches the speed of the base-state current at the critical level. In both cases, the Reynolds stresses of an exponentially growing wave transfer momentum from the vicinity of the critical level to the zone between the crests and troughs of a surface wave.

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Critical layers Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 739 , 25 January 2014 , pp. 276 - 307
Copyright
©2013 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

Bakas, N. & Ioannou, P. 2009 Modal and nonmodal growths of inviscid planar perturbations in shear flows with a free surface. Phys. Fluids 21, 024102.CrossRefGoogle Scholar
Bühler, O. 2009 Waves and Mean Flows. Cambridge University Press.CrossRefGoogle Scholar
Caponi, E., Caponi, M., Saffman, P. & Yuen, H. 1992 A simple-model for the effect of water shear on the generation of waves by wind. Proc. R. Soc. Lond. A 438 (1902), 95–101.Google Scholar
Caponi, E., Yuen, H., Milinazzo, F. & Saffman, P. 1991 Water-wave instability induced by a drift layer. J. Fluid Mech. 222, 207–213.CrossRefGoogle Scholar
Dimas, A. A. & Triantafyllou, G. S. 1994 Nonlinear-interaction of shear-flow with a free-surface. J. Fluid Mech. 260, 211–246.CrossRefGoogle Scholar
Engevik, L. 2000 A note on the instabilities of a horizontal shear flow with a free surface. J. Fluid Mech. 406, 337–346.CrossRefGoogle Scholar
van Gastel, K., Janssen, P. & Komen, G. 1985 On phase velocity and growth rate of wind-induced gravity–capillary waves. J. Fluid Mech. 161, 199–216.CrossRefGoogle Scholar
Hristov, T., Miller, S. & Friehe, C. 2003 Dynamical coupling of wind and ocean waves through wave-induced air flow. Nature 422 (6927), 55–58.CrossRefGoogle ScholarPubMed
Hughes, T. H. & Reid, W. H. 1965 On the stability of the asymptotic suction boundary-layer profile. J. Fluid Mech. 23, 715–735.CrossRefGoogle Scholar
Itoh, K., Inoue, M., Kumamaru, H. & Kukita, Y. 2007 Linear stability analysis on free-surface liquid jet with different simplification of velocity profile. J. Fluid Sci. Technol. 2 (2), 417–428.CrossRefGoogle Scholar
Janssen, P. 2004 The Interaction of Ocean Waves and Wind. Cambridge University Press.CrossRefGoogle Scholar
Kawai, S. 1979 Generation of initial wavelets by instability of a coupled shear flows and their evolution to wind waves. J. Fluid Mech. 93, 661–703.CrossRefGoogle Scholar
Lighthill, M. 1962 Physical interpretation of the mathematical theory of wave generation by wind. J. Fluid Mech. 14 (3), 385–398.CrossRefGoogle Scholar
Lighthill, M. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Longuet-Higgins, M. S. 1998 Instabilities of a horizontal shear flow with a free surface. J. Fluid Mech. 364, 147–162.CrossRefGoogle Scholar
Melville, W. K., Shear, R. & Veron, F. 1998 Laboratory measurements of the generation and evolution of langmuir circulations. J. Fluid Mech. 364, 31–58.CrossRefGoogle Scholar
Miles, J. W. 1957 On the generation of surface waves by shear flow. J. Fluid Mech. 3, 185–204.CrossRefGoogle Scholar
Miles, J. W. 1962 On the generation of surface waves by shear flow. Part 4. J. Fluid Mech. 13, 433–448.CrossRefGoogle Scholar
Miles, J. W. 2001 A note on surface waves generated by shear-flow instability. J. Fluid Mech. 447, 173–177.CrossRefGoogle Scholar
Morland, L. C. & Saffman, P. G. 1993 Effect of wind-profile on the instability of wind blowing over water. J. Fluid Mech. 252, 383–398.CrossRefGoogle Scholar
Morland, L. C., Saffman, P. G. & Yuen, H. C. 1991 Waves generated by shear layer instabilities. Proc. R. Soc. Lond. A 433 (1888), 441–450.Google Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
Shrira, V. I. 1993 Surface waves on shear currents: solution of the boundary-value problem. J. Fluid Mech. 252, 565–584.CrossRefGoogle Scholar
Stern, M. E. & Adam, Y. A. 1974 Capillary waves generated by a shear current in water. In Fifth Liège Colloquium on Ocean Hydrodynamics (ed. Nihoul, J.).Google Scholar
Taylor, G. I. 1915 Eddy motion in the atmosphere. Phil. Trans. R. Soc. Lond. 215, 1–26.Google Scholar
Valenzuela, G. 1976 The growth of gravity–capillary waves in a coupled shear flow. J. Fluid Mech. 76, 229–250.CrossRefGoogle Scholar
Veron, F. & Melville, W. K. 2001 Experiments on the stability and transition of wind-driven water surfaces. J. Fluid Mech. 446, 25–65.CrossRefGoogle Scholar
Wheless, G. & Csanady, G. 1993 Instability waves on the air–sea interface. J. Fluid Mech. 248, 363–381.CrossRefGoogle Scholar
Yih, C.-S. 1972 Surface waves in flowing water. J. Fluid Mech. 51, 209–220.CrossRefGoogle Scholar
Zeisel, A., Stiassnie, M. & Agnon, Y. 2008 Viscous effect on wave generation by strong winds. J. Fluid Mech. 597, 343–369.CrossRefGoogle Scholar
Zhang, X. 2005 Short surface waves on surface shear. J. Fluid Mech. 541, 345–370.CrossRefGoogle Scholar