Instability analysis of random inhomogeneous gravity waves with depth-uniform current

Starting from Zakharov integral equation, an equation that governs the evolution of a random field of nonlinear gravity waves (known as the sprctral transport equation) with depth-uniform current in deep water has been deduced. In the narrow-band approximation limit of this evolution equation, we have examined the modulational instability in the perturbed wavenumber space. The main objective of this work is to draw the stability diagrams and to study the effect of depth-uniform current on the growth rate of weakly nonlinear gravity waves. It is found that the effect of randomness reduces the growth rate of instability (GRI) and the extent of instability. The coflowing current decreases the growth rate of the instability, whereas the counterflowing current has the opposite effect. Also, we have regained the deterministic instability growth rate for vanishing spectral bandwidth.