Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-20T11:38:59.551Z Has data issue: false hasContentIssue false
  • English
  • Français

On phase velocity and growth rate of wind-induced gravity-capillary waves

Published online by Cambridge University Press:  21 April 2006

Klaartje van Gastel ,
Peter A. E. M. Janssen  and
Gerbrand J. Komen
Klaartje van Gastel
Affiliation:
Royal Netherlands Meteorological Institute, De Bilt, The Netherlands
Peter A. E. M. Janssen
Affiliation:
Royal Netherlands Meteorological Institute, De Bilt, The Netherlands
Gerbrand J. Komen
Affiliation:
Royal Netherlands Meteorological Institute, De Bilt, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

The generation and growth of gravity–capillary waves (λ ≈ 1 cm) by wind are reconsidered using linear instability theory to describe the process. For all friction velocities we solve the resulting Orr–Sommerfeld equation using asymptotic methods. New elements in our theory, compared with the work of Benjamin (1959) and Miles (1962), are more stress on mathematical rigour and the incorporation of the wind-induced shear current. We find that the growth rate of the initial wavelets, the first waves to be generated by the wind, is proportional to u*3.

We also study the effect of changes in the shape of the profiles of wind and wind-induced current. In doing this we compare results of Miles (1962), Larson & Wright (1975), Valenzuela (1976), Kawai (1979), Plant & Wright (1980) and our study. We find that the growth rate is very sensitive to the shape of the wind profile while the influence of changes in the current profile is much smaller. To determine correctly the phase velocity, the value of current and current shear at the interface are very important, much more so than the shape of either wind or current profile.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 161 , December 1985 , pp. 199 - 216
Copyright
© 1985 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

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Batchelor, G. K. 1981 An Introduction to Fluid Dynamics. Cambridge University Press.
Benjamin, T. B. 1959 Shearin flow over a wavy boundary. J. Fluid Mech. 6, 161–205.Google Scholar
Drazin, P. G. & Reid, W. H. 1982 Hydrodynamic Stability. Cambridge University Press.
Hopf, L. 1914 Der Verlauf kleiner Schwingungen auf einer Strömung reibender Flüssigkeit. Ann. Phys., Lpz. (4) 44, 1–60.Google Scholar
Janssen, P. A. E. M. & Peeck, H. H. 1985 On the quasilinear evolution of the coupled air-flow, water wave system. Submitted for publication.
Kawai, S. 1977 On the generation of wind waves relating to the shear flow in water - a preliminary study. Tôhoku University Series 5, vol. 24, nos 1/2.
Kawai, S. 1979a Generation of initial wavelets by instability of a coupled shear flow and their evolution to wind waves. J. Fluid Mech. 93, 661–703.Google Scholar
Kawai, S. 1979b Discussion on the critical wind speed for wind-wave generation on the basis of shear-flow instability theory. J. Ocean. Soc. Japan 35, 179–186.Google Scholar
Larson, T. R. & Wright, J. W. 1975 Wind-generated gravity capillary waves: laboratory measurements of temporal growth rates using microwave backscatter. J. Fluid Mech. 70, 417–436.Google Scholar
Miles, J. W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3. 185–204.Google Scholar
Miles, J. W. 1962 On the generation of surface waves by shear flows. Part 4. J. Fluid Mech. 13, 433–448.Google Scholar
von Mises, R. 1912a Beitrag zum Oszillationsproblem. In: Festschrift H. Weber, pp. 252–282. Teubner.
von Mises, R. 1912b Kleine Schwingungen und Turbulenz. Jber. Deutsch. Math.-Verein 21, 241–248.Google Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics: Mechanics of Turbulence, vol. I. M.I.T. Press.
Plant, W. J. & Wright, J. W. 1980 Phase speeds of upwind and downwind travelling short gravity waves. J. Geophys. Res. 85, 3304–3310.Google Scholar
Raney, R. K., Hasselmann, K., Plant, J. W., Alpers, W., Shuchman, R. A., Jain, A. & Shemdin, O. H. 1985 Theory of SCAR Ocean Wave Imaging: the Marsen Consensus, to be published.
Valenzuela, G. R. 1976 The growth of gravity-capillary waves in the coupled shear flow. J. Fluid Mech. 76, 229–250.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.