We demonstrate that two surface waves propagating at a small angle $2\theta$ to each other generate large-scale (compared to the wavelength) vertical vorticity owing to hydrodynamic nonlinearity in a viscous fluid. The horizontal geometric structure of the induced flow coincides with the structure of the Stokes drift in an ideal fluid, but its steady-state amplitude is larger and it penetrates deeper into the fluid volume as compared to the Stokes drift. In an unbounded fluid, the steady-state amplitude and penetration depth are increased by the factor of $1/sin\theta$, and the evolution time of the induced flow can be estimated as $1/(4\nu k^2{sin}^2\theta )$, where $\nu$ is the fluid kinematic viscosity and $k$ is the wavenumber. Also, we study how the finite depth of the fluid and a thin insoluble liquid film that possibly covers the fluid surface due to contamination effect the generation of large-scale vorticity, and discuss the physical consequences of this phenomenon in the context of recent experiments.

中文翻译：

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相互作用的表面波产生的大规模垂直涡

我们证明了两个以小角度传播的表面波 $2\theta$由于粘性流体中的流体动力学非线性，彼此之间会产生大规模的（与波长相比）垂直涡旋。诱导流的水平几何结构与理想流体中斯托克斯漂移的结构相吻合，但是与斯托克斯漂移相比，其稳态振幅更大，并且更深地渗透到流体体积中。在无界流体中，稳态振幅和穿透深度增加了以下因素：$\mathrm{1个}/罪\theta$，诱导流的演化时间可以估算为 $\mathrm{1个}/（4\nu {ķ}^2{罪}^2\theta ）$，在哪里 $\nu$ 是流体运动粘度， $ķ$是波数。此外，我们研究了流体的有限深度以及由于污染而可能覆盖流体表面的不溶性液体薄膜如何影响大规模涡旋的产生，并在最近的实验中讨论了这种现象的物理后果。