## The Free Interface Model
### Free Interface Model with Source Term  
A non-averaged thin-layer model for flows described by Equations (1) and (2) with a static/flowing interface has been derived in [40]. It is formulated for the coordinates \(X\) in the direction tangent to the topography and \(Z\) normal to the topography. The topography is described by its angle \(\theta(X)\) with respect to the horizontal (see Figure 4 for a flat topography). For the case of simple shear (no dependency on \(X\)), no smallness assumption is necessary ([40] Section 2.3) and the model can be written as the following free interface problem with a source term. The unknowns are \(U(t,Z)\) (velocity in the direction tangent to the topography) defined for \(b(t)<Z<h\), where \(h>0\) is the given constant thickness of the layer and \(b(t)\) the unknown position of the static/flowing interface, \(0<b(t)<h\). They obey the following equations,  
\[\partial_{t}U(t,Z)+S(t,Z)-\partial_{Z}\big{(}\
u\partial_{Z}U(t,Z)\big{)}=0 \quad\text{ for all }Z\in(b(t),h), \tag{3}\]with the following boundary conditions for all \(t>0\),  
\[U =0 \text{at }Z=b(t), \tag{4a}\] \[\
u\partial_{Z}U =0 \text{at }Z=b(t),\] (4b) \[\
u\partial_{Z}U =0 \text{at }Z=h. \tag{4c}\]  
The source \(S(t,Z)\) is assumed to be given and represents the opposite of the net force, excluding the viscous force (see [41] and (10) below). The viscosity \(\
u\) can a priori depend on \(t\) and \(Z\) in the model. The velocity is extended by defining \(U(t,Z)=0\) for \(0\leq Z\leq b(t)\), corresponding to the static domain. The initial condition is formulated as  
\[U(0,Z)=U^{0}(Z)\quad\text{ for all }Z\in[0,h], \tag{5}\]  
where \(U(0,Z)\) is the limit of \(U(t,Z)\) as \(t\downarrow 0\). The initial velocity \(U^{0}(Z)\) is given for \(Z\in[0,h]\), such that for some \(b^{0}\in(0,h)\) (initial position of the interface), the function \(U^{0}\) satisfies \(U^{0}(Z)=0\) for \(Z\in[0,b^{0}]\), and \(\partial_{Z}U^{0}>0\) for \(Z\in(b^{0},h)\). The system is completed by the static equilibrium condition  
\[S(t,b(t))\geq 0\quad\text{ for all }t\in(0,T], \tag{6}\]  
that states that the net force must be resistive in the static layer. The velocity \(U\) obtained when solving (3)-(6) is expected to satisfy \(\partial_{Z}U>0\) for \(b(t)<Z<h\). As proved in [41], this is the case as soon as \(\partial_{Z}S\leq 0\) and \(\
u>0\). The unusual nature of the system (3)-(6) lies in the fact that the interface position \(b(t)\) is free, meaning that it has no given equation relating \(\dot{b}(t)\equiv db/dt\) to other quantities. Its evolution is instead implicitly governed by the boundary conditions (4a) and (4b). An interesting property of the model (3)-(6) is that a differential equation can nevertheless be derived for the time evolution of the position of the interface, valid under some conditions, as follows (see [41]). * If \(\
u>0\) then \[\partial_{Z}\big{(}\
u\partial_{Z}U\big{)}(t,b(t))=\
u\partial_{ZZ}^{2}U(t,b(t ))=S(t,b(t)),\] (7) and whenever \(S(t,b(t))\
eq 0\), one has \[\dot{b}(t)=\left(\frac{\partial_{Z}S(t,b(t))-\partial_{ZZ}^{2}(\
u\partial_{Z} U)(t,b(t))}{S(t,b(t))}\right)\
u.\] (8)
* If \(\
u=0\) and \(\partial_{Z}U(t,b(t))\
eq 0\), then \[\dot{b}(t)=\frac{S(t,b(t))}{\partial_{Z}U(t,b(t))}.\] (9)  
These formulas show the strong interrelationship between the velocity profile in the direction perpendicular to the inclined plane and the evolution of the static/flowing interface position. Erosion, i.e., penetration of the static/flowing interface into the erodible bed, occurs when \(\dot{b}(t)<0\), and deposition when \(\dot{b}(t)>0\). The full model of [40] is indeed more general than (3)-(6), since it includes a possible dependency on \(X\). It is derived under the assumptions that the thickness of the layer, the curvature of the topography, and the viscosity are small, the internal friction angle is close to the slope angle, the velocity is small, and the pressure is convex with respect to the normal coordinate \(Z\).