关于波和涡的拟流函数形式
On The Quasi-streamfunction Formalism for Waves and Vorticity
摘要：
准流函数（Psi）形式主义提出的Kim等。等（J.W. Kim，K.J. Bai，R.C. Ertekin，W.C. Webster，J.Eng.Math.40，17（2001））提供了一个自然的框架，系统地研究零涡波在任意水深。这里开发的修改后的PSI形式主义放弃了原来的零涡度的限制，允许垂直涡度，这是海岸动力学最感兴趣的情况下。这个问题是重新制定的两个动力学方程的边界上补充一个方程，表示在内部的域的运动约束。在这个框架中，运动学约束可以被求解以用正则共轭变量来表示Psi。的形式主义证明了水平均匀流在温和的地形，其中渐近公式的哈密顿和拉格朗日函数推导出的基础上的Helmholz-Hodge分解。对于势流，哈密顿量的渐近形式与以前的结果相同。拉格朗日函数也表示为一个扩展的表面高度及其时间导数，并与扎哈罗夫的形式主义，其中发现协议一维波散射。
英文摘要：
The quasi-streamfunction (Psi) formalism proposed by Kim et. al. (J.W. Kim, K.J. Bai, R.C. Ertekin, W.C. Webster, J. Eng. Math. 40, 17 (2001)) provides a natural framework for systematically studying zero-vorticity waves over arbitrary bathymetry. The modified Psi-formalism developed here discards the original constraints of zero-vorticity by allowing for vertical vorticity which is the case of most interest for coastal dynamics. The problem is reformulated in terms of two dynamical equations on the boundary supplemented by one equation that represents a kinematic constraint in the interior of the domain. In this framework, the kinematic constraint can be solved to express Psi in terms of canonically-conjugated variables. The formalism is demonstrated for horizontally homogeneous flows over mild topography, where asymptotic formulations for the Hamiltonian and Lagrangian functions are derived based on the Helmholz-Hodge decomposition. For potential flows, the asymptotic form of the Hamiltonian is identical to previous results. The Lagrangian function is also expressed as an expansion in terms of the surface height and its time derivative and compared with Zakharov's formalism where agreement is found for one-dimensional wave scattering.
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