Nonlinear interactions between surface and internal gravity waves in a two-layer system are studied using explicit second-order nonlinear evolution equations in Hamiltonian form. Motivated by the detailed experiment of Lewis, Lake & Ko ( J. Fluid Mech. , vol. 63, 1974, pp. 773–800), our focus is on surface wave modulation by the group resonance mechanism that corresponds to near-resonant triad interactions between a long internal wave and short surface waves. Our numerical solutions show good agreement with laboratory measurements of the local wave amplitude and slope, and confirm that the surface modulation becomes significant when the group velocity of the surface waves matches the phase speed of the internal wave, as the linear modulation theory predicts. It is shown, however, that, after the envelope amplitude is increased sufficiently, the surface and internal waves start to exchange energy through near-resonant triad interactions, which is found to be crucial to accurately describe the long-term surface wave modulation by an internal wave. The reduced amplitude equations are also adopted to validate this observation. For oceanic applications, numerical solutions are presented for a density ratio close to one and it is found that significant energy exchanges occur through primary and successive resonant triad interactions.

中文翻译：

---

两层系统中表面和内部重力波之间的群共振相互作用

使用哈密顿形式的显式二阶非线性演化方程研究了双层系统中表面和内部重力波之间的非线性相互作用。受 Lewis, Lake & Ko (J. Fluid Mech., vol. 63, 1974, pp. 773–800) 的详细实验的启发，我们的重点是通过对应于近共振三元组的群共振机制进行表面波调制长内波和短表面波之间的相互作用。我们的数值解与本地波振幅和斜率的实验室测量结果非常吻合，并确认当表面波的群速度与内波的相位速度匹配时，表面调制变得显着，正如线性调制理论所预测的那样。然而，结果表明，在包络幅度充分增加后，表面波和内波开始通过近共振的三元组相互作用交换能量，这对于准确描述内波的长期表面波调制至关重要。还采用缩减幅度方程来验证这一观察结果。对于海洋应用，提出了密度比接近 1 的数值解，并且发现显着的能量交换通过初级和连续共振三元组相互作用发生。