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An empirical model to estimate the propagation of random breaking and nonbreaking waves over vegetation fields
Coastal Engineering 51 (2004) 103 – 118 www.elsevier.com/locate/coastaleng
An empirical model to estimate the propagation of random breaking and nonbreaking waves over vegetation fields Fernando J. Mendez *, Inigo J. Losada 1 Ocean and Coastal Research Group, Universidad de Cantabria, Dpto. de Ciencias y Te´cnicas del Agua y del Medio Ambiente, Av. de los Castros s/n 39005, Santander, Spain Received 12 March 2003; received in revised form 29 September 2003; accepted 5 November 2003
Abstract In this work, a model for wave transformation on vegetation fields is presented. The formulation includes wave damping and wave breaking over vegetation fields at variable depths. Based on a nonlinear formulation of the drag force, either the transformation of monochromatic waves or irregular waves can be modelled considering geometric and physical characteristics of the vegetation field. The model depends on a single parameter similar to the drag coefficient, which is parameterized as a function of the local Keulegan – Carpenter number for a specific type of plant. Given this parameterization, determined with laboratory experiments for each plant type, the model is able to reproduce the root-mean-square wave height transformation observed in experimental data with reasonable accuracy. D 2004 Elsevier B.V. All rights reserved. Keywords: Wave propagation; Vegetation field; Kelp bed; Drag force; Wave damping; Breaking waves; Empirical model; Dissipation
1. Introduction The importance of aquatic vegetation such as belts of seaweeds and seagrass as a biological and physical component of the coastal system has been widely recognised but is far from being understood. Aquatic vegetation is known to be food and shelter for many organisms, to control biogeochemical cycles in the coastal zone and to dissipate wave energy and turbulence protecting the shore from erosion. (f.i., Ifuku and Hayashi, 1998; Gacia and Duarte, 2001; Fonseca * Corresponding author. Tel.: +34-942-201810; fax: +34-942201860. E-mail addresses: [email protected] (F.J. Mendez), [email protected] (I.J. Losada). 1 Tel.: +34-942-201810; fax: +34-942-201860. 0378-3839/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2003.11.003
and Calahan, 1992). Furthermore, aquatic vegetation can influence coastal hydrodynamics, dependent upon its size, location, density, distribution and morphology. On the other hand, as pointed out by Koehl (1984), hydrodynamics influence the structure of the underwater plant communities by setting distribution limits related to wave and current exposure. Hydrodynamics also play a major role in controlling the dynamics of the underwater vegetation fields by dispersing spores and propagules and by mediating the availability of nutrients. The interaction of the water flow with the underwater vegetation is dynamic such that the structure of aquatic plant fields changes with time and is exposed to variable physical forcing of the water flow. Yet, this interaction also has some limits related to the capacity of the plants to resist the stress forces derived from the
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physical forcing. Drag forces and breaking forces from wind waves can remove plants (f.i., Seymour et al., 1989; Mork, 1996). Morphological and functional adaptations of marine plants can, to some extent, counteract the negative influence of hydrodynamic forces (Koehl, 1984). There are numerous examples showing the importance of shallow water wave-exposed aquatic vegetation for economical as well as for biological reasons: for instance, the harvesting of kelp Laminaria hyperborea in the Norwegian coast (Dubi and Torum, 1995; Mork, 1996), the effects of Macrocystis pyrifera kelp beds on the Californian coast (Elwany et al., 1995; Elwany and Flick, 1996), the shore protection of Posidonia oceanica seagrass meadows along the Mediterranean coast (Gacia and Duarte, 2001), the wave energy dissipation of Spartina alterniflora fields in English salt marshes (Moller et al., 1996, 1999) or the high biological activity of Zostera marina (Fonseca and Calahan, 1992; Ifuku and Hayashi, 1998). Wave attenuation depends on the characteristics of the plant (geometry, buoyancy, density, stiffness, degrees of freedom and spatial configuration) as well as wave parameters (mainly wave height, period and direction). The variability of wave damping is very large and trying to define a generalized behaviour of the ‘‘plant-induced dissipation’’ is absolutely impossible. For example, experimental data in L. hyperborea kelp beds in the coast of Norway show a damping of more than 60% of wave height (Mork, 1996) or a reduction of almost 40% of wave height was measured in emerged wetland plants in Lake Ontario, Canada (Tschirky et al., 2001). On the other hand, extensive field measurements carried out seaward and leeward of a very robust M. pyrifera kelp bed along the Californian coast assure that wave energy is practically unaffected by these plants (Elwany et al., 1995). Therefore, it is clear that an adequate modelling on wave transformation along vegetation fields would be highly desirable. As a consequence, several authors have carried out theoretical and numerical work mainly concentrated on wave transformation induced by a vegetation field. In general, the theoretical models try to estimate wave-induced forces on the plants. It must be noted that the validity of each model depends on the geometrical and biomechanical characteristics of the plants: on the one hand, if the plants
are subsurface, short or if its stiffness in the lower part is strong (f.i., L. hyperborea kelp beds, P. oceanica seagrass meadows, Spartina marshes), the plants can be considered as vertical cylinders and a drag force model can be considered to be valid. On the other hand, if the plants have a large number of degrees of freedom, the stiffness is reduced and the buoyancy is high (f.i., M. pyrifera kelp beds); this approach is not correct and a more complex modellization for the force would be required (Seymour, 1996). This paper is concerned about the flexible vertical cylinder-like plant, limiting its potential applications to the first category of plants. Initial models were based on neglecting plant motion and expressed in terms of a wave shear stress friction coefficient (Teeter et al., 2001) or the drag force acting on the vegetation (i.e. Dalrymple et al., 1984; Kobayashi et al., 1993). Asano et al. (1993) extend this solution to include the vegetation motion, coupling the flow field and swaying motion of the plants and comparing the solution with artificial seaweed experiments carried out in the lab. Dubi (1995) presents further laboratory experiments, analyzing the sensitivity of the damping rate to several parameters. Mendez et al. (1999a) extend previous work modelling wave height evolution, vegetation and fluid motion and forces and moments on the vegetation under irregular wave conditions and analyzing the consequences on wave transformation. However, all of these models, based on linearizing the drag force acting on the plant, have been developed for horizontal bottom and do not include wave breaking which are important shortcomings to analyzing real vegetation fields. Recently, Lovas (2000) has developed a wave transformation model in the surf zone based on the Larson (1995) model that takes into account the effect of kelp. Lovas’ model includes variable depth and wave breaking; however, the force is still linearized and the approach used limits the potential application of the model. Therefore, it cannot be applied as a predictive tool. In this work, an empirical model for wave transformation on vegetation fields is carried out. The presented formulation includes wave damping and wave breaking over vegetation fields on variable depths. Based on a nonlinear formulation of the drag force, either the transformation of monochromatic waves or random waves can be modelled considering
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geometric and physical characteristics of the vegetation field. The model depends on a single parameter similar to the drag coefficient which has been calibrated for a specific type of plant (L. hyperborea kelp). Results comparing numerical and experimental data will show the accuracy of the model. The paper is organized as follows. First, a review of regular wave transformation on vegetation fields is presented for constant depth as well as for a plane sloping beach. Later, the formulation for random waves is derived. The complete model for breaking and nonbreaking normally incident random waves is given in the next section. After the theoretical approach, the validation of the model for a specific type of vegetation follows based on experimental data. Next, the model is used to analyze the influence of several parameters on the wave transformation processes. Finally, some conclusions are given.
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where h is the water depth and ah is the mean vegetation height, the overbar stands for time average in a wave period, F=( Fx, 0, Fz) is the force acting on the vegetation per unit volume and u=(u, 0, w) is the velocity for the 2D case. Substituting into the last equation yields ev ¼
Z
hþah
ðFx u þ Fz wÞdz
ð3Þ
h
It is usually assumed (f.i., Kobayashi et al., 1993; Dubi and Torum, 1997; Mendez et al., 1999a) that in an anisotropic dissipative media such as the vegetation field, the term Fzw is negligible in comparison with Fxu. Therefore, the time-averaged rate of energy dissipation per unit horizontal area ev can be expressed as ev ¼
Z
hþah
Fx udz
ð4Þ
h
2. Nonbreaking wave transformation model 2.1. Wave energy dissipation Waves propagating through vegetation (seaweed, kelp beds and marsh grass) lose energy due to the work carried out on the vegetation. Assuming that linear wave theory is valid and considering regular waves normally incident on a coastline with straight and parallel contours, the conservation of energy equation is reduced to BEcg ¼ ev ð1Þ Bx where E=(1/8)qgH 2 is the energy density, H is the wave height, g is the acceleration of gravity, q is the water pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi density, cg is the group velocity, c ¼ ðg=kÞtanhkh is the wave celerity, r is the wave angular frequency, n=(1/2)[1+(2kh/sinh2kh)], k is the wave number, x is the onshore coordinate and ev is the time-averaged rate of energy dissipation per unit horizontal area induced by the vegetation. For a given vegetation field, the conventional definition for the depth-integrated and time-averaged energy dissipation per horizontal area unit is given by: Z hþah ev ¼ Fudz ð2Þ h
The correct calculation of the force Fx should take into account the relative motion between the fluid and the plant and both inertial and drag forces, considering a huge amount of geometrical and biomechanical parameters. Although in the past few years, several models have been proposed that consider the plant swaying (f.i., Asano et al., 1993; Dubi and Torum, 1995; Mendez et al., 1999a), in this work, plant motion has been neglected in order to obtain an empirical model. For this reason, and according to several authors (Dalrymple et al., 1984; Kobayashi et al., 1993), plant-induced forces acting on the fluid are expressed in terms of a Morison-type equation neglecting swaying motion and inertial force. Therefore, the horizontal force per unit volume is given by Fx ¼
1 qCD bv NuAuA 2
ð5Þ
where u is the horizontal velocity in the vegetation region due to the wave motion, bv is the plant area per unit height of each vegetation stand normal to u, N is the number of vegetation stands per unit horizontal area and CD is a depth-averaged drag coefficient. It must be noted that the correct estimation of the nonlinear force Fx should be calculated by using the relative velocity ur between fluid and plant instead of u (Mendez et al., 1999a). For this reason, Eq. (5) is taken to be valid not only for rigid plants, but also for
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flexible ones considering a different value of a bulk drag coefficient CD to cover our ignorance of the plant motion (Dalrymple et al., 1984). A number of authors (f.i., Kobayashi et al., 1993; Dubi and Torum, 1995; Losada et al., 1996; Mendez et al., 1999a) have solved the boundary problem of waves propagating over a dissipative media analytically linearizing the friction terms and obtaining a complex wave number KR + K iI that considers that waves propagate proportionally to exp( KIx)cos(KRx). In this case, the linearization of the force (in terms of H) is needed. Nevertheless, in this paper, a more general model which takes the nonlinear force into account is presented. It has been assumed that the linear wave theory for waves propagating over an impermeable bottom is valid to calculate u not only for the water region, but also within the vegetation area. Dalrymple et al. (1984) expressed the energy dissipation for waves propagating through a vegetation field as ev ¼
where Ao ¼
8 sinh3 kah þ 3sinhkah CD bv Nk 9p ðsinh2kh þ 2khÞsinhkh
Solving the linear differential equation and assuming that the wave height at the seaward limit of the vegetation field is H(x = 0) = Ho, the wave height evolution is equal to Ho ¼ Kv Ho 1 þ bx
H¼
ð9Þ
where Kv is the damping coefficient 1 1 þ bx
Kv ¼
ð10Þ
and b is b¼
Ao Ho 4 CD bv NHo k ¼ 9p 2
3 2 kg sinh3 kah þ 3sinhkah 3 qCD bv N H ð6Þ 3p 2r 3kcosh3 kh
A similar approach was presented by Kobayashi et al. (1993) which considered the real part of the complex wave number KR instead of k. However, it can be demonstrated that both numbers (KR and k) are practically the same for the typical values of dissipation of a vegetation field. It must be noted that Eq. (6) does not consider the reflection induced by the plants. However, the importance of this factor is very limited in terms of wave energy. For example, if we consider a reflection of, say, ARA = 20%, this represents an energy reflection of 4% since energy is proportional to jRj2. Therefore, Eq. (6) can be considered to be valid.
ð8Þ
sinh3 kah þ 3sinhkah : ðsinh2kh þ 2khÞsinhkh
ð11Þ
2.2.2. Plane sloping beach An analytical solution can be obtained in shallow water for waves propagating normally through a vegetation field on a plane sloping beach. Starting with the energy flux conservation Eq. (1) and describing the energy dissipation using Eq. (6) gives BH 2 h1=2 H3 ¼ A1 1=2 Bx h
ð12Þ
where A1 ¼
2CD bv N a 3p
ð13Þ
For a plane sloping beach, the bottom depth can be expressed as
2.2. Monochromatic wave height transformation
h ¼ ho mx
ð14Þ
2.2.1. Constant depth For the case of horizontal bottom, using Eq. (6), i.e. nonlinear drag force, Eq. (1) can be rewritten as (Dalrymple et al., 1984)
where m is the slope and ho is the seaward water depth. By defining y = H2h1/2, Eq. (12) is rewritten as
BH 2 ¼ Ao H 3 Bx
By A1 y3=2 ¼ Bh m h5=4
ð7Þ
ð15Þ
F.J. Mendez, I.J. Losada / Coastal Engineering 51 (2004) 103–118
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Fig. 1. Dependence of A1Ho /m for shallow water waves propagating over a vegetation field.
Eq. (15) can be easily integrated assuming a constant value for A1: 2 2A1 þ C ð16Þ y¼ mh1=4 Applying a boundary condition H = Ho in h = ho, ð17Þ
H ¼ Ho Kv Ks where the shoaling is governed by Green’s law 1=4
ho h1=4 and the damping coefficient is Ks ¼
Kv ¼
1þ2
A1 m
1 Ho ðKs 1Þ
ð18Þ
ð19Þ
The asymptotic case for a flat slope m ! 0 is Kv ¼
1 1 Ho x 1 þ A2h o
ð20Þ
since 1 lim m!0 m
!
1=4
ho
ðho mxÞ
1=4
1
¼
x 4ho
ð21Þ
Eq. (20) is the shallow water approximation of Eq. (10) for uniform depth. On the other hand, for null damping (A1 = 0), the damping coefficient Kv is unity. The dimensionless wave evolution H/Ho is plotted versus the normalized water depth h/ho for several values of the damping coefficient A1Ho/m in Fig. 1. Breaking is not considered. As can be seen, an increasing damping coefficient results in a greater reduction in wave height. No wave height variation is found for an approximate value of A1Ho/m = 0.5. This means that even though shoaling effects are important, the energy dissipation induced by the vegetation field cancels out the wave height growth. Depending on the value of A1Ho/m, the curvature of the wave evolution does not always correspond to the known exponential-decay-form for constant depth.
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2.3. Random wave transformation model
we obtain
Waves can be described by a joint distribution of wave height, period (or frequency) and direction. To simplify the analysis, it is usually assumed that the waves are very narrow-banded in frequency and come from the same direction, such that all wave heights are related to an average peak period Tp and mean direction h¯. A Rayleigh distribution often gives a satisfactory characterization of the random variation in wave height. The Rayleigh probability density function (pdf) is described by the root-mean-square wave height, Hrms, according to " # 2H H 2 pðHÞ ¼ 2 exp Hrms Hrms
ð22Þ
Z
l
H 2 pðHÞdH
ð23Þ
0
The model developed here describes the transformation of the wave height distribution assuming an unmodified Rayleigh distribution. For straight and parallel bottom contours, the average energy flux balance for normally incident waves is given by BEcg ¼ hev i Bx where Ecg ¼
1 qgcp np 8
ð24Þ Z
sinh3 kah þ 3sinhkah 3 pffiffiffi 3kcosh kh 3 p 3 Hrms ð28Þ 4 The derivation of Eq. (28) is not strictly correct since the drag coefficient may depend on the wave height. An average drag coefficient C˜D is proposed to overcome this problem. The evaluation of C˜D will be shown in the validation section. For a flat bottom, an expression similar to Eq. (7) is found
2 BHrms 3 ¼ Bo Hrms Bx where
where 2 ¼ Hrms
3 2 B 18 qgHrms cn 2 ˜ kg ¼ qCD bv N 3p 2r Bx
ð29Þ
2 sinh3 kah þ 3sinhkah Bo ¼ pffiffiffi C˜ D bv Nk ðsinh2kh þ 2khÞsinhkh 3 p
ð30Þ
Solving Eq. (29) and applying the boundary condition Hrms(x = 0) = Hrms,o, the root-mean-square wave height evolution is equal to Hrms;o Hrms ¼ ð31Þ ˜ 1 þ bx where Bo Hrms;o 1 ¼ pffiffiffi C˜ D bv NHrms;o k b˜ ¼ 2 3 p
l
H 2 pðHÞdH
ð25Þ
0
sinh3 kah þ 3sinhkah ðsinh2kh þ 2khÞsinhkh
ð32Þ
and hev i ¼
kp g 3 sinh3 kp ah þ 3sinhkp ah 2 ˜ qC D b v N 3p 2rp 3kp cosh3 kp h Z l H 3 pðHÞdH ð26Þ 0
where the magnitudes with subscript p are associated with the peak period Tp. For simplicity, in the following, the subscript p will be neglected. Applying Eq. (23) and pffiffiffi Z l 3 p 3 Hrms H 3 pðHÞdH ¼ ð27Þ 4 0
3. Random breaking waves model 3.1. Theoretical model The transformation of random waves, including shoaling, dissipation due to breaking and damping by a vegetation field, is described by an energy flux balance model. Lovas (2000) has recently developed a wave transformation model in the surf zone based on Larson (1995) that takes into account the effect of kelp. In that work, the damping due to vegetation in a horizontal cell of size Dx is modelled using
F.J. Mendez, I.J. Losada / Coastal Engineering 51 (2004) 103–118
Kv = exp( KIDx), where KI has been previously fitted as a function of the local water depth h for a specific type of kelp. However, this model cannot be used as a predictable tool since KI does not depend on the geometrical properties of the vegetation field or wave characteristics. In this paper, a similar approach has been carried out; however, the damping depends on wave and plant parameters. For straight and parallel contours, the energy conservation equation for normally incident waves, including breaking hebi and dissipation due to the vegetation hevi, is given by
109
In this model, Eq. (33) is solved together with the mean water level variations g¯, calculated following the balance of x-momentum for a vegetation field BSxx Bg¯ ¼0 ð36Þ þ qgh Bx Bx in which d is the water depth for the still water level, h = d + g¯ is the mean water level and the radiation stress component Sxx is calculated taking into account the effects of the vegetation (Mendez et al., 1999b). The results are not sensitive to the setup model used. 3.2. Numerical model
BEcg ¼ heb i hev i Bx
ð33Þ
The average rate of energy dissipation per unit area for wave breaking is (Thornton and Guza, 1983) pffiffiffi B 3 fp 7 3 p heb i ¼ qg 4 5 Hrms 16 cb h
ð34Þ
where B and cb are adjusting parameters and fp is an average frequency corresponding to the peak period Tp. The dissipation due to vegetation is obtained from Eq. (28) assuming an invariant Rayleigh distribution, 3 1 kg ˜ hev i ¼ pffiffiffi qCD bv N 2r 2 p
sinh3 kah þ 3sinhkah 3 Hrms 3kcosh3 kh
ð35Þ
The major assumption in this model is the linear summation of hebi and hevi. This has been done due to the difficulty of physically identifying the contribution of each of the terms when a wave is breaking along a vegetation field. However, as will be shown in the model validation, this approximation is good enough for describing the wave energy transformation whenever the drag coefficient C˜D is previously calibrated. It must be noted that the assumption of an invariant Rayleigh-like wave height distribution is adopted to obtain the average rate of energy dissipation per unit area. The analysis of the shape of the transformed wave height distribution is not considered in this work.
For the solution of Eqs. (33) and (36), numerical integration must be performed by substituting Eqs. (34) and (35) and solving from offshore to the shoreline. A forward stepping scheme is used assuming cells of size Dx and constant depth, locally. In the calibration of the drag coefficient shown below, C˜D appears as a function of the orbital velocity at the top of the plant, which depends on Hrms. Therefore, an iterative procedure to calculate C˜D is needed. In the application of the model, the coefficients B = 1 and cb = 0.6 have been adopted as fixed values as recommended in Mase and Kirby (1993). The initial condition for the integration of Eq. (36) is the set-down calculated from linear theory for a progressive wave. An iterative procedure is needed since the energy dissipation depends on h. Three iterations have been found to be enough for the cases considered. 3.3. Shallow water approximation In order to analyze the importance of each term, the shallow water approximation of Eq. (33) is obtained. For kh ! 0, the energy flux balance yields 2 3 BHrms h1=2 H7 Hrms ¼ C1 rms A1 1=2 5 Bx h h
where A1 was defined in Eq. (13) and pffiffiffi 3 pB3 C1 ¼ 4 2cb Tp g 1=2
ð37Þ
ð38Þ
Fig. 2 shows, for a plane-sloping beach, the evolution of the root-mean-square wave height for shoaling
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Fig. 2. Nondimensional root-mean-square wave height evolution over a plane sloping beach for different theories.
(A1 = 0, C1 = 0), shoaling and vegetation (A1 = 0.01, C1 = 0), shoaling and breaking (A1 = 0, C1 = 1) and shoaling, vegetation and breaking (A1 = 0.01, C1 = 1). 7 3 As can be seen, both C1H rms /h5 and A1H rms /h1/2 contribute to the energy dissipation. Results show how the damping caused by vegetation tends to reduce the ratio Hrms/h and therefore inducing wave breaking farther onshore.
4. Model validation 4.1. Nonbreaking random waves The theoretical solution for random waves is compared to the experimental results for an artificial kelp field given by Dubi (1995). The experiment was carried out in a 33-m-long, 1-m-wide and 1.6-m-high wave flume. The width of the channel was partitioned to give a width of 0.5 m. At the end of the wave flume, a wave absorber was installed to reduce reflection. The artificial kelp models were L. hyperborea with a plant area per unit height of bv = 0.025 m and a height of dv = 0.2 m.
This kelp plant consists of a thin stipe and a very flexible frond split in many segments, with the stipe being about half of the total height. The vegetation field, located at the center of the flume, had a total width of b = 9.3 m (Dubi and Torum, 1997, Fig. 1). The number of uniformly distributed plants per unit horizontal area was N = 1200 units/m2. A total of 154 runs were carried out with varying water depth h = 0.4 –1 m, wave peak periods, Tp = 1.26 –4.42 s, and root-meansquare wave heights, Hrms = 0.045 – 0.17 m. The input for irregular waves was the Joint North Sea Wave Project (JONSWAP) spectrum with shape parameter cj = 3.3. Free surface oscillations were measured by eight wave gauges at x = 1.15, . . ., 8.15 m, with x = 0 m at the front face of the vegetation field. In some runs, the last two gauges were not functioning properly. For each of the runs, the best fit to the experimental data was obtained using a least-squares method considering C˜D the single calibration parameter. Prior to any results analysis, it must be pointed out that, for simplicity, the model presented does not consider the reflection process. The linear solution for the boundary value problem including reflection and evanescent modes was solved by Mendez et al. (1999a).
F.J. Mendez, I.J. Losada / Coastal Engineering 51 (2004) 103–118
Fig. 3 shows the comparisons between the experimental data and the theoretical root-mean-square wave height described by Eq. (31) for six different runs covering the range of parameters. As can be seen, the agreement is pretty good, although a slight modulation in the data is present due to reflection. It is clear that the success of this model as a predictive tool requires the knowledge of the bulk drag coefficient C˜D. The value of this parameter depends on the flow around the plants and the plant
111
motion, which are functions of the hydrodynamic and biomechanical characteristics. However, the correct modelling of swaying motion is far from trivial not only for the difficulty of solving a nonlinear dynamic oscillator with n degrees of freedom, but also for the complexity and randomness of geometric and biomechanical properties of the plants. Several attempts for uniform flow (f.i., Kouwen and Unny, 1973; Kouwen, 1988; Carollo et al., 2002; Ghisalberti and Nepf, 2002) as well as for oscillatory flow (f.i., Asano et al., 1992;
Fig. 3. Numerical and experimental results of Hrms evolution.
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Dubi and Torum, 1995; Mendez et al., 1999a) have been carried out to model the biomechanical properties as well as the plant motion. Nevertheless, all these models need physical experimentation in order to calibrate the drag coefficient. For that reason, in this paper, a simplified analysis has been performed focusing mainly on the hydrodynamic processes and avoiding the plant motion. Therefore, a relation between C˜D and some nondimensional flow parameters is desirable to characterize hydrodynamically the L. hyperborea model plants for predictable purposes. In recent studies (Kobayashi et al., 1993; Mendez et al., 1999a), it has been found that the drag coefficient depends directly on the Reynolds number. However, in this paper, a more suitable relation is found using the Keulegan – Carpenter number, K, defined as K = ucTp/bv, where uc is a characteristic velocity acting on the plant and defined as the maximum horizontal velocity at the middle of the vegetation field x = b/2
Fig. 5. Estimated versus calculated drag coefficient C˜D.
Fig. 4. Calibrated value of drag coefficient C˜D as function of Keulegan – Carpenter number K and the relative plant height a.
F.J. Mendez, I.J. Losada / Coastal Engineering 51 (2004) 103–118
and z = h + ah. The velocity uc is defined using Hrms and Tp as the wave height and the wave period corresponding to a monochromatic wave train.
113
Fig. 4 shows the calibrated values of C˜D as a function of K for the 154 runs. For this case, the C˜D coefficient varies between 0.01 and 0.52. The follow-
Fig. 6. Calculated and measured root-mean-square wave height.
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ing relationship with the Keulegan– Carpenter number can be found
4.2. Breaking random waves
C˜ D ¼ 0:47expð0:052KÞ 3VKV59
The laboratory data of Lovas (2000) are used to validate the numerical model for the transformation of random waves including dissipation by vegetation and breaking. The experimental setup was carried out in a 40-m-long and 5-m-wide wave flume at SINTEF (Norway). The width of the channel was partitioned to give a width of 0.6 m. Five thousand L. hyperborea model plants were prepared on a sandy 1:30 slope. The sand used in the experiment had D50 = 0.22 mm and its density was 2650 kg/m3. The vegetation field, located at the center of the flume, had a total width of b = 7.27 m (Lovas, 2000, Fig. 4.43; Lovas and Torum, 2001, Fig. 2). The maximum number of uniformly distributed plants per unit horizontal area was N = 1200 units/m2. For the validation of the root-mean-square wave height evolution, 13 runs were analyzed, with varying water depth h = 0.69– 0.77 m, wave peak periods T p = 2.5 and 3.5 s, significant wave heights Hmo = 1.416Hrms = 0.12 –0.22 m and varying N (0, 600 and 1200 units/m2). For each run, free surface oscillations were measured at 10 wave gauges, the
ð39Þ
with a 76% correlation. Several parameters have been considered in order to understand the scattering in the data. It has been found that the drag coefficient depends not only on K, but also on the relative height of the plants, a, see Fig. 4. Although the formulation for the energy dissipation takes into account the relative height of the plants, the force induced by the plants depends strongly on the swaying motion, which is not considered in Eq. (6). Besides, the horizontal displacement of each strip is a function of a as pointed out by Mendez et al. (1999a). The best relation for C˜D as a function of K and a has been obtained using a modified Keulegan –Carpenter parameter Q = K/a0.76 and is given by expð0:0138QÞ C˜ D ¼ Q0:3
7VQV172
ð40Þ
The calculated values of C˜D versus the estimated values using Eq. (40) are plotted in Fig. 5 showing that the correlation obtained (92%) is pretty good.
Fig. 7. Measured and calculated root-mean-square wave height for several energy dissipation models.
F.J. Mendez, I.J. Losada / Coastal Engineering 51 (2004) 103–118
first defining the offshore wave conditions and the other nine over the sloping beach. The drag coefficient C˜D has been previously fitted since the plants are the artificial kelp
115
models L. hyperborea used by Dubi (1995) [see Eq. (40)]. Therefore, the validation of the numerical model does not depend on any unknown vegetation parameters. Thornton and Guza (1983)
Fig. 8. Evolution of the root-mean-square wave height for b = 25 and 100 m, with dv = 0, 1 and 3 m.
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parameters are assumed to be constant (B = 1 and cb = 0.6). Fig. 6 shows the results of the significant wave height evolution for three runs with and without kelp. The solid and the dashed lines correspond to the numerical results for no kelp as well as for kelp presence, respectively. The crosses stand for the experimental data for no kelp, and the boxes, for the kelp cases. The ability of the model to reproduce damping induced by vegetation over a sloping beach under nonbreaking waves can be seen in the bottom case. Besides, for breaking conditions, the numerical model performs very well, showing that the damping induced by the vegetation field cannot be considered to be negligible (center and upper panels). In order to analyze the relative contribution of hebi and hevi, the numerical results for energy dissipation induced separately by breaking (solid line), vegetation (dotted line) and both terms together (dashed line) are plotted in Fig. 7. The experimental data for this case are also included. Both energy dissipation terms contribute to the total wave decay, again showing good agreement with the experimental data.
5. Results In order to carry out an analysis of the influence of plant height and vegetation field width on the propagation, Fig. 8 shows the evolution of the root-meansquare wave height over a Dean’s shape profile, h = 0.25(300 x)2/3, with x = 0 in the offshore boundary. The incident wave conditions are given by Hrms,o = 2.5 m and Tp = 10 s. Two vegetation field widths have been considered, b = 25 and 100 m. Furthermore, for these vegetation fields, two different plant heights are considered, dv = 1 and 3 m. The number of plants per square meter is N = 20 units/m2, and the plant area per unit height of each vegetation is bv = 0.25 m. For all results, it has been assumed that the drag coefficient C˜D = 0.2. It is shown that the influence of the vegetation on the wave propagation depends not only on the plant height, but also on the vegetation field width. Increasing the plant height results in a higher dissipation since the drag force (or the energy dissipation term) increases. Moreover, wider vegetation fields result in higher wave reduc-
tion. Besides, the damping caused by vegetation reduces the root-mean-square wave height inducing wave breaking farther onshore. As can be seen, the geometrical properties of the vegetation field play an important role in wave transformation.
6. Conclusions In this paper, an empirical model to estimate the transformation of waves induced by a vegetation field is presented. Starting from Dalrymple et al.’s (1984) theory for damping induced by vegetation for nonbreaking regular waves and a horizontal bottom, several extensions have been carried out to take into account the bottom variations, the randomness of waves and to include both vegetation and breaking dissipation in the surf zone. The wave transformation along a vegetation field depends on wave characteristics, geometrical and physical properties of the plants and a drag coefficient that must be previously calibrated for each specific type of plant. Although a number of assumptions have been made (plant-induced reflection is not considered, the movement of the plants is not modelled), results show that the theory is appropriate to represent wave height transformation over a vegetation field and the damping depending on the bulk drag coefficient C˜D. This single calibration parameter, which has been parameterized as a function of the local Keulegan – Carpenter number, is good enough to properly define the energy dissipation in vegetation fields. The model has been validated for an artificial kelp model L. hyperborea (Dubi, 1995) obtaining an expression for the drag coefficient C˜D given by Eq. (40). The model is able to reproduce the root-meansquare wave height transformation with reasonable accuracy. Once the wave transformation induced by the vegetation is studied, a numerical model for the transformation of random waves including dissipation by both vegetation hevi and breaking hebi is developed. The energy dissipation term induced by breaking is obtained following the Thornton and Guza (1983) model. The comparison of the experimental data (Lovas, 2000) and the numerical results is pretty good showing that both terms, hebi and hevi, contribute to the total energy dissipation. This kind of model can be implemented easily in standard
F.J. Mendez, I.J. Losada / Coastal Engineering 51 (2004) 103–118
wave propagation numerical models in order to incorporate dissipation induced by vegetation fields and therefore to be able to estimate wave-induced circulation as well as sediment transport in a vegetation field and in its vicinity. More laboratory as well as field experiments for different plant types are highly desirable to explore the possible scale effects and to fully validate this parameterization in terms of the local Keulegan – Carpenter number for each specific plant.
urms x z w a b eb ev cb
Notation b vegetation field width (m) bv the plant area per unit height of each vegetation stand normal to u (m) B parameter of the Thornton and Guza (1983) model c wave celerity (m s 1) cg group velocity (m s 1) CD drag coefficient dv plant height (m) E energy density (N m 1) fp wave frequency associated to peak period (s 1) F force vector per unit volume, F=( Fx, 0, Fz) (N m 3) g acceleration of gravity (m s 2) h depth (m) H wave height (m) Hrms root-mean-square wave height (m) k water wave number (m 1) kp water wave number associated with peak period (m 1) K vegetation wave numbers, K = K R + K iI (m 1) K Keulegan – Carpenter number Ks shoaling coefficient Kv damping coefficient N number of vegetation stands per unit horizontal area (m 2) R reflection coefficient t time (s) T wave period (s) Tp peak wave period (s) u velocity vector, u=(u, 0, w) (m s 1) u horizontal velocity (m s 1) ur relative horizontal velocity (m s 1)
cj q r
117
root-mean-square horizontal velocity (m s 1) horizontal coordinate (m) vertical coordinate (m) vertical velocity (m s 1) relative vegetation height damping parameter time-averaged rate of energy dissipation induced by breaking (N m 1 s 1) time-averaged rate of energy dissipation induced by vegetation (N m 1 s 1) parameter of the Thornton and Guza (1983) model spectrum width parameter water density (kg/m3) wave frequency (rad/s)
Acknowledgements FJM is indebted to the Ministerio de Ciencia y Tecnologia for the funding provided in the ‘‘Programa de Investigacion Ramon y Cajal’’. The authors gratefully acknowledge financial support provided by the European EESD-ENV-99 Program under the contract EVK3-CT-2000-00037 (HUMOR). They also would like to thank Alfonse Dubi, Alf Torum and Stig Magnar Lovas for graciously providing the laboratory data used in this work. The constructive and helpful comments of Prof. Robert Guza and an anonymous reviewer are gratefully acknowledged.
References Asano, T., Deguchi, H., Kobayashi, N., 1993. Interaction between water waves and vegetation. In: Edge, B.L. (Ed.), Proceedings of the Twenty-Third Coastal Engineering Conference. Am. Soc. of Civil Eng., New York, pp. 2710 – 2723. Carollo, F.G., Ferro, V., Termini, D., 2002. Flow velocity measurements in vegetated channels. J. Hydraul. Eng. 128 (7), 664 – 673. Dalrymple, R.A., Kirby, J.T., Hwang, P.A., 1984. Wave diffraction due to areas of energy dissipation. J. Waterw. Port Coast. Ocean Eng. 110 (1), 67 – 79. Dubi, A., 1995. Damping of water waves by submerged vegetation: A case study on Laminaria hyperborea, PhD thesis, University of Trondheim, The Norwegian Institute of Technology, Trondheim, Norway. Dubi, A., Torum, A., 1995. Wave damping by kelp vegetation. In: Edge, B.L. (Ed.), Proceedings of the Twenty-Fourth Coastal
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Engineering Conference. Am. Soc. of Civil Eng., New York, pp. 142 – 156. Dubi, A., Torum, A., 1997. Wave energy dissipation in kelp vegetation. In: Edge, B.L. (Ed.), Proceedings of the Twenty-Fifth Coastal Engineering Conference. Am. Soc. of Civil Eng., New York, pp. 2626 – 2639. Elwany, M.H.S., Flick, R.E., 1996. Relationship between kelp beds and beach width in Southern California. J. Waterw. Port Coast. Ocean Eng. 122 (1), 34 – 37. Elwany, M.H.S., O’Reilly, W.C., Guza, R.T., Flick, R.E., 1995. Effects of Southern California kelp beds on waves. J. Waterw. Port Coast. Ocean Eng. 121 (2), 143 – 150. Fonseca, M.S., Calahan, J.A., 1992. A preliminary evaluation of wave attenuation by four species of seagrass. Estuar. Coast. Shelf Sci. 35, 565 – 576. Gacia, E., Duarte, C.M., 2001. Sediment retention by a Mediterranean Posidonia oceanica meadow: the balance between deposition and resuspension. Estuar. Coast. Shelf Sci. 52, 505 – 514. Ghisalberti, M., Nepf, H.M., 2002. Mixing layers and coherent structures in vegetated aquatic flows. J. Geophys. Res. 107 (C2) (3), 1 – 11. Ifuku, M., Hayashi, H., 1998. Development of eelgrass Zostera marina bed utilizing sand drift control mats. Coast. Eng. J. 40 (3), 223 – 239. Kobayashi, N., Raichle, A.W., Asano, T., 1993. Wave attenuation by vegetation. J. Waterw. Port Coast. Ocean Eng. 119 (1), 30 – 48. Koehl, M.A.R., 1984. How do benthic organisms withstand moving water? Am. Zool. 24, 57 – 70. Kouwen, N., 1988. Field estimation of the biomechanical properties of grass. J. Hydraul. Res. 26 (5), 559 – 568. Kouwen, N., Unny, T.E., 1973. Flexible roughness in open channels. J. Hydraul. Div., ASCE 99 (HY5), 713 – 728. Larson, M., 1995. Model for decay of random waves in surf zone. J. Waterw. Port Coast. Ocean Eng. 121 (1), 1 – 12. Losada, I.J., Silva, R., Losada, M.A., 1996. Interactions of nonbreaking directional random waves with submerged breakwaters. Coast. Eng. 28 (1 – 4), 249 – 266. Lovas, S.M., (2000) Hydro-physical conditions in kelp forests and the effect on wave damping and dune erosion: A case study on Laminaria hyperborea, PhD thesis, University of Trondheim, The Norwegian Institute of Technology, Trondheim, Norway. Lovas, S.M., Torum, A., 2001. Effect of the kelp Laminaria hyper-
borea upon sand dune erosion and water particle velocities. Coast. Eng. 44, 37 – 63. Mase, H., Kirby, J.T., 1993. Modified frequency-domain KdV equation for random wave shoaling. In: Edge, B.L. (Ed.), Proceedings of the Twenty-Third Coastal Engineering Conference. Am. Soc. of Civil Eng., New York, pp. 485 – 504. Mendez, F.J., Losada, I.J., Losada, M.A., 1999a. Hydrodynamics induced by wind waves in a vegetation field. J. Geophys. Res. 104 (C8), 18383 – 18396. Mendez, F.J., Losada, I.J., Dalrymple, R.A., Losada, M.A., 1999b. Effects of wave reflection and dissipation on wave-induced second order magnitudes. In: Edge, B.L. (Ed.), Proceedings of the Twenty-Sixth Coastal Engineering Conference. Am. Soc. of Civil Eng., Virginia, pp. 537 – 550. Moller, I., Spencer, T., French, J.R., 1996. Wind wave attenuation over saltmarsh surfaces: preliminary results from Norfolk England. J. Coast. Res. 12 (4), 1009 – 1016. Moller, I., Spencer, T., French, J.R., Leggett, D.J., Dixon, M., 1999. Wave transformation over salt marshes: a field and numerical modelling study from North Norfolk England. Estuar. Coast. Shelf Sci. 49, 411 – 426. Mork, M., 1996. Wave attenuation due to bottom vegetation. Waves and Nonlinear Processes in Hydrodynamics. Kluwer Academic Publishing, Oslo, Norway, pp. 371 – 382. Seymour, R.J., 1996. Discussion of ‘‘Effects of southern California kelp beds on waves’’, Elwany, H.S., O’Reilly, W.C., Guza, R.T., Flick, R.E.. J. Waterw. Port Coast. Ocean Eng. 122 (4), 207 – 208. Seymour, R.J., Tegner, M.J., Dayton, P.K., Parnell, P.E., 1989. Storm wave-induced mortality of giant kelp, Macrocystis pyrifera, in Southern California. Estuar. Coast. Shelf Sci. 28, 277 – 292. Teeter, A.M., Johnson, B.H., Berger, C., Stelling, G., Scheffner, N.W., Garcia, M.H., Parchure, T.M., 2001. Hydrodynamic and sediment transport modeling emphasis on shallow-water, vegetated areas (lakes, reservoirs, estuaries and lagoons). Hydrobiologia 444, 1 – 23. Thornton, E.B., Guza, R.T., 1983. Transformation of wave height distribution. J. Geophys. Res. 88 (C10), 5925 – 5938. Tschirky, P., Hall, K., Turcke, D., 2001. Wave attenuation by emergent wetland vegetation. In: Edge, B.L. (Ed.), Proceedings of the Twenty-Seventh Coastal Engineering Conference. Am. Soc. of Civil Eng., Virginia, pp. 865 – 877.
An empirical model to estimate the propagation of random breaking and nonbreaking waves over vegetation fields Fernando J. Mendez *, Inigo J. Losada 1 Ocean and Coastal Research Group, Universidad de Cantabria, Dpto. de Ciencias y Te´cnicas del Agua y del Medio Ambiente, Av. de los Castros s/n 39005, Santander, Spain Received 12 March 2003; received in revised form 29 September 2003; accepted 5 November 2003
Abstract In this work, a model for wave transformation on vegetation fields is presented. The formulation includes wave damping and wave breaking over vegetation fields at variable depths. Based on a nonlinear formulation of the drag force, either the transformation of monochromatic waves or irregular waves can be modelled considering geometric and physical characteristics of the vegetation field. The model depends on a single parameter similar to the drag coefficient, which is parameterized as a function of the local Keulegan – Carpenter number for a specific type of plant. Given this parameterization, determined with laboratory experiments for each plant type, the model is able to reproduce the root-mean-square wave height transformation observed in experimental data with reasonable accuracy. D 2004 Elsevier B.V. All rights reserved. Keywords: Wave propagation; Vegetation field; Kelp bed; Drag force; Wave damping; Breaking waves; Empirical model; Dissipation
1. Introduction The importance of aquatic vegetation such as belts of seaweeds and seagrass as a biological and physical component of the coastal system has been widely recognised but is far from being understood. Aquatic vegetation is known to be food and shelter for many organisms, to control biogeochemical cycles in the coastal zone and to dissipate wave energy and turbulence protecting the shore from erosion. (f.i., Ifuku and Hayashi, 1998; Gacia and Duarte, 2001; Fonseca * Corresponding author. Tel.: +34-942-201810; fax: +34-942201860. E-mail addresses: [email protected] (F.J. Mendez), [email protected] (I.J. Losada). 1 Tel.: +34-942-201810; fax: +34-942-201860. 0378-3839/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2003.11.003
and Calahan, 1992). Furthermore, aquatic vegetation can influence coastal hydrodynamics, dependent upon its size, location, density, distribution and morphology. On the other hand, as pointed out by Koehl (1984), hydrodynamics influence the structure of the underwater plant communities by setting distribution limits related to wave and current exposure. Hydrodynamics also play a major role in controlling the dynamics of the underwater vegetation fields by dispersing spores and propagules and by mediating the availability of nutrients. The interaction of the water flow with the underwater vegetation is dynamic such that the structure of aquatic plant fields changes with time and is exposed to variable physical forcing of the water flow. Yet, this interaction also has some limits related to the capacity of the plants to resist the stress forces derived from the
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physical forcing. Drag forces and breaking forces from wind waves can remove plants (f.i., Seymour et al., 1989; Mork, 1996). Morphological and functional adaptations of marine plants can, to some extent, counteract the negative influence of hydrodynamic forces (Koehl, 1984). There are numerous examples showing the importance of shallow water wave-exposed aquatic vegetation for economical as well as for biological reasons: for instance, the harvesting of kelp Laminaria hyperborea in the Norwegian coast (Dubi and Torum, 1995; Mork, 1996), the effects of Macrocystis pyrifera kelp beds on the Californian coast (Elwany et al., 1995; Elwany and Flick, 1996), the shore protection of Posidonia oceanica seagrass meadows along the Mediterranean coast (Gacia and Duarte, 2001), the wave energy dissipation of Spartina alterniflora fields in English salt marshes (Moller et al., 1996, 1999) or the high biological activity of Zostera marina (Fonseca and Calahan, 1992; Ifuku and Hayashi, 1998). Wave attenuation depends on the characteristics of the plant (geometry, buoyancy, density, stiffness, degrees of freedom and spatial configuration) as well as wave parameters (mainly wave height, period and direction). The variability of wave damping is very large and trying to define a generalized behaviour of the ‘‘plant-induced dissipation’’ is absolutely impossible. For example, experimental data in L. hyperborea kelp beds in the coast of Norway show a damping of more than 60% of wave height (Mork, 1996) or a reduction of almost 40% of wave height was measured in emerged wetland plants in Lake Ontario, Canada (Tschirky et al., 2001). On the other hand, extensive field measurements carried out seaward and leeward of a very robust M. pyrifera kelp bed along the Californian coast assure that wave energy is practically unaffected by these plants (Elwany et al., 1995). Therefore, it is clear that an adequate modelling on wave transformation along vegetation fields would be highly desirable. As a consequence, several authors have carried out theoretical and numerical work mainly concentrated on wave transformation induced by a vegetation field. In general, the theoretical models try to estimate wave-induced forces on the plants. It must be noted that the validity of each model depends on the geometrical and biomechanical characteristics of the plants: on the one hand, if the plants
are subsurface, short or if its stiffness in the lower part is strong (f.i., L. hyperborea kelp beds, P. oceanica seagrass meadows, Spartina marshes), the plants can be considered as vertical cylinders and a drag force model can be considered to be valid. On the other hand, if the plants have a large number of degrees of freedom, the stiffness is reduced and the buoyancy is high (f.i., M. pyrifera kelp beds); this approach is not correct and a more complex modellization for the force would be required (Seymour, 1996). This paper is concerned about the flexible vertical cylinder-like plant, limiting its potential applications to the first category of plants. Initial models were based on neglecting plant motion and expressed in terms of a wave shear stress friction coefficient (Teeter et al., 2001) or the drag force acting on the vegetation (i.e. Dalrymple et al., 1984; Kobayashi et al., 1993). Asano et al. (1993) extend this solution to include the vegetation motion, coupling the flow field and swaying motion of the plants and comparing the solution with artificial seaweed experiments carried out in the lab. Dubi (1995) presents further laboratory experiments, analyzing the sensitivity of the damping rate to several parameters. Mendez et al. (1999a) extend previous work modelling wave height evolution, vegetation and fluid motion and forces and moments on the vegetation under irregular wave conditions and analyzing the consequences on wave transformation. However, all of these models, based on linearizing the drag force acting on the plant, have been developed for horizontal bottom and do not include wave breaking which are important shortcomings to analyzing real vegetation fields. Recently, Lovas (2000) has developed a wave transformation model in the surf zone based on the Larson (1995) model that takes into account the effect of kelp. Lovas’ model includes variable depth and wave breaking; however, the force is still linearized and the approach used limits the potential application of the model. Therefore, it cannot be applied as a predictive tool. In this work, an empirical model for wave transformation on vegetation fields is carried out. The presented formulation includes wave damping and wave breaking over vegetation fields on variable depths. Based on a nonlinear formulation of the drag force, either the transformation of monochromatic waves or random waves can be modelled considering
F.J. Mendez, I.J. Losada / Coastal Engineering 51 (2004) 103–118
geometric and physical characteristics of the vegetation field. The model depends on a single parameter similar to the drag coefficient which has been calibrated for a specific type of plant (L. hyperborea kelp). Results comparing numerical and experimental data will show the accuracy of the model. The paper is organized as follows. First, a review of regular wave transformation on vegetation fields is presented for constant depth as well as for a plane sloping beach. Later, the formulation for random waves is derived. The complete model for breaking and nonbreaking normally incident random waves is given in the next section. After the theoretical approach, the validation of the model for a specific type of vegetation follows based on experimental data. Next, the model is used to analyze the influence of several parameters on the wave transformation processes. Finally, some conclusions are given.
105
where h is the water depth and ah is the mean vegetation height, the overbar stands for time average in a wave period, F=( Fx, 0, Fz) is the force acting on the vegetation per unit volume and u=(u, 0, w) is the velocity for the 2D case. Substituting into the last equation yields ev ¼
Z
hþah
ðFx u þ Fz wÞdz
ð3Þ
h
It is usually assumed (f.i., Kobayashi et al., 1993; Dubi and Torum, 1997; Mendez et al., 1999a) that in an anisotropic dissipative media such as the vegetation field, the term Fzw is negligible in comparison with Fxu. Therefore, the time-averaged rate of energy dissipation per unit horizontal area ev can be expressed as ev ¼
Z
hþah
Fx udz
ð4Þ
h
2. Nonbreaking wave transformation model 2.1. Wave energy dissipation Waves propagating through vegetation (seaweed, kelp beds and marsh grass) lose energy due to the work carried out on the vegetation. Assuming that linear wave theory is valid and considering regular waves normally incident on a coastline with straight and parallel contours, the conservation of energy equation is reduced to BEcg ¼ ev ð1Þ Bx where E=(1/8)qgH 2 is the energy density, H is the wave height, g is the acceleration of gravity, q is the water pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi density, cg is the group velocity, c ¼ ðg=kÞtanhkh is the wave celerity, r is the wave angular frequency, n=(1/2)[1+(2kh/sinh2kh)], k is the wave number, x is the onshore coordinate and ev is the time-averaged rate of energy dissipation per unit horizontal area induced by the vegetation. For a given vegetation field, the conventional definition for the depth-integrated and time-averaged energy dissipation per horizontal area unit is given by: Z hþah ev ¼ Fudz ð2Þ h
The correct calculation of the force Fx should take into account the relative motion between the fluid and the plant and both inertial and drag forces, considering a huge amount of geometrical and biomechanical parameters. Although in the past few years, several models have been proposed that consider the plant swaying (f.i., Asano et al., 1993; Dubi and Torum, 1995; Mendez et al., 1999a), in this work, plant motion has been neglected in order to obtain an empirical model. For this reason, and according to several authors (Dalrymple et al., 1984; Kobayashi et al., 1993), plant-induced forces acting on the fluid are expressed in terms of a Morison-type equation neglecting swaying motion and inertial force. Therefore, the horizontal force per unit volume is given by Fx ¼
1 qCD bv NuAuA 2
ð5Þ
where u is the horizontal velocity in the vegetation region due to the wave motion, bv is the plant area per unit height of each vegetation stand normal to u, N is the number of vegetation stands per unit horizontal area and CD is a depth-averaged drag coefficient. It must be noted that the correct estimation of the nonlinear force Fx should be calculated by using the relative velocity ur between fluid and plant instead of u (Mendez et al., 1999a). For this reason, Eq. (5) is taken to be valid not only for rigid plants, but also for
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flexible ones considering a different value of a bulk drag coefficient CD to cover our ignorance of the plant motion (Dalrymple et al., 1984). A number of authors (f.i., Kobayashi et al., 1993; Dubi and Torum, 1995; Losada et al., 1996; Mendez et al., 1999a) have solved the boundary problem of waves propagating over a dissipative media analytically linearizing the friction terms and obtaining a complex wave number KR + K iI that considers that waves propagate proportionally to exp( KIx)cos(KRx). In this case, the linearization of the force (in terms of H) is needed. Nevertheless, in this paper, a more general model which takes the nonlinear force into account is presented. It has been assumed that the linear wave theory for waves propagating over an impermeable bottom is valid to calculate u not only for the water region, but also within the vegetation area. Dalrymple et al. (1984) expressed the energy dissipation for waves propagating through a vegetation field as ev ¼
where Ao ¼
8 sinh3 kah þ 3sinhkah CD bv Nk 9p ðsinh2kh þ 2khÞsinhkh
Solving the linear differential equation and assuming that the wave height at the seaward limit of the vegetation field is H(x = 0) = Ho, the wave height evolution is equal to Ho ¼ Kv Ho 1 þ bx
H¼
ð9Þ
where Kv is the damping coefficient 1 1 þ bx
Kv ¼
ð10Þ
and b is b¼
Ao Ho 4 CD bv NHo k ¼ 9p 2
3 2 kg sinh3 kah þ 3sinhkah 3 qCD bv N H ð6Þ 3p 2r 3kcosh3 kh
A similar approach was presented by Kobayashi et al. (1993) which considered the real part of the complex wave number KR instead of k. However, it can be demonstrated that both numbers (KR and k) are practically the same for the typical values of dissipation of a vegetation field. It must be noted that Eq. (6) does not consider the reflection induced by the plants. However, the importance of this factor is very limited in terms of wave energy. For example, if we consider a reflection of, say, ARA = 20%, this represents an energy reflection of 4% since energy is proportional to jRj2. Therefore, Eq. (6) can be considered to be valid.
ð8Þ
sinh3 kah þ 3sinhkah : ðsinh2kh þ 2khÞsinhkh
ð11Þ
2.2.2. Plane sloping beach An analytical solution can be obtained in shallow water for waves propagating normally through a vegetation field on a plane sloping beach. Starting with the energy flux conservation Eq. (1) and describing the energy dissipation using Eq. (6) gives BH 2 h1=2 H3 ¼ A1 1=2 Bx h
ð12Þ
where A1 ¼
2CD bv N a 3p
ð13Þ
For a plane sloping beach, the bottom depth can be expressed as
2.2. Monochromatic wave height transformation
h ¼ ho mx
ð14Þ
2.2.1. Constant depth For the case of horizontal bottom, using Eq. (6), i.e. nonlinear drag force, Eq. (1) can be rewritten as (Dalrymple et al., 1984)
where m is the slope and ho is the seaward water depth. By defining y = H2h1/2, Eq. (12) is rewritten as
BH 2 ¼ Ao H 3 Bx
By A1 y3=2 ¼ Bh m h5=4
ð7Þ
ð15Þ
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107
Fig. 1. Dependence of A1Ho /m for shallow water waves propagating over a vegetation field.
Eq. (15) can be easily integrated assuming a constant value for A1: 2 2A1 þ C ð16Þ y¼ mh1=4 Applying a boundary condition H = Ho in h = ho, ð17Þ
H ¼ Ho Kv Ks where the shoaling is governed by Green’s law 1=4
ho h1=4 and the damping coefficient is Ks ¼
Kv ¼
1þ2
A1 m
1 Ho ðKs 1Þ
ð18Þ
ð19Þ
The asymptotic case for a flat slope m ! 0 is Kv ¼
1 1 Ho x 1 þ A2h o
ð20Þ
since 1 lim m!0 m
!
1=4
ho
ðho mxÞ
1=4
1
¼
x 4ho
ð21Þ
Eq. (20) is the shallow water approximation of Eq. (10) for uniform depth. On the other hand, for null damping (A1 = 0), the damping coefficient Kv is unity. The dimensionless wave evolution H/Ho is plotted versus the normalized water depth h/ho for several values of the damping coefficient A1Ho/m in Fig. 1. Breaking is not considered. As can be seen, an increasing damping coefficient results in a greater reduction in wave height. No wave height variation is found for an approximate value of A1Ho/m = 0.5. This means that even though shoaling effects are important, the energy dissipation induced by the vegetation field cancels out the wave height growth. Depending on the value of A1Ho/m, the curvature of the wave evolution does not always correspond to the known exponential-decay-form for constant depth.
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2.3. Random wave transformation model
we obtain
Waves can be described by a joint distribution of wave height, period (or frequency) and direction. To simplify the analysis, it is usually assumed that the waves are very narrow-banded in frequency and come from the same direction, such that all wave heights are related to an average peak period Tp and mean direction h¯. A Rayleigh distribution often gives a satisfactory characterization of the random variation in wave height. The Rayleigh probability density function (pdf) is described by the root-mean-square wave height, Hrms, according to " # 2H H 2 pðHÞ ¼ 2 exp Hrms Hrms
ð22Þ
Z
l
H 2 pðHÞdH
ð23Þ
0
The model developed here describes the transformation of the wave height distribution assuming an unmodified Rayleigh distribution. For straight and parallel bottom contours, the average energy flux balance for normally incident waves is given by BEcg ¼ hev i Bx where Ecg ¼
1 qgcp np 8
ð24Þ Z
sinh3 kah þ 3sinhkah 3 pffiffiffi 3kcosh kh 3 p 3 Hrms ð28Þ 4 The derivation of Eq. (28) is not strictly correct since the drag coefficient may depend on the wave height. An average drag coefficient C˜D is proposed to overcome this problem. The evaluation of C˜D will be shown in the validation section. For a flat bottom, an expression similar to Eq. (7) is found
2 BHrms 3 ¼ Bo Hrms Bx where
where 2 ¼ Hrms
3 2 B 18 qgHrms cn 2 ˜ kg ¼ qCD bv N 3p 2r Bx
ð29Þ
2 sinh3 kah þ 3sinhkah Bo ¼ pffiffiffi C˜ D bv Nk ðsinh2kh þ 2khÞsinhkh 3 p
ð30Þ
Solving Eq. (29) and applying the boundary condition Hrms(x = 0) = Hrms,o, the root-mean-square wave height evolution is equal to Hrms;o Hrms ¼ ð31Þ ˜ 1 þ bx where Bo Hrms;o 1 ¼ pffiffiffi C˜ D bv NHrms;o k b˜ ¼ 2 3 p
l
H 2 pðHÞdH
ð25Þ
0
sinh3 kah þ 3sinhkah ðsinh2kh þ 2khÞsinhkh
ð32Þ
and hev i ¼
kp g 3 sinh3 kp ah þ 3sinhkp ah 2 ˜ qC D b v N 3p 2rp 3kp cosh3 kp h Z l H 3 pðHÞdH ð26Þ 0
where the magnitudes with subscript p are associated with the peak period Tp. For simplicity, in the following, the subscript p will be neglected. Applying Eq. (23) and pffiffiffi Z l 3 p 3 Hrms H 3 pðHÞdH ¼ ð27Þ 4 0
3. Random breaking waves model 3.1. Theoretical model The transformation of random waves, including shoaling, dissipation due to breaking and damping by a vegetation field, is described by an energy flux balance model. Lovas (2000) has recently developed a wave transformation model in the surf zone based on Larson (1995) that takes into account the effect of kelp. In that work, the damping due to vegetation in a horizontal cell of size Dx is modelled using
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Kv = exp( KIDx), where KI has been previously fitted as a function of the local water depth h for a specific type of kelp. However, this model cannot be used as a predictable tool since KI does not depend on the geometrical properties of the vegetation field or wave characteristics. In this paper, a similar approach has been carried out; however, the damping depends on wave and plant parameters. For straight and parallel contours, the energy conservation equation for normally incident waves, including breaking hebi and dissipation due to the vegetation hevi, is given by
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In this model, Eq. (33) is solved together with the mean water level variations g¯, calculated following the balance of x-momentum for a vegetation field BSxx Bg¯ ¼0 ð36Þ þ qgh Bx Bx in which d is the water depth for the still water level, h = d + g¯ is the mean water level and the radiation stress component Sxx is calculated taking into account the effects of the vegetation (Mendez et al., 1999b). The results are not sensitive to the setup model used. 3.2. Numerical model
BEcg ¼ heb i hev i Bx
ð33Þ
The average rate of energy dissipation per unit area for wave breaking is (Thornton and Guza, 1983) pffiffiffi B 3 fp 7 3 p heb i ¼ qg 4 5 Hrms 16 cb h
ð34Þ
where B and cb are adjusting parameters and fp is an average frequency corresponding to the peak period Tp. The dissipation due to vegetation is obtained from Eq. (28) assuming an invariant Rayleigh distribution, 3 1 kg ˜ hev i ¼ pffiffiffi qCD bv N 2r 2 p
sinh3 kah þ 3sinhkah 3 Hrms 3kcosh3 kh
ð35Þ
The major assumption in this model is the linear summation of hebi and hevi. This has been done due to the difficulty of physically identifying the contribution of each of the terms when a wave is breaking along a vegetation field. However, as will be shown in the model validation, this approximation is good enough for describing the wave energy transformation whenever the drag coefficient C˜D is previously calibrated. It must be noted that the assumption of an invariant Rayleigh-like wave height distribution is adopted to obtain the average rate of energy dissipation per unit area. The analysis of the shape of the transformed wave height distribution is not considered in this work.
For the solution of Eqs. (33) and (36), numerical integration must be performed by substituting Eqs. (34) and (35) and solving from offshore to the shoreline. A forward stepping scheme is used assuming cells of size Dx and constant depth, locally. In the calibration of the drag coefficient shown below, C˜D appears as a function of the orbital velocity at the top of the plant, which depends on Hrms. Therefore, an iterative procedure to calculate C˜D is needed. In the application of the model, the coefficients B = 1 and cb = 0.6 have been adopted as fixed values as recommended in Mase and Kirby (1993). The initial condition for the integration of Eq. (36) is the set-down calculated from linear theory for a progressive wave. An iterative procedure is needed since the energy dissipation depends on h. Three iterations have been found to be enough for the cases considered. 3.3. Shallow water approximation In order to analyze the importance of each term, the shallow water approximation of Eq. (33) is obtained. For kh ! 0, the energy flux balance yields 2 3 BHrms h1=2 H7 Hrms ¼ C1 rms A1 1=2 5 Bx h h
where A1 was defined in Eq. (13) and pffiffiffi 3 pB3 C1 ¼ 4 2cb Tp g 1=2
ð37Þ
ð38Þ
Fig. 2 shows, for a plane-sloping beach, the evolution of the root-mean-square wave height for shoaling
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Fig. 2. Nondimensional root-mean-square wave height evolution over a plane sloping beach for different theories.
(A1 = 0, C1 = 0), shoaling and vegetation (A1 = 0.01, C1 = 0), shoaling and breaking (A1 = 0, C1 = 1) and shoaling, vegetation and breaking (A1 = 0.01, C1 = 1). 7 3 As can be seen, both C1H rms /h5 and A1H rms /h1/2 contribute to the energy dissipation. Results show how the damping caused by vegetation tends to reduce the ratio Hrms/h and therefore inducing wave breaking farther onshore.
4. Model validation 4.1. Nonbreaking random waves The theoretical solution for random waves is compared to the experimental results for an artificial kelp field given by Dubi (1995). The experiment was carried out in a 33-m-long, 1-m-wide and 1.6-m-high wave flume. The width of the channel was partitioned to give a width of 0.5 m. At the end of the wave flume, a wave absorber was installed to reduce reflection. The artificial kelp models were L. hyperborea with a plant area per unit height of bv = 0.025 m and a height of dv = 0.2 m.
This kelp plant consists of a thin stipe and a very flexible frond split in many segments, with the stipe being about half of the total height. The vegetation field, located at the center of the flume, had a total width of b = 9.3 m (Dubi and Torum, 1997, Fig. 1). The number of uniformly distributed plants per unit horizontal area was N = 1200 units/m2. A total of 154 runs were carried out with varying water depth h = 0.4 –1 m, wave peak periods, Tp = 1.26 –4.42 s, and root-meansquare wave heights, Hrms = 0.045 – 0.17 m. The input for irregular waves was the Joint North Sea Wave Project (JONSWAP) spectrum with shape parameter cj = 3.3. Free surface oscillations were measured by eight wave gauges at x = 1.15, . . ., 8.15 m, with x = 0 m at the front face of the vegetation field. In some runs, the last two gauges were not functioning properly. For each of the runs, the best fit to the experimental data was obtained using a least-squares method considering C˜D the single calibration parameter. Prior to any results analysis, it must be pointed out that, for simplicity, the model presented does not consider the reflection process. The linear solution for the boundary value problem including reflection and evanescent modes was solved by Mendez et al. (1999a).
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Fig. 3 shows the comparisons between the experimental data and the theoretical root-mean-square wave height described by Eq. (31) for six different runs covering the range of parameters. As can be seen, the agreement is pretty good, although a slight modulation in the data is present due to reflection. It is clear that the success of this model as a predictive tool requires the knowledge of the bulk drag coefficient C˜D. The value of this parameter depends on the flow around the plants and the plant
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motion, which are functions of the hydrodynamic and biomechanical characteristics. However, the correct modelling of swaying motion is far from trivial not only for the difficulty of solving a nonlinear dynamic oscillator with n degrees of freedom, but also for the complexity and randomness of geometric and biomechanical properties of the plants. Several attempts for uniform flow (f.i., Kouwen and Unny, 1973; Kouwen, 1988; Carollo et al., 2002; Ghisalberti and Nepf, 2002) as well as for oscillatory flow (f.i., Asano et al., 1992;
Fig. 3. Numerical and experimental results of Hrms evolution.
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Dubi and Torum, 1995; Mendez et al., 1999a) have been carried out to model the biomechanical properties as well as the plant motion. Nevertheless, all these models need physical experimentation in order to calibrate the drag coefficient. For that reason, in this paper, a simplified analysis has been performed focusing mainly on the hydrodynamic processes and avoiding the plant motion. Therefore, a relation between C˜D and some nondimensional flow parameters is desirable to characterize hydrodynamically the L. hyperborea model plants for predictable purposes. In recent studies (Kobayashi et al., 1993; Mendez et al., 1999a), it has been found that the drag coefficient depends directly on the Reynolds number. However, in this paper, a more suitable relation is found using the Keulegan – Carpenter number, K, defined as K = ucTp/bv, where uc is a characteristic velocity acting on the plant and defined as the maximum horizontal velocity at the middle of the vegetation field x = b/2
Fig. 5. Estimated versus calculated drag coefficient C˜D.
Fig. 4. Calibrated value of drag coefficient C˜D as function of Keulegan – Carpenter number K and the relative plant height a.
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and z = h + ah. The velocity uc is defined using Hrms and Tp as the wave height and the wave period corresponding to a monochromatic wave train.
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Fig. 4 shows the calibrated values of C˜D as a function of K for the 154 runs. For this case, the C˜D coefficient varies between 0.01 and 0.52. The follow-
Fig. 6. Calculated and measured root-mean-square wave height.
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ing relationship with the Keulegan– Carpenter number can be found
4.2. Breaking random waves
C˜ D ¼ 0:47expð0:052KÞ 3VKV59
The laboratory data of Lovas (2000) are used to validate the numerical model for the transformation of random waves including dissipation by vegetation and breaking. The experimental setup was carried out in a 40-m-long and 5-m-wide wave flume at SINTEF (Norway). The width of the channel was partitioned to give a width of 0.6 m. Five thousand L. hyperborea model plants were prepared on a sandy 1:30 slope. The sand used in the experiment had D50 = 0.22 mm and its density was 2650 kg/m3. The vegetation field, located at the center of the flume, had a total width of b = 7.27 m (Lovas, 2000, Fig. 4.43; Lovas and Torum, 2001, Fig. 2). The maximum number of uniformly distributed plants per unit horizontal area was N = 1200 units/m2. For the validation of the root-mean-square wave height evolution, 13 runs were analyzed, with varying water depth h = 0.69– 0.77 m, wave peak periods T p = 2.5 and 3.5 s, significant wave heights Hmo = 1.416Hrms = 0.12 –0.22 m and varying N (0, 600 and 1200 units/m2). For each run, free surface oscillations were measured at 10 wave gauges, the
ð39Þ
with a 76% correlation. Several parameters have been considered in order to understand the scattering in the data. It has been found that the drag coefficient depends not only on K, but also on the relative height of the plants, a, see Fig. 4. Although the formulation for the energy dissipation takes into account the relative height of the plants, the force induced by the plants depends strongly on the swaying motion, which is not considered in Eq. (6). Besides, the horizontal displacement of each strip is a function of a as pointed out by Mendez et al. (1999a). The best relation for C˜D as a function of K and a has been obtained using a modified Keulegan –Carpenter parameter Q = K/a0.76 and is given by expð0:0138QÞ C˜ D ¼ Q0:3
7VQV172
ð40Þ
The calculated values of C˜D versus the estimated values using Eq. (40) are plotted in Fig. 5 showing that the correlation obtained (92%) is pretty good.
Fig. 7. Measured and calculated root-mean-square wave height for several energy dissipation models.
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first defining the offshore wave conditions and the other nine over the sloping beach. The drag coefficient C˜D has been previously fitted since the plants are the artificial kelp
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models L. hyperborea used by Dubi (1995) [see Eq. (40)]. Therefore, the validation of the numerical model does not depend on any unknown vegetation parameters. Thornton and Guza (1983)
Fig. 8. Evolution of the root-mean-square wave height for b = 25 and 100 m, with dv = 0, 1 and 3 m.
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parameters are assumed to be constant (B = 1 and cb = 0.6). Fig. 6 shows the results of the significant wave height evolution for three runs with and without kelp. The solid and the dashed lines correspond to the numerical results for no kelp as well as for kelp presence, respectively. The crosses stand for the experimental data for no kelp, and the boxes, for the kelp cases. The ability of the model to reproduce damping induced by vegetation over a sloping beach under nonbreaking waves can be seen in the bottom case. Besides, for breaking conditions, the numerical model performs very well, showing that the damping induced by the vegetation field cannot be considered to be negligible (center and upper panels). In order to analyze the relative contribution of hebi and hevi, the numerical results for energy dissipation induced separately by breaking (solid line), vegetation (dotted line) and both terms together (dashed line) are plotted in Fig. 7. The experimental data for this case are also included. Both energy dissipation terms contribute to the total wave decay, again showing good agreement with the experimental data.
5. Results In order to carry out an analysis of the influence of plant height and vegetation field width on the propagation, Fig. 8 shows the evolution of the root-meansquare wave height over a Dean’s shape profile, h = 0.25(300 x)2/3, with x = 0 in the offshore boundary. The incident wave conditions are given by Hrms,o = 2.5 m and Tp = 10 s. Two vegetation field widths have been considered, b = 25 and 100 m. Furthermore, for these vegetation fields, two different plant heights are considered, dv = 1 and 3 m. The number of plants per square meter is N = 20 units/m2, and the plant area per unit height of each vegetation is bv = 0.25 m. For all results, it has been assumed that the drag coefficient C˜D = 0.2. It is shown that the influence of the vegetation on the wave propagation depends not only on the plant height, but also on the vegetation field width. Increasing the plant height results in a higher dissipation since the drag force (or the energy dissipation term) increases. Moreover, wider vegetation fields result in higher wave reduc-
tion. Besides, the damping caused by vegetation reduces the root-mean-square wave height inducing wave breaking farther onshore. As can be seen, the geometrical properties of the vegetation field play an important role in wave transformation.
6. Conclusions In this paper, an empirical model to estimate the transformation of waves induced by a vegetation field is presented. Starting from Dalrymple et al.’s (1984) theory for damping induced by vegetation for nonbreaking regular waves and a horizontal bottom, several extensions have been carried out to take into account the bottom variations, the randomness of waves and to include both vegetation and breaking dissipation in the surf zone. The wave transformation along a vegetation field depends on wave characteristics, geometrical and physical properties of the plants and a drag coefficient that must be previously calibrated for each specific type of plant. Although a number of assumptions have been made (plant-induced reflection is not considered, the movement of the plants is not modelled), results show that the theory is appropriate to represent wave height transformation over a vegetation field and the damping depending on the bulk drag coefficient C˜D. This single calibration parameter, which has been parameterized as a function of the local Keulegan – Carpenter number, is good enough to properly define the energy dissipation in vegetation fields. The model has been validated for an artificial kelp model L. hyperborea (Dubi, 1995) obtaining an expression for the drag coefficient C˜D given by Eq. (40). The model is able to reproduce the root-meansquare wave height transformation with reasonable accuracy. Once the wave transformation induced by the vegetation is studied, a numerical model for the transformation of random waves including dissipation by both vegetation hevi and breaking hebi is developed. The energy dissipation term induced by breaking is obtained following the Thornton and Guza (1983) model. The comparison of the experimental data (Lovas, 2000) and the numerical results is pretty good showing that both terms, hebi and hevi, contribute to the total energy dissipation. This kind of model can be implemented easily in standard
F.J. Mendez, I.J. Losada / Coastal Engineering 51 (2004) 103–118
wave propagation numerical models in order to incorporate dissipation induced by vegetation fields and therefore to be able to estimate wave-induced circulation as well as sediment transport in a vegetation field and in its vicinity. More laboratory as well as field experiments for different plant types are highly desirable to explore the possible scale effects and to fully validate this parameterization in terms of the local Keulegan – Carpenter number for each specific plant.
urms x z w a b eb ev cb
Notation b vegetation field width (m) bv the plant area per unit height of each vegetation stand normal to u (m) B parameter of the Thornton and Guza (1983) model c wave celerity (m s 1) cg group velocity (m s 1) CD drag coefficient dv plant height (m) E energy density (N m 1) fp wave frequency associated to peak period (s 1) F force vector per unit volume, F=( Fx, 0, Fz) (N m 3) g acceleration of gravity (m s 2) h depth (m) H wave height (m) Hrms root-mean-square wave height (m) k water wave number (m 1) kp water wave number associated with peak period (m 1) K vegetation wave numbers, K = K R + K iI (m 1) K Keulegan – Carpenter number Ks shoaling coefficient Kv damping coefficient N number of vegetation stands per unit horizontal area (m 2) R reflection coefficient t time (s) T wave period (s) Tp peak wave period (s) u velocity vector, u=(u, 0, w) (m s 1) u horizontal velocity (m s 1) ur relative horizontal velocity (m s 1)
cj q r
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root-mean-square horizontal velocity (m s 1) horizontal coordinate (m) vertical coordinate (m) vertical velocity (m s 1) relative vegetation height damping parameter time-averaged rate of energy dissipation induced by breaking (N m 1 s 1) time-averaged rate of energy dissipation induced by vegetation (N m 1 s 1) parameter of the Thornton and Guza (1983) model spectrum width parameter water density (kg/m3) wave frequency (rad/s)
Acknowledgements FJM is indebted to the Ministerio de Ciencia y Tecnologia for the funding provided in the ‘‘Programa de Investigacion Ramon y Cajal’’. The authors gratefully acknowledge financial support provided by the European EESD-ENV-99 Program under the contract EVK3-CT-2000-00037 (HUMOR). They also would like to thank Alfonse Dubi, Alf Torum and Stig Magnar Lovas for graciously providing the laboratory data used in this work. The constructive and helpful comments of Prof. Robert Guza and an anonymous reviewer are gratefully acknowledged.
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