Capillary‐gravity solitary waves on water of finite depth interacting with a linear shear current

Abstract

AbstractThe problem of two‐dimensional capillary‐gravity waves on an inviscid fluid of finite depth interacting with a linear shear current is considered. The shear current breaks the symmetry of the irrotational problem and supports simultaneously counter‐propagating waves of different types: Korteweg de‐Vries (KdV)‐type long solitary waves and wave‐packet solitary waves whose envelopes are associated with the nonlinear Schrödinger equation. A simple intuition for the broken symmetry is that the current modifies the Bond number differently for left‐ and right‐propagating waves. Weakly nonlinear theories are developed in general and for two particular resonant cases: the case of second harmonic resonance and long‐wave/short‐wave interaction. Traveling‐wave solutions and their dynamics in the full Euler equations are computed numerically using a time‐dependent conformal mapping technique, and compared to some weakly nonlinear solutions. Additional attention is paid to branches of elevation generalized solitary waves of KdV type: although true embedded solitary waves are not detected on these branches, it is found that periodic wavetrains on their tails can be arbitrarily small as the vorticity increases. Excitation of waves by moving pressure distributions and modulational instabilities of the periodic waves in the resonant cases described above are also examined by the fully nonlinear computations.