Phase-averaged equation for water waves
Phase-averaged equation for water waves
复制标题
水波的相位平均方程
DOI：
10.1017/jfm.2012.609
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发表时间：
2013
影响因子：
3.7
通讯作者：
M. Stiassnie
中科院分区：
文献类型：
--
作者：
O. Gramstad;M. Stiassnie
摘要研究了描述色散水波在弱非线性四重奏相互作用下谱演化的相位平均方程。与Hasselmann的动力学方程相反，我们包括了近共振四重奏相互作用的影响，导致“快速”$O({\epsilon}^{- 2})$时间尺度上的光谱演化，其中$\epsilon $是波陡度。这种相位平均方程是由Annenkov & Shrira （J.流体力学）提出的。， vol. 561, 2006b, pp. 181-207)。在本文中，我们重新推导了他们的方程，考虑到与频率的斯托克斯校正有关的一些额外的高阶效应。我们还推导了相位平均方程的运动不变量。提出了相位平均方程的数值求解方法，并对其收敛性和不变量守恒性进行了验证。进行了一维和二维光谱演化的数值模拟。结果表明，相位平均方程很好地描述了光谱在$O({\epsilon}^{- 2})$时间尺度上的“快速”演化，与使用Zakharov方程的蒙特卡罗模拟很好地吻合，并且与已知的一维和二维光谱演化特征在定性上一致。我们认为，在波场演化的初始阶段，以及在发生“快速”场演化的情况下，相位平均方程可能是动能方程的合适替代品。
Abstract We investigate phase-averaged equations describing the spectral evolution of dispersive water waves subject to weakly nonlinear quartet interactions. In contrast to Hasselmann’s kinetic equation, we include the effects of near-resonant quartet interaction, leading to spectral evolution on the ‘fast’ $O({\epsilon }^{- 2} )$ time scale, where $\epsilon $ is the wave steepness. Such a phase-averaged equation was proposed by Annenkov & Shrira (J. Fluid Mech., vol. 561, 2006b, pp. 181–207). In this paper we rederive their equation taking some additional higher-order effects related to the Stokes correction of the frequencies into account. We also derive invariants of motion for the phase-averaged equation. A numerical solver for the phase-averaged equation is developed and successfully tested with respect to convergence and conservation of invariants. Numerical simulations of one- and two-dimensional spectral evolution are performed. It is shown that the phase-averaged equation describes the ‘fast’ evolution of a spectrum on the $O({\epsilon }^{- 2} )$ time scale well, in good agreement with Monte-Carlo simulations using the Zakharov equation and in qualitative agreement with known features of one- and two-dimensional spectral evolution. We suggest that the phase-averaged equation may be a suitable replacement for the kinetic equation during the initial part of the evolution of a wave field, and in situations where ‘fast’ field evolution takes place.