### 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 language | English |
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Pages (from-to) | 131-167 |

Number of pages | 37 |

Journal | Studies in Applied Mathematics |

Volume | 94 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Feb 1995 |

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### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Studies in Applied Mathematics*,

*94*(2), 131-167. https://doi.org/10.1002/sapm1995942131

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

Research output: Contribution to journal › Article

*Studies in Applied Mathematics*, vol. 94, no. 2, pp. 131-167. https://doi.org/10.1002/sapm1995942131

}

TY - JOUR

T1 - Resonant Interactions between Vortical Flows and Water Waves. Part I

T2 - Deep Water

AU - Milewski, P. A.

AU - Benney, D. J.

PY - 1995/2/1

Y1 - 1995/2/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=21844516767&partnerID=8YFLogxK

U2 - 10.1002/sapm1995942131

DO - 10.1002/sapm1995942131

M3 - Article

VL - 94

SP - 131

EP - 167

JO - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

SN - 0022-2526

IS - 2

ER -