Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-18T10:32:12.416Z Has data issue: false hasContentIssue false
  • English
  • Français

An asymptotic theory for the generation of nonlinear surface gravity waves by turbulent air flow

Published online by Cambridge University Press:  26 April 2006

Cornelis A. Van Duin
Cornelis A. Van Duin
Affiliation:
Department of Oceanography, Royal Netherlands Meteorological Institute, 3730 AE De Bilt, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

Based on a previous linear theory (van Duin & Janssen 1992), turbulent air flow over a surface gravity wave of finite amplitude is studied analytically by the methods of matched asymptotic expansions and multiple-scale analysis. In particular, an initial-value problem for weakly nonlinear waves is solved, where the initial conditions are prescribed by a single Stokes wave, displacing the water surface. The water is inviscid and incompressible, and there is no mean shear current. Wave–wave interactions are not taken into account. The validity of the theory is restricted to slow waves and small drag coefficient.

We investigate in detail the change of the mean air flow with the evolution of the wave, with a prescribed order of magnitude of the initial wave slope. The rate of change of this flow is fully determined by an evolution equation for the wave slope, which is obtained from the continuity condition for the normal stress at the air-water interface. This equation also determines the amplitude-dependent rate of growth or damping of the wave, for which a closed-form expression is derived. It turns out that nonlinear effects reduce the rate of energy transfer from the mean air flow to the growing wave, which implies that nonlinearity has a stabilizing effect.

For sufficienty large time scales, the slope of the growing wave becomes so large that the original evolution equation, which is approximately a Landau equation, ceases to be valid. For such relatively large wave slope, an alternative evolution equation is derived, which presumably describes the further evolution of the wave until the occurrence of wave breaking. The relative effects of nonlinearity, which can be characterized by a single parameter, increase with increasing wave slope and decreasing wave frequency.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 320 , 10 August 1996 , pp. 287 - 304
Copyright
© 1996 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. 1964 Handbook of Mathematical Functions. Natl Bureau of Standards.
Al-Zanaidi, M. A. & Hui, W. H. 1984 Turbulent airflow over water waves—a numerical study. J. Fluid Mech. 148, 225–246.Google Scholar
Akylas, T. R. 1982 A nonlinear theory for the generation of water waves by wind. Stud. Appl. Maths 67, 1–24.Google Scholar
Belcher, S. E. & Hunt, J. C. R. 1993 Turbulent shear flow over slowly moving waves. J. Fluid Mech. 251, 109–148.Google Scholar
Belcher, S. E., Harris, J. A. & Street, R. L. 1994 Linear dynamics of wind waves in coupled turbulent air—water flow. Part 1. Theory. J. Fluid Mech. 271 119–151.Google Scholar
Benjamin, T. B. 1959 Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161–205.Google Scholar
Britter, R. E., Hunt, J. C. R. & Richards, K. J. 1981 Air flow over a 2-d hill: studies of velocity speed-up, roughness effects and turbulence. Q. J. R. Met. Soc. 107 91–110.Google Scholar
Burgers, G. J. H. & Makin, V. K. 1993 Boundary-layer model results for wind-sea growth. J. Phys. Oceanogr. 23 372–385.Google Scholar
Chalikov, D. V. 1976 A mathematical model of wind-induced waves. Dokl. Akad. Nauk. SSSR 229, 1083–1086.Google Scholar
Chalikov, D. V. 1978 The numerical simulation of wind-wave interaction. J. Fluid Mech. 87, 561–582.Google Scholar
Charnock, H. 1955 Wind stress on a water surface. Q. J. R. Met. Soc. 81, 639–640.Google Scholar
Davis, R. E. 1972 On prediction of the turbulent flow over a wavy boundary. J. Fluid Mech. 52, 287–306.Google Scholar
Davis, R. E. 1974 Perturbed turbulent flow, eddy viscosity and generation of turbulent stresses. J. Fluid Mech. 63, 673–693.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Duin, C. A. van 1994 An asymptotic theory for the generation of nonlinear surface gravity waves by turbulent air flow. Internal memorandum of KNMI, De Bilt.
Duin, C. A. van 1996 An analytical model of the generation of nonlinear water waves by turbulent air flow. In The Air-Sea Interface (ed. M. A. Donelan, W. H. Hui & W. J. Plant). The University of Toronto Press, Toronto (in press).
Duin, C. A. van & Janssen, P. A. E. M. 1992 An analytic model of the generation of surface gravity waves by turbulent air flow. J. Fluid Mech. 236 197–215 (referred to herein as VDJ).Google Scholar
Fabrikant, A. L. 1976 Quasilinear theory of wind-wave generation. Izv. Atmos. Oceanic Phys. 12, 524–526.Google Scholar
Gent, P. R. & Taylor, P. A. 1976 A numerical model of the air flow above water waves. J. Fluid Mech. 77, 105–128.Google Scholar
Jacobs, S. J. 1987 An asymptotic theory for the turbulent flow over a progressive water wave. J. Fluid Mech. 174, 69–80.Google Scholar
Janssen, P. A. E. M. 1982 Quasi-linear approximation for the spectrum of wind-genrated water waves. J. Fluid Mech. 117, 493–506.Google Scholar
Jenkins, A. D. 1992 A quasi-linear eddy-viscosity model for the flux of energy and momentum to wind waves. using conservation-law equations in a curvilinear coordinate system. J. Phys. Oceanogr. 22, 843–858.Google Scholar
Knight, D. 1977 Turbulent flow over a wavy boundary. Boundary Layer Met. 11, 205–222.Google Scholar
Makin, V. K. 1989 Numerical approximation of the wind wave interaction parameter. Meteorolgica i Gidrologia No. 10, 106–108. (Engl. transl. Sov. Met. Hydrol., No. 10.)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. 1959 On the generation of surface waves by shear flows. Part 4. J. Fluid Mech. 13, 568–582.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
Miles, J. W. 1993 Surface-wave generation revisited. J. Fluid Mech. 256 427–441.Google Scholar
Sykes, R. I. 1980 An asymptotic theory of incompressbile turbulent boundary-layer flow over a small hump. J. Fluid Mech. 101, 647–670.Google Scholar
Townsend, A. A. 1972 Flow in a deep turbulent boundary layer over a surface distorted by water waves. J. Fluid Mech. 55, 719–735.Google Scholar