# Parallel-Computing Two-Way Grid-Nested Storm Surge Model with a Moving Boundary Scheme and Case Study of the 2013 Super Typhoon Haiyan

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Governing Equation of the Storm Surge Model

^{2}/s), $H$ is the total water depth ($H=h+\eta $; unit: m), $h$ is the still water depth (unit: m), g is the gravitational acceleration (=9.81 m/s

^{2}), $f$ is the Coriolis parameter ($f=2\omega sin\phi $; unit: 1/s), $\omega $ is the Earth angular velocity (=7.2921 × 10

^{−5}rad/s), ${P}_{a}$ is the sea-level air pressure (unit: N/m

^{2}), (${\tau}_{sx}$, ${\tau}_{sy}$) are the wind shear stresses (unit: N/m

^{2}), (${\tau}_{bx}$, ${\tau}_{by}$) are the bottom frictional shear stresses (unit: N/m

^{2}), ${\rho}_{w}$ is the water density (unit: kg/m

^{3}), and ${A}_{h}$ is the horizontal eddy diffusion coefficient (unit: m

^{2}/s). Here we note that Equations (2) and (3) ignore the advection terms where the nonlinear effect becomes insignificant in deep waters. Thus, the linear momentum equations are shown below:

^{1/3}).

#### 2.2. Discretization

^{−5}m in this study. Following a similar procedure in calculating the upwind-discretized advection terms, the total water depth threshold is also adopted.

#### 2.3. Grid Nesting in Time and Space

#### 2.4. Moving Boundary Scheme

#### 2.5. OpenMP Parallel Computing

## 3. Model Validation—Solitary Wave Runup on a Circular Island

#### 3.1. Introduction

#### 3.2. Computational Setting

^{1/3}) are adopted in Grid 02 for the regions near the circular island. The grid size of Grid 01 is 0.1 m, and the grid size ratio between Grid 01 and Grid 02 is 3:1. The time step of Grid 01 is 0.01 s, and the time step ratio between Grid 01 and Grid 02 is 2:1. The boundary conditions for the left and right sides of Grid 01 are the wall-boundary condition, and the downstream side of Grid 01 is the radiation boundary condition [36]. Grid 02 will accept the volume-flux components of Grid 01 as its boundary conditions, and it is noted that the two-way grid-nesting function is activated in time and space between Grids 01 and 02. The water density used in this benchmark simulation is 1000.0 kg/m

^{3}(i.e., the reference density of pure water).

#### 3.3. Computed Free Surface Elevations

#### 3.4. Time History of Free Surface Elevations

#### 3.5. Runup Height and Inundation Area

## 4. Case Study of Storm Surges—2013 Super Typhoon Haiyan

#### 4.1. Introduction of 2013 Typhoon Haiyan

#### 4.2. Computational Setting

**Figure 9.**Computational domains of the three-layer nested grids in Leyte Gulf and San Pedro Bay. (

**a**) Domain D01 with Domain D02. The color shading indicates the water depths in m. The dashed black line indicates Domain D02. The dashed white line indicates the 50-m water-depth contour. (

**b**) Domain D02 with Domain D03. The color shading indicates the water depths in m. The solid black line indicates Domain D03. The dashed blue line indicates the 10-m water-depth contour. The green circles show the numerical gauge locations (see Table 3).

**Table 2.**Nested-grid computational domains, grid sizes, and time steps for calculating storm surges induced by the 2013 Super Typhoon Haiyan: D01-coarse grid, D02-medium grid, and D03-fine grid.

Domain | Longitude (Unit: °E) | Latitude (Unit: °N) | $\mathbf{Grid}\mathbf{Size}(\mathbf{\Delta}\mathit{x}$$,\mathbf{\Delta}\mathit{y})$ (Unit: m) | $\mathbf{Time}\mathbf{Step}\left(\mathbf{\Delta}\mathit{t}\right)$ (Unit: s) |
---|---|---|---|---|

D01 | 124.9–126.0 | 10.4–11.4 | (437.24, 445.28) | 0.4 |

D02 | 124.98–125.15 | 10.85–11.30 | (145.66, 148.42) | 0.2 |

D03 | 124.99–125.04 | 11.18–11.26 | (48.52, 49.48) | 0.1 |

Station Name | Longitude (Unit: °E) | Latitude (Unit: °N) | Domain |
---|---|---|---|

Basey | 125.0523 | 11.2723 | D02 |

Tacloban | 125.0004 | 11.2538 | D03 |

Palo | 125.0118 | 11.1561 | D02 |

Tanauan | 125.0264 | 11.1016 | D02 |

Dulag | 125.0413 | 10.9465 | D02 |

^{2}/s [50]; the water density is the reference density of seawater (=1025 kg/m

^{3}) [51]; the Manning’s coefficient is 0.025 s/m

^{1/3}[12].

#### 4.3. Storm Surges and Storm-Induced Currents

#### 4.4. Maximum Storm Surges and Flood Depths

#### 4.5. Time Series of Storm Surges

## 5. Numerical Experiments

#### 5.1. Linear/Nolinear Equations with a Fixed or Moving Shoreline

#### 5.2. Parallel-Computing Efficiency

## 6. Conclusions and Future Work

#### 6.1. Conclusions

#### 6.2. Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sketch of a nested-grid domain in the grid size ratio of 3:1 (

**a**) at the upper left corner of a fine grid; (

**b**) at the lower right corner of a fine grid. Circles indicate the free surface elevations (black–the coarse grid; blue–the fine grid). Arrows show the volume-flux components of the x- and y-directions (black–the coarse grid; blue–the fine grid).

**Figure 2.**Grid-nesting procedure between the outer and inner grid in $\mathsf{\Delta}{t}_{out}/\mathsf{\Delta}{t}_{in}=2$. We note that $\mathsf{\Delta}{t}_{out}$ and $\mathsf{\Delta}{t}_{in}$ indicate the time step of the outer and inner grids, respectively. $\eta $ is the free surface elevation, ($P,Q$ ) are the volume-flux components, and the superscript $n$ denotes the time label. This figure is replotted from [26].

**Figure 3.**One-dimensional illustration for the moving boundary scheme: (

**a**) the shoreline stops between the grid cells i and i+1; (

**b**) the shoreline stops between the grid cells i+1 and i+2. Circles indicate the cell centers (blue–wet cells; black–dry cells); rectangles imply the cell edges for volume-flux components (green–nonzero flux; red–zero flux); the blue-shaded object identifies the water body.

**Figure 4.**(

**a**) Top view of the wave basin and the circular island. The blue asterisks indicate the locations of the wave gauges. The black arrow implies the incident wave direction. The dashed red line shows the domain of Grid 02. (

**b**) Side view of the circular island on the x-z plane along the centerline of the circular island. The blue line indicates the still water surface.

**Figure 5.**Snapshots for the computed free surface elevations of Grid 02 on the frontal side of the circular island (A/h = 0.091) at t = (

**a**) 8.5, (

**b**) 9.0, (

**c**) 9.5, (

**d**) 10.0, (

**e**) 10.5, (

**f**) 11.0, (

**g**) 11.5, and (

**h**) 12.0 sec. The incident solitary wave propagates along the +y-direction. The color bar indicates the free surface elevations in the unit of m. The gray-shaded object shows the circular island.

**Figure 6.**Snapshots for the computed free surface elevations of Grid 02 on the lee side of the circular island (A/h = 0.091) at t = (

**a**) 11.0, (

**b**) 11.5, (

**c**) 12.5, (

**d**) 13.5, (

**e**) 14.0, (

**f**) 14.5, (

**g**) 15.0, and (

**h**) 16.0 sec. The incident solitary wave propagates along the +y-direction. The color bar indicates the free surface elevations in the unit of m. The gray-shaded object shows the circular island.

**Figure 7.**Comparisons between the computed free surface elevations (red lines) and measured water levels (black dots) (A/h = 0.091) at wave gauges G6, G9, G16, and G22, respectively. The wave gauge locations can be found in Table 1.

**Figure 8.**Comparison between the maximum computed and measured runup heights (A/h = 0.091). Both numerical results and measured data are projected onto the circular island. Note that the incident solitary wave propagates along the +y-direction from the bottom boundary. The red line shows the computed maximum runup heights; the blue asterisks indicate the measured runup heights.

**Figure 10.**Computed storm surges of Domain D01: (

**a**) at 21:00 UTC on 7 November 2013; (

**b**) at 23:30 UTC on 7 November 2013; (

**c**) at 00:00 UTC on 8 November 2013; (

**d**) at 00:30 UTC on 8 November 2013. The color shading indicates the free surface elevations in m, and the black arrows show the 10-m wind directions.

**Figure 11.**Computed storm surges of Domain D03: (

**a**) at 23:00 UTC on 7 November 2013; (

**b**) at 23:30 UTC on 7 November 2013; (

**c**) at 00:00 UTC on 8 November 2013; (

**d**) at 00:30 UTC on 8 November 2013. The color shading indicates the free surface elevations in m. The blue circle shows the location of the Tacloban Airport.

**Figure 12.**Computed storm-induced currents of Domain D03: (

**a**) at 23:00 UTC on 7 November 2013; (

**b**) at 23:30 UTC on 7 November 2013; (

**c**) at 00:00 UTC on 8 November 2013; (

**d**) at 00:30 UTC on 8 November 2013. The color shading indicates the flow speed in m/s. The black arrows show the flow velocities in m/s. The red circle shows the location of the Tacloban Airport.

**Figure 13.**(

**a**) Maximum storm surges of Domain D02. The color shading indicates the maximum storm surges in m. The solid black line indicates the coastline in the numerical model. (

**b**) Maximum flood depth of Domain D03. The color shading indicates the maximum flood depth in m. The solid black line indicates the coastline in the numerical model. The blue circle shows the location of the Tacloban DZR Airport. The green circles imply the locations of measured flood depths (see Table 5). The yellow rectangles with the dashed black line mark inundation areas of [44].

**Figure 14.**Time series of computed storm surges at specified numerical gauges: Basey, Tacloban, Palo, Tanauan, and Dulag. The y-axis indicates storm surge height in m, and the x-axis shows the time from 12:00 UTC on 7 November 2013 to 12:00 UTC on 8 November 2013. The locations of these numerical gauges can be found in Table 3.

**Figure 15.**Maximum storm surges of the (

**a**) nonlinear equation model with the moving shoreline; (

**b**) nonlinear equation model with the fixed shoreline; and (

**c**) linear equation with the fixed shoreline. The color shading indicates the maximum storm surges in m. The solid black line shows the shoreline of COMCOT-SURGE.

**Figure 16.**Storm-induced current fields by the nonlinear equation model with the fixed shoreline at (

**a**) 00:00 UTC on 8 November 2013 and (

**b**) 00:30 UTC on 8 November 2013. Storm-induced current fields by the linear equation model with the fixed shoreline at (

**c**) 00:00 UTC on 8 November 2013 and (

**d**) 00:30 UTC on 8 November 2013. The color shading indicates the flow speed in m/s. The black arrows show the flow velocities in m/s.

**Figure 17.**Clock time versus thread numbers of the workstation. It is noted that 2 threads are equal to 1 CPU. The blue rectangles indicate the clock time using different numbers of threads. The red line is the curve-fitted result: $y=42.16+196.3\times \mathrm{exp}\left(-0.365x\right)$, and x and y here imply the x and y-axis of this figure.

Gauge Number | X Location (Unit: m) | Y Location (Unit: m) |
---|---|---|

G6 | 15.00 | 9.40 |

G9 | 15.00 | 10.40 |

G16 | 17.58 | 13.00 |

G22 | 15.00 | 15.60 |

**Table 4.**Switches for advection term, forcing terms, and moving boundary scheme. O indicates the switch is turned on; X indicates the switch is turned off.

Domain | Sea-Level Pressure/Wind Shear Stress | Advection Term | Coriolis Force Term | Bottom Friction | Horizontal Eddy Diffusion Term | Moving Boundary Scheme |
---|---|---|---|---|---|---|

D01 | O | X | O | O | O | X |

D02 | O | O | O | O | O | X |

D03 | O | O | X | O | X | O |

**Table 5.**Measured and predicated flood depths at P1, P2, and P3. The measured flood depths are from [44].

Field Survey Points | Longitude (Unit: °E) | Latitude (Unit: °N) | Measured Flood Depth (Unit: m) | Predicted Flood Depth (Unit: m) |
---|---|---|---|---|

P1 | 125.0247 | 11.2005 | 5.9 | 3.095 |

P2 | 125.0224 | 11.2271 | 3.5 | 3.783 |

P3 | 125.0004 | 11.2457 | 3.5 | 0.891 |

**Table 6.**Clock time with the corresponding usage of the thread number in the workstation. It is noted here that 1 CPU has 2 threads.

Thread Number | 1 | 2 | 4 | 8 | 12 | 16 | 20 | 24 |

Clock Time (Unit: min) | 185.47 | 123.58 | 90.87 | 66.17 | 37.22 | 43.95 | 39.83 | 40.11 |

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**MDPI and ACS Style**

Tsai, Y.-L.; Wu, T.-R.; Yen, E.; Lin, C.-Y.; Lin, S.C.
Parallel-Computing Two-Way Grid-Nested Storm Surge Model with a Moving Boundary Scheme and Case Study of the 2013 Super Typhoon Haiyan. *Water* **2022**, *14*, 547.
https://doi.org/10.3390/w14040547

**AMA Style**

Tsai Y-L, Wu T-R, Yen E, Lin C-Y, Lin SC.
Parallel-Computing Two-Way Grid-Nested Storm Surge Model with a Moving Boundary Scheme and Case Study of the 2013 Super Typhoon Haiyan. *Water*. 2022; 14(4):547.
https://doi.org/10.3390/w14040547

**Chicago/Turabian Style**

Tsai, Yu-Lin, Tso-Ren Wu, Eric Yen, Chuan-Yao Lin, and Simon C. Lin.
2022. "Parallel-Computing Two-Way Grid-Nested Storm Surge Model with a Moving Boundary Scheme and Case Study of the 2013 Super Typhoon Haiyan" *Water* 14, no. 4: 547.
https://doi.org/10.3390/w14040547