## 1. Introduction

[2] Many geophysical flows occur as dense granular avalanches that are driven downslope under balance between gravity and friction, such as debris flows, snow avalanches, lahars and dense pyroclastic flows [*Savage and Hutter*, 1989; *Iverson*, 1997; *Pudasaini and Hutter*, 2007]. Finite-sized granular avalanches are time-varying processes. They involve transitions from a rapid dilute avalanche front to a long-lasting avalanche tail, as investigated in detail by*Bartelt et al.* [2007], with a possible quasi-steady avalanche body. When such a flow interacts with a wall-like obstacle, grains are trapped upstream of the wall and form a stagnant zone that can co-exist with an inertial zone above with a smooth variation of the free-surface in the vicinity of the wall (in contrast to the granular bore that implies a sharp variation of the free-surface, as is reported in*Gray et al.* [2003], *Pudasaini et al.* [2007] or *Pudasaini and Kroener* [2008]). The transition from the dead zone process to the granular bore, and reciprocally, is mainly controlled by the Froude number of the incident flow, , and the wall height *H* relative to the thickness *h* of the incoming undisturbed flow, where is the depth-averaged velocity,*θ* the slope angle and *g* the gravity. For extended and generalized *Fr* that include the depth variations of flow properties, we refer to *Pudasaini and Domnik* [2009] and *Domnik and Pudasaini* [2012].

[3] The critical height, *H*_{c}, below which the granular bore does no longer exist can be expressed as *H*_{c}/*h* = *h*_{2}/*h*_{1} − *Fr*^{2/3} [*Hákonardóttir*, 2004], where *h*_{2}/*h*_{1} is the ratio of the heights before and after the jump: . Based on the pioneering work of *Savage* [1979] on granular jumps, a general expression of *χ* can be derived for frictional fluids down an inclined slope *θ*: . Here, *μ*_{e} is the effective friction coefficient, the length of the granular jump (see inset in Figure 1) and a shape-coefficient of the jump. This expression is derived from a balance between (i) the momentum flux across the jump, (ii) the pressure force gradient across the jump, (iii) the weight*W* of the jump ( with the assumption that the density is unchanged before and after the jump: ) and (iv) the friction force (*μ*_{e}*Wcosθ*) with the bottom and side-walls, as detailed in*Savage* [1979]. As an example, Figure 1 displays the ratio *H*/*h* over time measured by *Caccamo et al.* [2012] and the corresponding ratio *H*_{c}/*h* with *χ* = 1 or *χ* = 1.27. Here, *χ* = 1 refers to horizontal channels and frictionless fluids or is likely to stand for flow conditions for which *μ*_{e} would exactly balance *tanθ.* Also, note that, *χ* = 1.27 is calculated with *θ* = 30° (slope corresponding to tests of *Caccamo et al.* [2012] drawn in Figure 1) and the following typical values: *μ*_{e} = 0.36 (*μ*_{e} = *tanδ* where *δ* = 20° is the typical friction angle between walls and grains) and , as prescribed by *Savage* [1979]in lack of the precise geometry of the jump. The cross-comparison of*H*/*h* and *H*_{c}/*h* in Figure 1 shows that granular bores are expected to be formed at very short times (*H* > *H*_{c} for *t* < *t*_{1}: first impact of the dilute avalanche front) and at final times just before the avalanche comes to rest (*t* > *t*_{2}) while a dead zone process is likely to occur during most of the time of the flow-wall interaction (*H* < *H*_{c} for *t*_{1} < *t* < *t*_{2}). A detailed experimental analysis of the transition at time *t*_{2} was provided by *Caccamo et al.* [2011] and demonstrated that this analysis was relevant and promising. Investigating the transition at time *t*_{2} from a *dead zone process*(solid-like, stagnant zone that co-exists with a fluid-like, inertial zone above) towards a*granular jump* remains a great challenge. Detecting the transition from a granular jump to a dead zone process and measuring the time *t*_{1} is also a challenge which is made more difficult by the nature of the dilute avalanche front. This letter aims to lighten a recent analytical model that was developed by *Chanut et al.* [2010] to predict the force on the wall between times *t*_{1} and *t*_{2}. This model has important applications for geophysical flows which are likely to hit structures in relatively low slope runout zones while granular jumps are not systematically formed.

[4] Section 2 discusses how a scaling law can be derived from the theory described in *Chanut et al.* [2010]. The robustness of the scaling law is demonstrated by revisiting small-scale numerical simulations in two dimensions (section 3) and laboratory data (section 4) on granular flows past walls. Finally, we conclude on how the present study opens the path to future theoretical studies and could be applied to full-scale avalanche-flows.