The effects of randomness on the stability of two-dimensional surface wavetrains

Abstract

A simplified nonlinear spectral transport equation, for narrowband Gaussian random surface wavetrains, slowly varying in space and time, is derived fron the weakly nonlinear equations of Davey & Stewartson. The stability of an initially homogeneous wave spectrum, to small oblique wave perturbations is studied for a range of spectral bandwidths, resulting in an integral equation for the amplification rate of the disturbance. It is shown for random deep water waves that instability of the wavetrain can exist, as in the corresponding deterministic Benjamin-Feir (B-F) problem, provided that the normalized spectral bandwidth σ / k 0 is less than twice the root mean square wave slope, multiplied by a function of the perturbation wave angle ϕ = arctan ( m/l ). A further condition for instability is that the angle ϕ be less than 35.26°. It is demonstrated that the amplification rate, associated with the B-F type instability, diminishes and then vanishes as the correlation length scale of the random wave field ( ca . 1/ σ )is reduced to the order of the characteristic length scale for modulational instability of the wave system.