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Waves, circulation and vertical dependence

机译：波浪，循环和垂直依赖性

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Longuet-Higgins and Stewart (J Fluid Mech 13:481–504, 1962; Deep-Sea Res 11:529–562, 1964) and later Phillips (1977) introduced the problem of waves incident on a beach, from deep to shallow water. From the wave energy equation and the vertically integrated continuity equation, they inferred velocities to be Stokes drift plus a return current so that the vertical integral of the combined velocities was nil. As a consequence, it can be shown that velocities of the order of Stokes drift rendered the advective term in the momentum equation negligible resulting in a simple balance between the horizontal gradients of the vertically integrated elevation and wave radiation stress terms; the latter was first derived by Longuet-Higgins and Stewart. Mellor (J Phys Oceanogr 33:1978–1989, 2003a), noting that vertically integrated continuity and momentum equations were not able to deal with three-dimensional numerical or analytical ocean models, derived a vertically dependent theory of wave–circulation interaction. It has since been partially revised and the revisions are reviewed here. The theory is comprised of the conventional, three-dimensional, continuity and momentum equations plus a vertically distributed, wave radiation stress term. When applied to the problem of waves incident on a beach with essentially zero turbulence momentum mixing, velocities are very large and the simple balance between elevation and radiation stress gradients no longer prevails. However, when turbulence mixing is reinstated, the vertically dependent radiation stresses produce vertical velocity gradients which then produce turbulent mixing; as a consequence, velocities are reduced, but are still larger by an order of magnitude compared to Stokes drift. Nevertheless, the velocity reduction is sufficient so that elevation set-down obtained from a balance between elevation gradient and radiation stress gradients is nearly coincident with that obtained by the aforementioned papers. This paper includes four appendices. The first appendix demonstrates the numerical process by which Stokes drift is excluded from the turbulence stress parameterization in the momentum equation. A second appendix determines a bottom slope criterion for the application of linear wave relations to the derivation of the wave radiation stress. The third appendix explores the possibility of generalizing results by non-dimensionalization. The final appendix applies the basic theory to a problem introduced by Bennis and Ardhuin (J Phys Oceanogr 41:2008–2012, 2011).
机译：Longuet-Higgins和Stewart（J Fluid Mech 13：481-504，1962; Deep-Sea Res 11：529-562，1964）和后来的Phillips（1977）提出了从深水到浅水的海滩上入射波的问题。从波能方程和垂直积分连续方程，他们推断出的速度为斯托克斯漂移加返回电流，因此组合速度的垂直积分为零。结果，可以证明斯托克斯漂移量级的速度使得动量方程中的对流项可以忽略不计，从而导致了垂直积分仰角的水平梯度和波辐射应力项之间的简单平衡；后者最早由Longuet-Higgins和Stewart派生。 Mellor（J Phys Oceanogr 33：1978–1989，2003a）指出，垂直积分的连续性和动量方程无法处理三维数值或解析海洋模型，他推导出了垂直依赖的波-环流相互作用理论。此后已部分修订，此处对修订进行了审核。该理论由常规的三维连续性和动量方程式以及垂直分布的波辐射应力项组成。当应用于湍流动量混合基本为零的入射在海滩上的波浪问题时，速度非常大，并且海拔和辐射应力梯度之间的简单平衡不再占优势。但是，当恢复湍流混合时，垂直方向的辐射应力会产生垂直速度梯度，然后会产生湍流混合。结果，速度降低了，但与斯托克斯漂移相比仍增加了一个数量级。然而，速度降低是足够的，使得从仰角梯度和辐射应力梯度之间的平衡获得的仰角下降几乎与上述论文所获得的一致。本文包括四个附录。第一个附录演示了数值过程，通过该过程可以将动量方程中的湍流应力参数化排除斯托克斯漂移。第二个附录确定了将线性波关系应用于波辐射应力推导的底坡准则。第三个附录探讨了通过无量纲化来概括结果的可能性。最后的附录将基本理论应用于Bennis和Ardhuin提出的问题（《物理学海洋》 41：2008–2012，2011）。

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