A nonlinear Schrödinger equation for gravity waves slowly modulated by linear shear flow

doi:10.1088/1674-1056/ab53cf

中国物理B ›› 2019, Vol. 28 ›› Issue (12): 124701-124701.doi: 10.1088/1674-1056/ab53cf

• ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS • 上一篇下一篇

A nonlinear Schrödinger equation for gravity waves slowly modulated by linear shear flow

Shaofeng Li(李少峰), Juan Chen(陈娟), Anzhou Cao(曹安州), Jinbao Song(宋金宝)

Ocean College, Zhejiang University, Zhoushan 316000, China

收稿日期:2019-09-09 修回日期:2019-10-18 出版日期:2019-12-05 发布日期:2019-12-05
通讯作者: Anzhou Cao, Jinbao Song E-mail:caoanzhou@zju.edu.cn;songjb@zju.edu.cn
基金资助:
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).

A nonlinear Schrödinger equation for gravity waves slowly modulated by linear shear flow

Shaofeng Li(李少峰), Juan Chen(陈娟), Anzhou Cao(曹安州), Jinbao Song(宋金宝)

Ocean College, Zhejiang University, Zhoushan 316000, China

Received:2019-09-09 Revised:2019-10-18 Online:2019-12-05 Published:2019-12-05
Contact: Anzhou Cao, Jinbao Song E-mail:caoanzhou@zju.edu.cn;songjb@zju.edu.cn
Supported by:
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).

摘要/Abstract

摘要： 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.

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.

中图分类号: (Ocean waves and oscillations)

92.10.Hm

47.35.Bb (Gravity waves) 47.35.De (Shear waves)

李少峰, 陈娟, 曹安州, 宋金宝. A nonlinear Schrödinger equation for gravity waves slowly modulated by linear shear flow[J]. 中国物理B, 2019, 28(12): 124701-124701.

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[J]. Chin. Phys. B, 2019, 28(12): 124701-124701.

参考文献 32

[1]	Davey A and Stewartson K 1974 Proc. R. Soc. Lond. A 338 101 doi: 10.1098/rspa.1974.0076
[2]	Turpin F M, Benmoussa C and Mei C C 1983 J. Fluid Mech. 132 1
[3]	Djordjevic V D and Redelopp L G 1978 Z. Angew. Math. Phys. 29 950 doi: 10.1007/BF01590816
[4]	Peregrine D H 1976 Adv. Appl. Mech. 16 9 doi: 10.1016/S0065-2156(08)70087-5
[5]	Longuet-Higgins M S and Stewart R W 1961 J. Fluid Mech. 10 529 doi: 10.1017/S0022112061000342
[6]	Kantardgi I 1995 Coast Eng. 26 195 doi: 10.1016/0378-3839(95)00021-6
[7]	Maciver R D, Simons R R and Thomas G P 2006 J. Geophys. Res. 111 C03009
[8]	Choi W 2009 Math. Comput. Simul. 80 29 doi: 10.1016/j.matcom.2009.06.021
[9]	Ma Y X, Dong G H, Perlin M, Ma X Z, Wang G and Xu J 2010 J. Fluid Mech. 661 108 doi: 10.1017/S0022112010002880
[10]	Ma Y X, Ma X Z, Perlin M and Dong G H 2013 Phys. Fluids 25 114109 doi: 10.1063/1.4832715
[11]	Sedletsky Y V 2005 Phys. Lett. A 343 293 doi: 10.1016/j.physleta.2005.04.076
[12]	Toffoli A, Waseda T, Houtani H, Cavaleri L, Greaves D and Onorato M 2015 J. Fluid Mech. 769 277 doi: 10.1017/jfm.2015.132
[13]	Thomas R, Kharif C and Manna M 2012 Phys. Fluids 24 127102 doi: 10.1063/1.4768530
[14]	Smith R 1976 J. Fluid Mech. 77 417 doi: 10.1017/S002211207600219X
[15]	Henderson K L, Peregrine D H and Dold J W 1999 Wave Motion 29 341 doi: 10.1016/S0165-2125(98)00045-6
[16]	Dysthe K B and Trulsen K 1999 Phys. Scr. T82 48 doi: 10.1238/Physica.Topical.082a00048
[17]	Ma Y C 1979 Stud. Appl. Math. 60 43 doi: 10.1002/sapm197960143
[18]	Peregrine D H 1983 Austral. Math. Soc. Ser. B 25 16 doi: 10.1017/S0334270000003891
[19]	Tao Y S, He J S and Porsezian K 2013 Chin. Phys. B 22 074210 doi: 10.1088/1674-1056/22/7/074210
[20]	Kharif C and Pelinovsky E 2003 Eur. J. Mech. B Fluids 22 603
[21]	Liao B, Dong G, Ma Y and Gao J L 2017 Phys. Rev. E. 96 043111 doi: 10.1103/PhysRevE.96.043111
[22]	Touboul J, Charl, J, Rey V and Belibassakis K 2016 Coast. Eng. 116 77 doi: 10.1016/j.coastaleng.2016.06.003
[23]	Quinn B E, Toledo Y and Shrira V I 2017 Ocean Modell. 112 33 doi: 10.1016/j.ocemod.2017.03.003
[24]	Weng S C and Yu Z W 1985 Wave Theory Calculation Principle (Beijing: Sci. Press) Chap 1 p. 31 (in Chinese)
[25]	Song S Y, Wang J and Meng J M 2010 Acta Phys. Sin. 59 1123 (in Chinese)
[26]	Zhao L C and Ling L M 2016 J. Opt. Soc. Am. B 33 850 doi: 10.1364/JOSAB.33.000850
[27]	Ling L M, Zhao L C, Yang Z Y and Guo B L 2017 Phys. Rev. E 96 022211 doi: 10.1103/PhysRevE.96.022211
[28]	Baronio F, Conforti M, Degasperis A, Lombardo S, Onorato M and Wabnitz S 2014 Phys. Rev. Lett. 113 034101 doi: 10.1103/PhysRevLett.113.034101
[29]	Baronio F, Chen S H, Grelu P, Wabnitz S and Conforti M 2015 Phys. Rev. A 91 033804 doi: 10.1103/PhysRevA.91.033804
[30]	Xie T, Shen T, William P, Chen W and Kuang H L 2010 Chin. Phys. B 19 054102 doi: 10.1088/1674-1056/19/5/054102
[31]	Xie T, Zou G H, William P, Kuang H L and Chen W 2010 Chin. Phys. B 19 059201 doi: 10.1088/1674-1056/19/5/059201
[32]	Didenkulova I I, Nikolkina I F and Pelinovsky E N 2013 JETP Lett. 97 194 doi: 10.1134/S0021364013040024

A nonlinear Schrödinger equation for gravity waves slowly modulated by linear shear flow

A nonlinear Schrödinger equation for gravity waves slowly modulated by linear shear flow

RichHTML

PDF (PC)

赞

可视化

摘要/Abstract

引用本文

使用本文

参考文献 32

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	. [J]. 中国物理B, 2020, 29(12): 124702-.
[2]	谢涛, 方贺, 赵立, 于文金, 何宜军. Effective dielectric constant model of electromagnetic backscattering from stratified air-sea surface film-sea water medium[J]. 中国物理B, 2017, 26(5): 54102-054102.
[3]	谢涛, William Perrie, 赵尚卓, 方贺, 于文金, 何宜军. Electromagnetic backscattering from one-dimensional drifting fractal sea surface II:Electromagnetic backscattering model[J]. 中国物理B, 2016, 25(7): 74102-074102.
[4]	谢涛, 赵尚卓,,方贺, 于文金, 何宜军. Electromagnetic backscattering from one-dimensional drifting fractal sea surface I: Wave-current coupled model[J]. 中国物理B, 2016, 25(6): 64101-064101.
[5]	李近元, 方念乔, 张吉, 薛玉龙, 王雪木, 袁晓博. (2+1)-dimensional dissipation nonlinear Schrödinger equation for envelope Rossby solitary waves and chirp effect[J]. 中国物理B, 2016, 25(4): 40202-040202.
[6]	陈耀登, 杨红卫, 高玉芳, 尹宝树, 冯兴如. A new model for algebraic Rossby solitary waves in rotation fluid and its solution[J]. 中国物理B, 2015, 24(9): 90205-090205.
[7]	崔红, 何海伦, 刘晓辉, 李熠. Effect of oceanic current on typhoon-wave modeling in the East China Sea[J]. 中国物理B, 2012, 21(10): 109201-109201.
[8]	杨红卫, 尹宝树, 杨徳周, 徐振华. Forced solitary Rossby waves under the influence of slowly varying topography with time[J]. 中国物理B, 2011, 20(12): 120203-120203.
[9]	王运华, 张彦敏, 郭立新. Investigation on the Doppler shifts induced by 1-D ocean surface wave displacements by the first order small slope approximation theory: comparison of hydrodynamic models[J]. 中国物理B, 2010, 19(7): 74103-074103.
[10]	张彦敏, 王运华, 郭立新. Study of scattering from time-varying Gerstners sea surface using second-order small slope approximation[J]. 中国物理B, 2010, 19(5): 54103-054103.
[11]	谢涛, 邹光辉, WilliamPerrie, 旷海兰, 陈伟. A two scale nonlinear fractal sea surface model in a one dimensional deep sea[J]. 中国物理B, 2010, 19(5): 59201-059201.
[12]	谢涛, 何超, William Perrie, 旷海兰, 邹光辉, 陈伟. Numerical study of electromagnetic scattering from one-dimensional nonlinear fractal sea surface[J]. 中国物理B, 2010, 19(2): 24101-024101.
[13]	钟兰花, 吴福根, 钟会林. Effects of orientation and shape of holes on the band gaps in water waves over periodically drilled bottoms[J]. 中国物理B, 2010, 19(2): 20301-020301.
[14]	谢涛, 旷海兰, WilliamPerrie, 邹光辉, 南撑峰, 何超, 沈涛, 陈伟. Numerical method of studying nonlinear interactions between long waves and multiple short waves[J]. 中国物理B, 2009, 18(7): 3090-3098.
[15]	宋健, 杨联贵. Modified KdV equation for solitary Rossby waves with β effect in barotropic fluids[J]. 中国物理B, 2009, 18(7): 2873-2877.