Hadadpour, S.; Paul, M., and Oumeraci, H., 2019. Numerical investigation of wave attenuation by rigid vegetation based on a porous media approach. In: Silva, R.; Martínez, M.L.; Chávez, V., and Lithgow, D. (eds.), Integrating Biophysical Components in Coastal Engineering Practices. Journal of Coastal Research, Special Issue No. 92, pp. 92–100. Coconut Creek (Florida), ISSN 0749-0208.
Coastal areas are very complex and sensitive regions, which are extremely important in terms of economic, social and environmental values. Providing protection against coastal erosion is thus a significant issue and considerable research has been dedicated to the development of shore protection solutions. The importance of coastal vegetation and its role in wave attenuation and coastal protection in this context is still not fully understood. This study focuses on wave attenuation by coastal vegetation. For this purpose, numerical analysis is used to determine whether the vegetation field can be considered as a porous medium and whether a porous media based approach can be applied to describe the flow in a vegetation field. The computational fluid dynamic (CFD) solver “PorousWaveFoam” in the frame of OpenFOAM, which solves the Volume Averaged Navier–Stokes (VRANS) equations, is used for the simulation of flow in porous media. The model is calibrated and a new equivalent porosity (neq) based on leaf area index (LAI) is developed and implemented in “PorousWaveFoam”. The model is validated by various laboratory experiments of wave propagation through rigid vegetation, showing a good agreement between the measured and calculated wave height dissipation. It is concluded that the presented porous media approach performs well in simulating wave attenuation by a rigid vegetation field. Moreover, by using the validated model, it is confirmed that for a given water depth, wave attenuation depends on the plant characteristics (plant density, height and length of vegetation field). A higher density and longer vegetation field leads to higher attenuation rates. Wave attenuation decreases if the submergence ratio increases.
Given the importance of coasts as one of the most valuable natural resources, the development of shore protection solutions becomes one of the most crucial issues. In this scope, the common approach using hard coastal protection structures such as conventional breakwaters and revetments along the shorelines to dissipate and reflect the waves is not only costly and inappropriate to adapt to climate changes. More importantly, these hard structures may disrupt water circulation, regional and local sediment transport, alter the near-shore hydrodynamics and aquatic habitats, and may thus affect the dynamic balance of coastal ecosystems (Bray, Carter, and Hooke, 1995).
Recently, coastal vegetation has been considered as an alternative to or a support of hard structures for shore protection (Guannel et al., 2015; Mendez and Losada, 2004; Temmerman et al., 2013). There is a growing interest in utilizing the vegetation along the coast and also a combination of hard and soft structures to control wave energy and sediment transport (Sorensen, 1997; Tschirky, 2000). However, the use of vegetation may also result in negative consequences and risks. For example, vegetation can show spatial and temporal variation due to environmental parameters leading to uncertainty. Moreover, non-native species may have a negative effect on the local ecosystem.
Many experimental, field and numerical studies have been carried out to enhance the knowledge of wave-vegetation interaction. The results of field measurements revealed that the rates of wave energy dissipation are almost three times higher over saltmarshes (ca. 80%) than over a sand flat (ca. 30%) irrespective of the water depth (Möller et al., 1999). Paul and Amos (2011) assessed the effects of the seagrass Zostera noltii on wave attenuation in a field study. It was observed that a minimum shoot density is needed to induce a noticeable wave attenuation. Though field observations can provide valuable data under realistic conditions, they are difficult to perform and also have some limitations such as controlling hydrodynamic conditions and vegetation characteristics or replicating the tests.
In order to study wave-vegetation interaction under controlled conditions, several laboratory studies have been carried out using real vegetation (Bouma et al., 2005; Bouma, de Vries, and Herman, 2010; Maza et al., 2015) or surrogates (Augustin, Irish, and Lynett, 2009; Stratigaki et al., 2011). These include attempts to quantify the effects of stiff and flexible vegetation in emergent and submerged conditions. In the laboratory experiments, the main drawback is designing appropriate plant mimics, which are able to properly simulate the mechanical properties of plants, or keeping real vegetation healthy and alive in the flume. In addition, scale effects are also presented as an important limitation of laboratory experiments due to the difficulty in reproducing simultaneously realistic hydrodynamic and plant conditions at reduced spatial scale.
Additionally, various numerical models have been developed to study wave-vegetation interaction, which can provide useful results with less costs and efforts than laboratory testing. Numerical models are usually based on the Reynolds averaged Navier–Stokes (RANS) equations (Li and Yan, 2007; Marsooli and Wu, 2014; Maza, Lara, and Losada, 2013), the Boussinesq wave equations (Augustin et al., 2009), or the shallow-water equations (Wu and Marsooli, 2012). In some studies, the vegetation effect is modelled by the Manning equation expression and vegetation is treated as additional flow resistance added to the bed roughness (Zhang, Li, and Shen, 2013). However, in most of the existing models, the resistance of vegetation, considered as a group of vertical cylinders, to water flow is represented by an internal source of resistant drag force and introduced as a sink term into the governing equation (Fischer-Antze et al., 2001; Husrin, 2013; Marsooli and Wu, 2014; Maza, Lara, and Losada, 2013). Moreover, in some cases, the presence of vegetation is accounted for using methods in which vegetation is treated as a porous medium (Hoffmann, 2004; Zinke, 2012). Zinke (2012) determined the flow resistance using a pore Reynolds number based on the mean void nearest neighbour distance between vegetation elements and the mean pore velocity. The porous media approach was described based on typical macroscopic pore structure parameters, consisting of porosity η, solid fraction s=1- η, specific surface area SP and specific permeability K. However, investigations were restricted to steady uniform flow through emergent canopies. More details of the simplified momentum equation and the expressions for different terms as well as the formulations for pore friction factor and pore Reynolds number can be seen in Zinke (2012). Although the mentioned studies show good results, a universal expression for the parameterization of vegetation flow resistance based on the measurable parameters for real vegetation is still missing, and also due to a very few test cases, more research is needed to fully understand whether using porous media based approaches can be applied to reliably assess wave height dissipation through a vegetation field.
This study aims to improve the modelling of wave height attenuation by vegetation. The main objective of this study is to investigate whether the vegetation field can be considered as a porous medium in numerical modelling and the flow resistance of submerged rigid vegetation can be described using the porous media flow concept. Although the Volume Averaged Navier–Stokes (VRANS) equations contain a porosity term, the effect of porosity is often neglected due to the high porosity of vegetation (often more than 0.97). In order to avoid uncertainty resulting from this description and to obtain a more practical parameterization, an equivalent porosity (neq) is developed based on the leaf area index (LAI), which is a physically well-defined parameter of the vegetation field, to make the model feasible for canopy scale. Our porous media based approach relies on easily measurable vegetation parameters such as plant density, height and diameter, which can be included directly in the computational model, instead of the common use of approximate resistance factors. Moreover, by representing the vegetation field as a porous block there is no need to generate a complicated mesh to model the plants in this approach, which speeds up computational time.
In this study, a computational fluid dynamic (CFD) model by applying a porous media approach for the rigid vegetation field in the frame of OpenFOAM is adjusted and validated for wave attenuation using experimental data from the literature. In this scope, a parameterization of plant meadows is carried out and the required parameters of pore structure are defined for a meadow. The validated model is then applied to additional cases in order to test the applicability of the new approach for modelling wave attenuation by rigid vegetation.
OpenFOAM involves the numerical schemes to discretise the governing equation based on the Finite Volume Method (FVM) by applying the Volume Of Fluid (VOF) method for tracking the free surface. This study is carried out by using the “PorousWaveFoam” solver, which solves the Volume Averaged Navier–Stokes (VRANS) equations for the simulation of flow in porous media without representing the exact geometry of the pores forming the porous media (for more details see Jensen et al. (2014)). This is due to the fact that if the general form of the incompressible Navier–Stokes equation (NS-model) would be solved in porous media, the knowledge of the geometry of the pores forming the porous medium is required. In addition, the computational mesh should be fine enough to capture this geometry, which in most cases is not feasible. The applied continuity (Equation 1) and momentum (Equation 2) equations for the incompressible NS-model are:denotes the intrinsic volume averaged (pore) pressure over the pore volume only, n is the porosity given as the ratio of the pore volume to the total volume, gj is the jth component of the gravitational vector, t is the time, Cm is the added mass coefficient, μe = μt + μ is the effective dynamic viscosity (the sum of molecular dynamic (μ) and turbulent dynamic eddy viscosity (μt)), Fi is the resistance force term, and xi and xj are the Cartesian coordinates.
The extended Darcy–Forchheimer equation including linear and non-linear forces (through Fi ), as well as inertia forces (through Cm ), was applied as sink terms in the momentum equation:1995) applied in the model, and ϕp is a nondimensional empirical coefficient, which takes the value of 0.34. 2014) according to the parameter investigation for Forchheimer, transitional, and turbulent flow regimes. They concluded that the proposed coefficients could be applied for different porous structures with different pore parameters, and therefore different flow regimes.
The turbulent dynamic eddy viscosity can be calculated by the turbulence models available in the OpenFOAM modelling system. For the current simulations, the k-ω-SST (Shear Stress Transport model) turbulence model, first presented by Menter (1994), was applied in a RANS framework, which was successfully implemented by del Jesus, Lara, and Losada (2012) for porous media. The k-ω-SST turbulence model is a two-equation eddy-viscosity model based on a simultaneous use of the k-ϵ and the kω models. Hereby, the model accuracy is improved by using k-ϵ in the free-flow region far from the walls and k-ω for the boundary layer flow with high performance in these regions.
Furthermore, tracking of the fluid interface is required to simulate the fluid flow at the free surface. This is conducted using a revised VOF approach to track the free surface interface in porous media (Jensen et al., 2014). This revised version is based on the introduction of an additional convective term in the transport equation for phase fraction, resulting in a sharper interface resolution (Berberović et al., 2009).is the relative velocity between fluid and air. A more detailed description of the method is given in Berberović et al. (2009).
Vegetation Parameters for Porous Media
In order to apply a porous media approach for a vegetation field, a plant meadow is parameterized and the porosity properties for the PorousWaveFoam model are defined. Dullien (1979) showed that some concepts from conduit flow can be successfully used for the description of the flow in porous media. Therefore, the distance between plants ΔS is defined as the characteristic length scale of the unit cell of the porous media, assuming flow in only one direction through an isotropic porous medium. From the flow perspective, a porous medium is mostly characterized by its porosity, i.e. the fraction of the total volume occupied by void or pore space. For a vegetation field, an equivalent porosity (neq) is developed as a function of the 1-sided leaf area index (LAI = leaf length × leaf width × density, m2m-2). LAI is a dimensionless quantity that characterizes plant canopies including both leaf length and shoot density (i.e. the number of plants per square meter). For this purpose, the model is set up in similarity to the experimental conditions, and calibrated and validated using the data from these laboratory experiments.
Model Calibration and Validation
In order to calibrate the model, two sets of experiments with both artificial plant meadows (Paul, Bouma, and Amos, 2012) and natural Spartina anglica plants (Bouma et al., 2005) are used. The experiments of Paul, Bouma, and Amos (2012) were carried out in a racetrack wave flume at the NIOO-CEME (Centre for Estuarine & Marine Ecology) in Yerseke, the Netherlands, which was 0.6 m wide and had a straight working section of 10.8 m. Plant mimics were used to investigate the impact of vegetation characteristics such as plant density, flexibility and leaf length on wave attenuation. The dimensions of the mimics were chosen based on the natural size and density ranges of Zostera noltii in the natural environment. The stiff mimic meadow was 3 m long constructed using cable ties with a density of 1000 and 4000 shoots/m2 and a width of 2 mm. To explore the effect of submergence ratio (defined as the ratio of water depth to vegetation height) on wave attenuation, two vegetation heights (0.10 m and 0.15 m) were used and the water depth was set to 0.3 m which allowed a submergence ratio of 3: 1 and 2: 1, respectively. Regular waves with a height of 0.10 m and a period of 1.0 s were generated by the wave paddle (more details in Paul, Bouma, and Amos (2012)).
Experiments of Bouma et al. (2005) were conducted in a wave flume at WL|Delft Hydraulics (Delft, Netherlands). They applied regular waves with a wave height of 0.05 m and a period of 1 s generated by a piston type wave maker at the end of the flume. The flume was 13.5 m long and 0.5 m wide. The vegetation meadow was 3 m long. The tests included wave attenuation by Spartina anglica with different densities as well as by stiff mimics (cable ties) with comparable dimensions (0.1 m long and 0.005 m wide) (see Table 1).
Plant parameters, leaf area index (LAI), equivalent porosity (neq), and wave conditions in different cases for calibration, validation and model test.
In both aforementioned experiments, the dissipated wave height ΔH per meter of plant patch was calculated assuming linear wave dissipation along the patch:Paul, Bouma, and Amos (2012), and within the vegetation at 0.22 m (xH1) and 2.22 m (xH2) in Bouma et al. (2005), respectively, and Δx = xH1 - xH2 denotes the horizontal distance between wave height measurements in the direction of wave propagation. Although the relationship between wave dissipation and distance into a meadow is non-linear, linearity is a valid simplification for vegetation with relatively small wave attenuating capacity over short distances (Paul, Bouma, and Amos, 2012).
The model calibration was carried out iteratively minimizing the simulation error of dissipated wave height ΔH per meter (Equation 8) to find the most appropriate equivalent porosity (neq) for each case. After calibration of the model for the stiff vegetation and introduction of the new equivalent porosity (neq), as the best fit function to all calibration data points, the CFD model is validated for wave attenuation.
For model validation, the experiments conducted in a wave flume of the Fluid Mechanics Laboratory at Delft University of Technology by Hu et al. (2014) are used. The wave flume was 40 m long and 0.8 m wide, and the mimic canopies were constructed using stiff wooden rods with a height of 0.36 m and a diameter of 0.01 m (see Table 1). The canopy was 6 m long and the stems were distributed uniformly in space with different densities.
Error metrics are used for quantitative evaluation of the models' performance including Bias, Root Mean Square Error (RMSE), correlation coefficient (R) and Willmott Index given as:and represent their mean value and N shows the total number of data points.
After successful validation of the model for wave attenuation based on the assumption of linear wave dissipation along the patch, its performance to reproduce the wave height evolution over the vegetation field and damping coefficient has been tested. For this purpose, a set of experiments by Bouma, de Vries, and Herman (2010) were used. The experiments were conducted in a wave flume at WL|Delft Hydraulics (Delft, Netherlands) with a water level of 0.2 m in the flume and regular waves with a height of 0.07 m and a period of 1 s. The wave height was measured at the beginning, within and at the end of the vegetation section. 3 m long meadows of relatively stiff Spartina anglica Hubbard using three different densities were tested to investigate the effect of plant density on wave attenuation. In all cases, the vegetation field covered the full width of the respective flume, which has been modelled as a porous zone, and the model is set up in similarity to the experimental conditions including vegetation characteristics and wave parameters, which are summarized in Table 1.
In order to assess the ability of the model to simulate wave attenuation by stiff vegetation, wave attenuation is assessed as a function of the damping coefficient using the wave height damping formulation presented by Dalrymple, Kirby, and Hwang (1984) for regular waves:
Effect of Rigid Vegetation Parameters on Wave Attenuation
The validated CFD model was then used to investigate the effect of vegetation characteristics (plant density, height and length of the vegetation field) on wave attenuation by rigid submerged vegetation. The model was setup with a vegetation field starting 4.8 m behind the forcing boundary with varying parameters (Table 2). It was forced with constant wave parameters (water depth h = 0.3 m, wave height H = 0.1 m, wave period T = 1 s) in all cases.
Plant parameters, leaf area index (LAI), equivalent porosity (neq ), and wave conditions in different cases for calibration, validation and model test.
The percentage of wave height reduction within the vegetation field was determined as:
In this section, the performance of the proposed porous media based approach is demonstrated through the test of the model applicability for modelling wave attenuation by rigid vegetation considering the vegetation field as a porous medium.
Model Calibration and Validation
The model is calibrated to develop an equivalent porosity (neq) as a function of the leaf area index (LAI) for a stiff plant meadow (see Figure 1):
According to Figure 2 and the error indices in the embedded table, the wave attenuation ΔH calculated using the calibrated CFD model and experimental data are in good agreement which suggests that the model performs well with respect to the simulation of wave attenuation for rigid vegetation.
To quantify model performance, the calibrated model was applied to the experiments of Bouma, de Vries, and Herman (2010) (see Table 1) for reproducing wave height evolution over the Spartina anglica vegetation field and assessing the damping coefficient based on the Equation 13 (Figure 3).
As seen from Figure 3, there is a relatively good agreement between the measured and calculated wave height and therefore damping coefficient, which indicates that the proposed CFD model can correctly simulate the wave height dissipation over a stiff vegetation field. A slight overestimation of attenuation by the CFD model (around 6-7%) is seen from this figure, which might be caused by the uncertainties of the considered input parameters.
Effect of Rigid Vegetation Parameters on Wave Attenuation
The effect of vegetation characteristics, including plant density, height and length of the vegetation field (see Table 2), on wave attenuation is investigated and wave height evolution is assessed for different vegetation fields (Figure 4).
It can be stated that the vegetation did not cause an effect upstream of the vegetation field, which can be attributed to the high porosity of vegetation field. Wave height decreased slowly and almost linearly over the length of the vegetation field in all cases. Behind the vegetation field wave height remained constant, but at a lower level than in front of the vegetation field.
The transmitted wave heights decreased as the plant density D (stems/m2) increased due to the increase in the wave energy dissipation rate of the vegetation field (Figure 4a). This finding agrees with previous studies in which plant density was identified as an important factor affecting wave attenuation and which concluded that wave attenuation increases with higher density (Bouma et al., 2005; Maza et al., 2015; Paul, Bouma, and Amos, 2012).
Regarding the effect of plant density, wave height dissipation over the 3 m rigid vegetation field (Equation 14) increased from 30% reduction in the case of 1000 (stems/m2) to 37% and 41.5% for 2000 and 3000 (stems/m2), respectively (Figure 5).
The transmitted wave heights also decreased with decreasing submergence ratio (Figure 4b), which is in good agreement with the results found in the literature (Augustin et al., 2009; Bouma et al., 2005; Fonseca and Cahalan, 1992). The submergence ratio significantly affects wave height dissipation. The submergence ratio of 3 led to wave height reduction of 25.4% which increased to 39.4% and 51.6% reduction by decreasing the submergence ratio to 2 and 1.5, respectively (Figure 5). It can be stated that doubling the height of vegetation under a constant water depth may result in doubled reduction.
The transmitted wave heights decreased as the relative length of the vegetation field (B/Li) increased (Figure 4c). This result confirms previous findings from a coastal forest, where the relative forest length affects the rate of the force reduction within the forest due to the greater energy dissipation over a longer wave propagation distance (Hadadpour et al., 2016; Husrin, 2013). Wave height dissipation (Equation 14) increased from 25.4% reduction to 30% reduction in the case that the vegetation filed length was doubled with the same wavelength (i.e. from B/L=1 to B/L=2). However, the reduction did not increase notably with increasing the length of the vegetation field more than double (Figure 5).
This study investigates numerically the effects of vegetation on wave attenuation using a new porous media approach. Firstly, it was examined whether a porous media based approach can be applied to describe the flow in vegetation and assess wave height dissipation through a rigid vegetation field. The model was calibrated using laboratory data to develop an equivalent porosity (neq ) as a function of dimensionless leaf area index (LAI). After validating the developed model based on the simplifying assumption of linear wave height damping along the meadow, its performance has been tested for reproducing the wave attenuation as a function of the damping coefficient over the vegetation field using experimental cases from the literature. The results showed that the model performs well, yet leads to slight over predictions of wave height attenuation inside the vegetation field. Finally, the validated model was used to determine the effect of vegetation characteristics (plant density, plant height and the length of vegetation field) on wave reduction by a rigid vegetation field to assess its applicability for coastal protection.
According to the test cases, the wave attenuation calculated using the developed CFD model and experimental data are nearly similar; therefore, it might be concluded that the proposed CFD model considering vegetation as a porous medium with the new equivalent porosity performs well in simulating wave attenuation by a rigid vegetation field. Hence, the modified CFD model and the developed new formula of equivalent porosity (neq ) are applicable to compute the wave height reduction through a rigid vegetation field. However, the study is based on a limited number of data sets and there are not enough cases to be exactly matched with the assumptions of the model. More and better measurement data is required to further develop and generalize the proposed approach.
Though in some studies the vegetation field has been modelled as a porous medium, the performance of previous models cannot be accurately compared with the present model due to their different approaches and purposes. For example, Zinke (2012) investigated the flow resistance for steady uniform flow through emergent canopies, but the present model focuses on the wave attenuation by submerged rigid vegetation. However, there are some benefits of the present model which should be highlighted: (i) the model is easy to use and relatively fast, because the vegetation field is presented as a porous block in this model and there is no need to generate a complicated mesh to model the plants, (ii) the model calibration is based on only one parameter (i.e. equivalent porosity (neq )) which is determined as a function of LAI, an easily measurable parameter of the vegetation field, (iii) the model can easily be calibrated for further experimental studies and a new relationship for the equivalent porosity could be proposed to reproduce many conditions which are not possible to test due to the limitation of the laboratory facilities as well as save the money and time.
It should be taken into account that the model is calibrated for a stiff plant meadow with LAI values less than 2.5, and hence, it could be utilized for the meadows in this range of LAI values.
The slight overestimations by the present model observed in the test case of Bouma, de Vries, and Herman (2010) might be caused by the uncertainties of some assumptions. Especially, not considering the Spartina anglica plants movement and their probable bending may lead to an overestimation of energy dissipation.
For applying the model in coastal protection, this kind of overestimation is critical to be taken into account, due to the fact that the transmitted wave height is a very important parameter in coastal protection purposes, and waves higher than the expected ones may result in substantial damages. However, detailed analysis based on additional test data is required to further improve the model.
Moreover, for simplification, the same vertical distribution for biomass and leaf area was assumed (as well as Zinke (2012)) but in some of the test cases the natural saltmarshes with uneven distribution of biomass were used. Hence, further investigations using more realistic vertical biomass and plant area profiles are necessary to determine whether the assumption of same profiles could be applied to species that have more biomass in the upper or lower part of the canopy.
Regarding the effect of plant characteristics on wave attenuation, as expected, the higher plant density enhances the reduction of wave height (Bouma et al., 2005; Maza et al., 2015; Paul, Bouma, and Amos, 2012). But there is no linear relationship between the plant density and wave attenuation. For example, the difference of wave height reduction between density of 1000 and 2000 (stems/m2) is 2.5% higher than that between density of 2000 and 3000 (stems/m2). It can be concluded that there may be an optimal density for each condition above which attenuation will not increase further.
The submergence ratio (h/hp), which determines the influence of plant length, strongly affects the wave damping. In this study, the effect of submergence ratio on wave attenuation is investigated by changing the vegetation height under a constant water depth matching the approach by Paul, Bouma, and Amos (2012). However, in the most of previous studies (e.g., Augustin et al., 2009; Bouma et al., 2005; Fonseca and Cahalan, 1992), a constant vegetation height was used and water depth was varied to achieve different ratios. Since a change in water depth typically results in modified wave parameters (i.e. wave height and/or wave period), this approach may influence the results.
It was seen that wave height decreased slowly and without a rapid change inside the vegetation field in all cases. Therefore, a sufficient length of vegetation field is needed to maximize the effect of vegetation on wave height dissipation. To investigate the effect of the length of the vegetation field on wave attenuation, the relative vegetation field length (B/L) was used to make results applicable across hydrodynamic conditions. Generally, a longer vegetation field resulted in higher wave height attenuation, but there is an optimal length of the vegetation field above which additional length does not increase wave attenuation markedly. Here, we found that the maximum reduction (30%) was produced in the case of B/L=2, and hence, the vegetation fields longer than that (e.g., B/L=3) did not lead to a higher wave dissipation. Knowledge of this optimal vegetation length may improve planning for coastal protection purposes.
The results indicated that the developed porous media model successfully reproduced the wave height attenuation inside a rigid vegetation field. However, the study is based on a limited number of data sets and, therefore, more and better measurement data would be valuable to further develop our new approach. Despite the complicated structure of the vegetation and despite some simplifying assumptions, the use of the leaf area index (LAI) in the relationship for the equivalent porosity represents an important step in the proposed modelling approach for rigid vegetation. In fact, the calibrated and validated model performed very well in reproducing various laboratory tests, so that it might be concluded that the CFD model, when well-calibrated and validated, is able to be applied for more systematic parameter studies to extend the range of conditions tested in the laboratory for rigid vegetation.
Regarding the influence of vegetation on wave attenuation, a good agreement with the findings in the literature is found, which may prove the model reliability. However, the proposed porous media based model with the new equivalent porosity concept are for rigid vegetation. Further research within the frame of the PhD of the first author is ongoing to clarify the effect of flexibility on wave attenuation due to the existence of flexible meadows in many coastal areas, which focuses on the extension of the model to account for the flexibility of the vegetation. In conclusion, these findings and the proposed porous media approach may help coastal communities to better assess the coastal protection offered by different meadows.
The authors are grateful to the Centro Mexicano de Innovación en Energía del Océano (CEMIE-Océano) and the German Academic Exchange Service (DAAD), Excellence Center for Development Cooperation, Sustainable Water Management (EXCEED/SWINDON) for their financial and technical support, enabling participation at the Summer School on Integrating Ecosystems in Coastal Engineering Practice (INECEP).
The financial support by the Deutscher Akademischer Austauschdienst (DAAD) of this research in the frame of the PhD study by the first author is gratefully acknowledged. M. Paul acknowledges funding from the German Science Foundation (grant no. PA 2547/1-1). We do appreciate Prof. Dr. Rodolfo Silva Casarín and Prof. Dr. M. Luisa Martínez for all the time and effort they put in initiating and managing this Special Issue, and their helpful comments as Guest Editors. V. Chávez Cerón and D. Lithgow are appreciated for the time and effort they put into organizing EXCEED-SWINDON Summer School INECEP in Mexico, 2017.