Difference between revisions of "Wave damping by vegetation"
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| − | Vegetation can make an important contribution to shore protection by damping incident waves, see [[Nature-based shore protection]] and [[Shore protection vegetation]]. Wave damping by vegetation is generally expressed by means of a friction drag coefficient <math>C_D</math>. This coefficient depends on plant characteristics and wave parameters. Vegetation and water flow interact in coupled, nonlinear ways<ref name="Koch">Koch, E.W., Sanford, L.P., Chen, S-N., Shafer, D.J., Mckee Smith, J., 2006. Waves in seagrass systems: Review and Technical recommendations. US Army Corps of Engineers®. Technical Report, ERDC TR-06-15. This interaction is dynamic since the structure of aquatic plant fields changes with time and is exposed to variable physical forcing of the water flow<ref>Mendez, F.J. | + | Vegetation can make an important contribution to shore protection by damping incident waves, see [[Nature-based shore protection]] and [[Shore protection vegetation]]. Wave damping by vegetation is generally expressed by means of a friction drag coefficient <math>C_D</math>. This coefficient depends on plant characteristics and wave parameters. Vegetation and water flow interact in coupled, nonlinear ways<ref name="Koch">Koch, E.W., Sanford, L.P., Chen, S-N., Shafer, D.J., Mckee Smith, J., 2006. Waves in seagrass systems: Review and Technical recommendations. US Army Corps of Engineers®. Technical Report, ERDC TR-06-15</ref>. This interaction is dynamic since the structure of aquatic plant fields changes with time and is exposed to variable physical forcing of the water flow<ref name=ML>Mendez, F.J. and Losada, I.J. 2004. An empirical model to estimate the propagation of random breaking and nonbreaking waves over vegetation fields. Coastal Engineering 51: 103-118</ref>. The damping effect of vegetation is stronger for incident waves with high frequencies than for incident waves with the low frequencies<ref>Dermentzoglou, D., Tissier, M., Muller, J.R.M., Hofland, B., Lakerveld, S., Borsje, B.W. and Antonini, A. 2026. Transformation of 𝐻𝑚0 and 𝑇𝑚−1,0 over a model salt marsh. Coastal Engineering 204, 104900</ref>. |
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<math>E_{dis} \approx ½ \rho C_D \Large\frac{b l}{s^2}\normalsize <|u_0 \sin \omega t|^3 > \approx \rho C_D \Large\frac{2}{3 \pi}\frac{bl}{s^2} (\frac{a c}{h})^3\normalsize \, .\qquad (A1)</math>. | <math>E_{dis} \approx ½ \rho C_D \Large\frac{b l}{s^2}\normalsize <|u_0 \sin \omega t|^3 > \approx \rho C_D \Large\frac{2}{3 \pi}\frac{bl}{s^2} (\frac{a c}{h})^3\normalsize \, .\qquad (A1)</math>. | ||
| − | The factor <math>\; bl/s^2 \;</math> is the plant frontal surface per unit seabed surface. Assumptions used in the formula (A1) are: (1) non-broken waves, (2) the shallow water approximation <math>kh << 1</math>, and (3) the wave orbital velocity in the canopy is not much smaller than the surface wave orbital velocity. For irregular waves an expression for the energy dissipation <math>E_{dis}</math> similar to (A1) can be derived<ref>Mendez, F.J. and Losada, I.J. 2004. An empirical model to estimate the propagation of random breaking and non-breaking waves over vegetation fields. Coast. Eng. 52: 103–118</ref> with the replacements <math>a \rightarrow \frac{1}{2} H_{rms}</math> and <math>\dfrac{2}{3 \pi} \rightarrow \dfrac{1}{2 \sqrt{\pi}}</math>. | + | The factor <math>\; bl/s^2 \;</math> is the plant frontal surface per unit seabed surface. Assumptions used in the formula (A1) are: (1) non-broken waves, (2) the shallow water approximation <math>kh << 1</math>, and (3) the wave orbital velocity in the canopy is not much smaller than the surface wave orbital velocity. For irregular waves an expression for the energy dissipation <math>E_{dis}</math> similar to (A1) can be derived<ref name=ML>Mendez, F.J. and Losada, I.J. 2004. An empirical model to estimate the propagation of random breaking and non-breaking waves over vegetation fields. Coast. Eng. 52: 103–118</ref> with the replacements <math>a \rightarrow \frac{1}{2} H_{rms}</math> and <math>\dfrac{2}{3 \pi} \rightarrow \dfrac{1}{2 \sqrt{\pi}}</math>. It should be noted that this substitution does not take into account the dependence of the dissipation on the wave frequency. |
For intermediate water (<math>kh \sim 1</math>) a factor <math>\dfrac{4 k h^2}{3l}\dfrac{\sinh^3 kl + 3 \sinh kl}{(\sinh 2kh + 2kh)\sinh kh}</math> must be included in the r.h.s. of Eq. (A1). | For intermediate water (<math>kh \sim 1</math>) a factor <math>\dfrac{4 k h^2}{3l}\dfrac{\sinh^3 kl + 3 \sinh kl}{(\sinh 2kh + 2kh)\sinh kh}</math> must be included in the r.h.s. of Eq. (A1). | ||
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An approach to avoid these complications, proposed by Luhar and Nepf (2016<ref name=L16/>), consists of the replacing the rigid stem length <math>l</math> in the wave energy dissipation equation (A1) with a reduced length <math>l_e</math>. They also proposed an empirical formula in which <math>l_e</math> depends on the blade flexibility through the elastic Young modulus <math>E</math>. While this approach worked well for individual seagrass plants, a different behavior was observed in seagrass meadows. In seagrass meadows, the blades are bending more strongly, thus opposing less resistance to the wave orbital flow. For strong steady or tidal currents, it was observed that bending of the blades strongly reduces the drag force exerted by the seagrass meadow<ref>Monismith, S. G., Hirsh, H., Batista, N., Francis, H., Egan, G. and Dunbar, R. B. 2019. Flow and drag in a seagrass bed. Journal of Geophysical Research: Oceans 124: 2153–2163</ref>. However, this is not the case in wave orbital flow, where, according to flume experiments, the drag coefficients for rigid and flexible vegetation are similar<ref>Lei, J. and Nepf, H. 2019. Wave damping by flexible vegetation: Connecting individual blade dynamics to the meadow scale. Coast. Eng. 147: 138–148</ref>. Reis et al. (2024<ref name=R24/>) suggested that this may be explained by the influence of the swaying motion of flexible blades on the acceleration of the wave orbital flow. | An approach to avoid these complications, proposed by Luhar and Nepf (2016<ref name=L16/>), consists of the replacing the rigid stem length <math>l</math> in the wave energy dissipation equation (A1) with a reduced length <math>l_e</math>. They also proposed an empirical formula in which <math>l_e</math> depends on the blade flexibility through the elastic Young modulus <math>E</math>. While this approach worked well for individual seagrass plants, a different behavior was observed in seagrass meadows. In seagrass meadows, the blades are bending more strongly, thus opposing less resistance to the wave orbital flow. For strong steady or tidal currents, it was observed that bending of the blades strongly reduces the drag force exerted by the seagrass meadow<ref>Monismith, S. G., Hirsh, H., Batista, N., Francis, H., Egan, G. and Dunbar, R. B. 2019. Flow and drag in a seagrass bed. Journal of Geophysical Research: Oceans 124: 2153–2163</ref>. However, this is not the case in wave orbital flow, where, according to flume experiments, the drag coefficients for rigid and flexible vegetation are similar<ref>Lei, J. and Nepf, H. 2019. Wave damping by flexible vegetation: Connecting individual blade dynamics to the meadow scale. Coast. Eng. 147: 138–148</ref>. Reis et al. (2024<ref name=R24/>) suggested that this may be explained by the influence of the swaying motion of flexible blades on the acceleration of the wave orbital flow. | ||
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Wave forces on rigid stems are dealt with in [[Shallow-water wave theory#Vertical Piles]]. | Wave forces on rigid stems are dealt with in [[Shallow-water wave theory#Vertical Piles]]. | ||
| + | The above descriptions assume that wave attenuation in vegetation areas is the sum of the wave attenuation by individual plants. In reality, wave attenuation is also influenced by the composition and structure of the plant community. Therefore, field observations remain essential for reliable estimates of the wave-attenuating effect of vegetation areas.<ref>Anderson, M.E., McKee Smith, J. and McKay, S.K. 2011. Wave Dissipation by Vegetation. Technical note USACE, ERDC/CHL CHETN-I-82</ref>. | ||
Latest revision as of 12:25, 5 December 2025
Vegetation can make an important contribution to shore protection by damping incident waves, see Nature-based shore protection and Shore protection vegetation. Wave damping by vegetation is generally expressed by means of a friction drag coefficient [math]C_D[/math]. This coefficient depends on plant characteristics and wave parameters. Vegetation and water flow interact in coupled, nonlinear ways[1]. This interaction is dynamic since the structure of aquatic plant fields changes with time and is exposed to variable physical forcing of the water flow[2]. The damping effect of vegetation is stronger for incident waves with high frequencies than for incident waves with the low frequencies[3].
Contents
Parameters relevant for wave-plant interaction
Wave attenuation by vegetation depends on many parameters referring to hydrographic conditions (mainly the parameters wave height, wave period and wave incidence direction) and vegetation characteristics (plant geometry, buoyancy, density, stiffness, degrees of freedom and vegetation sensity and spatial configuration). Relevant parameters are represented by the symbols listed below[4].
| Parameter | Symbol | Typical value, range | Parameter | Symbol | Typical value, range | Parameter | Symbol | Typical value, range |
|---|---|---|---|---|---|---|---|---|
| wave amplitude [[math]m[/math]] | [math]a[/math] | 0.2-1 | water depth [[math]m[/math]] | [math]h[/math] | 0.5-3 | wave orbital velocity amplitude [[math]m s^{-1}[/math]] | [math]u_0 \approx a c / h[/math] | 0.2-1 |
| blade / stem width [[math]m[/math]] |
[math]b[/math] | 0.003-0.3 | wave damping coefficient [[math]m^{-2}[/math]] | [math]K_D[/math] | 0.001-0.01 | vegetation cross-shore length [[math]m[/math]] | [math]x[/math] | 10-1000 |
| wave celerity [[math]m s^{-1}[/math]] | [math]c \approx \sqrt{gh}[/math] | 2-5 | Keulegan-Carpenter number | [math]KC = u_0 T/ b[/math] | 5-2000 | water density [[math]kg m^{-3}[/math]] | [math]\rho[/math] | 1025 |
| friction drag coefficient | [math]C_D[/math] | 0.5-3 | submerged blade / stem length [[math]m[/math]] | [math]l[/math] | (0.2-1) h | wave radial frequency [[math]s^{-1}[/math]] |
[math]\omega = 2 \pi / T[/math] | 0.5-2 |
| gravitational acceleration [[math]m s^{-2}[/math]] | [math]g[/math] | 9.8 | plant spacing [m] | [math]s[/math] | 0.02 - 0.07 |
Wave energy dissipation equation
The wave energy dissipation [math]E_{dis}[/math] per unit seabed surface [[math] kg s^{-3}[/math]] due to the drag force of rigid vegetation, can be approximated for regular waves by[5]
[math]E_{dis} \approx ½ \rho C_D \Large\frac{b l}{s^2}\normalsize \lt |u_0 \sin \omega t|^3 \gt \approx \rho C_D \Large\frac{2}{3 \pi}\frac{bl}{s^2} (\frac{a c}{h})^3\normalsize \, .\qquad (A1)[/math].
The factor [math]\; bl/s^2 \;[/math] is the plant frontal surface per unit seabed surface. Assumptions used in the formula (A1) are: (1) non-broken waves, (2) the shallow water approximation [math]kh \lt \lt 1[/math], and (3) the wave orbital velocity in the canopy is not much smaller than the surface wave orbital velocity. For irregular waves an expression for the energy dissipation [math]E_{dis}[/math] similar to (A1) can be derived[2] with the replacements [math]a \rightarrow \frac{1}{2} H_{rms}[/math] and [math]\dfrac{2}{3 \pi} \rightarrow \dfrac{1}{2 \sqrt{\pi}}[/math]. It should be noted that this substitution does not take into account the dependence of the dissipation on the wave frequency.
For intermediate water ([math]kh \sim 1[/math]) a factor [math]\dfrac{4 k h^2}{3l}\dfrac{\sinh^3 kl + 3 \sinh kl}{(\sinh 2kh + 2kh)\sinh kh}[/math] must be included in the r.h.s. of Eq. (A1).
Wave dissipation causes a decrease of the incident wave energy flux [math]dF/dx \; [kg s^{-3}][/math] given by
[math]dF/dx \approx ½ \rho g c \Large\frac{d a^2}{dx}\normalsize \, ,\qquad (A2)[/math]
where [math]a[/math] is the wave amplitude. This formula assumes a horizontal seabed and shallow water where the wave group velocity [math]c_g[/math] can be approximated by [math]c \approx \sqrt{gh}[/math].
The decay of the wave amplitude [math]a(x)[/math] can be found by equating [math]dF/dx = E_{dis}[/math] with the result
[math]a(x) = \dfrac{a_0}{1 + K_D a_0 x} , \quad K_D = \dfrac{2}{3 \pi} C_D \dfrac{b l}{h^2 s^2} \, , \qquad (A3)[/math]
where [math]a_0[/math] is the amplitude of the wave entering the vegetated area.
Empirical expressions of the friction drag coefficient [math]C_D[/math]
Many empirical expressions for the coefficient [math]C_D[/math] can be found in the literature, based on flume experiments or field observations, with rigid or flexible stems and synthetic or natural plants[6]. Most formulas relate the friction drag coefficient to the Reynolds number or to the Keulegan-Carpenter number [math]KC[/math]. For example, Chen et al. (2018[7]) propose
[math]C_D \approx 1.2 + 13 \, KC^{-1.25} \qquad (A4)[/math]
For emerging vegetation (e.g. mangroves), the length [math]l[/math] has to be taken equal to the water depth [math]h[/math]. Typical values of [math]C_D[/math] for vegetation with rigid stems are in the range 0.5-3.
The expression (A3) assumes submerged vegetation with rigid stems. For vegetation with flexible stems and leaves (e.g. seagrass), the equations describing the energy dissipation are more complex, involving the elastic restoring force due to blade stiffness[8][9].
An approach to avoid these complications, proposed by Luhar and Nepf (2016[8]), consists of the replacing the rigid stem length [math]l[/math] in the wave energy dissipation equation (A1) with a reduced length [math]l_e[/math]. They also proposed an empirical formula in which [math]l_e[/math] depends on the blade flexibility through the elastic Young modulus [math]E[/math]. While this approach worked well for individual seagrass plants, a different behavior was observed in seagrass meadows. In seagrass meadows, the blades are bending more strongly, thus opposing less resistance to the wave orbital flow. For strong steady or tidal currents, it was observed that bending of the blades strongly reduces the drag force exerted by the seagrass meadow[10]. However, this is not the case in wave orbital flow, where, according to flume experiments, the drag coefficients for rigid and flexible vegetation are similar[11]. Reis et al. (2024[9]) suggested that this may be explained by the influence of the swaying motion of flexible blades on the acceleration of the wave orbital flow.
Wave forces on rigid stems are dealt with in Shallow-water wave theory#Vertical Piles.
The above descriptions assume that wave attenuation in vegetation areas is the sum of the wave attenuation by individual plants. In reality, wave attenuation is also influenced by the composition and structure of the plant community. Therefore, field observations remain essential for reliable estimates of the wave-attenuating effect of vegetation areas.[12].
Related articles
- Nature-based shore protection
- Seagrass meadows
- Salt marshes
- Mangroves
- Shore protection vegetation
- Climate adaptation measures for the coastal zone
References
- ↑ Koch, E.W., Sanford, L.P., Chen, S-N., Shafer, D.J., Mckee Smith, J., 2006. Waves in seagrass systems: Review and Technical recommendations. US Army Corps of Engineers®. Technical Report, ERDC TR-06-15
- ↑ 2.0 2.1 Mendez, F.J. and Losada, I.J. 2004. An empirical model to estimate the propagation of random breaking and nonbreaking waves over vegetation fields. Coastal Engineering 51: 103-118 Cite error: Invalid
<ref>tag; name "ML" defined multiple times with different content - ↑ Dermentzoglou, D., Tissier, M., Muller, J.R.M., Hofland, B., Lakerveld, S., Borsje, B.W. and Antonini, A. 2026. Transformation of 𝐻𝑚0 and 𝑇𝑚−1,0 over a model salt marsh. Coastal Engineering 204, 104900
- ↑ Vettori, D., Pezzutto, P., Bouma, T.J., Shahmohammadi, A. and Manes, C. 2024. On the wave attenuation properties of seagrass meadows. Coastal Engineering 189, 104472
- ↑ Dalrymple, R.A., Kirby, J.T. and Hwang, P.A. 1984. Wave diffraction due to areas of energy dissipation. J. Waterw. Port Coast. Ocean Eng. 110: 67–79
- ↑ Yin, K., Xu, S. Huang, W., Xu, H., Lu, Y. and Ma, M. 2024. A study on the drag coefficient of emergent flexible vegetation under regular waves. Ocean Modelling 191, 102422
- ↑ Chen, H., Ni, Y., Li, Y., Liu, F., Ou, S., Su, M., Peng, Y., Hu, Z., Uijttewaal, W. and Suzuki, T. 2018. Deriving vegetation drag coefficients in combined wave-current flows by calibration and direct measurement methods. Adv. Water Resour. 122: 217–227
- ↑ 8.0 8.1 Luhar, M. and Nepf, H.M. 2016. Wave-induced dynamics of flexible blades. J. Fluids Struct. 61: 20–41
- ↑ 9.0 9.1 Reis, R.A., Fortes, C.J.E.M., Rodrigues, J.A., Hu, Z. and Suzuki, T. 2024. Experimental study on drag coefficient of flexible vegetation under non-breaking waves. Ocean Engineering 296, 117002
- ↑ Monismith, S. G., Hirsh, H., Batista, N., Francis, H., Egan, G. and Dunbar, R. B. 2019. Flow and drag in a seagrass bed. Journal of Geophysical Research: Oceans 124: 2153–2163
- ↑ Lei, J. and Nepf, H. 2019. Wave damping by flexible vegetation: Connecting individual blade dynamics to the meadow scale. Coast. Eng. 147: 138–148
- ↑ Anderson, M.E., McKee Smith, J. and McKay, S.K. 2011. Wave Dissipation by Vegetation. Technical note USACE, ERDC/CHL CHETN-I-82
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