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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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