Free alternate bars are large-scale, downstream-migrating bedforms characterized by an alternating sequence of three-dimensional depositional fronts and scour holes that frequently develop in rivers as the result of an intrinsic instability of the erodible bed. Theoretical models based on two-dimensional shallow water and Exner equations have been successfully employed to capture the bar instability phenomenon, and to estimate bar properties such as height, wavelength and migration rate. However, the mathematical complexity of the problem hampered the understanding of the key physical mechanisms that sustain bar formation. To fill this gap, we considered a simplified version of the equations, based on neglecting the deformation of the free surface, which allows us to: (a) provide the first complete explanation of the bar formation mechanism as the result of a simple bond between variations of the water weight and flow acceleration; (b) derive a simplified, physically based formula for predicting bar formation in a river reach, depending on channel width-to-depth ratio, Shields number and relative submergence. Comparison with an unprecedented large set of laboratory experiments reveals that our simplified formula appropriately predicts alternate bar formation in a wide range of conditions. Noteworthy, the hypothesis of negligible free surface effect also implies that bar formation is fully independent of the Froude number. We show that this intriguing property is intimately related to the three-dimensional nature of river bars, which allows for a gentle lateral deviation of the flow without significant deformation of the water surface.

中文翻译：

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河流中的自由交替棒：关键物理机制和简单的形成标准

自由交替条带是大规模的下游迁移床型，其特征是三维沉积前沿和冲刷孔的交替序列，由于可蚀床的内在不稳定性，这些河床经常在河流中形成。基于二维浅水和 Exner 方程的理论模型已成功用于捕捉棒不稳定现象，并估计棒的特性，如高度、波长和迁移率。然而，该问题的数学复杂性阻碍了对维持柱线形成的关键物理机制的理解。为了填补这一空白，我们考虑了方程的简化版本，基于忽略自由表面的变形，这使我们能够：(a) 首次完整地解释了作为水重变化和流动加速度变化之间简单联系的结果的条形形成机制；(b) 推导出一个简化的、基于物理的公式，用于预测河段中坝的形成，具体取决于河道宽深比、盾牌数和相对淹没。与前所未有的大量实验室实验的比较表明，我们的简化公式可以适当地预测各种条件下的交替棒形成。值得注意的是，自由表面效应可忽略不计的假设也意味着棒的形成完全独立于弗劳德数。我们表明，这种有趣的特性与河堤的三维特性密切相关，