非线性表面重力波哈密顿理论中的耦合包络演化方程
On coupled envelope evolution equations in the Hamiltonian theory of nonlinear surface gravity waves
摘要：
本文提出了一种新的非线性表面重力波的哈密顿理论框架。在框架中引入了水面高程包络和自由水面速度势，它们是一对新的正则变量。以这两个包络为主要未知量，基于微扰展开导出了耦合包络演化方程。类似于高阶谱方法，CEEE可以推导出任意阶的波陡。相比之下，它们具有与波谱的变化率一样慢的时间尺度，并且允许在计算（空间）域上规定的波场具有大得多的尺寸并且具有比特征波长更长的间隔，而不以精度和数值效率为代价。基于CEEEs推导了能量平衡方程。CEEEs中的非线性项是波谐波分离的形式，因此单个项在是否能够强迫服从色散关系的自由波方面具有明确的物理意义。这两个非线性项，只能导致强迫的束缚波和这些能够强迫自由波的演示，后者通过分析的四重和五重共振相互作用的线性波。建立了CEEEs与其他两个现有理论框架之间的关系，包括波陡高达二阶的Stokes波训练理论[Fenton，J. waterway，Port，Coast. &海洋工程，111，2，1985]和Li & Li [Phys.Fluids，33，7，2021]的具有任意带宽和大方向扩展的三维弱非线性表面波的半分析框架。
英文摘要：
This paper presents a novel theoretical framework in the Hamiltonian theory of nonlinear surface gravity waves. The envelope of surface elevation and the velocity potential on the free water surface are introduced in the framework, which are shown to be a new pair of canonical variables. Using the two envelopes as the main unknowns, coupled envelope evolution equations (CEEEs) are derived based on a perturbation expansion. Similar to the High Order Spectral method, the CEEEs can be derived up to arbitrary order in wave steepness. In contrast, they have a temporal scale as slow as the rate of change of a wave spectrum and allow for the wave fields prescribed on a computational (spatial) domain with a much larger size and with spacing longer than the characteristic wavelength at no expense of accuracy and numerical efficiency. The energy balance equation is derived based on the CEEEs. The nonlinear terms in the CEEEs are in a form of the separation of wave harmonics, due to which an individual term is shown to have clear physical meanings in terms of whether or not it is able to force free waves which obey the dispersion relation. Both the nonlinear terms that can only lead to the forcing of bound waves and these which are capable of forcing free waves are demonstrated, with the latter through the analysis of the quartet and quintet resonant interactions of linear waves. The relations between the CEEEs and two other existing theoretical frameworks are established, including the theory for a train of Stokes waves up to second order in wave steepness [Fenton, J. waterway, Port, Coast. & Ocean Eng., 111, 2, 1985] and a semi-analytical framework for three-dimensional weakly nonlinear surface waves with arbitrary bandwidth and large directional spreading by Li & Li [Phys. Fluids, 33, 7, 2021].
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