Two-dimensional (2D) depth-averaged shallow water equations (SWE) are widely used to model unsteady free surface flows, such as flooding processes, including those due to dam-break or levee breach. However, the basic hypothesis of small bottom slopes may be far from satisfied in certain practical circumstances, both locally at geometric singularities and even in wide portions of the floodable area, such as in mountain regions. In these cases, the classic 2D SWE might provide inaccurate results, and the steep-slope shallow water equations (SSSWE), in which the restriction of small bottom slopes is relaxed, are a valid alternative modeling option. However, different 2D formulations of this set of equations can be found in the geophysical flow literature, in both global horizontally-oriented and local bottom-oriented coordinate systems. In this paper, a new SSSWE model is presented in which water depth is defined along the vertical direction and flow velocity is assumed parallel to the bottom surface. This choice of the dependent variables combines the advantages of considering the flow velocity parallel to the bottom, as can be expected in gradually varied shallow flow, and handling vertical water depths consistent with elevation data, usually available as digital terrain models. The pressure distribution is assumed linear along the vertical direction and flow curvature effects are neglected. A new formulation of the 2D depth-averaged SSSWE is derived, in which the two dynamic equations represent momentum balances along two spatial directions parallel to the bottom, whose horizontal projections are parallel to two fixed orthogonal coordinate directions. The analysis of the mathematical properties of the new SSSWE equations shows that they are strictly hyperbolic for wet bed conditions and reduce to the conventional 2D SWE when bottom slopes are small. Finally, it is shown that the SSSWE predict a slower flow compared with the conventional SWE in the theoretical case of a 1D dam-break on a frictionless channel with fixed slope. The capabilities of the proposed model are demonstrated in a companion paper on the basis of numerical and experimental tests.

二维 (2D) 深度平均浅水方程 (SWE) 广泛用于模拟不稳定的自由表面流动，例如洪水过程，包括由于溃坝或堤坝破裂造成的洪水过程。然而，小底坡的基本假设在某些实际情况下可能远不能满足，无论是在几何奇点的局部，甚至在可淹区域的广泛部分，例如在山区。在这些情况下，经典的 2D SWE 可能提供不准确的结果，而放宽了小底坡限制的陡坡浅水方程 (SSSWE) 是一种有效的替代建模选项。然而，这组方程的不同二维公式可以在地球物理流动文献中找到，在全球水平定向和局部底部定向坐标系中。在本文中，提出了一种新的 SSSWE 模型，其中沿垂直方向定义水深，并假设流速与底面平行。这种因变量的选择结合了考虑平行于底部的流速的优点，正如在逐渐变化的浅水流中可以预期的那样，以及处理与高程数据一致的垂直水深，通常作为数字地形模型提供。假设压力分布沿垂直方向呈线性，忽略流动曲率效应。推导了二维深度平均 SSSWE 的新公式，其中两个动力学方程表示沿平行于底部的两个空间方向的动量平衡，其水平投影平行于两个固定的正交坐标方向。对新的 SSSWE 方程的数学性质的分析表明，它们在湿床条件下是严格的双曲线，当底部坡度较小时，它们会简化为传统的 2D SWE。最后，结果表明，在具有固定坡度的无摩擦通道上的一维溃坝的理论情况下，与传统 SWE 相比，SSSWE 预测的流量较慢。在数值和实验测试的基础上，所提出的模型的能力在配套论文中得到了证明。