三层水流：Dirichlet-Neumann算子及其近似
Three-layer water flows: Dirichlet-Neumann operators and approximations
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中文总结 AI 辅助
本文针对三层流体水流建立哈密顿形式，推导含Dirichlet-Neumann算子的非线性与线性方程，给出传播速度的实解、界值与近似公式，结果适用于内波研究及相关实验。
中文摘要 AI 辅助
本文研究对象为具有分段常数密度分层的二维无粘性水流的非线性运动方程，该水流存在于具有平坦底部、自由表面和两个界面的三层流体中。我们为该设定下的非线性控制方程建立了哈密顿形式，系统的哈密顿量以及表面和界面的运动方程借助各层引入的Dirichlet-Neumann（DN）算子来表达。随后，从表面和界面高程小振幅的主导阶线性方程中推导出色散关系的双三次方程，并对其解进行分析。可能的传播速度共有六个实解（三个正解对应向右传播的波，三个负解对应向左传播的波），其量级各不相同，我们还根据方程的系数给出了上述根的上下界，进而推导出传播速度的近似公式。我们通过对具有平坦表面（刚性盖）的三层模型的单独分析，进一步阐明了DN算子的重要性：完整的非线性演化方程再次用DN算子表达，且通过对DN算子的适当展开，推导出了线性 regime 和弱非线性传播 regime（Boussinesq近似）下的方程，同时也得到了两层自由表面模型的极限情况。所得结果适用于湖泊和海洋中的内波，以及三层叠加流体的实验室实验。
英文摘要
The object of investigation in this paper are the nonlinear equations of motion for two-dimensional inviscid water flows with piecewise constant density stratification in a three-layer fluid with a flat bottom, a free surface and two interfaces. We establish a Hamiltonian formulation for the nonlinear governing equations in this setup. The Hamiltonian of the system and the equations of motion of the surface and of the interfaces are expressed with the help of the Dirichlet-Neumann (DN) operators, which are introduced for each of the layers. Then, the linear equations for small amplitudes of the elevation of the surface and of the interfaces in the leading order are derived from which a bi-cubic equation for the dispersion relation is obtained, whose solutions are analysed. The six real solutions for the possible propagation speeds (three positive, related to right-moving waves and three negative, related to left-moving waves) have magnitudes of different order. Upper and lower bounds for the previously mentioned roots are also given in terms of the coefficients of the equation. Subsequently, approximate formulae for the propagation speeds are derived. The importance of the DN operators is further illustrated in a separate analysis of the three-layer model with flat surface (rigid lid). The full nonlinear evolution equations are expressed again in terms of the DN operators, and the equations in the linear regime and the weakly nonlinear propagation regime (the Boussinesq approximation) are derived by a proper expansion of the DN operators. Limits to the two-layer free surface model are obtained as well. The obtained results are applicable to internal waves in lakes and in the ocean as well as to laboratory experiments with three superimposed fluid layers.
