TY - JOUR
T1 - Resonant Interactions between Vortical Flows and Water Waves. Part I
T2 - Deep Water
AU - Milewski, P. A.
AU - Benney, D. J.
N1 - Publisher Copyright:
© 2015 Wiley Periodicals, Inc., A Wiley Company.
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.
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U2 - 10.1002/sapm1995942131
DO - 10.1002/sapm1995942131
M3 - Article
AN - SCOPUS:21844516767
SN - 0022-2526
VL - 94
SP - 131
EP - 167
JO - Studies in Applied Mathematics
JF - Studies in Applied Mathematics
IS - 2
ER -