## 5 Discussion and Conclusions
For the very simple case of flows over a constant slope and if the pressure is assumed to be hydrostatic (\(p=g\cos\theta(h-Z)\)), taking into account the \(X\)-variations is the same as (according to [41]) considering once again (3)-(6) with \(S\) including an additional term,  
\[S=g\cos\theta\left(\tan\delta-\tan\theta+\partial_{X}h\right), \tag{49}\]  
and as considering a thickness \(h=h(t,X)\) that evolves according to (2.2) in [41]. Then \(U=U(t,X,Z)\) and \(b=b(t,X)\). However \(S=S(t,X)\) in (49) is still independent of \(Z\). Then Equations (8) and (9) read as follows with \(S(t,  
\[\dot{b}(t,X)=\frac{S(t,X)}{\partial_{Z}U(t,X,b(t))}. \tag{51}\]  
The model with \(0<b<h\) is valid as long as \(S(t,X)\geq 0\), so as to satisfy (6). Indeed, violating this condition leads to instantaneous flow of the whole layer i.e., \(b=0\), and full erosion of the bed. In this case, (3) still has to be solved (with \(b=0\) and \(S<0\)), but with boundary conditions (4a) and (4c) only. If at some later time \(S\) becomes positive again, \(b\) can rise above the value \(b=0\). Thus when \(S(t,X)\) can change sign, there is in general a succession of progressive depositions and sudden full erosions of the bed. As discussed previously, the experimental granular flow is not perfectly uniform, as illustrated for example for the granular collapse at \(\theta=22^{\circ}\) in Figure 17. In this case, \(\tan\delta-\tan\theta=0.0837\), so that near the front, where the \(h\)-gradient is strongest (\(\partial_{X}h(t=0.66\,\mathrm{s},X=90\,\mathrm{cm})\simeq 0.125\) as shown in Figure 17), \(S\) is negative and thus even the inviscid model predicts that erosion occurs. Later, when the front flows beyond the position \(X=90\) cm, \(\partial_{X}h\) decreases with values around \(\simeq 5\times 10^{-3}\). Therefore \(S\) becomes positive, and with the inviscid model, the static/flowing interface rises (i.e., deposition occurs). For the viscous model, for the case \(S\geq 0\) studied in this paper, occurrence of erosion (\(b<0\)) or deposition (\(b>0\)) is related according to (50) to the change of sign of \(\partial_{ZZ}^{2}(v\partial_{Z}U)(t,X,b(t))\), i.e., positive and negative, respectively. Therefore, when the \(X\) variations are considered and under the hydrostatic assumption, the occurrence of erosion or deposition depends on the signs of both \(S\) and \(\partial_{ZZ}^{2}(v\partial_{Z}U)(t,X,b(t))\). The influence of a \(Z\)-dependency of \(S\) on the static/flowing interface dynamics and on the erosion process in the free interface model (3)-(6) is studied in [41]. It would be interesting to extend the approach proposed here to 2D and possibly 3D, so as to capture the static/flowing interface in thin-layer models, as proposed in [40]. The orders of magnitude assumed in [40] are indeed satisfied in the experiments discussed here, because the typical length is \(L=1\) m, the typical time is \(\tau=0.33\) s, (so that \(L/\tau^{2}=g\)), \(h=0.02\) m, and \(\
u=5\times 10^{-5}\) m\({}^{2}\)s\({}^{-1}\), leading to \(\varepsilon\equiv h/L=0.02\), \(\tan\delta-\tan\theta=O(\varepsilon)\), and the normalized viscosity \(\
u\tau/L^{2}\simeq 10^{-5}\) is of the order of \(\varepsilon^{2}\) or \(\varepsilon^{3}\). The primary models of [40] with dependency on \(X\) and without hydrostatic assumption (giving \(S\) depending also on \(Z\)) lead however to severe nonlinearities and ill-posedness, so that the numerical treatment of these extensions with all flow-aligned variations represents a major challenge. An important issue is also to summarize the dynamics of the normal velocity profile using a finite number of parameters (for example the interface position \(b\), the thickness \(h\), and the shear rate), in order to keep computational costs low enough to simulate natural situations. This could lead to a depth-averaged model, which has until now seemed inaccessible because of the dependency on the third normal derivative in the differential Equation (8).