## 1 Introduction  
Understanding and theoretically describing the static/flowing transition in dense granular flows is a central issue in research on granular materials, with strong implications for industry and geophysics, in particular in the study of natural gravity-driven flows. Such flows (e.g., landslides or debris avalanches) play a key role in erosion processes on the Earth's surface and represent major natural hazards. In recent years, significant progress in the mathematical, physical, and numerical modeling of gravity-driven flows has made it possible to develop and use numerical models for the investigation of geomorphological processes and assess risks related to such natural hazards. However, severe limitations prevent us from fully understanding the processes acting in natural flows and predicting landslide dynamics and deposition, see e.g., [1]. In particular, a major challenge is to accurately describe complex natural phenomena such as the static/flowing transition. Geophysical, geotechnical, and physical studies have shown that the static/flowing transition related to the existence of no-flow and flow zones within the mass plays a crucial role in most granular flows and provides a key to understanding their dynamics in a natural context, see [1] for a review. This transition occurs in erosion-deposition processes when a layer of particles flows over a static layer or near the destabilization and stopping phases. Note that natural flows often travel over deposits of past events, which may or may not be made of the same grains, and entrain material from the initially static bed. Even though erosion processes are very difficult to measure in the field, e.g., [2; 3; 4], entrainment of underlying material is known to significantly change the flow dynamics and deposition, e.g., [5; 6; 7; 8; 9; 10]. Experimental studies have provided some information on the static/flowing dynamics in granular flows, showing for example that the presence of a very thin layer of erodible material lying on an inclined bed may increase the maximum runout distance of a granular avalanche flowing down the slope by up to 40% and change the flow regimes [9; 11; 12]. In these experiments that mimic natural flows over initially static beds (Figure 1), quasi-uniform flows develop when the slope angle \(\theta\) is a few degrees lower than the typical friction angle \(\delta\) of the involved material [12]. Figures 2 and 3 show new data extracted from the experiments performed by [12] on the change with time of the static/flowing interface position \(b\) and the velocity profiles \(U(Z)\) within the granular mass, where \(Z\) is the direction normal to the bed. At a given position \(X\) along the plane, the flow is shown to excavate the initially static layer immediately when the front reaches this position. The static/flowing interface rapidly penetrates into the static layer, reaching a lowest position (that depends on the slope angle) for significantly high slopes and then rises almost linearly or exponentially toward the free surface until the whole mass of material stops (Figure 2). A theoretical description of these observations is still lacking. In particular, questions remain as to what controls the change with time of the static/flowing interface position and the velocity profiles and how these characteristics are related to the rheology of the granular material and the initial and boundary conditions. Figure 1: Experimental setup from [9; 12] to study granular column collapse over inclined planes covered by an initially static layer made of the same grains as those released in the column. For slope angles \(\theta\) a few degrees smaller than the typical friction angle \(\delta\) of the involved material, a quasi-uniform flow develops behind the front.