Fourth-order stability analysis for capillary-gravity waves on finite-depth currents with constant vorticity

Abstract

We derive a fourth-order nonlinear evolution equation (NLEE) for narrow-banded Stokes wave in finite depth in the presence of surface tension and a mean flow with constant vorticity. The two-dimensional capillary-gravity wave motion on the surface of finite depth is considered here. The analysis is limited to one horizontal dimension, parallel to the direction of wave propagation, in order to take advantage of a formulation using potential flow theory. This evolution equation is then employed to examine the effect of vorticity on the Benjamin–Feir instability (BFI) of the Stokes capillary-gravity wave trains. It is found that the vorticity modifies significantly the modulational instability and in the case of finite depth, the combined effect of vorticity and capillarity is to enhance the instability growth rate influenced by capillarity when the vorticity is negative. The key point is that the present fourth-order analysis exhibits considerable deviations in the stability properties from the third-order analysis and gives better results consistent with the exact numerical results. Furthermore, the influence of linear shear current on Peregrine breather (PB) is studied.