Abstract

Any weak, steady vortical flow is a solution to leading order of the inviscid fluid equations with a free surface, so long as this flow has horizontal streamlines coinciding with the undisturbed free surface. This work considers the propagation of irrotational surface gravity waves when such a vortical flow is present. In particular, when the vortical flow and the irrotational surface waves are both periodic, resonant interactions can occur between the various components of the flow. The periodic vortical component of the flow is proposed as a model for more complicated vortical flows that would affect surface waves in the ocean, such as the turbulence in the wake of a ship. These resonant interactions are studied in two dimensions, both in the limit of deep water (Part I) and shallow water (Part II). For deep water, the resonant set of surface waves is governed by "triad-like" ordinary differential equations for the wave amplitudes, whose coefficients depend on the underlying rotational flow. These coefficients are calculated explicitly and the stability of various configurations of waves is discussed. The effect of three dimensionality is also briefly mentioned.

Original languageEnglish
Pages (from-to)131-167
Number of pages37
JournalStudies in Applied Mathematics
Volume94
Issue number2
DOIs
Publication statusPublished - 1 Feb 1995

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Water waves
Water Waves
Surface waves
Water
Surface Waves
Interaction
Rotational flow
Gravity waves
Free Surface
Steady flow
Ordinary differential equations
Wave propagation
Ships
Turbulence
Gravity Waves
Shallow Water
Coefficient
Streamlines
Wake
Fluids

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Resonant Interactions between Vortical Flows and Water Waves. Part I : Deep Water. / Milewski, P. A.; Benney, D. J.

In: Studies in Applied Mathematics, Vol. 94, No. 2, 01.02.1995, p. 131-167.

Research output: Contribution to journalArticle

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