# Comparisons of Dynamic Landslide Models on GIS Platforms
## 2 Models
### Descriptions
#### 2.1.2 Models Based on the Eulerian Description  
In the continuum systems, the Eulerian description is best for simulating motion, and the Navier-Stokes equations are the basis of modeling [32]:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\rho\mathbf{v}\right)=0 \tag{5}\]  
\[\rho(\partial\mathbf{v}/\partial t+\mathbf{v}\cdot\
abla\mathbf{v})=\mathbf{S _{f}} \tag{6}\]  
where \(\rho\) is the mass density, \(t\) is time, \(\mathbf{v}\) is the velocity, \(\mathbf{S_{f}}\) is the field which includes gravity, friction, and other forces. The best way to solve these equations is through numerical analysis due to lacking a smooth analytical solution. Traditional CFDs such as Openfoam(r) use these equations to simulate the landslide dynamics. In these programs, the geometric modeling process is complex. To describe flowing grain-fluid mixture, [PERSON] proposed a mixture theory [33], and the \(\mathbf{S_{f}}\) is:  
\[\mathbf{S_{f}}=-\
abla\cdot\left(\mathbf{T_{s}}+\mathbf{T_{f}}+\mathbf{T^{ \prime}}\right)+\rho\mathbf{g} \tag{7}\]  
in which  
\[\rho=\rho_{s}v_{s}+\rho_{v}v_{v} \tag{8}\]  
\[\mathbf{v}=\left(\rho_{s}v_{s}\mathbf{v}_{s}+\rho_{f}v_{f}\mathbf{v}_{f} \right)/\rho. \tag{9}\]  
Here, \(\rho\) is the mixture mass density, \(\mathbf{v}\) is the mixture velocity, \(\mathbf{v_{s}}\) is the velocity in the solid phase and \(\mathbf{v_{f}}\) is the velocity in the fluid phase, \(v_{s}\) is the volume fraction of solid, \(v_{f}\) is the volume fraction of fluid, \(\mathbf{g}\) is the gravitational acceleration, \(\mathbf{T_{s}}\) is the solid stress, \(\mathbf{T_{f}}\) is the fluid stress, and \(\mathbf{T^{\prime}}\) is a contribution to the mixture stress. The stress \(\mathbf{T^{\prime}}\) can be avoided by using an approximation suitable for many debris flows. To reduce the difficulty of traditional CFDs, [PERSON] and [PERSON] proposed the depth-averaged theory [34]. This model allows the GIS to integrate simulation codes. Depth averaging is one of the steps to eliminate the calculation in the \(z\)-direction. The averaged velocities and resultant forces are calculated on the \(x\)-\(y\) plane [32; 34; 35]. In this method, the assumption is that \(\rho\) is constant and the landslide depth is shallow. Thus, the equations [32; 36] are  
\[\frac{\partial h}{\partial t}+\frac{\partial h\overline{v}_{x}}{\partial x}+ \frac{\partial h\overline{v}_{y}}{\partial y}=0 \tag{10}\]  
\[\rho\left[\frac{\partial(h\overline{v}_{x})}{\partial t}+\frac{\partial(h \overline{v}_{x}{}^{2})}{\partial x}+\frac{\partial\big{(}h\overline{v}_{x} \overline{v}_{y}\big{)}}{\partial y}\right]=-\int_{0}^{h}\left[\frac{\partial T _{xx}}{\partial x}+\frac{\partial T_{yx}}{\partial y}+\frac{\partial T_{zx}} {\partial z}-\rho g_{x}\right]dz \tag{11}\]  
\[\rho\left[\frac{\partial(h\overline{v}_{y})}{\partial t}+\frac{\partial\big{(} h\overline{v}_{y}{}^{2}\big{)}}{\partial y}+\frac{\partial\big{(}h \overline{v}_{x}\overline{v}_{y}\big{)}}{\partial x}\right]=-\int_{0}^{h} \left[\frac{\partial T_{xy}}{\partial x}+\frac{\partial T_{yy}}{\partial y}+ \frac{\partial T_{zy}}{\partial z}-\rho g_{y}\right]dz. \tag{12}\]  
where \(\overline{v}_{x}=\frac{1}{h}\int_{0}^{h}v_{x}dz\), \(\overline{v}_{y}=\frac{1}{h}\int_{0}^{h}v_{y}dz\). As shown in the equations, the velocity in the \(z\)-direction is averaged. Based on the depth-averaged theory, a depth-averaged mixture model was proposed by [PERSON] and [PERSON] [32]. Subsequently, two-phase and multiphase depth-averaged models were proposed to describe distinct mechanical responses and dynamic behaviors of material [35; 37]. Some studies considered the erosion process, a mechanical process by which the bed material is mobilized by the flow and dominant mechanical processes in geophysical mass flows [38]. Erosion determines enhanced or reduced mobility but is not understood thoroughly. In the dynamic model, the erosion rate \(E=-\partial b/\partial t\) and erosion velocity \(u_{b}\) are two important parameters, which change mass and momentum productions \(E\) and \(u_{b}E\)[38], respectively.