Hamilton系统理论在内波研究中的应用
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摘要
分层流体域中的内波理论不仅在海洋工程方面具有重要意义,同时也是非线性色散方程模型的重要来源。本文运用Dirichlet-Neumann算子给出的Dirichlet积分的表达式,推导了两层和三层流体域中有关Zakharov在Hamilton算子方面的公式。通过此公式,本文应用Hamilton摄动理论对重要的长波尺度形式,Boussinesq尺度形式和KdV尺度形式,进行了系统的分析。本文得到的有关公式不仅在摄动计算方面有着重要意义,也为数值模拟提供了基础保证。
论文的主要结果如下:
第一,本文得出了周期底部边界条件下两层密度成层流体中2-维非线性长波问题的Hamilton公式,其中振幅的变化与流体深度是同阶的。从这个公式出发,应用Hamilton摄动理论,本文导出了底地形短尺度变化下描述双向长波运动的有效Boussinesq方程和描述单向长波运动的近似KdV方程。这些结果的推导都是在多重尺度算子渐近分析理论框架下完成的。另外,为了公式推导的方便简洁,本文将上层流体的自由面假设为刚盖条件。当然,借助第二章附录中给出的自由面非刚盖条件下上层流体Dirichlet-Neumann算子的泰勒展式,本文的结果完全可以推广到非刚盖情形。
第二,在刚盖边界条件下,本文研究了三层密度成层流体内内波的长波展开,给出了流体域中2-维非线性长波问题的Hamilton公式。应用此公式,本文导出了界面的线性化方程和相应的色散关系,得知三层密度成层流体界面内波有两个运动模态,它们分别对应于两个界面波的传播方式。同时,应用Hamilton摄动理论得到了界面波动的耦合Boussinesq方程;依据界面振幅差距的大小,本文分两种情况给出了内波单向传播的KdV方程(一个界面)和耦合KdV方程组(两个界面)。这些结果的推导都是在多重尺度算子渐近分析理论框架下完成的。
The theory of internal waves in stratified layers of fluid is important both for its interest to ocean engineering, and as a source of numerous interesting mathematical model equations which exhibit nonlinearity and dispersion. In this paper, using an expression for Dirichlet integral in terms of the Dirichlet-Neumann operator, we derive a Hamiltonian formulation of the fluid in terms of Zakharov's Hamiltonian. From the formulation we carry out a systematic analysis of the principal long wave scaling regimes. Our considerations include the Boussinesq and the KdV regimes. Our formulation of the fluid is shown to be very effective for perturbation calculations, and as well it holds promise as a basis for numerical simulations.
The main conclusions of this dissertation are described as follows.
1. We derive a Hamiltonian formulation for two-dimensional nonlinear long waves between two bodies of immiscible fluid with a periodic bottom. From the formulation, using the Hamiltonian perturbation theory, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves and unidirectional equations that are similar to the KdV equation for the case in which the bottom possesses short length scale. The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators. In addition, for the convenience and simplicity of formulas' deducing, we suppose the upper layer is rigid lid. Certainly, using the Taylor expansions of the Dirichlet-Neumann operators for the upper fluid domains in the appendix of chapter 2, our methods can be extended to the free surface case.
2. We study the long-wave asymptotic regime for internal waves in three bodies of immiscible fluid with rigid lid upper boundary conditions. We derive a Hamiltonian formulation for two-dimensional nonlinear long waves. From the Formulation, we obtain the linearized free interfaces equation. Then we get the corresponding dispersion relation and know that there are two different modes of waves motion, namely two interfaces displacements. In addition, using the Hamiltonian perturbation theory, we get the fully coupled effective Boussinesq equations of two interfaces. According to the difference between the interfaces, we derive respectively the KdV equation of a interface and the coupled KdV equation of the two interface in regime emphasizing one-way propagation. The computations for the results are performed in the framework of an asymptotic analysis of multiple scale. The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators.
论文的主要结果如下:
第一,本文得出了周期底部边界条件下两层密度成层流体中2-维非线性长波问题的Hamilton公式,其中振幅的变化与流体深度是同阶的。从这个公式出发,应用Hamilton摄动理论,本文导出了底地形短尺度变化下描述双向长波运动的有效Boussinesq方程和描述单向长波运动的近似KdV方程。这些结果的推导都是在多重尺度算子渐近分析理论框架下完成的。另外,为了公式推导的方便简洁,本文将上层流体的自由面假设为刚盖条件。当然,借助第二章附录中给出的自由面非刚盖条件下上层流体Dirichlet-Neumann算子的泰勒展式,本文的结果完全可以推广到非刚盖情形。
第二,在刚盖边界条件下,本文研究了三层密度成层流体内内波的长波展开,给出了流体域中2-维非线性长波问题的Hamilton公式。应用此公式,本文导出了界面的线性化方程和相应的色散关系,得知三层密度成层流体界面内波有两个运动模态,它们分别对应于两个界面波的传播方式。同时,应用Hamilton摄动理论得到了界面波动的耦合Boussinesq方程;依据界面振幅差距的大小,本文分两种情况给出了内波单向传播的KdV方程(一个界面)和耦合KdV方程组(两个界面)。这些结果的推导都是在多重尺度算子渐近分析理论框架下完成的。
The theory of internal waves in stratified layers of fluid is important both for its interest to ocean engineering, and as a source of numerous interesting mathematical model equations which exhibit nonlinearity and dispersion. In this paper, using an expression for Dirichlet integral in terms of the Dirichlet-Neumann operator, we derive a Hamiltonian formulation of the fluid in terms of Zakharov's Hamiltonian. From the formulation we carry out a systematic analysis of the principal long wave scaling regimes. Our considerations include the Boussinesq and the KdV regimes. Our formulation of the fluid is shown to be very effective for perturbation calculations, and as well it holds promise as a basis for numerical simulations.
The main conclusions of this dissertation are described as follows.
1. We derive a Hamiltonian formulation for two-dimensional nonlinear long waves between two bodies of immiscible fluid with a periodic bottom. From the formulation, using the Hamiltonian perturbation theory, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves and unidirectional equations that are similar to the KdV equation for the case in which the bottom possesses short length scale. The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators. In addition, for the convenience and simplicity of formulas' deducing, we suppose the upper layer is rigid lid. Certainly, using the Taylor expansions of the Dirichlet-Neumann operators for the upper fluid domains in the appendix of chapter 2, our methods can be extended to the free surface case.
2. We study the long-wave asymptotic regime for internal waves in three bodies of immiscible fluid with rigid lid upper boundary conditions. We derive a Hamiltonian formulation for two-dimensional nonlinear long waves. From the Formulation, we obtain the linearized free interfaces equation. Then we get the corresponding dispersion relation and know that there are two different modes of waves motion, namely two interfaces displacements. In addition, using the Hamiltonian perturbation theory, we get the fully coupled effective Boussinesq equations of two interfaces. According to the difference between the interfaces, we derive respectively the KdV equation of a interface and the coupled KdV equation of the two interface in regime emphasizing one-way propagation. The computations for the results are performed in the framework of an asymptotic analysis of multiple scale. The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators.
引文
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[7]B.R.Sutherland,M.R.Flynn,K.Dohan.Internal wave excitation from a collapsing mixed region.Deep-Sea Research Ⅱ.2004,51:2889-2904
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[29]郭仲衡等.近代数学与力学.北京:北京大学出版社.1987,1-37
[30]谷超豪等.孤立子理论及其应用.上海:复旦大学出版社.1989,252-269
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[32]张宝善,卢东强,戴世强,程友良.非线性水波Hamilton系统理论与应用研究进展.力学进展.1998,28(4):521-531
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[34]R.Salmon.Hamiltonian fluid mechanics.Ann.Rev.Fluid Mech.1988,20:225-256
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