Abstract Assume that a fluid is inviscid, incompressible, and irrotational. A nonlinear Schrödinger equation (NLSE) describing the evolution of gravity waves in finite water depth is derived using the multiple-scale analysis method. The gravity waves are influenced by a linear shear flow, which is composed of a uniform flow and a shear flow with constant vorticity. The modulational instability (MI) of the NLSE is analyzed, and the region of the MI for gravity waves (the necessary condition for existence of freak waves) is identified. In this work, the uniform background flows along or against wave propagation are referred to as down-flow and up-flow, respectively. Uniform up-flow enhances the MI, whereas uniform down-flow reduces it. Positive vorticity enhances the MI, while negative vorticity reduces it. Hence, the influence of positive (negative) vorticity on MI can be balanced out by that of uniform down (up) flow. Furthermore, the Peregrine breather solution of the NLSE is applied to freak waves. Uniform up-flow increases the steepness of the free surface elevation, while uniform down-flow decreases it. Positive vorticity increases the steepness of the free surface elevation, whereas negative vorticity decreases it.

Received: 09 September 2019
Revised: 18 October 2019
Accepted manuscript online:

Fund: Project supported by the National Key Research and Development Program of China (Grant Nos. 2016YFC1401404 and 2017YFA0604102) and the National Natural Science Foundation of China (Grant No. 41830533).

Corresponding Authors:
Anzhou Cao, Jinbao Song
E-mail: caoanzhou@zju.edu.cn;songjb@zju.edu.cn

Cite this article:

Shaofeng Li(李少峰), Juan Chen(陈娟), Anzhou Cao(曹安州), Jinbao Song(宋金宝) A nonlinear Schrödinger equation for gravity waves slowly modulated by linear shear flow 2019 Chin. Phys. B 28 124701

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