Depth‐averaged thin‐layer models are commonly used to model rapid gravity‐driven flows such as debris flows or debris avalanches. However, the formal derivation of thin‐layer equations for general topographies is not straightforward. The curvature of the topography results in a force that maintains the velocity tangent to the topography. Another curvature term appears in the bottom friction force with frictional rheologies. In this work, we present the main lines of the mathematical derivation for these curvature terms that are proportional to the square velocity. With the SHALTOP numerical model, we quantify their influence on flow dynamics and deposits over synthetic and real topographies. This is done by comparing simulations in which these terms are exact, disregarded or approximated. With the Coulomb rheology, for slopes θ = 10 and for friction coefficients below μ = tan (5°), neglecting the curvature force increases the simulated travel times by up to 10% and 30%, for synthetic and real topographies respectively. When the curvature in the friction force is neglected, the travel distance may be increased by several hundred meters on real topographies, whatever the topography slopes and friction coefficients. We observe similar effects on a synthetic channel with slope θ = 25° and μ = 15°, with a 50% increase of the kinetic energy. Finally, approximations of curvature in the friction force can break the noninvariance of the equations and decelerate the flow. With the Voellmy rheology, these discrepancies are less significant. Curvature effects can thus have significant impact for model calibration and for overflows prediction, both being critical for hazard assessment.

中文翻译：

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薄层模型中重力驱动流的薄层曲率效应，无床蚀

深度平均薄层模型通常用于对重力驱动的快速流（例如泥石流或碎石雪崩）进行建模。但是，对于一般形貌的薄层方程式的形式推导并不简单。形貌的曲率会产生一个力，该力保持与形貌相切的速度。另一个曲率项出现在具有摩擦流变学的底部摩擦力中。在这项工作中，我们介绍了这些曲率项与平方速度成比例的数学推导的主线。使用SHALTOP数值模型，我们可以量化它们对合成和实际地形上的水流动力学和沉积物的影响。通过比较其中精确，无视或近似这些术语的模拟来完成此操作。库仑流变学，用于斜坡θ  = 10且对于小于μ  = tan（5°）的摩擦系数，忽略曲率力，对于合成地形和实际地形，分别将模拟行进时间增加多达10％和30％。当忽略摩擦力的曲率时，无论地形坡度和摩擦系数如何，在真实地形上的行进距离都可以增加几百米。我们在斜率θ  = 25°和μ的合成通道上观察到类似的影响 = 15°，动能增加50％。最后，摩擦力的曲率近似值可以打破方程的不变性，并降低流量。使用Voellmy流变学，这些差异不那么明显。因此，曲率效应可能对模型校准和溢流预测产生重大影响，而这对于危害评估至关重要。