# Three-Dimensional Modeling of Tsunami Waves Triggered by Submarine Landslides Based on the Smoothed Particle Hydrodynamics Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Approach

#### 2.1. SPH Theory

**x**denotes the coordinates of the SPH particle, m refers to the particle mass, ρ represents the density, W represents the kernel function, h represents the smoothing length, and N represents the total number of neighboring particles.

_{D}is a parameter with a value of 7/4π in two dimensions and 21/16π in three dimensions. This formulation ensures that the Wendland function provides a smooth and accurate interpolation within a limited support radius.

#### 2.2. Governing Equations

**v**is the velocity vector, p represents the pressure of the fluid,

**F**is the external force acting on the fluid, and

**Θ**represents the diffusion term.

**g**represents the gravitational acceleration acting on the particles, which is a form of the external force

**F**in Equation (2).

**Θ**in the momentum equation can be calculated as:

_{a}= ρ

_{a}v

_{0}, μ

_{b}= ρ

_{b}v

_{0}, and v

_{0}is the dynamic viscosity of the fluid.

**τ**over superscripts i and j is defined according to

_{SPS}is the eddy viscosity determined by

_{l}is taken to be 0.0066 in this work. C

_{s}= 0.12 is the Smagorinsky constant. Δ is the initial particle spacing. $\left|\overline{\mathbf{S}}\right|$ is the local strain rate given by

**S**

^{ij}is the Favre-filtered rate of strain tensor:

#### 2.3. Material Model

_{y}is reached. The constitutive law for a Bingham fluid is given by:

**τ**is the shear stress tensor, η is the dynamic viscosity coefficient, and

**D**is the rate of strain tensor defined by:

**v**denotes the velocity gradient tensor, and the superscript T denotes its transpose tensor.

## 3. Validation of the SPH Model

#### 3.1. Benchmark Problem 1: 2D Submarine Landslide Test

^{3}and 1950 kg/m

^{3}, respectively. The viscosity of the water is set to η

_{w}= 0.001 Pa·s. The rheological parameters of the sediment cannot be measured directly in the experiment. According to a trial-and-error approach, we set η

_{s}= 0.15 Pa·s and τ

_{y}= 750 Pa in this study. In this case, the particle spacing D

_{p}= 0.01 m, the smoothing length h = 0.021 m, and the timestep D

_{t}= 1.0 × 10

^{−4}s.

#### 3.2. Benchmark Problem 2: 3D Submarine Landslide Test

^{2}, and the volume V is 0.0039 m

^{3}. The special gravity of the sliding granular material γ is 1900 kg/m

^{3}, and the total weight W is 7.41 kg. The slope angle of the sliding bed θ is 45°. The water depth in the wave tank h

_{0}is 0.966 m, and the initial submergence of slide h

_{c}is 0.025 m. A wave gauge was set above the initial position of the slide block. In this simulation, the particle spacing D

_{p}= 0.005 m, the smoothing length h = 0.011 m, and the timestep D

_{t}= 1.0 × 10

^{−5}s.

## 4. 3D Modeling of Baiyun Submarine Landslide

#### 4.1. Baiyun Landslide in the South China Sea

^{2}, can be documented in the PRMB. Figure 10 shows the distribution of the seismic faults and submarine landslides, as well as the cities on the coastline of the SCS which have been affected by landslide tsunamis in history. It is reported that the risk of landslide tsunamis is very high in the PRMB [55,56], which poses a great threat to China’s marine construction projects in the SCS. Therefore, it is important to investigate the tsunami characteristics generated by submarine landslides in this area.

^{2}, with a total removal volume of ~1035 km

^{3}of sediment, including a significant amount of mud, gravel, and coral reef blocks [57,58]. The landslide resulted in significant changes to the submarine topography and hydrogeological environment of the SCS, causing severe damage to the marine ecosystem and fisheries resources. It also made a significant impact on China’s maritime security strategy and territorial sovereignty in the SCS. Some previous studies have investigated the tsunamigenic potential of the Baiyun landslide and highlighted the devastating waves generated by the Baiyun landslide [59,60,61]. However, the key features of the landslide propagation and tsunami generation were not reproduced.

#### 4.2. Numerical Simulation of Baiyun Landslide

_{w}= 1000 kg/m

^{3}, and the viscosity is set to η

_{w}= 0.001 Pa·s. The landslide body is simulated as a Bingham fluid, with a density of ρ

_{s}= 1860 kg/m

^{3}, a viscosity of η

_{s}= 0.15 Pa·s, and a yield stress of τ

_{y}= 750 Pa. To balance the computational efficiency and accuracy, the particle spacing D

_{p}is 50 m in this simulation, h = 1.35 D

_{p}, and D

_{t}= 5.0 × 10

^{−2}s.

#### 4.3. Discussion

#### 4.3.1. Effect of Landslide Volume

_{1}= 250 km

^{3}, V

_{2}= 500 km

^{3}, and V

_{3}= 1000 km

^{3}) were carried out in this work. Figure 16 shows the tsunami waves at different moments generated by submarine landslides with different volumes. Figure 17 shows the relationship between the tsunami amplitude and the landslide volume. The axis of abscissas represents the maximum height of the head wave generated by the submarine landslide. The axis of ordinates represents the volume of the landslide body. In the simulations, the rheological parameters of the submarine sediment were constant. The simulated results show that the maximum heights of the head waves were about 7.6 m, 15.1 m, and 18.4 m when the landslide volumes were 250 km

^{3}, 500 km

^{3}, and 1000 km

^{3}, respectively. Therefore, it can be concluded that the tsunami amplitude becomes higher as the landslide volume is increased.

^{3}, 500 km

^{3}, and 1000 km

^{3}, the length of the leading waves are about 19.4 km, 22.1 km, and 26.3 km, and the periods range from 24.6 to 28.5 min, respectively. Therefore, it can be concluded that a slide with a larger volume can generate a tsunami with a larger wavelength and lower frequency.

#### 4.3.2. Effect of Water Depth

_{1}= 2000 m, H

_{2}= 2500 m, and H

_{3}= 3000 m) of the landslide head scarp are conducted in this work. Figure 18 shows the tsunami waves triggered by the Baiyun landslide (V = 1000 km

^{3}) with different water depths at a time of 1.0 h after landslide initiation. The simulation results show that the tsunami amplitude is 18.4 m, 14.5 m, and 7.3 m when the water depth is 2000 m, 2500 m, and 3000 m, respectively. Figure 19 shows the relationship between the tsunami amplitude and water depth, which indicates that a submarine landslide in a shallower water area may result in a larger tsunami.

#### 4.3.3. Limitations of the Presented SPH Model

## 5. Conclusions

- (1)
- A 3D numerical model based on the SPH method was established in this work to simulate a submarine landslide’s movement across complex submarine terrain and the near-field characteristics of the resulting tsunami waves.
- (2)
- To validate the SPH model, two physical model experiments, in both 2D and 3D, which have been recorded in the literature were simulated and analyzed. The water pressure distribution and velocity vector of the fluid were obtained. The simulated landslide configurations and surface water profiles were compared to the experimental data. The presented results show that despite some discrepancies, the SPH model established in this paper is capable of simulating the soil–water interaction and predicting landslide-generated tsunami events with satisfactory accuracy. The benchmark problem was simulated using the SPH model with different particle resolutions. The results show that the SPH model with finer particle resolution can obtain more accurate results. Therefore, high particle resolutions are necessary in SPH simulations to ensure sufficient computational accuracy.
- (3)
- The Baiyun submarine landslide in the South China Sea was simulated using the presented SPH model. The entire motion process of the landslide and the generation of tsunami waves were reproduced. The propagation direction of the leading wave basically agreed with the dominant landslide direction. The effects of water depth and slide volume on the landslide-generated tsunami waves were investigated. The simulation results show that landslides with a larger volume generate larger tsunamis with higher amplitudes, longer wavelengths, and lower frequencies. A landslide in a shallower water area can result in a larger tsunami. These relationships can be used for the rapid prediction of a tsunami disaster.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Simulated slope configuration and water profile at different moments. (

**a**) t = 0.4 s and (

**b**) t = 0.8 s.

**Figure 3.**Simulated water pressure field (first row) and velocity vector of fluid phase (second row) at (

**a**) t = 0.4 s and (

**b**) t = 0.8 s.

**Figure 4.**Comparison of the simulated and tested slope configurations at different times. (

**a**) t = 0.4 s and (

**b**) t = 0.8 s.

**Figure 5.**Comparison of simulated and recorded water profiles at different times. (

**a**) t = 0.4 s and (

**b**) t = 0.8 s.

**Figure 6.**Simulated slope configuration at different moments. (

**a**) t = 0.0 s, (

**b**) t = 0.3 s, and (

**c**) t = 0.6 s. The experimental scenarios in the figures are from reference [12].

**Figure 7.**Simulated slope configuration at different moments. (

**a**) t = 0.1 s, (

**b**) t = 0.3 s, and (

**c**) t = 1.2 s.

**Figure 9.**Distribution of sediment accumulation in the South China Sea [50].

**Figure 10.**Distribution of faults, submarine landslides, and cities effected by the tsunami around the South China Sea.

**Figure 14.**The 3D simulated results for the motion process of the Baiyun submarine landslide. (

**a**) t = 0.5 h, (

**b**) t = 1.0 h, (

**c**) t = 1.5 h, and (

**d**) t = 2.0 h.

**Figure 15.**Tsunami waves triggered by the Baiyun landslide at different moments. (

**a**) t = 0.5 h, (

**b**) t = 1.0 h, (

**c**) t = 1.5 h, and (

**d**) t = 2.0 h.

**Figure 16.**Tsunami waves triggered by Baiyun landslides with different volumes. (

**a**) V = 250 km

^{3}, (

**b**) V = 500 km

^{3}, and (

**c**) V = 1000 km

^{3}.

**Figure 18.**Tsunami waves triggered by Baiyun landslides with different water depths. (

**a**) H

_{1}= 2000 m, (

**b**) H

_{2}= 2500 m, and (

**c**) H

_{3}= 3000 m.

Density of sediment | ρ_{s} (kg/m^{3}) | 1950 |

Viscosity coefficient of sediment | η_{s} (Pa·s) | 0.15 |

Yield stress of sediment | τ_{y} (Pa) | 750 |

Density of water | ρ_{w} (kg/m^{3}) | 1000 |

Viscosity coefficient of water | η_{w} (Pa·s) | 1.0 × 10^{−3} |

Gravity acceleration | g (m/s^{2}) | 9.8 |

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**MDPI and ACS Style**

Dai, Z.; Li, X.; Lan, B.
Three-Dimensional Modeling of Tsunami Waves Triggered by Submarine Landslides Based on the Smoothed Particle Hydrodynamics Method. *J. Mar. Sci. Eng.* **2023**, *11*, 2015.
https://doi.org/10.3390/jmse11102015

**AMA Style**

Dai Z, Li X, Lan B.
Three-Dimensional Modeling of Tsunami Waves Triggered by Submarine Landslides Based on the Smoothed Particle Hydrodynamics Method. *Journal of Marine Science and Engineering*. 2023; 11(10):2015.
https://doi.org/10.3390/jmse11102015

**Chicago/Turabian Style**

Dai, Zili, Xiaofeng Li, and Baisen Lan.
2023. "Three-Dimensional Modeling of Tsunami Waves Triggered by Submarine Landslides Based on the Smoothed Particle Hydrodynamics Method" *Journal of Marine Science and Engineering* 11, no. 10: 2015.
https://doi.org/10.3390/jmse11102015