Fourth order nonlinear evolution equation for capillary gravity waves in deep water in the presence of a thin thermocline including the effect of wind

A fourth order non-linear evolution equation is derived for a capillary-gravity wave packet in deep water in the presence of a thin thermocline including the effect of wind and viscous dissipation in water. In deriving this equation it has been assumed that the wind induced basic current in water is exponential and the effect of shear in air flow and viscous dissipation in water is accounted for by including a term in the evolution equation. The nonlinear evolution equation is used to study the stability of a uniform capillary-gravity wave train. Expressions for the maximum growth rate of instability and wave number at marginal stability are obtained. From results shown graphically it is found that the inclusion of wind effect increases the growth rate of instability irrespective of the presence of a thin thermocline. For waves with a small wave number, a thin thermocline has a stabilizing influence both in the presence and in the absence of wind input and the maximum growth rate of instability decreases with the increase of thermocline depth. But for waves with a large wave number a thin thermocline has no influence.