The one-dimensional numerical model is based on the SWEs, and they can be expressed in vector form as follows (Maleki and Khan, 2016):

where t is the time; while U, F and S are the vectors of conserved variables, fluxes, and source term, respectively. It can be defined as follows:

where h is the flow depth, η is the water surface elevation, u is the depth-averaged velocity, g is the gravitational acceleration, n is the Manning roughness coefficient, and f is the vegetation effect force on the wave.

This numerical model uses a one-dimensional explicit FVM to solve the equations, which can be integrated at a control volume:

After discretizing, Eq. (3) becomes

where $F{(U)_{i + 1/2}}$ is the flux at face (i+1/2), $F{(U)_{i - 1/2}}$ is the flux at face (i-1/2), $\Delta t$ is time step, and $\Delta {x_i}$ is the cell length at cell i.

The HLL approximate Riemann solver is a suitable choice to handle discontinuous solutions. It is used to determine the fluid mass and momentum fluxes at an interface as:

where ${ F}_{\rm L} $ and ${ F}_{\rm R} $ are the fluxes evaluated at the left-hand and the right-hand sides of each cell interface. Here, the HLL approach provides the approximate expression for estimating ${ F} $* for a series of the wetting-drying problems. The detailed procedures can be found in previous works (Liang et al., 2015; Maleki and Khan, 2016; Kuiry et al., 2012). In considering the drag force caused by coastal vegetation, an additional drag force sink term was implemented into the momentum equations. This approach has been proven to be successful in representing the interaction of waves and plants (Kuiry et al., 2012; Zhang et al., 2013).

Solitary wave is one type of long wave, and it can propagate long distances without changes in wave characteristics (Tang et al., 2017). When it reaches the sloping beach, the wave run-up and run-down play an important role in the nearshore dynamics and sediment transport, which have attracted an increasing amount of attention in long-wave modeling. Synolakis (1986) carried out a series of laboratory experiments to investigate the non-breaking wave propagation on sloping beach, as shown in Fig. 1. In Synolakis’s experiment, the constant water depth was 1 m, solitary wave propagated from left to right on a 1:19.85 sloping beach, and its height was 0.018 5 m. In the present numerical experiment, the space step is 0.1 m, the Manning coefficient is 0.01, and the variable time step is needed to require a Courant-Friedrichs-Lewy (CFL) condition. Figure 2 shows the comparison between simulated results and experimental data in wave run-up and run-down for non-breaking solitary wave. For data analysis, we employed dimensionless time, length, and wave height. The water depth normalizes the wave height and length scale, while the time scale is normalized as ${t^ * } = t\sqrt {g/d} $. It can be seen that the simulated results are in agreement with the experimental data; the wave run-up and run-down processes are generally well reproduced in this simulation. The results confirm the capability of the presented model in reproducing wave propagation processes with wet-dry moving boundary.

A laboratory experiment was conducted in Saitama University to explore the propagation of long periodic waves on a vegetation sloping beach (Thuy et al., 2018). In this experiment, the wave tank was 15 m long and 0.4 m wide, and the bed terrain consisted of various slopes (S =1:25, 1:4.7, and 1:20.5), as depicted in Fig. 3. The still water depth was 0.44 m. The vegetation diameter was 0.005 m, the vegetation density was 2 200 stem/m², and the vegetation zone ranged from x = 10.36 to 11.36 m, as shown in Fig. 3. The drag coefficient C_D of the vegetation was calibrated as 2.5 in the present case. An incident sinusoidal wave with a period of T = 20 s and wave height of 0.16 cm was implemented at the right boundary. In the present simulation case, the grid spacing is 0.01 m, the Manning coefficient is 0.012, and the time step is restricted by means of a Courant number of 0.7. Figure 4 compares the time series of velocities behind the vegetation, the simulated results in flow velocity are in agreement with the measured values, indicating that the a drag force from vegetation coupled with the momentum equation is reasonable for expressing the resistance effect on wave propagation.

Coastal vegetations growing on sloping beaches can act as a biological buffer zone, effectively reducing the tsunami run-up and mitigating natural disasters in coastal areas caused by tsunamis. A number of studies have emphasized the potential of coastal vegetation as a buffer zone to reduce damage caused by tsunamis (Tanaka, 2009; Thuy et al., 2010). As shown in Fig. 5, a coastal topography including four slopes (S = 1/10, 1/100, 1/50 and 1/500) was designed to investigate the impact of forests on tsunami wave propagation. The incident tsunami wave was represented in this work by a single sinusoidal wave with wave height of 6 m and wave period of 1 200 s, coming from the left side of open sea (Thuy et al., 2010). The still water depth was 100 m below the datum level, the tide level at the attack of the tsunami was considered to be 2 m. As shown in Fig. 5, the coastal vegetation was planted at the sloping beach with a slope of 1/500, where the ground was 4 m above the datum level (Thuy et al., 2010). The coastal forest was P. odoratissimus vegetation with complex aerial root structure. In the presence of forest cases, the tree height was 8 m, the reference diameter was 0.195 m, the vegetation density was 0.226 trees/m², and the vegetation width was 100 m. The drag force coefficient of P. odoratissimus varied with water depth and was derived from field investigation (Tanaka, 2009). In this study, the computational domain is subdivided into uniform meshes with a size of 5 m, the time interval is 0.01 s, and the bottom roughness coefficient is 0.025. The tsunami force (F) is an important factor for the evaluation of the impact characteristics of tsunami, and it is defined and assessed as the following equation (Thuy et al., 2018):

where ρ is water density, and u is calculated from present model.

Figure 6 shows the spatial distributions of maximum water level of the first run-up wave in the absence and presence of forest. In front of the vegetation domain, the water level with forest is significantly higher than that without forest, the water level drops more dramatically in the forest zone and maintains a lower water level behind the forest compared to that in absence of forest. Figure 7 shows the simulated results of water depth, current speed, and tsunami force in front of and behind forest zone for the non-vegetation case. Except for time delay, these results indicate that there are no significant differences in magnitude for the three variables in two locations (before forest and behind forest). Figure 8 shows the comparison of the simulated variables for the non-vegetated and vegetated cases. The numerical results indicate that the reflection and attenuation effect from the coastal forest on water depth is remarkable. The current speed and tsunami force dramatically reduce because of the vegetation resistance effect; the tsunami forces behind the vegetated domain decline by 75% compared to the simulated results in the absence of forest. These results show that this forest in a sloping beach environment can significantly reduce energy from tsunami waves, having an obvious effect of mitigating the natural disasters caused by tsunami waves.

Vegetation density is an important factor influencing tsunami wave attenuation in coastal beach covered with forest. In this numerical experiment, six different vegetation densities (0.1, 0.15, 0.2, 0.226, 0.3, and 0.35 tree/m²) are designed to investigate the interaction mechanism between coastal vegetation and tsunamis. Figure 9 shows the results comparing the front and back of the vegetated domain. In front of the forest domain, an increasing trend of water depth with rise of vegetation density is presented, but it is precisely the opposite behind the vegetated area. Compared to that in front of the vegetation zone, the maximum velocity behind the vegetation domain sharply declines with the rise of vegetation density. The tsunami force in front of and behind the vegetated area significantly decrease with rise of vegetation density, these conclusions can also be confirmed in Fig. 10. The conclusion can be drawn that the increase in vegetation density can more effectively mitigate the damage caused by the tsunami.

Seven wave heights (1.5, 2, 2.5, 3, 3.5, 4, and 4.5 m) are also selected to investigate the effect of coastal forest on tsunami wave mitigation. As shown in Fig. 11, we compare the differences in water depth, current speed, and tsunami force in different conditions of wave heights. Whether in front of the forest or behind it, three variables increase with an increase of wave height from 1.5 m to 4.5 m. Further, their maximum amplitude reaches ahead of time – the current speed and tsunami force in front of forest belt become steeper. As can be observed in Fig. 12, the maximum water depth and velocity behind the forest are noticeably smaller than those in front of the forest belt under condition of different wave heights, signifying that the presence of vegetation significantly contributes to the energy attenuation of tsunami waves. In addition, the increase of maximum depth and velocity with increasing wave height is seemingly linear in front of and behind the forest; the maximum value of tsunami force represents a nonlinear change with increasing incident wave height. In Fig. 13, as the wave periods (10, 20, 30, 40, and 50 min) increase, the maximum values for three variables in front of and behind the vegetated area are significantly delayed. Six sets of different Manning coefficients (0.01, 0.015, 0.02, 0.025, 0.03, and 0.035) are established in this numerical experiment.. The simulated results are shown in Fig. 14, water depth in front of forest decreases as the increasing of the Manning coefficient, while it increases behind forest. In the meanwhile, other variables in velocity and tsunami force decrease in front of and behind of vegetation area. In Fig. 15, the maximum water depth in front of the vegetation belt monotonously decreases with the increasing Manning coefficients, but there is a slightly increase behind the vegetation belt, the reductions of velocity and tsunami force are obvious in front of and behind of forest. The results indicate that the increment in Manning coefficients can more effectively mitigate the damage caused by the tsunami. Furthermore, in this simulation, three different slopes – typical (1:500), mild (1:250), and relatively steep (1:100) – extending from the shoreline were selected to identify the wave propagation on the topography variations with forest coverage. The simulated results are provided in Fig. 16. It can be observed that when the bed slope increases the maximum values for water depth, current velocity and tsunami force are almost steady in front of the forest, allowing them to clearly decrease behind the forest as the bed slope increases. It was determined that the slope effect (or gravitational effect) is a significant factor impacting tsunami energy dissipation on a sloping vegetation beach.