Modeling Attenuation of Storm Surge over Deformable Vegetation: Methodology and Verification

Coastal wetlands play an important role in protecting coastal communities and stabilizing shorelines (Costanza et al. 2008; Gedan et al. 2011; Shepard et al. 2011). It is commonly accepted that vegetation can attenuate not only short waves (e.g., Mendez et al. 1999; Chen and Zhao 2012; Jadhav et al. 2013) but also long waves such as tsunami waves. For instance, it was observed that mangrove swamps effectively attenuated tsunami waves and protected a sheltered community while communities without the protection of mangroves were damaged severely (Latief and Hadi 2007; Alongi 2008; Teo et al. 2009). However, whether coastal wetlands can effectively attenuate forced long waves such as storm surges remains under debate (Resio and Westerink 2008; Feagin et al. 2010). Since Hurricane Katrina struck New Orleans, Louisiana, in 2005, more attention has been drawn to the potential benefits of coastal wetlands for reducing storm surge. Field measurements and numerical simulations in recent years are in support of vegetation’s role in storm surge reduction (e.g., Loder et al. 2009; Wamsley et al. 2009; Sheng et al. 2012). Wetland restoration is advocated from both the ecological and flood-reduction perspectives (Walton et al. 2006; Day et al. 2007).

Typically, surface waves that may cause damages to coastal communities include short waves (wind- or boat-generated), tsunamis, and storm surges. Short waves are generated by winds or moving boats with wave periods of seconds. Tsunamis often result from seismic activity or landslides on the ocean floor, and consist of a series of waves with the period ranging from minutes to hours. Storm surges as a gradual rise of water lasting from hours to days are built up as a result of a combination of wind setup, low atmospheric pressure, wave setup, and interaction with tidal conditions. Tropical cyclones and extratropical storms produce storm surges with surge heights ranging 1–9 m depending on the wind intensity, the size of the storm, proximity to the landfall location, as well as local bathymetry and geometry (e.g., Chen et al. 2008). Small strips of coastal wetlands and forests that effectively attenuate wind waves and tsunami waves are typically insufficient for reducing storm surge. Potential benefits of coastal wetlands for reducing storm surge heights depend on the wetland size and vegetation properties. With the urgent need of coastal restoration and hurricane protection along the Louisiana coast, a number of science-based programs have been launched to sustain a coastal ecosystem that provides support and protection to the environment and economy of southern Louisiana and beyond [Louisiana Coastal Area Science & Technology Program (LCASTO) 2010; Louisiana Applied Coastal Engineering and Science Division (LACES) 2012; Louisiana Coastal Protection and Restoration Authority (CPRA) 2012]. Both numerical models and field measurements are primary tools to investigate the role of coastal wetlands in storm surge and wave reduction. Numerical simulations using increased bottom friction resulting from vegetation have shown that coastal wetlands together with other landscape features are able to attenuate storm surge and waves to some extent (Suhayda 1997; Loder et al. 2009; Wamsley et al. 2009, 2010). The dominant vegetative resistance to the flow in various circulation or storm surge models is parameterized as an analog of the bottom friction using an enhanced, static Manning’s coefficient ($n$), which may not be adequately accurate according to recent studies (e.g., Kouwen and Li 1980; Wu et al. 1999; Freeman et al. 2000; Wilson and Horritt 2002; Carollo et al. 2005; Wilson 2007).

Vegetation-induced drag force, as an extra force exerted on the flow, was originally explored in the literature primarily for determining the discharge capacity of an open channel with submerged or emergent vegetation. For a given energy slope, the vegetative flow resistance becomes dominant as vegetation density increases. Lopez and Garcia (1998) showed that the bed shear stress of a vegetation field, with a 0.3 ratio of the stem frontal area to the substrate area where the stem shoots are rooted, is only 20% of the value experienced by a bare bed. The diminished bed shear stress entrains fewer bottom sediments (Lopez and Garcia 1998) because the rooting soil has been strengthened physically and biologically by vegetation (Micheli and Kirchner 2002). Therefore, the bed stress resulting from the bottom friction is often neglected (Fenzl and Davis 1964; Carollo et al. 2005). The drag force caused by vegetation is composed of the form drag, inertial force, and skin friction, while the latter two are often neglected in a time-averaged model. Therefore, the vegetation-induced drag force is simply considered as the form drag.

The vegetative drag force is commonly computed using the quadratic friction law, where the hydraulic relationship is utilized to determine the flow resistance coefficient given a value of the Manning’s roughness coefficient, $n$, and a hydraulic radius (Chow 1959). This relationship is widely adopted for flood plains, coastal plains, and other aquatic environments with the hydraulic radius replaced by a flow depth (e.g., Guardo and Tomasello 1995; Copeland 2000; Kouwen and Fathi-Maghadam 2000; Doncker et al. 2009; Bunya et al. 2010; Dietrich et al. 2011). A reliable estimation of $n\text{-values}$ for a given type of vegetation is critical for investigating the hydrodynamics and ecology of a wetland environment (Lee et al. 2000, 2004; Schaffranek 2004). The USGS had a guide for selecting $n\text{-values}$ (Arcement and Schneider 1989) and suggested adding a constant $n$ to a base value for a channel bottom with a certain type of growing plants in addition to other adjustments resulting from the channel irregularity, variation of cross section area, and so on. Nevertheless, it was also stated that the effects of vegetation on $n$ depend on the flow depth, the population density of vegetation, the degree to which the vegetation is flattened by strong currents, and the alignment of vegetation relative to the flow. As those influencing factors were identified, various laboratory experiments and field measurements have been undertaken to quantify the relationships with $n$. Manning’s $n$ was found to vary seasonally as a result of the seasonal variation of vegetation biomechanical properties and aboveground biomass (e.g., Shih and Rahi 1982; Doncker et al. 2009). More importantly, $n$ varies with the flow stage. Typically, as water depth increases, $n$ increases for emergent plants and decreases for submerged plants (Wu et al. 1999; Wilson and Horritt 2002; Wilson 2007). When considering the flexibility of vegetation, $n$ decreases as the flow velocity increases because a fast flow would bend the flexible vegetation and reduce the effective roughness height (Kouwen and Fathi-Maghadam 2000; Lee et al. 2000). The variation of $n$ with the flow depth and flow velocity leads to a relationship between $n$ and the product of flow velocity and flow depth proposed in Palmer (1945) and denoted as the $n\text{-}\mathit{VR}$ curve. However, because of the large discrepancy in $n\text{-}\mathit{VR}$ curves (e.g., Kouwen and Li 1980; Wilson and Horritt 2002; Carollo et al. 2005), this relationship is physically meaningful yet not practical owing to the absence of the vegetation biomechanical properties.

On the other hand, a drag coefficient is often used, especially when the underlying physics of flow-vegetation interaction are concerned. A large number of laboratory experiments have been performed to investigate the drag coefficient of an individual plant shoot within a group of the same kind and the corresponding drag force resulting from both rigid and flexible vegetation (live or artificial) under either emergent or submerged conditions (e.g., Tsujimoto et al. 1996; Lopez and Garcia 1997; Nepf 1999; Stone and Shen 2002; Wilson et al. 2003). Note that drag coefficients were defined with respect to different reference flow velocities. In addition to the well-known drag coefficient of an infinitely long isolated circular cylinder, a stem layer drag coefficient was introduced in Stone and Shen (2002) with respect to a spatially averaged velocity within the stem layer; a bulk drag coefficient was introduced in Nepf (1999) with a pore velocity averaged over the entire water column for a flow through an emergent canopy field; and a bulk drag coefficient with respect to the depth-averaged discharge velocity is useful for horizontal two-dimensional (2D) models. These drag coefficients are different in their reference velocities but similarly represent an array of rigid circular cylinders, where the flow is affected by adjacent cylinders.

Nepf (1999) conducted a series of experiments in a flume covered with emergent cylinders. A dimensionless parameter of population density was used to represent the volume concentration of emergent stems. It was found that the bulk drag coefficient decreases as the population density increases for a stem Reynolds number (${\mathsf{R}}_d$) larger than 200. The reduction of the drag coefficient is caused by the velocity reduction in the affected wake structure and by the delayed point of separation at the downstream stem resulting from the upstream wake induced turbulence (Nepf 2004). However, the sheltering effects of the upstream cylinders on downstream elements are less significant for a lower stem density. Experimental data showed that the bulk drag coefficient remains nearly constant up to a population density of 0.01 (Nepf 1999) and the bulk drag coefficient can be approximated from that of a single cylinder (Stone and Shen 2002; Nepf 2004). The drag coefficient of a single cylinder depends on the stem Reynolds number, ${\mathsf{R}}_d$, for a laminar regime, and is virtually equal to 1 up to ${\mathsf{R}}_d=3\times {10}^5$ (Naot et al. 1996; Nepf 2004; Wilson et al. 2006; Zhang et al. 2010). Stone and Shen (2002) conducted an extensive set of flume experiments on flows through both emergent and submerged rigid cylinders and showed that the drag coefficient of a single cylinder equals 1.05 with a small standard deviation for a wide range of population density, stem diameter, and ${\mathsf{R}}_d$.

The drag for a submerged, flexible canopy field is likely to be overestimated if plants are assumed to be rigid (Wilson and Horritt 2002), because a lower drag is expected as a result of the bending and streamlining of deformable plants. In reality, flexible plants are common, although they may act like rigid cylinders under weak flow conditions. A stiffness parameter, $\mathit{EI}$, which is the product of the modulus of elasticity and the second moment of the stem cross section, was introduced as a measure of the plant flexibility and can be related to the deflected vegetation height and flow resistance (Kouwen and Unny 1973; Kouwen and Li 1980; Tsujimoto et al. 1996) for submerged plants. Fathi-Maghadam and Kouwen (1997) presented more evidence showing the reduction of the flow resistance resulting from vegetation flexibility under emergent conditions. Wilson et al. (2003, 2006) emphasized the importance of vegetation deformation in determining the mean velocity of flow through submerged vegetation.

In addition to plant flexibility, the degree of submergence is another key parameter that has a close relation with the vegetation drag force, especially during a hurricane event when the total water depth changes with the rising and falling surge water level at different stages of the event. Submergence is defined as the ratio of flow depth to the vegetation height. The flow through a canopy field can be classified as emergent, submerged, and deeply submerged, as the dominant driving forces of the canopy flow vary with the submergence (Nepf 2004). The flow over a deeply submerged canopy consists of three layers including a logarithmic velocity profile up to the water surface, a stem flow layer inside the canopy, and a mixing layer in between. The logarithmic layer becomes dominant as the submergence increases ($>10$) and the deeply submerged flow eventually resembles the unconfined flow (Nepf 2004). As the submergence decreases, the logarithmic layer becomes less important while the mixing layer persists. When the submergence decreases below 1.5, the mixing layer takes up the nonvegetation layer up to the water surface (Nepf and Vivoni 2000) until the plants are emergent with the water column occupied simply by the canopy. With different degrees of submergence, the drag force varies. The dependencies of the vegetative flow resistance on vegetation properties, flow depth, flow velocity, and the combination have been widely revealed in laboratory scale experiments, yet less recognized in numerical modeling. A vegetative flow resistance module considering these dependencies is desirable for a horizontal 2D storm surge model to (1) take into account effects of vegetation on storm surge reduction by wetlands, (2) quantify the flood risk reduction benefits, and (3) provide guidance for restoration project design. The purpose of this study is threefold: (1) to review different flow resistance relationships resulting from rigid and flexible vegetation; (2) to extend the existing formula to account for both the rigidity of vegetation and varying degrees of submergence; and (3) to develop a vegetation-surge dynamically coupled model for predicting the effect of vegetation on the mean flow, surge levels, and wind waves. The results from the study are presented in this paper. This paper is focused on model development and verification using laboratory data. A future paper will document the study using simplified field conditions and hurricane forcing, which reveals the spatial and temporal variability of the equivalent Manning’s coefficient in coastal wetlands during a hurricane and provides insight into surge-vegetation-wave interactions.

The paper is organized as follows. After the introduction, the “Methodology” section reviews the flow resistance laws proposed by Stone and Shen (2002) and Kouwen and Unny (1973), and integrates them for a full range of submergence and a variety of vegetation rigidity. Next, the integrated formulations are implemented into a storm surge model and the numerical algorithm is explained. Then the vegetative flow resistance module is tested against three sets of laboratory experimental data for emergent vegetation, submerged rigid vegetation, and submerged deformable vegetation. The findings are summarized in the “Summary and Conclusions” section.

The resistance force induced by vegetation has been included in the momentum equation as an extra term in many numerical studies (e.g., Lopez and Garcia 2001; Stoesser et al. 2003; Li and Yan 2007). In the current study, the barotropic (horizontal 2D) mode of an existing coastal ocean circulation model, ECOMSED (HydroQual 2002), is adopted, and a separate subroutine is developed to calculate the vegetal shear stress. The governing equations of the depth-integrated continuity equation and momentum equations with the extra term of vegetation shear stress are written aswhere $t$ = time; $\eta$ = water surface elevation above the still water datum; $v_x$ and $v_y$ = depth-averaged velocities in the $x\text{-}$ and $y\text{-directions}$, respectively; $h$ = total water depth; $F_c$ = Coriolis force coefficient; $g$ = acceleration due to gravity; $F_{\mathit{dif}\text{,}x}$, and $F_{\mathit{dif}\text{,}y}$ are the horizontal diffusion terms in the $x\text{-}$ and $y\text{-directions}$, respectively; ${\tau }_0$, ${\tau }_b$, and ${\tau }_v$ = wind stress, bottom shear stress, and the vegetal shear stress, respectively; and their subscripts represent the components in the $x\text{-}$ and $y\text{-directions}$. The vegetal shear stress is calculated in a separate module using the methodology and algorithm described in the previous section. It is then passed to the momentum equations at each time step where the velocity components and surface elevation are solved. Three approaches are implemented to account for rigid plants, unyieldingly deformable plants, and yieldingly deformable plants.

In Approach 1, Eq. (6) is utilized to calculate the friction factor for rigid plants. In Approach 2, Eq. (12) extended from the formula for rigid vegetation and Eq. (9), the plant deformation relationship are employed for deformable plants with a certain degree of rigidity. This procedure is valid for a wider range of submergence, hereafter known as the SS-extended (Stone and Shen extended) approach. In Approach 3, Eq. (11) developed for highly flexible plants and the plant deformation relationship of Eq. (9) are employed for flexible plants, hereafter known as the KL-extended (Kouwen and Li extended) approach. Because Eq. (11) was developed from experiments of submerged vegetation, the KL-extended approach is valid only for submerged conditions. Therefore, during the transition from the submerged to emergent conditions, a linear interpolation is introduced for the continuity of the formulation and completeness of the approach. The KL-extended approach is applied for $s\ge s_d$ (given sufficient submergence, e.g., $s_d=1.5$), while the friction factor at $s=s_c$ is computed using the SS-extended approach. In the transition zone ($s_c<s<1.5$), the friction factor is obtained by linear interpolation.

Although different equations are implemented for deformable plants, the procedure of solving each set of equations is similar. Fig. 4 illustrates the iteration process using Eqs. (9) and (11), where Eq. (1) and the definition of the friction velocity [$V_{\ast }={\left({\tau }_v/\rho \right)}^{0.5}$] are introduced to rewrite those equations in terms of $V_{\ast }$ and $H_s$, as Eqs. (14a) and (14b). The deflected vegetation height and the friction velocity are solved iteratively.In Fig. 4, the intersection of the two curves is the solution, while the circles connected by the thin, straight lines are quasi-solutions at iterative steps that gradually approach the final solution. The number of iterations is usually less than 10.

The required input parameters for the vegetal shear stress module include $H_v$, $B_v$, $\mathit{EI}$, $N$, and $C_D$, (or $C_0$, $C_1$ if the KL-extended approach is used). The value of $C_D$ for a single cylinder within an array is used. Other parameters are determined by laboratory or field measurements of the vegetation. The numerical model incorporating the vegetation module is tested against three laboratory measurements. Specific boundary conditions are given in accordance with the laboratory experimental setup.

The paper has extended and unified resistance formulations for rigid, flexural rigid, and flexible plants under both emergent and submerged conditions. Three approaches were examined in detail and implemented into a storm surge model.

First, the flow resistance formulations for rigid plants were critically reviewed. By introducing the plant deformation relations, the formulation for rigid plants developed by Stone and Shen (2002) was extended to flexural rigid (or unyieldingly flexible) plants, namely the SS-extended approach, in which the deflected vegetation height substitutes the erect height of rigid vegetation. Both the rigid formulation and the SS-extended approach are valid continuously for a wide range of submergence from emergent to submerged conditions. The formula for rigid plants was tested against laboratory measurements while the SS-extended approach was only checked analytically owing to the lack of observation data for unyieldingly flexible plants. As the plant stiffness parameter becomes sufficiently large, the solution converges to that of the rigid formula. In Fig. 2, the friction factor for deformable plants (dashed line) predicted by the SS-extended will shift upward and approach the solution of rigid plants (solid line) for large plant rigidity.

Another flow resistance formulation directly derived from submerged, yieldingly flexible plants by Kouwen and Li (1980) was also examined and extended, namely, the KL-extended approach. This approach essentially covers only submerged conditions. However, to avoid discontinuity and numerical instability, near emergent to fully submerged conditions were considered using the linear interpolation in the transition zone. The degree of full submergence ($s_d$) is recommended to be 1.5 according to Nepf and Vivoni (2000), above which a logarithmic nonvegetation flow starts developing beyond the mixing layer. Both the KL-extended and SS-extended approaches solve a set of two equations, that is, a deformation relation and a resistance law, iteratively, in a similar fashion. The three approaches have been implemented into a storm surge model as three options. Good agreement with laboratory measurements has been found for the approach of rigid plants and the KL-extended approach. The SS-extended approach analytically tested is promising because it accounts for both the flexibility of natural plants and a full range of submergence. However, further tests against observations are desirable for future research.

In summary, this study has developed a physics-based procedure to incorporate the effects of deformable vegetation into a numerical storm surge model and unified formulations for emergent, near-emergent, and fully submerged rigid or flexible vegetation. A submodel of vegetal stress and deflected vegetation height was developed and tested against laboratory experiments. To improve the prediction of surge reduction by wetland vegetation, quantifying the spatial and temporal variations of deflected vegetation heights and equivalent Manning's coefficient under realistic field conditions is of significance and will be discussed in an upcoming paper.

The flow depth, depth-averaged velocity, and bed slope are available in the experimental data sets. The bed friction factor is estimated by using the method provided in Stone and Shen (2002).

Funding provided by the National Science Foundation (NSF Grant Nos. EPS-1010640 and DMS-1115527), the Louisiana Sea Grant (LSG) College Program, and the Northern Gulf Institute (NGI) is greatly acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the NSF, LSG, or NGI.