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Wave- and Wind-Driven Flow in Water of Finite Depth

Zhigang XuDepartment of Oceanography, Dalhousie University, Halifax, Nova Scotia, Canada

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A. J. BowenDepartment of Oceanography, Dalhousie University, Halifax, Nova Scotia, Canada

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Abstract

The authors first derive both Coriolis-induced and viscosity-induced stresses for arbitrary water depth and arbitrary wave direction. Opportunity is taken here to succinctly and rigorously derive the Longuet-Higgins virtual tangential stress due to wave motion. It is shown that the virtual stress is a projection on the surface slope of two viscous normal stresses acting on the vertical and horizontal planes. Then a simple Eulerian model is presented for the steady flow driven by waves and by waves and winds This simple Eulerian model demonstrates that the wave forcing can he easily incorporated with other conventional forcing, rather than resorting to a complicated and lengthy perturbation analysis of the Lagrangian equations of motion. A further focus is given to the wave-driven flow when the various limits of the wave-driven steady flow are discussed. The wave-driven steady flow given by the model yields a unified formula between Ursell and Hasselmann's inviscid but rotational theory and the Longuet-Higgins viscid but nonrotational theory, and it becomes an Eulerian counterpart of Madsen's deep-water solution when the deep-water limit is taken. The model is further expanded for the case of unsteady wave forcing, yielding a general formula for any type of time variation in the wave field. Two examples are considered: a suddenly imposed wave field that is then maintained steady and a suddenly imposed wave field that is then subject to internal and bottom frictional decay. The extension of these results to the case of random waves is briefly discussed. Finally, an example is presented that suggests the need to add surface wave forcing in classical shelf dynamics.

Abstract

The authors first derive both Coriolis-induced and viscosity-induced stresses for arbitrary water depth and arbitrary wave direction. Opportunity is taken here to succinctly and rigorously derive the Longuet-Higgins virtual tangential stress due to wave motion. It is shown that the virtual stress is a projection on the surface slope of two viscous normal stresses acting on the vertical and horizontal planes. Then a simple Eulerian model is presented for the steady flow driven by waves and by waves and winds This simple Eulerian model demonstrates that the wave forcing can he easily incorporated with other conventional forcing, rather than resorting to a complicated and lengthy perturbation analysis of the Lagrangian equations of motion. A further focus is given to the wave-driven flow when the various limits of the wave-driven steady flow are discussed. The wave-driven steady flow given by the model yields a unified formula between Ursell and Hasselmann's inviscid but rotational theory and the Longuet-Higgins viscid but nonrotational theory, and it becomes an Eulerian counterpart of Madsen's deep-water solution when the deep-water limit is taken. The model is further expanded for the case of unsteady wave forcing, yielding a general formula for any type of time variation in the wave field. Two examples are considered: a suddenly imposed wave field that is then maintained steady and a suddenly imposed wave field that is then subject to internal and bottom frictional decay. The extension of these results to the case of random waves is briefly discussed. Finally, an example is presented that suggests the need to add surface wave forcing in classical shelf dynamics.

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