A New Development of Reduced Hamiltonian Equations for Ocean Surface Waves: an Extension From Small to Finite Amplitude
-
摘要: 海洋表面波的3-波至5-波约化Hamilton方程由于其对称多项式简化结构以及保能量等独特优点,得到广泛应用.但是,据相关近似假设,其适用范围局限于波陡很小的弱非线性波.于是进一步探讨下述推广问题: 对一定范围内的有限幅非线性波,在足够精确意义上是否也能获得具对称多项式简化结构的约化Hamilton方程?由于涉及复杂非线性强耦合,在该重要方面至今尚未取得进展.提出基于Chebyshev(切比雪夫)多项式逼近处理精确水波方程强非线性耦合的新简化途径,导出具对称多项式简化结构的新约化Hamilton方程.新结果将波数与波陡之积为小量的弱非线性情形拓广到该积直至1.035的非线性情形.分析表明,在该范围内新结果的误差不超过5%,特别,当前述积邻近于0.9时新结果给出精确结果.
-
关键词:
- 海洋表面波 /
- 有限波幅 /
- 新简化途径 /
- 新约化Hamilton方程
Abstract: The reduced 3-wave and 4-wave Hamiltonian equations for ocean surface waves were widely used for the simplified structure with symmetric polynomial kernels and for the conservation of energy, etc. However, according to the related assumption for approximation in derivation, the range of applicability was limited to weakly nonlinear waves of small amplitude. Here the following issue was further studied: for nonlinear waves of finite amplitude within a certain range, was it also possible to obtain reduced Hamiltonian equations with symmetric polynomial kernels in a sense of sufficient accuracy? Because of complicated strongly nonlinear coupling, few development in this significant respect had been made as yet. A new approach was proposed based on the Chebyshev polynomials to best approximate the primitive water wave equations in the exact sense of strongly nonlinear coupling and derive new reduced Hamiltonian equations with symmetric polynomial kernels. The new results exhibit an extension from a weakly nonlinear case in which the product of the wave number and the wave steepness is small to a nonlinear case in which this product goes up to about 1.035. Moreover, within this range, the approximation errors are lower than 5%, and in particular, the new results prove exact whenever the said product lies close to 0.9. -
[1] Bouscasse B, Colagrossi A, Marrone S, Antuono M. Nonlinear water wave interaction with floating bodies in SPH[J].Journal of Fluid and Structures,2013,42: 112-129. [2] Belibassakis K A, Athanassoulis G A. A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions[J].Coastal Engineering,2011,58(4): 337-350. [3] TAO Long-bin, SONG Hao, Chakrabarti S. Nonlinear progressive waves in water of finite depth—an analytic approximation[J].Coastal Engineering,2007,54(11): 825-834. [4] LI Bin. A mathematical model for weakly nonlinear water wave propagation[J].Wave Motion,2010,47(5): 265-278. [5] Bateman W J D, Katsardi V, Swan C. Extreme ocean waves—part I: the practical application of fully nonlinear wave modeling[J].Applied Ocean Research,2012,34: 209-224. [6] Zakharov V. Stability of periodic waves of finite amplitude on the surface of a deep fluid[J].Journal of Applied Mechanics and Technical Physics,1968,9(2): 190-198. [7] Zakharov V E, Kharitonov V G. Instability of monochromatic waves on the surface of a liquid of arbitrary depth[J].Journal of Applied Mechanics and Technical Physics,1970,11(5): 741-751. [8] Miles J W. On Hamilton’s principle for surface waves[J].Journal of Fluid Mechanics,1977,83: 153-158. [9] Krasitskii V P. Canonical transformations in a theory of weakly nonlinear waves with a nondecay dispersion law[J].Soviet Physics JETP,1990,71(5): 921-927. [10] Krasitskii V P. On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves[J].Journal of Fluid Mechanics,1994,272: 1-20. [11] Stiassnie M, Shemer L. On modifications of the Zakharov equation for surface gravity waves[J].Journal of Fluid Mechanics,1984,143: 47-67. [12] Zakharov V E, Musher S L, Rubenchick A M. Hamiltonian approach to the description of non-linear plasma phenomena[J].Physics Reports,1985,129(5): 285-366. [13] Janssen P A E M. On some consequences of the canonical transformation in the Hamiltonian theory of water waves[J].Journal of Fluid Mechanics,2009,637: 1-44. [14] Benney D J, Roskes G J. Wave instabilities[J].Studies in Applied Mathematics,1969,48: 377-385. [15] Chu V H, Mei C C. On slowly-varying Stokes waves[J].Journal of Fluid Mechanics,1970,41(4): 873-887. [16] Davey A, Stewartson K. On three-dimensional packets of surface waves[J].Proceedings of the Royal Society of London(Series A: Mathematical and Physical Sciences),1974,338(1613): 101-110. [17] Lavrova O T. On the lateral instability of waves on the surface of a finite-depth fluid[J].Izvestiya Atmospheric and Oceanic Physics,1983,19: 807-810. [18] Newell A C, Rumpf B. Wave turbulence[J].Annual Review of Fluid Mechanics,2011,43: 59-78. [19] Tanaka M. Verification of Hasselmann’s energy transfer among surface gravity waves by direct numerical simulations of primitive equations[J].Journal of Fluid Mechanics,2001,444: 199-221. [20] Annenkov S Y, Shrira V I. Numerical modelling of water-wave evolution based on the Zakharov equation[J].Journal of Fluid Mechanics,2001,449: 341-371. [21] 黄虎. 近海波-流相互作用的缓坡方程理论体系[J]. 物理学报, 2010,59(2): 740-743.(HUANG Hu. A theoretical hierarchy of the mild-slope equations for wave-current interactions in coastal waters[J].Acta Physica Sinica,2010,59(2): 740-743.(in Chinese))
点击查看大图
计量
- 文章访问数: 1348
- HTML全文浏览量: 117
- PDF下载量: 685
- 被引次数: 0