This study aims at developing a new set of equations of mean motion in the presence of surface waves, which is practically applicable from deep water to the coastal zone, estuaries, and outflow areas. The generalized Lagrangian mean (GLM) method is employed to derive a set of quasi-Eulerian mean three-dimensional equations of motion, where effects of the waves are included through source terms. The obtained equations are expressed to the second-order of wave amplitude. Whereas the classical Eulerian-mean equations of motion are only applicable below the wave trough, the new equations are valid until the mean water surface even in the presence of finite-amplitude surface waves. A two-dimensional numerical model (2DV model) is developed to validate the new set of equations of motion. The 2DV model passes the test of steady monochromatic waves propagating over a slope without dissipation (adiabatic condition). This is a primary test for equations of mean motion with a known analytical solution. In addition to this, experimental data for the interaction between random waves and a mean current in both non-breaking and breaking waves are employed to validate the 2DV model. As shown by this successful implementation and validation, the implementation of these equations in any 3D model code is straightforward and may be expected to provide consistent results from deep water to the surf zone, under both weak and strong ambient currents.

中文翻译：

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广义拉格朗日均值方法的拟欧拉均值三维运动方程的建立和验证

这项研究的目的是在存在表面波的情况下开发一套新的平均运动方程，该方程组实际上适用于从深水到沿海地区，河口和流出区域的情况。采用广义拉格朗日均值（GLM）方法来导出一组准欧拉均值三维运动方程，其中通过源项将波的影响包括在内。所获得的方程表示为波振幅的二阶。尽管经典的欧拉均值运动方程式仅适用于波谷以下，但即使存在有限振幅的表面波，新的方程式也适用于平均水面。开发了二维数值模型（2DV模型）以验证新的运动方程组。2DV模型通过了在斜坡上传播且没有耗散（绝热条件）的稳定单色波的测试。这是使用已知解析解对平均运动方程进行的主要测试。除此之外，还采用了随机波与非连续波和连续波中平均电流之间相互作用的实验数据来验证2DV模型。如成功的实施和验证所示，在任何3D模型代码中实现这些方程式都非常简单，并且可以在弱电流和强电流作用下，从深水到冲浪区提供一致的结果。随机波和平均电流在不破裂和破裂波中相互作用的实验数据被用来验证2DV模型。如成功的实施和验证所示，在任何3D模型代码中实现这些方程式都非常简单，并且可以在弱电流和强电流作用下，从深水到冲浪区提供一致的结果。随机波和平均电流在不破裂和破裂波中相互作用的实验数据被用来验证2DV模型。如成功的实施和验证所示，在任何3D模型代码中实现这些方程式都非常简单，并且可以在弱电流和强电流作用下，从深水到冲浪区提供一致的结果。