Wave damping due to vegetation

SWAN has an option to include wave damping over a vegetation field (mangroves, salt marshes, etc.) at variable depths. A popular method of expressing the wave dissipation due to vegetation is the cylinder approach as suggested by Dalrymple et al. (1984). Here, energy losses are calculated as actual work carried out by the vegetation due to plant induced forces acting on the fluid, expressed in terms of a Morison type equation. In this method, vegetation motion such as vibration due to vortices and swaying is neglected. For relatively stiff plants the drag force is considered dominant and inertial forces are neglected. Moreover, since the drag due to friction is much smaller than the drag due to pressure differences, only the latter is considered. Based on this approach, the time-averaged rate of energy dissipation per unit area over the entire height of the vegetation is given by

  $$\varepsilon_{\rm v} = \frac{2}{3\pi} \rho C_{\rm D} b_{\rm v} N_{... ...ma} \right)^3 \frac{\sinh^3 k\alpha h + 3\sinh\, k\alpha h}{3k\cosh^3 kh} H^3$$ (2.139)

where $\rho$ is the water density, $C_{\rm D}$ is the drag coefficient, $b_{\rm v}$ is the stem diameter of cylinder (plant), $N_{\rm v}$ is the number of plants per square meter, $\alpha h$ is the vegetation height, $h$ is the water depth and $H$ is the wave height. This formula was modified by Mendez and Losada (2004) for irregular waves. The mean rate of energy dissipation per unit horizontal area due to wave damping by vegetation is given by

  $$<\varepsilon_{\rm v}> = \frac{1}{2\sqrt{\pi}} \rho {\tilde C}_{\r... ...)^3 \frac{\sinh^3 k\alpha h + 3\sinh\, k\alpha h}{3k\cosh^3 kh} H_{\rm rms}^3$$ (2.140)

with ${\tilde C}_{\rm D}$ is the bulk drag coefficient that may depend on the wave height. This is the only calibration parameter required for a given plant type.


To include wave damping due to vegetation in SWAN, Eq. (2.28) will be extended with $S_{\rm ds,veg}$ based on Eq. (2.141). A spectral version of the vegetation dissipation model of Mendez and Losada can be obtained by expanding Eq. (2.141) to include frequencies and directions as follows

  $$S_{\rm ds,veg} (\sigma,\theta) = \frac{D_{\rm tot}}{E_{\rm tot}} E(\sigma,\theta)$$ (2.141)

with

  $$D_{\rm tot} = -\frac{1}{2g\sqrt{\pi}} {\tilde C}_{\rm D} b_{\rm v... ...\alpha h + 3\sinh\, {\tilde{k}}\alpha h}{3k\cosh^3 {\tilde{k}}h} H_{\rm rms}^3$$ (2.142)

where the mean frequency $\tilde{\sigma}$, the mean wave number $\tilde{k}$ are given by Eqs. (2.47) and (2.48), respectively. With $H^2_{\rm rms} = 8E_{\rm tot}$, the final expression reads

  $$S_{\rm ds,veg} = -\sqrt{\frac{2}{\pi}} g^2 {\tilde C}_{\rm D} b_{... ...tilde{k}}\alpha h}{3k\cosh^3 {\tilde{k}}h} \sqrt{E_{\rm tot}} E(\sigma,\theta)$$ (2.143)

Apart from extending the Mendez and Losada's formulation to a full spectrum, the possibility to vary the vegetation vertically is included. The contribution of each vertical segment is calculated individually with the total energy dissipation equal to the sum of the dissipation in each layer up till the still water level. With this implementation of the differences in characteristics of each layer, plants such as mangrove trees may be conveniently input into the SWAN model. The layer-wise segmentation is implemented by integration of the energy dissipation over height as follows

  $$S_{\rm ds,veg} = \sum_{i=1}^{I} S_{{\rm ds,veg},i}$$ (2.144)

where $I$ the number of vegetation layers and $i$ the layer under consideration with the energy dissipation for layer $i$. First, a check is performed to establish whether the vegetation is emergent or submergent relative to the water depth. In case of submergent vegetation the energy contributions of each layer are added up for the entire vegetation height. In case of emergent vegetation only the contributions of the layers below the still water level are taken into account. The implementation of vertical variation is illustrated in Figure 2.3. The energy dissipation term for a given layer $i$ therefore becomes

$$S_{{\rm ds,veg},i} = -\sqrt{\frac{2}{\pi}} g^2 {\tilde C}_{{\rm D... ... v},i} \left( \frac{{\tilde{k}}}{{\tilde{\sigma}}} \right)^3 \sqrt{E_{\rm tot}}$$

$$\frac{\left(\sinh^3 {\tilde{k}}\alpha_i h - \sinh^3 {\tilde{k}}\a... ...h\, {\tilde{k}}\alpha_{i-1} h\right)} {3k\cosh^3 {\tilde{k}}h} E(\sigma,\theta)$$ (2.145)

The corresponding terms for each layer can then be added and the total integrated as described earlier to obtain the energy dissipation over the entire spectrum. Here $h$ is the total water depth and $\alpha_i$ the ratio of the depth of the layer under consideration to the total water depth up to the still water level, such that

  $$\sum_{i=1}^{I} \alpha_i \leq 1$$ (2.146)

Figure 2.3: Layer schematization for vegetation.

Finally, in addition to the vertical variation, the possibility of horizontal variation of the vegetation characteristics is included as well. This inclusion enables the vegetation in a given region to be varied so as to reflect real density variations in the field. Since, the parameters ${\tilde C}_{\rm D}$, $b_{\rm v}$ and $N_{\rm v}$ are used in a linear way, we can use $N_{\rm v}$ as a control parameter to vary the vegetation factor $V_f = {\tilde C}_{\rm D} b_{\rm v} N_{\rm v}$ spatially, by setting ${\tilde C}_{\rm D}=1$ and $b_{\rm v} = 1$, so that $V_f = N_{\rm v}$.


An alternative vegatation model is due to Jacobsen et al. (2019) for waves propagating over a canopy, with which the associated energy dissipation is frequency dependent. The spectral distribution of the dissipation is given by

  $$\delta_v = 2\Gamma S_u \sqrt{\frac{2m_{u,0}}{\pi}}$$ (2.147)

where $S_u$ is the velocity spectrum

  $$S_u = \left ( \frac{\sigma \cosh k(z+h)}{\sinh kh} \right )^2\, S_\eta$$ (2.148)

with $S_\eta$ the energy density spectrum, i.e. $E(\sigma)$. The zeroth moment of the velocity spectrum is computed as

  $$m_{u,0} = \int_0^\infty S_u\,df$$ (2.149)

Furthermore,

  $$\Gamma = \frac{1}{2} \rho C_{\rm D} b_{\rm v} N_{\rm v}\, \alpha_u^3$$ (2.150)

with $\alpha_u$ a velocity reduction factor.


The depth-integrated frequency-dependent dissipation is then found to be

  $$S_{\rm ds,veg} = -\frac{1}{\rho g}\,\int_{-h}^{-h+h_v} \delta_v\,dz$$ (2.151)

with $h_v$ the canopy height. The vertical integration is approximated using the Simpson's rule.