## 5 Scale Effect
### Scale Effect of Slump-Type Landslide Tsunamis
Figure 11: The similarity of water free-surfaces and slump profiles in different length scales, represented by water depth \(H_{1}\) (black color), \(H_{2}=10H_{1}\) (blue color), \(H_{3}=100H_{1}\) (green color), and \(H_{4}=1000H_{1}\) (red color) at \(t_{1}\) s, \(t_{2}=\sqrt{10}t_{1}\) s, \(t_{3}=10t_{1}\) s, and \(t_{4}=10\sqrt{10}t_{1}\) s, respectively. the Froude number, and the shape and movement of the slump are controlled by Bingham number. Moreover, the result of wave height from the laboratory scale can be applied to the field scale. The slump dynamics and kinematics in the laboratory scale, however, deviate significantly from the field scale. In the landslide tsunamis problems, the wave properties are governed by the landslide Froude number [62]. The velocity used in the landslide Froude number is the sliding speed of the landslide. [PERSON] et al. [63] undertook a series of experiments on landslide-induced tsunami under the landslide Froude number between \(\text{Fr}=0.125\) and \(\text{Fr}=0.375\). At relatively low Froude numbers (\(\text{Fr}=0.125\)), the offshore and onshore waves propagate freely after being excited by the initial acceleration of the slump [63]. On the other hand, at higher Froude numbers (\(\text{Fr}=0.375\)), the long effected duration of the landslide acceleration meant that the relatively high landslide velocity appears to increase the amplitudes of the offshore-propagating waves behind the landslide while decreasing the amplitudes of those waves passing over the landslide [63]. In this study, the Froude numbers of the numerical results vary between \(\text{Fr}=0.1785\) (case \(H_{1-5}\) in Table 2) and \(\text{Fr}=0.4016\) (case \(H_{1-6}\) in Table 2). The results show that, in the case of high yield viscosity \(\mu_{B}=5\times 10^{2}\) (Pa s) (case \(H_{1-5}\)), the slump velocity is small, and the landslide Froude number is small (\(\text{Fr}=0.1785\)). The effect of the slump deformation to the wave propagation is minimal. However, in the case of \(H_{1-6}\), the yield stress \(\tau_{y}=0\) (Pa). This means that the rheological property is ignored, and the slump deforms freely without the limitation from the yield stress. As a result, the slump velocity is fast, and the landslide Froude number is large (\(\text{Fr}=0.4016\)). With the limitation of our knowledge, the study on the scale effect of sliding a Bingham fluid is rare. [PERSON] et al. [61] is one of pioneer studies in this field. In their study, the Bingham number ranges from 0.014 to 30 [61]. However, [PERSON] et al. [61] only focused on a single Bingham fluid without considering the mixture with the ambient fluid. In the problem of Bingham flow in a channel driven by pressure gradient, the Bingham number ranges from 0 to 100 [64; 65]. At a small Bingham number, the viscous stress plays an important role, the fluid is more likely to behave like a Newtonian fluid. At a large Bingham numbers, the yield stress plays an important role. In this study, the Binghamnumber varies from 0 to 82.2815. In the case H\({}_{1-2}\), the Bingham number is infinite due to \(\mu_{\rm B}=0\). In the case H\({}_{1-6}\), the Bingham number is zero, the fluid is Newtonian fluid. In the case H\({}_{1-8}\), the Bingham number is high (Bm = 82.2815), due to the high yield stress (\(\tau_{\rm y}=2000\) Pa). The current conclusions are made under the scenarios of submarine slumps. However, the scale effect can probably be applied to subaerial slumps because no assumptions are made for the subaerial or submarine slumps. However, the detailed analysis shall be performed in the future study.