## 2 Methodology
### Theoretical Derivation: Particle Initiation Model  
Han [26] comprehensively considered the joint random action of water flow and particles, and gave the critical condition of particle initiation, displayed in Equation (4).  
\[|F_{ht}|(a+R\cos\theta)+|F_{ul}|(b+R\sin\theta)-\Big{(}|G|+|\Delta G|+P_{\mu} \Big{)}R\sin\theta=0 \tag{4}\]  
In Equation (4), \(a\), \(b\) and \(\theta\) are parameters related to the relative position of particles, and \(\Delta\mathbf{G}\) denotes the additional pressure caused by the layer of water.  
As shown in Figure 2, the horizontal thrust generated by water flow has a positive moment \(|\mathbf{F_{hl}}|(a+R\cos\theta)\) on particle under investigation, which is also affected by the horizontal thrust \(\mathbf{F_{ht}}\), position parameters \(a\) and \(\theta\), where \(a\) denotes the relative height between the horizontal thrusting point and the center of particle gravity. When the horizontal thrusting point is higher than the center of particle gravity, \(a\) is positive, otherwise \(a\) is negative. \(\theta\) denotes the angle between the center of the particle gravity and the connecting line of the contact point and the vertical direction. When \(\theta>90^{\circ}\), \(\cos\theta<0\), the effect of the horizontal thrust on the particles hinders the initiation of the particles. The uplift force generated by the water flow has a positive moment \(|\mathbf{F_{ul}}|(b+R\sin\theta)\) on the particle, which is affected by the uplift force \(\mathbf{F_{ul}}\), position parameters \(b\) and \(\theta\), where \(b\) denotes the horizontal distance between the position of the uplift force and the center of particle gravity, when the uplift force is on the opposite side of the fulcrum, \(b\) is positive, otherwise \(b\) is negative.  
Figure 2: Critical condition within the equilibrium of moment for the particle initiation.  
In previous studies on initiation mechanisms of debris flows, the particle initiation theory, typically adopted in river dynamics, was usually used, and the critical velocity or shear stress corresponding to debris flow particles was regarded as the critical condition of debris flow particle initiation. However, in the calculation of critical velocity or shear stress, the flow is usually regarded as uniform without velocity gradient along its path. This generalization effectively simplifies the complexity of the problem, which is very suitable for the study of the initiation mechanism of the frequent occurrence of debris flow. However, in the formation process of debris flow induced by dam-break, the depth of the fluid in front of the dam decreases sharply, and the flow pattern in front of the dam can no longer be regarded as uniform flow. Hence, considering the change of the flow conditions, the critical assumption of particle initiation will also change, accordingly.  
As shown in Figure 3, when velocities in regions A and B are approximately equal (\(V_{A}\approx V_{B}\)), there is nearly no gradient along the flow velocity and the force of the water on the particles can be simplified as \(\mathbf{F}_{0}=\mathbf{F_{d}}=C_{d}(\mathbf{u_{f}}-\mathbf{u_{s}})|\mathbf{u_ {f}}-\mathbf{u_{s}}|\). When \(V_{A}<V_{B}\), the velocity gradient of the erosion particles is \(\
abla\mathbf{u_{f}}>0\), the generalized mass force of the sediment \(F_{vmg}=(1/2)\alpha_{s}\rho/\big{(}1+2\alpha_{s}/\alpha_{f}\big{)}u_{f}\
abla u _{f}\) is in the direction of dotted line in Figure 3, and the generalized mass force will prevent particles from starting to move. When \(V_{A}>V_{B}\), \(\mathbf{F_{vmg}}\) is the direction of solid line in Figure 3 and it will enable the initial movement of the particles.