arXiv 2202.114052022-02-23physics.flu-dynphysics.ao-ph

斯托克斯漂移及其不满

Stokes drift and its discontents

Jacques Vanneste, William R. Young

DOI：: 10.1098/rsta.2021.0032

摘要：

Stokes速度$\mathbf{u}^\mathrm{S}$，由Stokes近似定义（1847，Trans. Camb.哲学学会，8，441-455），并完全通过广义拉格朗日平均，是发散的，即使在不可压缩流体。我们发现，斯托克斯速度可以自然地分解成螺线管组件，$\mathbf{u}^\mathrm{S}_\mathrm{sol}$，和其余的是小的波缓慢变化的振幅。我们进一步表明，$\mathbf{u}^\mathrm{S}_\mathrm{sol}$作为唯一的斯托克斯速度时，拉格朗日平均流被适当地重新定义，以确保其准确的不可压缩性。该构造是Soward和Roberts的glm理论（2010，J. Fluid Mech.，661，45-72），我们专门用于表面重力波，并有效地实现使用李级数展开。我们进一步证明了相应的拉格朗日平均动量方程与用$\mathbf{u}^\mathrm{S}_\mathrm{sol}$代替$\mathbf{u}^\mathrm{S}$的Craik-Leibovich方程在形式上是相同的，并讨论了与$\mathbf{u}^\mathrm{S}$和$\mathbf{u}^\mathrm{S}_\mathrm{sol}$相关的Stokes抽运的形式.

英文摘要：

The Stokes velocity $\mathbf{u}^\mathrm{S}$, defined approximately by Stokes (1847, Trans. Camb. Philos. Soc., 8, 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, $\mathbf{u}^\mathrm{S}_\mathrm{sol}$, and a remainder that is small for waves with slowly varying amplitudes. We further show that $\mathbf{u}^\mathrm{S}_\mathrm{sol}$ 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, J. Fluid Mech., 661, 45-72) which we specialise 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 equation with $\mathbf{u}^\mathrm{S}_\mathrm{sol}$ replacing $\mathbf{u}^\mathrm{S}$, and we discuss the form of the Stokes pumping associated with both $\mathbf{u}^\mathrm{S}$ and $\mathbf{u}^\mathrm{S}_\mathrm{sol}$.

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