- Email: [email protected]
Modeling of random wave transformation with strong wave-induced coastal currents
Water Science and Engineering, Mar. 2008, Vol. 1, No. 1, 18–26 ISSN 1674–2370, http://kkb.hhu.edu.cn, e-mail: [email protected]
Modeling of random wave transformation with strong wave-induced coastal currents Zheng Jinhai*1, H. Mase2, Li Tongfei1 1. College of Ocean, Hohai University, Nanjing 210098, P. R. China 2. Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto 611-0011, Japan Abstract: The propagation and transformation of multi-directional and uni-directional random waves over a coast with complicated bathymetric and geometric features are studied experimentally and numerically. Laboratory investigation indicates that wave energy convergence and divergence cause strong coastal currents to develop and inversely modify the wave fields. A coastal spectral wave model, based on the wave action balance equation with diffraction effect (WABED), is used to simulate the transformation of random waves over the complicated bathymetry. The diffraction effect in the wave model is derived from a parabolic approximation of wave theory, and the mean energy dissipation rate per unit horizontal area due to wave breaking is parameterized by the bore-based formulation with a breaker index of 0.73. The numerically simulated wave field without considering coastal currents is different from that of experiments, whereas model results considering currents clearly reproduce the intensification of wave height in front of concave shorelines. Key words: random wave; coastal current; spectral wave model; numerical simulation
1 Introduction In some coastal areas with complicated bathymetry, the convergence and divergence of wave energy inversely create a Doppler shift, influence wave refraction, reflection, and breaking, and significantly modify the overall redistribution of wave fields, causing strong coastal currents (Castelle et al. 2006; MacMahan et al. 2006). Under such circumstances, the effects of ambient currents on wave transformations should be taken into account in coastal wave prediction. Reliable wave predictions in coastal areas are crucial to coastal engineering applications associated with shore protection, coastal morphological evolution, harbor construction, navigation channel maintenance and maritime disaster reduction. The modeling of coastal wave transformation has therefore been a subject of considerable interest in the field of harbor, coastal and offshore engineering, and has advanced a great deal in the past few decades. Several models that can predict combined refraction, diffraction, reflection and dissipation are now used in practice (Berkhoff 1972; Nwogu 1993; Resio 1993; Kirby and Dalrymple 1994; Booij et al. 1999; Panchang and Demirbilek 2001; Lin et al. 2008). However, each of these models is accompanied by its own set of modeling difficulties. The purpose of this study is to test the ability of the WABED wave model to predict the transformation of uni-directional and multi-directional random waves over a bathymetrically and geometrically complicated coast where wave-induced coastal currents are well developed üüüüüüüüüüüüü This work was supported by the Program for New Century Excellent Talents in Universities (Grant No. NCET-07-0255). *Corresponding author (e-mail: [email protected]) Received Jan. 07, 2008; accepted Mar. 12, 2008
and affect the overall distribution of wave energy. It is a spectral wave model based on the wave action balance equation, with a diffraction term formulated from a parabolic approximation of wave theory (Mase et al. 2005). The mean energy dissipation rate per unit horizontal area due to wave breaking is parameterized by the bore-based formulation of Battjes and Janssen (1978) with a breaker height of 0.73 times of the water depth (Zheng et al. 2006).
2 Experimental setup Experimental studies were carried out in the Research and Development Department of the Kansai Electric Power Company, Inc., Japan. The wave basin was 20 m long and 38 m wide, and a model beach was built in the middle. The nearshore bottom topography is plotted in Figure 1. The coastal configuration is a concave curve with two headlands. An offshore rip current was therefore formed by the assembling of two longshore currents. Uni-directional and multi-directional Bretschneider-Mitsuyasu type wave spectra were generated by an irregular wave-maker with 60 wave paddles, each of which had a width of 30 cm. The wave directional spreading function was of the Mitsuyasu type with Smax=25. The model-to-prototype scale was 1:125. Using the electrical capacitance wave gauges and the electromagnetic type current meters, both wave and current data were measured at 127 locations in six regions, as shown in Figure 2. The data were collected for 400 s after wave paddles started to move, and the data of first 60 s were discarded in the analysis to avoid transient effects. Table 1 shows the test conditions with different return periods. Table 1 Test conditions Return period (years)
Significant wave height (cm)
Significant wave period (s)
1
1.86
0.68
5
5.14
1.00
10
5.75
1.06
20
6.26
1.11
100
7.26
1.21
Figure 3 displays experimental results of the transformation of multi-directional and uni-directional waves with a return period of 100 years, in which the contours denote the value of the local measured significant wave height being normalized by the incident one. The maximum measured current velocity induced by uni-directional waves in the prototype was 2.7 m/s. Measurements reveal that the wave-induced nearshore currents are very strong, resulting in wave height intensification in the vicinity of the concave coasts. On the other hand, the wave heights in front of the headland do not increase due to the wave-induced coastal currents moving in the same direction of wave transformation. It should be noted that the mass transport flux is not measured in the offshore area, since it is too small to affect the wave field. In the multi-directional wave case, there is a region around (9.0, 6.0) in which the wave height is 1.05 times of that of the incident wave. In the uni-directional wave case, there is a
Zheng Jinhai et al. Water Science and Engineering, Mar. 2008, Vol. 1, No. 1, 18–26
19
region around (9.0, 6.0) in which the wave height is 1.20 times of that of the incident wave. The wave height intensification area in the uni-directional wave case is far greater than that in the multi-directional wave case. Theoretically, the wave height should be smaller due to energy divergence in this region if no current occur, therefore, the wave height intensification here is obviously caused by the strong opposing current, as shown in Figure 3.
Figure 1 Bank line and nearshore water
Figure 2 Locations of wave gauges and
depth contour of physical model
(a) Multi-directional wave case
current meters
(b) Uni-directional wave case
Figure 3 Experimental results of normalized significant wave heights and wave-induced currents
3 Numerical simulations 3.1 Description of the WABED wave model In order to take the effect of ambient currents into account, the WABED wave model uses the action density rather than the energy density, since the action density is conservative in the presence of ambient currents whereas the energy density is not (Bretherton and Garrett 1968).
20
Zheng Jinhai et al. Water Science and Engineering, Mar. 2008, Vol. 1, No. 1, 18–26
The Doppler shift is considered in the solution of intrinsic frequency calculated by the wave dispersion equation. The action density against the absolute frequency is calculated directly. The wave action balance equation with diffraction in terms of three variables is used in the development of a practice-oriented random wave model for coastal engineering studies of inlets, navigation projects, and wave-structure interactions. In these applications, wave breaking, dissipation, reflection, diffraction, and wave-current interaction are important processes that need to be represented accurately for reliable estimation of wave properties in engineering design, maintenance, and operations. The governing equation of WABED wave model is ∂ ( Cx N ) ∂x
+
∂ (Cy N ) ∂y
+
∂ ( Cθ N ) ∂θ
=
κ ª 1 º CCg cos 2 θ N y ) − CCg cos 2 θ N yy » − ε b N ( « y 2σ ¬ 2 ¼
(1)
where N is the wave action density, defined as the wave energy-density E divided by the relative angular frequency σ (with respect to a current), (x, y) are the horizontal coordinates, and θ is the wave direction measured counterclockwise from the x-axis. The first term on the right-hand-side of Eq. (1) is the wave diffraction term formulated from a parabolic approximation of wave theory, in which the coefficient κ is a free parameter to be optimized to tune the diffraction effect. The recommended value is 2.5 (Mase 2001). C and Cg are wave celerity and group velocity, respectively, ε b is the parameterization of wave breaking energy dissipation rate per unit horizontal area, and Cx , C y , and Cθ are the characteristic velocity with respect to x, y and θ , respectively, which can be expressed as (2) Cx = Cg cos θ + u
C y = Cg sin θ + v Cθ =
§ ∂h ∂h · ∂u ∂u ∂v ∂v − cos θ ¸ + cos θ sin θ − cos 2 θ + sin 2 θ − sin θ cos θ ¨ sin θ sinh 2kh © ∂x ∂y ¹ ∂x ∂y ∂x ∂y
σ
(3) (4)
where u and v are current velocity components in the x and y directions and k is the wave number. The relationships between the relative angular frequency σ , the absolute angular frequency ω , the wave number vector k, the current velocity vector U, and the water depth h are as follows: (5) σ 2 = g k tanh k h (6) σ = ω − kU A forward-marching, first order upwind finite-difference method is used to solve the above wave-action balance equation with diffraction (Mase 2001; Mase et al. 2005). With given values at the offshore boundary, solutions in the direction of wave propagation are obtained at each forward marching step, and statistical quantities are calculated at each row before moving to the next row. If the seaward-reflection option is activated, the model will perform backward marching for seaward-reflection after forward-marching calculations are completed. The parameterization of the mean energy dissipation rate per unit horizontal area due to
Zheng Jinhai et al. Water Science and Engineering, Mar. 2008, Vol. 1, No. 1, 18–26
21
wave breaking, ε b , is derived from Battjes and Janssen’s formulation (1978). The dissipation was calculated for a bore of the same height, and the probability of occurrence of breaking waves was estimated from the Rayleigh distribution, with an upper cut-off determined by local depth in shallow water. Their formula is αρ g (7) D= Qb f H b2 4 where D is the bulk energy dissipation due to all breaking waves, ȡ is the density of water, g is the gravitational acceleration, α is a constant of order one, f is the mean frequency, H b is the breaker height, and Qb is the probability of the fraction of broken waves passing at a point, which can be estimated as the area of the truncated Rayleigh wave height probability distribution function under the Delta function at H b . Qb is determined by 2
§H · 1 − Qb (8) = − ¨ rms ¸ ln Qb © Hb ¹ where H rms is the root-mean-square wave height. Battjes and Janssen (1978) used a constant breaker parameter of 0.8 in Eq. (8) to determine the maximum possible individual wave height at the local water depth. The breaker parameter in SWAN version 40.01 and previous versions has been considered exclusively dependent on the local water depth, and expressed either as a constant value or as being bottom-slope dependent. With SWAN version 40.11, the bottom-slope dependency has been removed and improved predictions have been obtained with a constant value of 0.73 (Booij et al. 1999). Chen et al. (2005) also included this breaking criterion in their finite element coastal/harbor wave model based on an extended mild-slope wave-current equation. The parameterization of wave-breaking energy dissipation rate per unit horizontal area can be described as D (9) εb = ( ρ gH rms2 8) σ
3.2 Numerical modeling In the numerical simulations, the grid size was set at 0.2 m in both along-shore and cross-shore directions. The input wave spectrum was divided into 10 frequency bins and 36 direction bins. The velocity of the current in each cell was estimated by interpolation of the measured data so that the wave model would not be influenced by computation error in the numerical nearshore circulation model. Figures 4 and 5 show calculated wave height fields with and without currents for multi-directional and uni-directional incident waves. With currents included in the simulations, simulated results showed wave height intensification in front of the concave shoreline. This feature was absent in simulations made without accounting for currents. Figure 6 provides a measured and computed data comparison of the normalized significant wave heights. The predicted values are within 20% of the measured values. Figures
22
Zheng Jinhai et al. Water Science and Engineering, Mar. 2008, Vol. 1, No. 1, 18–26
7 and 8 show comparisons of the calculated and measured normalized significant wave heights along longitudinal and transverse transects for multi-directional and uni-directional waves, respectively. Results from simulations that incorporate the effects of currents are in good agreement with data on wave height intensification caused by strong opposing coastal currents.
(a) Without currents
(b) With currents
Figure 4 Simulated wave fields for the multi-directional wave case
(a) Without currents
(b) With currents
Figure 5 Simulated wave fields for the uni-directional wave case
Two statistical parameters were used to evaluate the overall performance of the WABED wave model. The first was the mean value of absolute relative errors for the normalized significant wave height, defined as
R=
1 N
N
¦ i =1
ª H1 / 3 º ª H1 / 3 º « » −« » ¬ ( H1 / 3 )0 ¼ ci ¬ ( H1 / 3 )0 ¼ mi × 100% ª H1 / 3 º « » ¬ ( H1 / 3 )0 ¼ mi
Zheng Jinhai et al. Water Science and Engineering, Mar. 2008, Vol. 1, No. 1, 18–26
(10)
23
where N is the total number of wave height data available in each experiment condition, (H1/3)0 is the incident significant wave height, H1/3 is the significant wave height at the locations of wave gauges, the subscripts c and m denote the computed and measured normalized significant wave height, respectively. Smaller values of R are indicators of agreement between computed and measured data. A value of zero implies a perfect match between computations and measurements. The second statistical measure was the correlation coefficient between computed and measured normalized wave heights.
Figure 6 Computed versus measured normalized wave height
Figure 7 Normalized wave height comparisons along different transects for multi-directional wave case
Figure 8 Normalized significant wave height comparisons along different transects for uni-directional wave case 24
Zheng Jinhai et al. Water Science and Engineering, Mar. 2008, Vol. 1, No. 1, 18–26
The statistics indicate that the mean values of absolute relative errors in wave height prediction for uni-directional and multi-directional incident waves are, respectively, 5.42% and 7.67%, and the correlation coefficients are 0.94 and 0.92. This demonstrates that the WABED wave model is capable of predicting random wave transformation with strong wave-induced coastal currents when an accurate current field is given. Additional studies are underway to develop a quasi-3D wave-induced coastal current model and couple it with the present WABED wave model.
4 Concluding remarks Laboratory investigations and numerical simulations were carried out to study the transformation of random waves over a coast with complicated bathymetric and geometric features. Wave energy convergence and divergence were found to cause strong nearshore currents and, inversely, to significantly modify the overall redistribution of wave fields in the experiments. The wave field generated by the WABED wave model is different from that of experiments in which current is excluded. The simulated results that consider currents accurately capture the phenomenon of wave height intensification in front of concave shorelines. The mean values of absolute relative errors in wave height prediction are 5.42% and 7.67%, and the correlation coefficients are 0.94 and 0.92, for uni-directional and multi-directional incident waves, respectively. The satisfactory performance of numerical simulations indicates that the effects of currents are indeed important in predictions of random wave transformation over complicated coastal bathymetry and geometry.
Acknowledgements The first author is grateful to the China Scholarship Council and the Ministry of Education, Culture, Sports, Science and Technology–Japan for their financial support during his stay (Oct. 2005–Sept. 2006) as a visiting scholar at the Disaster Prevention Research Institute of Kyoto University in Japan.
References Battjes, J. A., and Janssen, J. P. F. M. 1978. Energy loss and set-up due to breaking of random waves. Proceedings of 16th International Conference on Coastal Engineering. Hamburg: ASCE, 569–587. Berkhoff, J. C. W. 1972. Computation of combined refraction and diffraction. Proceedings of 13th International Conference on Coastal Engineering. Vancouver: ASCE, 745–747. Booij, N., Ris, R. C., and Holthuijsen, L. H. 1999. A third-generation wave model for coastal regions: 1. Model description and validation. Journal of Geophysical Research, 104 (C4), 7649–7666. Bretherton, F. P., and Garrett, C. J. R. 1968. Wave trains in inhomogeneous moving media. Proceedings of Royal Society, Ser. A, 302, 529–554. Castelle, B., Bonneton, P., Senechal, N., Dupuis, H., Butel, R., and Michel, D. 2006. Dynamics of wave-induced currents over an alongshore non-uniform multiple-barred sandy beach on the Aquitanian Coast, France. Continental Shelf Research, 26(1), 113–131. Chen, W., Panchang, V., and Demirbilek, Z. 2005. On the modeling of wave-current interaction using the elliptic mild-slope wave equation. Ocean Engineering, 32 (17–18), 2135–2164.
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Kirby, J. T., and Dalrymple, R. A. 1994. Combined Refraction/Diffraction Model REF/DIF 1. Version 2.5, Documentation and User’s Manual. Newark: Department of Civil Engineering, University of Delaware. Lin, L., Demirbilek, Z., Mase, H., Zheng, J. H., and Yamada, F. 2008. CMS-Wave: A Nearshore Spectral Wave Processes Model for Coastal Inlets and Navigation Projects. Vicksburg: U.S. Army Engineer Research and Development Center. MacMahan, J. H., Thornton, E. B., and Reniers, A. J. H. M. 2006. Rip current reviews. Coastal Engineering, 53(2), 191–208. Mase, H. 2001. Multidirectional random wave transformation model based on energy balance equation. Coastal Engineering Journal, 43 (4), 317–337. Mase, H., Amamori, H., and Takayama, T. 2005. Wave prediction model in wave-current coexisting field. Proceedings of 12th Canadian Coastal Conference (CD-ROM). Dartmouth: CSCE. Nwogu, O. 1993. Alternative form of Boussinesq equations for nearshore wave propagation. Journal of Waterway, Port, Coastal and Ocean Engineering, 119 (6), 618–638. Panchang, V., and Demirbilek, Z. 2001. Simulation of waves in harbors using two-dimensional elliptic equation models. Advances in Coastal and Ocean Engineering, 7, 125–162. Resio, D. T. 1993. STWAVE: Wave Propagation Simulation Theory, Testing and Application. Florida: Department of Oceanography, Ocean Engineering and Environmental Science, Florida Institute of Technology. Zheng, J. H., Mase, H., and Mimeta, T. 2006. Incorporation of different wave breaking formulas into multi-directional wave transformation model. Annual Journal of Coastal Engineering, 53, 31–35. (in Japanese)
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Modeling of random wave transformation with strong wave-induced coastal currents Zheng Jinhai*1, H. Mase2, Li Tongfei1 1. College of Ocean, Hohai University, Nanjing 210098, P. R. China 2. Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto 611-0011, Japan Abstract: The propagation and transformation of multi-directional and uni-directional random waves over a coast with complicated bathymetric and geometric features are studied experimentally and numerically. Laboratory investigation indicates that wave energy convergence and divergence cause strong coastal currents to develop and inversely modify the wave fields. A coastal spectral wave model, based on the wave action balance equation with diffraction effect (WABED), is used to simulate the transformation of random waves over the complicated bathymetry. The diffraction effect in the wave model is derived from a parabolic approximation of wave theory, and the mean energy dissipation rate per unit horizontal area due to wave breaking is parameterized by the bore-based formulation with a breaker index of 0.73. The numerically simulated wave field without considering coastal currents is different from that of experiments, whereas model results considering currents clearly reproduce the intensification of wave height in front of concave shorelines. Key words: random wave; coastal current; spectral wave model; numerical simulation
1 Introduction In some coastal areas with complicated bathymetry, the convergence and divergence of wave energy inversely create a Doppler shift, influence wave refraction, reflection, and breaking, and significantly modify the overall redistribution of wave fields, causing strong coastal currents (Castelle et al. 2006; MacMahan et al. 2006). Under such circumstances, the effects of ambient currents on wave transformations should be taken into account in coastal wave prediction. Reliable wave predictions in coastal areas are crucial to coastal engineering applications associated with shore protection, coastal morphological evolution, harbor construction, navigation channel maintenance and maritime disaster reduction. The modeling of coastal wave transformation has therefore been a subject of considerable interest in the field of harbor, coastal and offshore engineering, and has advanced a great deal in the past few decades. Several models that can predict combined refraction, diffraction, reflection and dissipation are now used in practice (Berkhoff 1972; Nwogu 1993; Resio 1993; Kirby and Dalrymple 1994; Booij et al. 1999; Panchang and Demirbilek 2001; Lin et al. 2008). However, each of these models is accompanied by its own set of modeling difficulties. The purpose of this study is to test the ability of the WABED wave model to predict the transformation of uni-directional and multi-directional random waves over a bathymetrically and geometrically complicated coast where wave-induced coastal currents are well developed üüüüüüüüüüüüü This work was supported by the Program for New Century Excellent Talents in Universities (Grant No. NCET-07-0255). *Corresponding author (e-mail: [email protected]) Received Jan. 07, 2008; accepted Mar. 12, 2008
and affect the overall distribution of wave energy. It is a spectral wave model based on the wave action balance equation, with a diffraction term formulated from a parabolic approximation of wave theory (Mase et al. 2005). The mean energy dissipation rate per unit horizontal area due to wave breaking is parameterized by the bore-based formulation of Battjes and Janssen (1978) with a breaker height of 0.73 times of the water depth (Zheng et al. 2006).
2 Experimental setup Experimental studies were carried out in the Research and Development Department of the Kansai Electric Power Company, Inc., Japan. The wave basin was 20 m long and 38 m wide, and a model beach was built in the middle. The nearshore bottom topography is plotted in Figure 1. The coastal configuration is a concave curve with two headlands. An offshore rip current was therefore formed by the assembling of two longshore currents. Uni-directional and multi-directional Bretschneider-Mitsuyasu type wave spectra were generated by an irregular wave-maker with 60 wave paddles, each of which had a width of 30 cm. The wave directional spreading function was of the Mitsuyasu type with Smax=25. The model-to-prototype scale was 1:125. Using the electrical capacitance wave gauges and the electromagnetic type current meters, both wave and current data were measured at 127 locations in six regions, as shown in Figure 2. The data were collected for 400 s after wave paddles started to move, and the data of first 60 s were discarded in the analysis to avoid transient effects. Table 1 shows the test conditions with different return periods. Table 1 Test conditions Return period (years)
Significant wave height (cm)
Significant wave period (s)
1
1.86
0.68
5
5.14
1.00
10
5.75
1.06
20
6.26
1.11
100
7.26
1.21
Figure 3 displays experimental results of the transformation of multi-directional and uni-directional waves with a return period of 100 years, in which the contours denote the value of the local measured significant wave height being normalized by the incident one. The maximum measured current velocity induced by uni-directional waves in the prototype was 2.7 m/s. Measurements reveal that the wave-induced nearshore currents are very strong, resulting in wave height intensification in the vicinity of the concave coasts. On the other hand, the wave heights in front of the headland do not increase due to the wave-induced coastal currents moving in the same direction of wave transformation. It should be noted that the mass transport flux is not measured in the offshore area, since it is too small to affect the wave field. In the multi-directional wave case, there is a region around (9.0, 6.0) in which the wave height is 1.05 times of that of the incident wave. In the uni-directional wave case, there is a
Zheng Jinhai et al. Water Science and Engineering, Mar. 2008, Vol. 1, No. 1, 18–26
19
region around (9.0, 6.0) in which the wave height is 1.20 times of that of the incident wave. The wave height intensification area in the uni-directional wave case is far greater than that in the multi-directional wave case. Theoretically, the wave height should be smaller due to energy divergence in this region if no current occur, therefore, the wave height intensification here is obviously caused by the strong opposing current, as shown in Figure 3.
Figure 1 Bank line and nearshore water
Figure 2 Locations of wave gauges and
depth contour of physical model
(a) Multi-directional wave case
current meters
(b) Uni-directional wave case
Figure 3 Experimental results of normalized significant wave heights and wave-induced currents
3 Numerical simulations 3.1 Description of the WABED wave model In order to take the effect of ambient currents into account, the WABED wave model uses the action density rather than the energy density, since the action density is conservative in the presence of ambient currents whereas the energy density is not (Bretherton and Garrett 1968).
20
Zheng Jinhai et al. Water Science and Engineering, Mar. 2008, Vol. 1, No. 1, 18–26
The Doppler shift is considered in the solution of intrinsic frequency calculated by the wave dispersion equation. The action density against the absolute frequency is calculated directly. The wave action balance equation with diffraction in terms of three variables is used in the development of a practice-oriented random wave model for coastal engineering studies of inlets, navigation projects, and wave-structure interactions. In these applications, wave breaking, dissipation, reflection, diffraction, and wave-current interaction are important processes that need to be represented accurately for reliable estimation of wave properties in engineering design, maintenance, and operations. The governing equation of WABED wave model is ∂ ( Cx N ) ∂x
+
∂ (Cy N ) ∂y
+
∂ ( Cθ N ) ∂θ
=
κ ª 1 º CCg cos 2 θ N y ) − CCg cos 2 θ N yy » − ε b N ( « y 2σ ¬ 2 ¼
(1)
where N is the wave action density, defined as the wave energy-density E divided by the relative angular frequency σ (with respect to a current), (x, y) are the horizontal coordinates, and θ is the wave direction measured counterclockwise from the x-axis. The first term on the right-hand-side of Eq. (1) is the wave diffraction term formulated from a parabolic approximation of wave theory, in which the coefficient κ is a free parameter to be optimized to tune the diffraction effect. The recommended value is 2.5 (Mase 2001). C and Cg are wave celerity and group velocity, respectively, ε b is the parameterization of wave breaking energy dissipation rate per unit horizontal area, and Cx , C y , and Cθ are the characteristic velocity with respect to x, y and θ , respectively, which can be expressed as (2) Cx = Cg cos θ + u
C y = Cg sin θ + v Cθ =
§ ∂h ∂h · ∂u ∂u ∂v ∂v − cos θ ¸ + cos θ sin θ − cos 2 θ + sin 2 θ − sin θ cos θ ¨ sin θ sinh 2kh © ∂x ∂y ¹ ∂x ∂y ∂x ∂y
σ
(3) (4)
where u and v are current velocity components in the x and y directions and k is the wave number. The relationships between the relative angular frequency σ , the absolute angular frequency ω , the wave number vector k, the current velocity vector U, and the water depth h are as follows: (5) σ 2 = g k tanh k h (6) σ = ω − kU A forward-marching, first order upwind finite-difference method is used to solve the above wave-action balance equation with diffraction (Mase 2001; Mase et al. 2005). With given values at the offshore boundary, solutions in the direction of wave propagation are obtained at each forward marching step, and statistical quantities are calculated at each row before moving to the next row. If the seaward-reflection option is activated, the model will perform backward marching for seaward-reflection after forward-marching calculations are completed. The parameterization of the mean energy dissipation rate per unit horizontal area due to
Zheng Jinhai et al. Water Science and Engineering, Mar. 2008, Vol. 1, No. 1, 18–26
21
wave breaking, ε b , is derived from Battjes and Janssen’s formulation (1978). The dissipation was calculated for a bore of the same height, and the probability of occurrence of breaking waves was estimated from the Rayleigh distribution, with an upper cut-off determined by local depth in shallow water. Their formula is αρ g (7) D= Qb f H b2 4 where D is the bulk energy dissipation due to all breaking waves, ȡ is the density of water, g is the gravitational acceleration, α is a constant of order one, f is the mean frequency, H b is the breaker height, and Qb is the probability of the fraction of broken waves passing at a point, which can be estimated as the area of the truncated Rayleigh wave height probability distribution function under the Delta function at H b . Qb is determined by 2
§H · 1 − Qb (8) = − ¨ rms ¸ ln Qb © Hb ¹ where H rms is the root-mean-square wave height. Battjes and Janssen (1978) used a constant breaker parameter of 0.8 in Eq. (8) to determine the maximum possible individual wave height at the local water depth. The breaker parameter in SWAN version 40.01 and previous versions has been considered exclusively dependent on the local water depth, and expressed either as a constant value or as being bottom-slope dependent. With SWAN version 40.11, the bottom-slope dependency has been removed and improved predictions have been obtained with a constant value of 0.73 (Booij et al. 1999). Chen et al. (2005) also included this breaking criterion in their finite element coastal/harbor wave model based on an extended mild-slope wave-current equation. The parameterization of wave-breaking energy dissipation rate per unit horizontal area can be described as D (9) εb = ( ρ gH rms2 8) σ
3.2 Numerical modeling In the numerical simulations, the grid size was set at 0.2 m in both along-shore and cross-shore directions. The input wave spectrum was divided into 10 frequency bins and 36 direction bins. The velocity of the current in each cell was estimated by interpolation of the measured data so that the wave model would not be influenced by computation error in the numerical nearshore circulation model. Figures 4 and 5 show calculated wave height fields with and without currents for multi-directional and uni-directional incident waves. With currents included in the simulations, simulated results showed wave height intensification in front of the concave shoreline. This feature was absent in simulations made without accounting for currents. Figure 6 provides a measured and computed data comparison of the normalized significant wave heights. The predicted values are within 20% of the measured values. Figures
22
Zheng Jinhai et al. Water Science and Engineering, Mar. 2008, Vol. 1, No. 1, 18–26
7 and 8 show comparisons of the calculated and measured normalized significant wave heights along longitudinal and transverse transects for multi-directional and uni-directional waves, respectively. Results from simulations that incorporate the effects of currents are in good agreement with data on wave height intensification caused by strong opposing coastal currents.
(a) Without currents
(b) With currents
Figure 4 Simulated wave fields for the multi-directional wave case
(a) Without currents
(b) With currents
Figure 5 Simulated wave fields for the uni-directional wave case
Two statistical parameters were used to evaluate the overall performance of the WABED wave model. The first was the mean value of absolute relative errors for the normalized significant wave height, defined as
R=
1 N
N
¦ i =1
ª H1 / 3 º ª H1 / 3 º « » −« » ¬ ( H1 / 3 )0 ¼ ci ¬ ( H1 / 3 )0 ¼ mi × 100% ª H1 / 3 º « » ¬ ( H1 / 3 )0 ¼ mi
Zheng Jinhai et al. Water Science and Engineering, Mar. 2008, Vol. 1, No. 1, 18–26
(10)
23
where N is the total number of wave height data available in each experiment condition, (H1/3)0 is the incident significant wave height, H1/3 is the significant wave height at the locations of wave gauges, the subscripts c and m denote the computed and measured normalized significant wave height, respectively. Smaller values of R are indicators of agreement between computed and measured data. A value of zero implies a perfect match between computations and measurements. The second statistical measure was the correlation coefficient between computed and measured normalized wave heights.
Figure 6 Computed versus measured normalized wave height
Figure 7 Normalized wave height comparisons along different transects for multi-directional wave case
Figure 8 Normalized significant wave height comparisons along different transects for uni-directional wave case 24
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The statistics indicate that the mean values of absolute relative errors in wave height prediction for uni-directional and multi-directional incident waves are, respectively, 5.42% and 7.67%, and the correlation coefficients are 0.94 and 0.92. This demonstrates that the WABED wave model is capable of predicting random wave transformation with strong wave-induced coastal currents when an accurate current field is given. Additional studies are underway to develop a quasi-3D wave-induced coastal current model and couple it with the present WABED wave model.
4 Concluding remarks Laboratory investigations and numerical simulations were carried out to study the transformation of random waves over a coast with complicated bathymetric and geometric features. Wave energy convergence and divergence were found to cause strong nearshore currents and, inversely, to significantly modify the overall redistribution of wave fields in the experiments. The wave field generated by the WABED wave model is different from that of experiments in which current is excluded. The simulated results that consider currents accurately capture the phenomenon of wave height intensification in front of concave shorelines. The mean values of absolute relative errors in wave height prediction are 5.42% and 7.67%, and the correlation coefficients are 0.94 and 0.92, for uni-directional and multi-directional incident waves, respectively. The satisfactory performance of numerical simulations indicates that the effects of currents are indeed important in predictions of random wave transformation over complicated coastal bathymetry and geometry.
Acknowledgements The first author is grateful to the China Scholarship Council and the Ministry of Education, Culture, Sports, Science and Technology–Japan for their financial support during his stay (Oct. 2005–Sept. 2006) as a visiting scholar at the Disaster Prevention Research Institute of Kyoto University in Japan.
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