Numerical simulation of the Benjamin-Feir instability and its consequences

Full nonlinear equations for one-dimensional potential surface waves were used for investigation of the evolution of an initially homogeneous train of exact Stokes waves with steepness A K = 0.01 – 0.42 . The numerical algorithm for the integration of nonstationary equations and the calculation of exact Stokes waves is described. Since the instability of the exact Stokes waves develops slowly, a random small-amplitude noise was introduced in initial conditions. The development of instability occurs in two stages: in the first stage the growth rate of disturbances was close to that established for small steepness by Benjamin and Feir [J. Fluid. Mech. 27, 417 (1967)] and for medium steepness [McLean, J. Fluid Mech. 114, 315 (1982)]. For any steepness, the Stokes waves disintegrate and create random superposition of waves. For A K < 0.13 , waves do not show a tendency to breaking, which is recognized by approaching a surface to non-single-value shape. Sooner or later, if A K > 0.13 , one of the waves increases its height, and finally it comes to the breaking point. For large steepness of A K > 0.35 the rate of growth is slower than for medium steepness. The data for spectral composition of disturbances and their frequencies are given.