Nonlinear water waves with shear

Various aspects of nonlinear inviscid gravity waves in the presence of shear in the air and water are investigated. The shear, which appears due to the presence of wind in the air and current in the water, is modeled by a piecewise linear velocity profile. The interaction of short and long gravity waves is studied numerically, using spectral methods, and analytically, using perturbation methods. Special attention is paid to the verification of observations and experimental results. It is confirmed that finite amplitude waves propagating in the same direction as the wind or current are more stable with respect to superharmonic infinitesimal perturbations than the waves moving against the wind or current. Infinitesimal perturbations in the form of side bands are also investigated both numerically and analytically. The nonlinear cubic Schrodinger equation for the wave envelope of a slowly varying wave train is derived. It is shown that depending on the direction of propagation (along or against the shear) of the finite amplitude waves, the effect of the shear on the stability is substantially different. In most cases, however, the shear strength increase first enhances the instability, but later suppresses it. Three-wave interactions of gravity waves with shear in the water are considered. The interaction equations are derived with the help of two different perturbation approaches. The question of stability is addressed for both resonant and near-resonant interactions. The regions of explosive and "pump-wave" instability are identified for various types of three-wave interactions. A new type of steady two-dimensional gravity waves with water shear is computed numerically. These waves appear at relatively low amplitudes and lack symmetry with respect to any crest or trough. A boundary integral formulation is used to obtain a one-parameter family of non-symmetric solutions through a symmetry-breaking bifurcation.</p