# Tsunami Inundation Modelling in a Built-In Coastal Environment with Adaptive Mesh Refinement: The Onagawa Benchmark Test

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## Abstract

**:**

## 1. Introduction

^{2}. This disaster had profound consequences, claiming more than 15,000 lives and ranking as the second-largest human loss event after the 2004 Indian Ocean tsunami [1,2]. The economic toll was staggering, with estimated losses reaching approximately USD 210 billion, making it the costliest natural disaster on record [3].

## 2. Materials and Methods

#### 2.1. Onagawa Town Physical and Numerical Experiments

#### 2.2. Open-Source Flow Solver

#### 2.2.1. Saint-Venant (SV) Solver

#### 2.2.2. Serre–Green–Naghdi (SGN) Solver

#### 2.2.3. Multilayer (ML) Solver

#### 2.3. Quadtree Adaptive Mesh Refinement (AMR)

#### 2.4. Dynamic Terrain Reconstruction

#### 2.5. Model Set-Up

- (1)
- The mesh refinement was set between min = 2
^{6}(approximately 14 cm) and max = 2^{9}(approximately 1.8 cm); - (2)
- A maximum simulation time of 100 s and a dry parameter of 0.1 mm were also applied;
- (3)
- In the case of the nonhydrostatic models, the CFL condition and the breaking parameter were defined;
- (4)
- The time series at WG2 was considered an incident wave profile following the same input wave used in the 2D and Q3D experiments [27];
- (5)
- The value of n was set to 0.025 for sea and 0.013 for land in wet cells, following the methods of Prasetyo et al. [27] for accurate comparison with the 2D and Q3D experiments.

#### 2.6. Statistical Metrics

## 3. Results

#### 3.1. Validation Using Experimental Results

^{6}→ 2

^{7}→ 2

^{8}→ 2

^{9}) with propagation inland in wet cells. When the leading wave reaches the wave gauge location, the resolution is 2

^{7}. Hence, the rough resolution at the front line affects accurate detection. Furthermore, the n value in the built-in area is 0.013, which poorly reflects the potential situation. The wave gauges in the southern section (WG4–WG8) are on a road with fewer obstacles (see Figure 2a), and the wave gauges in the northern section (WG10–WG13) are surrounded by many obstacles. In Figure 5b, the bias ranged between −0.4 cm and 0.5 cm for the SGN and SV, with mostly underestimation of the experimental wave at WG3, WG4, WG6, WG7, and WG9 and overestimation at WG2, WG5, WG8, WG10, WG11, WG12, and WG13. On the other hand, the Q3D model mainly underestimated the experimental wave, producing overly low values, as shown in Figure 5b; the values were −0.6 cm at WG6 and WG13, as depicted in the time series comparison in Figure 3.

#### 3.2. Nonhydrostatic Solutions

^{9}was used to resolve the propagation in the northern section. Applying a higher resolution on the order of 2

^{11}or more further improves the wave solution. However, the demanding computational capacity and the actual simulation time are drawbacks of this approach.

#### 3.3. Improved n Using the ML Solver

#### 3.4. Spatial Inundation

^{6}was uniformly structured for the entire domain. As the wave propagated inland, the mesh was automatically refined, capturing the leading edge of the wave. Figure 8 shows that high resolution (2

^{9}) continued to be obtained in the built-in area even after the leading edge of the wave had further propagated. The hydrodynamic solution of the ML solver is the reason behind this behaviour. Water oscillation in wet cells demands high refinement. Thus, increasing the resolution will require greater computational capacity.

#### 3.5. Computational Time Step

## 4. Discussion

^{6}and 2

^{9}refinement levels. Hence, the computational capacity will be concentrated at a specific location where the mesh is refined during the simulation to solve the wave propagation problem instead of consuming computational time and effort in the entire domain. Blaise et al. [48] developed an SV model with adaptive unstructured mesh on a spherical coordinate system. The developed model was validated with real tsunami events and emphasised the efficiency of the adaptive mesh in reducing the simulation time compared to a uniform grid. Another AMR adaptation was presented by Pons et al. [49,50]. Indeed, the AMR method enhanced the simulation efficiency with a reduction in computational time. However, the implementation of the AMR method with nonhydrostatic models to study tsunami inundation in complex coastal areas remains a thriving topic.

^{9}, and the second was conducted on a quadtree adaptive mesh with a resolution varying from 2

^{6}to 2

^{9}. Table 2 presents a comparison between the two experiments. The simulation with a constant grid consumed 28,024 time steps and lasted 6.8 h, while the simulation with AMR consumed 16,060 time steps and lasted 0.93 hr. The AMR method reduced the simulation time by approximately 86.3% compared to that of the constant grid. This efficiency makes the AMR method more dependable in real-time forecasting and early-warning systems. However, further comparisons with other numerical tsunami models regarding computational efficiency are recommended.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Map of Japan with the location of Miyagi Prefecture; (

**b**) Miyagi Prefecture, with Onagawa town in the northeast of the prefecture facing the Pacific Ocean; and (

**c**) an aerial image of Onagawa town, which was built between the mountains with a valley connecting to the Pacific Ocean, taken after the 2011 Tohoku Tsunami event.

**Figure 2.**(

**a**) The layout of the Onagawa physical model in the experimental wave flume along with the locations of the wave gauges, WG1–WG13, in the model. The incident wave profile for the hydraulic bore (HB) is imposed at the left boundary. The dashed box is the focused built-in area for the inundation study. (

**b**) A 3D layout of the Onagawa model used for numerical experiments (9.2 × 4 m), depicting the terrain surrounding the town and the slope toward the wavemaker.

**Figure 3.**Time series comparison between SGN, SV, 2D, and Q3D and the physical experiment at (

**a**) WG4, (

**b**) WG6, and (

**c**) WG12.

**Figure 4.**Scatter plots with correlation coefficient (R) values of all waves with respect to the experimental results: (

**a**) SGN, (

**b**) SV, (

**c**) 2D, and (

**d**) Q3D.

**Figure 5.**Comparisons of the (

**a**) RMSE and (

**b**) bias results among the SGN, SV, 2D, and Q3D methods at all the wave gauges.

**Figure 6.**Comparison of water level time series among the SGN, SV, ML, and experimental flume results at the locations of (

**a**) WG11, (

**b**) WG12, and (

**c**) WG13.

**Figure 7.**Comparison of water level time series among ML (before improving n), ML_F (after improving n), and the experiment at the locations of (

**a**) WG11, (

**b**) WG12, and (

**c**) WG13. (

**d**) The location of the three wave gauges in the northern section.

**Figure 8.**Snapshots of the wave propagation and the mesh refinement level with regard to the ML_F solver in the focused built-in area for the inundation study. The left column shows the tsunami wave propagation throughout the built-in area of the Onagawa model at 20, 50, 70, and 100 s. The right column illustrates the mesh refinement levels used to detect the change in the water surface precisely in the entire domain.

**Table 1.**Comparison between the SGN, SV, ML, and ML_F models in terms of the time steps and the simulation time needed to solve 100 s of tsunami propagation.

Model | Time Steps | Simulation Time (h) |
---|---|---|

SGN | 84,249 | 5.825 |

SV | 16,060 | 0.93 |

ML | 11,384 | 4.55 |

ML_F | 33,522 | 12.94 |

**Table 2.**Comparison of the computation times of the SV solver with a constant grid and an AMR in the case of the hydraulic bore.

Type of the Grid | Refinement Level | Time Steps | Simulation Time (h) |
---|---|---|---|

Constant grid | 2^{9} | 28,024 | 6.8 |

AMR | 2^{6} ≤ AMR ≤ 2^{9} | 16,060 | 0.93 |

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**MDPI and ACS Style**

Aljber, M.; Lee, H.S.; Jeong, J.-S.; Cabrera, J.S.
Tsunami Inundation Modelling in a Built-In Coastal Environment with Adaptive Mesh Refinement: The Onagawa Benchmark Test. *J. Mar. Sci. Eng.* **2024**, *12*, 177.
https://doi.org/10.3390/jmse12010177

**AMA Style**

Aljber M, Lee HS, Jeong J-S, Cabrera JS.
Tsunami Inundation Modelling in a Built-In Coastal Environment with Adaptive Mesh Refinement: The Onagawa Benchmark Test. *Journal of Marine Science and Engineering*. 2024; 12(1):177.
https://doi.org/10.3390/jmse12010177

**Chicago/Turabian Style**

Aljber, Morhaf, Han Soo Lee, Jae-Soon Jeong, and Jonathan Salar Cabrera.
2024. "Tsunami Inundation Modelling in a Built-In Coastal Environment with Adaptive Mesh Refinement: The Onagawa Benchmark Test" *Journal of Marine Science and Engineering* 12, no. 1: 177.
https://doi.org/10.3390/jmse12010177