Resonant triads of capillary–gravity waves in the presence of a current

Resonant triads of capillary–gravity waves propagating on the surface of inviscid fluid of infinite depth, in which the basic state is a plane, parallel current of the wind-drift type, are examined in the case that one of the waves propagates parallel to the current. The velocity profiles chosen for the current have nonconstant vorticity, necessitating numerical computation of the dispersion relation and the coefficients of the amplitude equations. A wave propagating in the direction of the current is found to participate in two families of triads, corresponding to those described by Simmons [Proc. R. Soc. London Ser. A 309, 551 (1969)] for a stagnant basic state. Only triads from one of the families will be excited, and this family only exists when the original wave has a sufficiently short wavelength. The presence of a current increases the maximum wavelength at which this family of resonances occurs and increases the maximum possible angle between the original wave and an oblique, excited, resonant wave. When a wave propagates against the current the maximum wavelength and the maximum angle are decreased by the presence of the current. The amplitude equations governing temporal modulation are derived, and in the cases examined, the coefficients are found to be of differing sign, which excludes the possibility of explosive instability.