School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3FD, UK.
Scripps Institution of Oceanography, University of California at San Diego, La Jolla, CA 92093-0213, USA.
Philos Trans A Math Phys Eng Sci. 2022 Jun 13;380(2225):20210032. doi: 10.1098/rsta.2021.0032. Epub 2022 Apr 25.
The Stokes velocity [Formula: see text], defined approximately by Stokes (1847, , , 441-455.), and exactly via the Generalized Lagrangian Mean, is divergent even in an incompressible fluid. We show that the Stokes velocity can be naturally decomposed into a solenoidal component, [Formula: see text], and a remainder that is small for waves with slowly varying amplitudes. We further show that [Formula: see text] arises as the sole Stokes velocity when the Lagrangian mean flow is suitably redefined to ensure its exact incompressibility. The construction is an application of Soward & Roberts's glm theory (2010, , , 45-72. (doi:10.1017/S0022112010002867)) which we specialize to surface gravity waves and implement effectively using a Lie series expansion. We further show that the corresponding Lagrangian-mean momentum equation is formally identical to the Craik-Leibovich (CL) equation with [Formula: see text] replacing [Formula: see text], and we discuss the form of the Stokes pumping associated with both [Formula: see text] and [Formula: see text]. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.
斯托克斯速度[公式：见文本]，由斯托克斯（1847 年，，，441-455.）近似定义，并通过广义拉格朗日平均精确定义，即使在不可压缩流体中也是发散的。我们证明，斯托克斯速度可以自然地分解为无散分量[公式：见文本]，以及一个对于振幅变化缓慢的波很小的剩余项。我们进一步证明，当合适地重新定义拉格朗日平均流以确保其精确不可压缩性时，[公式：见文本]是唯一的斯托克斯速度。该构造是 Soward 和 Roberts 的 glm 理论（2010 年，，，45-72. (doi:10.1017/S0022112010002867))的应用，我们将其专门应用于表面重力波，并使用 Lie 级数展开有效地实现了它。我们进一步证明，相应的拉格朗日平均动量方程与具有[公式：见文本]代替[公式：见文本]的 Craik-Leibovich (CL)方程在形式上是相同的，并且我们讨论了与[公式：见文本]和[公式：见文本]相关的斯托克斯泵送的形式。本文是主题为“物理流体动力学中的数学问题（第 1 部分）”的一部分。
