Hamiltonian long–wave expansions for water waves over a rough bottom
Hamiltonian long–wave expansions for water waves over a rough bottom
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粗糙底部水波的哈密顿长波扩展
DOI：
10.1098/rspa.2004.1367
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
C. Sulem
中科院分区：
文献类型：
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作者：
W. Craig;P. Guyenne;D. Nicholls;C. Sulem
本文研究的是底部周期性变化的流体自由表面的非线性波动问题。目的是描述长波渐进状态下的波传播特征，扩展 R. Rosales 和 G. Papanicolaou (1983 Stud. Appl. Math. 68, 89–102) 关于二维流的周期性底部的结果。我们采用依赖于小尺度参数的哈密顿系统的扰动观点，起点是扎哈罗夫的哈密顿量 (V. E. Zakharov) 1968 J. Mech. Phys. 9, 1990–1994) 水波的欧拉方程。我们考虑底部地形，它在短长度尺度上的水平变量是周期性的，其变化幅度与流体深度具有相同的数量级。底部也可能表现出与表面波波长数量级相同或更长的缓慢变化。我们不考虑随机底部变化的问题，这是 Rosales 和 Papanicolaou (1983) 中考虑的主题。在通道中波的二维情况下，我们给出了 Rosales 和 Papanicolaou (1983) 中获得的有效 Korteweg-de Vries (KdV) 方程的另一种推导。此外，我们获得了有效的布辛涅斯克方程来描述双向长波的运动，在底部同时具有短尺度和长尺度变化的情况下。在某些情况下，我们还可以获得类似于 KdV 方程的单向方程。在三维中，我们在 Boussinesq 尺度范围内获得有效的三维长波方程，并且在某些情况下在适当的单向极限下再次获得有效的 Kadomtsev-Petviashvili (KP) 系统。这些结果的计算是在多尺度算子渐近分析的框架中进行的。在本例中，这涉及流体域的狄利克雷-诺依曼算子，该算子考虑了底部地形的变化以及自由表面相对于平衡的变形。
This paper is a study of the problem of nonlinear wave motion of the free surface of a body of fluid with a periodically varying bottom. The object is to describe the character of wave propagation in a long–wave asymptotic regime, extending the results of R. Rosales & G. Papanicolaou (1983 Stud. Appl. Math. 68, 89–102) on periodic bottoms for two–dimensional flows.We take the point of view of perturbation of a Hamiltonian system dependent on a small scaling parameter, with the starting point being Zakharov's Hamiltonian (V. E. Zakharov 1968 J. Appl. Mech. Tech. Phys. 9, 1990–1994) for the Euler equations for water waves. We consider bottom topography which is periodic in horizontal variables on a short length–scale, with the amplitude of variation being of the same order as the fluid depth. The bottom may also exhibit slow variations at the same length–scale as, or longer than, the order of the wavelength of the surface waves. We do not take up the question of random bottom variations, a topic which is considered in Rosales & Papanicolaou (1983). In the two–dimensional case of waves in a channel, we give an alternate derivation of the effective Korteweg–de Vries (KdV) equation that is obtained in Rosales & Papanicolaou (1983). In addition, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves, in cases in which the bottom possesses both short and long–scale variations. In certain cases we also obtain unidirectional equations that are similar to the KdV equation. In three dimensions we obtain effective three–dimensional long–wave equations in a Boussinesq scaling regime, and again in certain cases an effective Kadomtsev–Petviashvili (KP) system in the appropriate unidirectional limit. The computations for these results are performed in the framework of an asymptotic analysis of multiple–scale operators. In the present case this involves the Dirichlet–Neumann operator for the fluid domain which takes into account the variations in bottom topography as well as the deformations of the free surface from equilibrium.