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Effect of the air on the drift velocity of water waves

Published online by Cambridge University Press:  26 April 2006

Jan Erik Weber  and
Even
Jan Erik Weber
Affiliation:
institute of Geophysics. University of Oslo, P.O. Box 1022 Blindern, N-0315 Oslo 3, Norway
Even
Affiliation:
Statoil. P.O. Box 300, N-4001 Stavanger, Norway
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Abstract

Mean drift currents due to damped, progressive, capillary-gravity waves at an air/water interface are investigated theoretically. The analysis is based on a Lagrangian description of motion. Both media are assumed to be semi-infinite, viscous, homogeneous fluids. The system rotates about the vertical axis with a constant angular velocity ½f, where f is the Coriolis parameter. Owing to viscous effects, the wave field attenuates in time or space. Linear analysis verifies the temporal decay rate reported by Dore (1978a). The nonlinear drift velocities are obtained by a series expansion of the solutions to second order in a parameter ε, which essentially is proportional to the wave steepness. The effect of the air on the drift current in the water is shown to depend on the values of the frequency ω, wavenumber k, density ρ, and kinematic viscosity ν through one single dimensionless parameter Q defined by \[ Q = \frac{\rho_1}{\rho_2}\bigg[\frac{\omega\nu_1}{2k^2\nu^2_2}\bigg]^{\frac{1}{2}}, \] where subscripts 1 and 2 refer to the air and the water, respectively. Dynamically, the increased shear near the interface due to the presence of the air leads to a higher value of the virtual wave stress (Longuet-Higgins 1969). This yields a tendency towards a higher (Eulerian) mean drift velocity as compared to the free-surface case. For temporally damped waves, this stress, due to increased damping, effectively acts over a shorter period of time. Accordingly, the mean current associated with such waves tends to be larger for short times and smaller for large times than that obtained with a vacuum above the water. For spatially damped waves, the virtual wave stress becomes independent of time. The Coriolis force is then needed to balance the wave stress in order to avoid infinitely large drift velocities as t → ∞. Furthermore, assessment of realistic values for the turbulent eddy viscosities in the air and the ocean is shown to bring the results closer to those obtained for a vacuum/water system.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 218 , September 1990 , pp. 619 - 640
Copyright
© 1990 Cambridge University Press

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References

Andrews, D. G. & McIntyre, M. E., 1978 An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.Google Scholar
Basset, A. B.: 1888 Hydrodynamics, vol. 2. Cambridge: Deighton and Bell.
Bye, J. A. T.: 1988 The coupling of wave drift and wind velocity profiles. J. Mar. Res. 46, 457–472.Google Scholar
Chang, M.-S.: 1969 Mass transport in deep-water long-crested random gravity waves. J. Geophys. Res. 74, 1515–1536.Google Scholar
Cheung, T. K. & Street, R. L., 1988 The turbulent layer in the water at an air-water interface. J. Fluid Mech. 194, 133–151.Google Scholar
Churchill, J. H. & Csanady, G. T., 1983 Near-surface measurements of quasi-Lagrangian velocities in open water. J. Phys. Oceanogr. 13, 1669–1680.Google Scholar
Craik, A. D. D.: 1982 The drift velocity of water waves. J. Fluid Mech. 116, 187–205.Google Scholar
Dore, B. D.: 1978a Some effects of the air-water interface on gravity waves. Geophys. Astrophys. Fluid Dyn. 10, 215–230.Google Scholar
Dore, B. D.: 1978b A double boundary-layer model of mass transport in progressive interfacial waves. J. Engng Maths 12, 289–301.Google Scholar
Førland, E.: 1989 Wave-induced currents in the ocean. Dr. Scient. dissertation, University of Oslo, Norway. 182 pp.
Gaster, M.: 1962 A note on the relation between temporally-icreasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222–224.Google Scholar
Grimshaw, R.: 1981 Mean flows generated by a progressing water wave packet. J. Austral. Math. Soc. B B22, 318–347.Google Scholar
Hasselmann, K.: 1970 Wave-driven inertial oscillations. Geophys. Fluid Dyn. 1, 463–502.Google Scholar
Jenkins, A. D.: 1986 A theory for steady and variable wind- and wave-induced currents. J. Phys. Oceanogr. 16, 1370–1377.Google Scholar
Jenkins, A. D.: 1987 Wind and wave induced currents in a rotating sea with depth-varying eddy viscosity. J. Phys. Oceanogr. 17, 938–951.Google Scholar
Lamb, H.: 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Longuet-Higgins, M. S.: 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A A245, 535–581.Google Scholar
Longuet-Higgins, M. S.: 1969 A nonlinear mechanism for the generation of sea waves. Proc. R. Soc. Lond. A 311, 371–389.Google Scholar
Madsen, O. S.: 1977 A realistic model of the wind-induced Ekman boundary layer. J. Phys. Oceanogr. 7, 248–255.Google Scholar
McLeish, W. L. & Putland, G. E., 1975 Measurements of wind-driven flow profiles in the top millimeter of water. J. Phys. Oceanogr. 5, 516–518.Google Scholar
Phillips, O. M.: 1985 Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J. Fluid Mech. 156, 505–531.Google Scholar
Pierson, W. J.: 1962 Perturbation analysis of the Navier-Stokes equations in Lagrangian form with selected linear solutions. J. Geophys. Res. 61, 3151–3160.Google Scholar
Pollard, R. T.: 1970 Surface waves with rotation: An exact solution. J. Geophys. Res. 75, 5895–5898.Google Scholar
Shemdin, O. H.: 1972 Wind-generated current and phase speed of wind waves. J. Phys. Oceanogr. 2, 411–419.Google Scholar
Stokes, G. G.: 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Weber, J. E.: 1983a Attenuated wave-induced drift in a viscous rotating ocean. J. Fluid Mech. 137, 115–129 (referred to as (I) herein).Google Scholar
Weber, J. E.: 1983b Steady wind- and wave-induced currents in the open ocean. J. Phys. Oceanogr. 13, 524–530.Google Scholar
Weber, J. E.: 1987 Wave attenuation and wave drift in the marginalice zone. J. Phys. Oceanogr. 17, 2351–2361.Google Scholar
Weber, J. E.: 1989 Eulerian versus Lagrangian approach to wave drift in a rotating ocean. Vetenskaps och Vitterhetssamhället, Göteborg, Acta: Geophys. 4, (in press).Google Scholar
Weber, J. E. & Førland, E. 1989 Effect of an insoluble surface film on the drift velocity of capillary-gravity waves. J. Phys. Oceanogr. 19, 952–961.Google Scholar
Wu, J.: 1975 Wind-induced drift currents. J. Fluid Mech. 68, 49–70.Google Scholar
Wu, J.: 1982 Wind-stress coefficients over sea surface from breeze to hurricane. J. Geophys. Res. 87, 9704–9706.Google Scholar