GRAVITY-CAPILLARY WAVES SLOWLY MODULATED BY UNIFORM FLOW IN FINITE WATER DEPTH

The higher harmonics are produced by the nonlinear terms, which react on the original low order harmonics, making the amplitude of waves changes slowly with space and time, thus producing slow modulation phenomenon. Based on the basic equations of water waves which are influenced by a uniform flow, under the assumptions of the fluid motion is inviscid, incompressible and irrotational, a nonlinear Schr\ddotodinger equation (NLSE) for gravity-capillary waves in finite water depth is derived by using the multiple scale analysis method. The modulational instability of the NLSE is analyzed, the conditions of modulational instability for gravity-capillary waves and the generation of bell solitary waves are proposed. The trend of dimensionless maximum instability growth rate with dimensionless water depth and surface tension is analyzed. At the same time, the dimensionless instability growth rate as a function of dimensionless perturbation wave number is also analyzed, it is shown that it increases from zero and then decreases to zero with the increase of the dimensionless perturbation wave number. In addition, it is found that uniform down-flow decreases the dimensionless growth rate and maximum growth rate, on the contrary, uniform up-flow increases them. Capillary waves generated by the surface tension and gravity-capillary waves generated by the gravity and surface tension which are modulated by uniform flow can change the surface roughness and the structure of the upper ocean flow field, and then affect the exchange of momentum, heat and water vapor at the air-sea interface. Hence, Understanding these short-waves dynamic mechanisms of the sea surface is of great significance to the accurate measurement of satellite remote sensing, the study of sea-air interaction and the improvement of sea-air coupling model.