The water motion computed using 3D and 2DH models in tidally dominated shallow waters can, in some cases, differ significantly. In 2DH models, bed friction is typically parametrised in terms of the depth-averaged velocity, whereas in 3D models, typically the near-bed velocity is used. This difference causes the bed shear stress in 2DH models to point towards the depth-averaged velocity, whereas in 3D models, it points towards the near-bed velocity, which are not necessarily the same. Focussing on linearised barotropic models, we derive an exact friction parametrisation for 2DH models such that the same depth-averaged dynamics are described as in the corresponding 3D model. The result is a convolutional friction formulation where the instantaneous friction depends on the present and past velocities, thus modifying the traditional 2DH friction formulation that only depends on the present depth-averaged velocity. In the case of harmonic (tidal) waves, this parametrisation has a clear physical interpretation and shows that the near-bed velocity should be parametrised as a rotated, deformed and phase shifted variant of the depth-averaged velocity. We demonstrate that in certain regions of the parameter space, it may be impossible to calibrate a 2DH model that uses a traditional friction law to reproduce the water levels from a 3D model, showing that the 3D friction formulation can be crucial to capture the 3D dynamics within a depth-averaged model. This phenomenon is explored in detail in a narrow well-mixed estuary.

中文翻译：

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潮汐主导浅水区线性化 3D 和 2DH 模型之间的关系

在某些情况下，在潮汐主导的浅水中使用 3D 和 2DH 模型计算的水运动可能存在显着差异。在 2DH 模型中，床摩擦通常根据深度平均速度进行参数化，而在 3D 模型中，通常使用近床速度。这种差异导致 2DH 模型中的床剪应力指向深度平均速度，而在 3D 模型中，它指向近床速度，两者不一定相同。专注于线性正压模型，我们推导了 2DH 模型的精确摩擦参数化，以便在相应的 3D 模型中描述相同的深度平均动力学。结果是一个卷积摩擦公式，其中瞬时摩擦取决于当前和过去的速度，从而修改了仅取决于当前深度平均速度的传统 2DH 摩擦公式。在谐波（潮汐）波的情况下，这种参数化具有清晰的物理解释，并表明近床速度应参数化为深度平均速度的旋转、变形和相移变体。我们证明，在参数空间的某些区域，可能无法校准使用传统摩擦定律从 3D 模型再现水位的 2DH 模型，这表明 3D 摩擦公式对于捕获 3D 动力学至关重要在深度平均模型内。在一个狭窄且混合良好的河口对这种现象进行了详细探讨。