## 2 Physically Based Dynamic Models for Modeling Complex Landslides
### The Depth-Averaged Mass Flow Model for Diffusive Rock Avalanche  
In the later stage of the landslide, the sliding mass is usually broken and disintegrated into smaller blocks in the friction and collision process with the basal bed. Landslides usually exhibit obvious diffusive and high mobility characteristics. Therefore, the depth-averaged models based on the principles of hydrodynamics or particle dynamics are more suitable. Depth-averaged dynamic models have been widely used in the analysis of landslide or debris flow events with high mobility [2; 43; 44]. When analyzing landslides with the Global Cartesian System (GCS) under steep terrain, the gravitational acceleration should be corrected [45; 46]. When the depth-averaged model is applied to the GIS system, the correction based on the terrain slope in the Global Cartesian System (GCS) is more convenient. In this case, the component of gravitational acceleration in the direction perpendicular to the basal bed can be expressed as  
\[\mathbf{g}\cdot\mathbf{n}=g\text{cos}\theta=\frac{g}{\sqrt{(\partial z_{b}/ \partial x)^{2}+(\partial z_{b}/\partial y)^{2}+1}}, \tag{18}\]  
Following the work of [PERSON] [47] on the non-erosive depth-averaged model under GCS and the theory of [PERSON] [36] about the erosional landslide, the depth-averaged model of erosive granular flows under GCS can be expressed as  
\[\begin{cases}\frac{\partial h}{\partial t}+\frac{\partial}{\partial x}(hu)+ \frac{\partial}{\partial y}(hv)=E\\ \frac{\partial}{\partial t}(hu)+\frac{\partial}{\partial x}\left(huu^{2}+ \frac{K_{\tau}}{2}g\text{{\small{$\circ$}}}h^{2}\right)+\frac{\partial}{ \partial y}(huv)=-g\text{{\small{$\circ$}}}h\frac{\partial z_{b}}{\partial x} -\frac{\tau}{\rho}\frac{s_{v}}{\text{cos}\theta}+2\lambda_{b}uE\\ \frac{\partial}{\partial t}(hv)+\frac{\partial}{\partial x}(huv)+\frac{ \partial}  
where the vector of conservation variables is \(\mathbf{U}=(h,\,hu,\,hv,\,z_{b})^{\mathrm{T}}\), with \(h\) representing the flow depth in the vertical \(z\) coordinate (Figure 2), and \((u,v)\) are the depth-averaged velocity components along the \(x,y\) directions, respectively.  
The vector of flux in the \(x\) and \(y\) directions is denoted by \(\mathbf{F}\) and \(\mathbf{G}\), respectively. Under global Cartesian coordinates, on the erodible bed, these fluxes are given by  
\[\mathbf{F}(\mathbf{U})=\left(hu,hu^{2}+\frac{1}{2}K_{x}g_{b}h^{2},huv,0\right) ^{\mathrm{T}}, \tag{21}\]  
\[\mathbf{G}(\mathbf{U})=\left(hv,huv,hv^{2}+\frac{1}{2}K_{y}g_{b}h^{2},0\right) ^{\mathrm{T}}, \tag{22}\]  
where the lateral pressure coefficients in the \(x\) and \(y\) directions are denoted as \(K_{x}\) and \(K_{y}\), respectively (refer to [26]). The corrected acceleration of gravity \(g_{0}=g\cos^{2}\theta\) takes into account the influence of steep terrain in global coordinates [45]. The variable \(\theta\) represents the angle between the normal of the bed and the vertical coordinate axis (see Figure 2). The source terms on the right hand of Equation (20) are given by  
\[\mathbf{S}_{b}=\left(0,-ggh\frac{\partial z_{b}}{\partial x},-ggh\frac{ \partial z_{b}}{\partial y},0\right)^{\mathrm{T}}, \tag{23}\]  
\[\mathbf{S}_{\mathrm{T}}=\left(0,-\frac{\tau_{b}}{\rho}\frac{u}{\parallel \mathbf{u}\parallel},-\frac{\tau_{b}}{\rho}\frac{v}{\parallel\mathbf{u} \parallel},0\right)^{\mathrm{T}}, \tag{24}\]  
\[\mathbf{S}_{E}=\left(E,2\lambda_{b}uE,2\lambda_{b}vE,-E\right)^{\mathrm{T}}, \tag{25}\]  
where \(\mathbf{S}_{b},\,\mathbf{S}_{\mathrm{T}}\), and \(\mathbf{S}_{E}\) represent the source term of topography, basal friction, and entrainment, respectively.