## 5 Scale Effect
### Tsunamis Excited by 3D Slumps in the Field Scale  
The scenario of the three-dimensional slump-type landslide tsunami is designed to present the 3D effect by comparing the result with the 2D case. The geometry and the slump rheology parameters are \(H_{4}\). The slump initial width is 650 m and the domain in the lateral direction is 8000 m as shown in Figure 13. From t = 0 s to 20 s, not only does the slump slide down, but also collapses on the slope. This phenomenon makes the thickness of the slump small, which can be observed from t = 20 s to t = 100 s. The slump reaches the bottom between t = 40 s and t = 60 s. After reaching the bottom, the slump is deaccelerated by the bottom friction. However, the upper part of the slump keeps on moving and forms a thicker slump in the front part at t = 80 s and t = 100 s. The profile can be easily observed in Figure 14. Figure 14 shows the slump and water profiles in the centerline cross-section (y = 0 m). The red lines are the 3D results and the black lines are the 2D results. The thickness of the slump is thinner in 3D and thicker in 2D. It is not surprising because the 2D setup indicates a slump with an infinite width. The moving speed of the sliding slump is nearly the same in the 2D and 3D cases. However, the local highest wave height at t = 20 s around x = -400 m is generated by the 3D landslide. This is caused by the rebounding waves from the edge of the sliding slump. The rebounding tsunami waves are concentrated in the center part of the slump and generated a water wave higher than that in the 2D case. This phenomenon was also described in [PERSON] et al. [47] for the case of slid landslide. Figure 15 shows the snapshots of the free-surface profiles. The results show that the tsunamis excited by the 3D slump propagate in every direction. A large negative wave is formed due to the drag force from the sinking slump. Then, this negative wave rebounds and generates a positive wave with the largest wave height, 49 m, at t = 20 s. After that, the waves keep on spreading out with a wave height reduced. Higher wave height can be observed in the shallow area around x = 0 m at t = 60, 80, and 100 s. This is due to the effect of wave refraction. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & **Water** & **Un-Yield Viscosity** & **Yield Viscosity** & **Yield Stress \(\tau_{\rm y}\)** & **Slump Velocity** & **Landslide** & **Bingham** \\  & **Depth (m)** & \(\mu_{A}\) & \(\mu_{B}\) & **(Pa)** & **(V)** & **(V)** & **Froude** & **Number,** \\  & **(Pa s)** & **(Pa s)** & **(Pa s)** & **(m/s)** & **Number, Fr** & **Bm** \\ \hline \(H_{1-1}\) & 1.6 & \(10^{2}\) & \(5\times 10^{1}\) & \(10^{3}\) & 0.9899 & 0.2499 & 32.3249 \\ \(H_{1-2}\) & 1.6 & \(10^{10}\) & 0 & \(10^{3}\) & 1.0607 & 0.2677 & Inf \\ \(H_{1-3}\) & 1.6 & \(10^{10}\) & \(5\times 10^{1}\) & \(10^{3}\) & 0.9899 & 0.2499 & 32.3249 \\ \(H_{1-4}\) & 1.6 & \(10^{10}\) & \(2\times 10^{2}\) & \(10^{3}\) & 0.7778 & 0.1963 & 10.2852 \\ \(H_{1-5}\) & 1.6 & \(10^{10}\) & \(5\times 10^{2}\) & \(10^{3}\) & 0.7071 & 0.1785 & 4.5255 \\ \(H_{1-6}\) & 1.6 & \(10^{10}\) & \(5\times 10^{1}\) & 0 & 1.5910 & 0.4016 & 0 \\ \(H_{1-7}\) & 1.6 & \(10^{10}\) & \(5\times 10^{1}\) & \(10^{2}\) & 1.3081 & 0.3302 & 2.4462 \\ \(H_{1-8}\) & 1.6 & \(10^{10}\) & \(5\times 10^{1}\) & \(2\times 10^{2}\) & 0.7778 & 0.1963 & 82.2815 \\ \(H_{2}\) & 1.6 & \(10^{10}\) & \(1.6\times 10^{3}\) & \(10^{4}\) & 3.1303 & 0.2499 & 32.3249 \\ \(H_{3}\) & 160 & \(10^{10}\) & \(5\times 10^{4}\) & \(10^{5}\) & 9.8990 & 0.2499 & 32.3249 \\ \(H_{4}\) & 1600 & \(10^{10}\) & \(1.6\times 10^{6}\) & \(10^{6}\) & 31.3034 & 0.2499 & 32.3249 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Froude number and landslide Bingham numbers in this study. Figure 13: Snapshot of slump shape and velocity profile of 3D \(H_{4}\) scenario with \(\left(\mu_{A},\,\mu_{B},\,\tau_{y}\right)=\left(10^{10},\,1.6\times 10^{6},\,10^{6}\right)\).