被均匀流缓慢调制的有限水深毛细重力波

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

非线性的存在会产生高次谐波,这些谐波又反作用于原来的低次谐波,使波幅发生缓慢变化,从而产生缓慢调制现象.这里从考虑均匀流作用下的毛细重力水波基本方程出发,在不可压缩、无旋、无黏条件假设下,使用多重尺度分析方法推导出了在均匀流影响下有限深水毛细重力波振幅所满足的非线性Schr¨odinger方程(NLSE).分析了NLSE解的调制不稳定性.给出了毛细重力波调制不稳定的条件和钟型孤立波产生的条件.分析了无量纲最大不稳定增长率随无量纲水深和表面张力的变化趋势.同时给出了无量纲不稳定增长率随无量纲微扰动波数变化的曲线,呈现出了先增大后减小的趋势.最后指出均匀顺流减小了无量纲不稳定增长率及最大增长率,逆流增大了它们.由表面张力作用产生的毛细波及重力与表面张力共同作用产生的毛细重力波,与流的相互作用可以改变海表粗糙度和海洋上层流场结构,进而影响海气界面动量、热量及水汽的交换.了解海表这些短波动力机制,对卫星遥感的精确测量、海气相互作用的研究及海气耦合模式的改进等有重要意义.

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?dinger 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.