# Numerical Investigations of Tsunami Run-Up and Flow Structure on Coastal Vegetated Beaches

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## Abstract

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## 1. Introduction

## 2. Numerical Method

#### 2.1. Governing Equations

_{t}is defined as the eddy viscosity coefficient calculated by ${\nu}_{t}=\alpha {u}_{\ast}h$ where α is empirical constant and ranges from 0.3 to 1.0, ${u}_{\ast}$ means the bed shear velocity, ${\tau}_{bx}$ and ${\tau}_{by}$ denote friction in the x and y directions, respectively, ${\tau}_{bx}=\mathrm{g}\frac{{n}^{2}u\sqrt{{u}^{2}+{v}^{2}}}{{h}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$, ${\tau}_{by}=\mathrm{g}\frac{{n}^{2}v\sqrt{{u}^{2}+{v}^{2}}}{{h}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$, where n stands for Manning’s roughness coefficient, f

_{c}indicates the Coriolis parameter, and f

_{x}and f

_{y}stand for the drag force caused by vegetation.

**U**,

**F**,

**G**,

**F**

_{d}and

**G**

_{d}are shown as follows:

#### 2.2. Vegetation Drag Force

_{v}indicates the diameter, and h

_{v}indicates the height of the vegetation. C

_{D}(h) indicates the depth-averaged equivalent drag coefficient which considers the vertical stand structures of a tree, defined by Tanaka et al. [1] as follows:

_{D-ref}is the reference drag coefficient of the trunk at z

_{g}(equal to 1.2 m in principle), α(z

_{g}) is considered as an additional coefficient to express the effects of cumulative width on drag force at each height z

_{g}, b(z

_{g}) means the projected width, and b

_{ref}indicates the reference projected width. β(z

_{g}) is expressed as an additional coefficient representing the effect of leaves or aerial roots on drag force, and C

_{D}(z

_{g}) is drag coefficient of a tree at the height z

_{g}above the ground surface.

#### 2.3. Finite Volume Method

_{i}stands for the domain of the ith. Basing Green’s theorem. Equation (11) can be rewritten as:

_{i}means the boundary of the V

_{i}. ${A}_{i}$ denotes the area of the ith cell,

**n**is the outward surface normal vector of L

_{i}, with $\mathbf{n}=({n}_{x},{n}_{y})=(\mathrm{cos}\varphi ,\mathrm{sin}\varphi )$, and $\varphi $ is the angle included between the x direction and the outward normal vector.

#### 2.4. Evaluation of Numerical Fluxes

#### 2.5. Treatment of Wetting and Drying Fronts

- Wet edge (see Figure 1a): two adjacent cells are wet, in which water depth of left cell h
_{L}> ε and water depth of right cell h_{R}> ε. - Partially wet edge (with flux), as presented in Figure 1b: a wet cell (left) links to a dry cell on the right, and the water level of the wet cell is higher than that of the dry cell, where h
_{L}> ε, h_{R}≤ ε and water level of left cell η_{L}> water level of left cell η_{R}. - Partially wet edge (no flux), as shown in Figure 1c: a wet cell (left) links to a dry cell on the right, and the water level of the wet cell is lower than that of the dry cell, where h
_{L}> ε, h_{R}≤ ε, and η_{L}< η_{R}. To eliminate the non-physical flux problem produced in the interface, the water level η_{R}and bed level Z_{bR}for the dry cell were temporarily replaced by a value which equaled to the water level η_{L}in the wet cell.

- Wet cell: all the edges of this cell consisted of a wet or partially wet edges (with flux) and all the nodes of the cell are flooded.
- Dry cell: all the edges of this cell consist of dry or partially wet edges (no flux).
- Partially wet cell: all other cells do not satisfy the criteria of either a wet or dry cell, as defined above.

## 3. Numerical Simulation and Experimental Validation

#### 3.1. Solitary Wave Run-up on a Bare Sloping Beach

_{w}means wave height, h

_{0}denotes initial water depth with a value of 1 m, X

_{0}stands for the position of initial wave crest and is located a half wave length from the toe of the sloping beach in the computing domain, and u is wave velocity.

_{w}/h

_{0}= 0.0185. For convenience, the results are presented in non-dimensional forms: ${\mathrm{x}}^{\ast}=\frac{x}{{h}_{0}}$, ${\eta}^{\ast}=\frac{\eta}{{h}_{0}}$ and ${\mathrm{t}}^{\ast}=t\sqrt{\frac{\mathrm{g}}{{h}_{0}}}$. Comparisons of the simulated and experimental free-surface evolutions are presented in Figure 3. As illustrated, the incident wave propagates on the sloping beach at the early stage (t

^{∗}= 25, 30, 35, 40, 45, and 50), and reaches maximum run-up height at about t

^{∗}= 55, at which point backwash occurs. The maximum run-down happens at around t

^{∗}= 70. Simulated free surface profiles show good agreement with experimental data. Figure 4 shows the comparison of simulated and experimental water surface processes from a breaking solitary wave with H

_{w}/h

_{0}= 0.3. In Figure 4, wave breaking is not well reproduced by the depth-averaged shallow model and the computed wave fronts are steeper and slightly earlier than the experimental results at t

^{∗}= 15 and t

^{∗}= 20, as the model does not consider a wave dispersion term [37,38]. However, the wave breaking is only in a small portion of the domain. At the next step (t

^{∗}= 25), wave breaking is reasonably simulated by the model as a collapse of the wave near the shoreline approximately which can be explained that the wave breaking may be weakened and become smaller portion of the domain. The breaking solitary wave reaches a maximum height around t

^{∗}= 45, and approaches the lowest position where a hydraulic jump is formed near the shoreline around t

^{∗}= 55. The simulated results agree with experimental data very well, which indicate that the proposed model can accurately predict the propagation of breaking and non-breaking solitary waves on a bare sloping beach.

#### 3.2. Propagation of Long Periodic Waves on a Partially Vegetated Sloping Beach

_{g}= 0, 0.07, and 0.4 m), while the vegetation density was set as 2200 stem/m

^{−2}, and the drag coefficient C

_{D}of wooden cylinders was calibrated as 2.5 in the present study. An incident sinusoidal wave with a period of T = 20 s and wave height of 0.16 cm was propagated from a 0.52 m long constant bottom segment to a sloping beach. In the current case, water surface elevations were measured using capacitance wave gauges at six positions (G1–G6) along the center of the flume in cases of B

_{g}= 0 and 0.4 m, and the flow velocities were measured using electromagnetic current meters at locations in the cross section passing behind vegetation domain G6 (see Figure 5b) in case of B

_{g}= 0.07 m. The run-up height above still water surface was measured by tracing the moving of water front by eyes with a scale on the slope. The computational domain was represented by uniform triangular cells with a grid spacing of 0.01 m and a time step of 0.002 s for the numerical model. The minimum depth criterion for a dry and wet bed was considered as 0.001 m, and the Manning coefficient was set as 0.012.

_{g}= 0.07 m, indicating that the peak flow velocity at the center of the gap exit can reach 0.42 m/s and is 3.07 times more than the peak flow velocity at behind the center of the vegetation zone. Figure 8 shows the distribution of the peak velocities averaged from five wave periods at steady state in a cross-section of Gage 6, indicating that vegetation plays an important role in the distribution of flow velocities. The results demonstrate that the present model is an effective tool to predict long periodic wave propagation on a partially vegetated sloping beach. Vegetation can effectively attenuate the wave propagation; however, an open gap in vegetation zone generates large flow and adverse effect at gap exit.

#### 3.3. Effects of Forest on Tsunami Run-up at Actual Scale

#### 3.3.1. Coastal Topography and Forest Conditions

_{v}of P. odoratissimus and C. equisetifolia are considered as 8 m and 11 m, respectively; the reference diameters b

_{ref}of P. odoratissimus and C. equisetifolia are set as 0.2 m and 0.15 m, respectively; and the density values N of P. odoratissimus and C. equisetifolia are 0.22 trees/m

^{2}and 0.4 trees/m

^{2}, respectively. The depth-averaged equivalent drag coefficient C

_{D}(h) are variable with vegetation parameters and inundation depth h, and they are modified by Tanaka et al. and Thuy et al. [22,26]. The coastal forest starts at a slope of 1/500 (see Figure 9).

#### 3.3.2. Effects of Forest with a Straight Open Gap on Tsunami Run-up

_{F}= 200 m, B

_{F}= 200 m, and B

_{G}= 15 m. In this section, tsunami run-up on the sloping beach is considered using a straight gap (Case 1) to investigate the effect on tsunami run-up.

#### 3.3.3. Effects of Arrangements of Vegetation Patches and Open Gaps

#### 3.3.4. Effects of Forest Parameters on Tsunami Run-up

_{0}) and normalized peak flow velocities (V/V

_{0}) at simulated gauges (G3 and G4) with different tree arrangements, respectively, where η

_{0}and V

_{0}denote the run-up height and the peak flow velocity in the absence of vegetation. The results indicate that C. equisetifolia as a new vegetation layer at the back of existing P. odoratissimus (Collocation 3) can minimize the amplification of the peak flow velocity through the open gap, while generating a slightly larger peak flow velocity at the G3 station. The run-up height is higher than that of Collocation 1 (pure P. odoratissimus) and smaller than that of Collocation 2. This is because the peak flow velocity is reduced significantly due greater resistance of P. odoratissimus when it passes the P. odoratissimus vegetation layer, and in back zone of C. equisetifolia vegetation with smaller drag force, the velocity increment in open gap is smaller compared to that of other tree configurations.In Collocation 1 (pure P. odoratissimus), Figure 18 indicates that normalized run-up height and normalized peak flow velocity at G3 decreased monotonously with increased densities, while the normalized peak flow velocity of G4 first increased (vegetation density ≤ 0.15 tree/m

^{2}) and then presented adverse variation. Figure 19 indicates that the increased vegetation height and diameter at different growth stages of P. odoratissimus generate positively correlated effects on normalized peak flow velocity at G4, and significant attenuation effects on normalized run-up height and normalized peak flow velocity at G3. Generally, changing forest parameters (e.g. vegetation collocations, densities and growth stages) could be a more cost-effective way to mitigate tsunami damage.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Run-up and run-down of H

_{w}/h

_{0}= 0.0185 non-breaking solitary wave on a 1:19.85 sloping beach.

**Figure 4.**Run-up and run-down of H

_{w}/h

_{0}= 0.3 breaking solitary wave on a 1:19.85 sloping beach.

**Figure 5.**Experimental setup of wave flume. (

**a**) Longitudinal section, (

**b**) plan view of vegetation zone and measurement points.

**Figure 6.**Comparison of measured and calculated wave crests, heights, and troughs. (

**a**) Case without vegetation, (

**b**) Case with full vegetation.

**Figure 7.**Time series of velocities, (

**a**) at the center of the gap exit, (

**b**) at the center of the end of the vegetation.

**Figure 10.**Maximum water surface elevations of the first wave without vegetation and with P. odoratissimus vegetation [26].

**Figure 12.**Flow structure in Case 1 at different times ((

**a**) T = 600 s, (

**b**) T = 660 s, (

**c**) T = 700 s).

**Figure 13.**Temporal variation of water surface elevations and flow velocities at four predicted stations of Case 1, (

**a**) G1, (

**b**) G2, (

**c**) G3, (

**d**) G4.

**Figure 15.**Temporal variation of surface water elevations and current velocities at (

**a**) = G3 and (

**b**) = G4.

**Figure 17.**Variation of normalized run-up heights and normalized peak flow velocities at G3 and G4 with different tree collocations, (

**a**) The normalized run-up heights, (

**b**) The normalized peak flow velocities.

**Figure 18.**Variation of normalized run-up heights and normalized peak flow velocities at G3 and G4 with different densities of pure P. odoratissimus. (

**a**) The normalized run-up heights, (

**b**) The normalized peak flow velocities.

**Figure 19.**Variation of normalized run-up heights and normalized peak flow velocities at G3 and G4 with different heights of pure P. odoratissimus; (

**a**) The normalized run-up heights, (

**b**) The normalized peak flow velocities.

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## Share and Cite

**MDPI and ACS Style**

Zhang, H.; Zhang, M.; Xu, T.; Tang, J.
Numerical Investigations of Tsunami Run-Up and Flow Structure on Coastal Vegetated Beaches. *Water* **2018**, *10*, 1776.
https://doi.org/10.3390/w10121776

**AMA Style**

Zhang H, Zhang M, Xu T, Tang J.
Numerical Investigations of Tsunami Run-Up and Flow Structure on Coastal Vegetated Beaches. *Water*. 2018; 10(12):1776.
https://doi.org/10.3390/w10121776

**Chicago/Turabian Style**

Zhang, Hongxing, Mingliang Zhang, Tianping Xu, and Jun Tang.
2018. "Numerical Investigations of Tsunami Run-Up and Flow Structure on Coastal Vegetated Beaches" *Water* 10, no. 12: 1776.
https://doi.org/10.3390/w10121776