Energy Balance of Wind Waves As a Function of the Bottom Friction Formulation R

Coastal Engineering 43Ž. 2001 131–148 www.elsevier.comrlocatercoastaleng

Energy balance of wind waves as a function of the bottom friction formulation R. Padilla-Hernandez´ ), J. Monbaliu Hydraulics Laboratory, Katholieke UniÕersiteit LeuÕen, Kasteelpark Arenberg 40, B-3001 HeÕerlee, Belgium Received 4 September 2000; received in revised form 14 February 2001; accepted 15 March 2001

Abstract

Four different expressions for wave energy dissipation by bottom friction are intercompared. For this purpose, the SWAN wave model and the wave data set of Lake GeorgeŽ. Australia are used. Three formulations are already present in SWAN Ž.ver. 40.01 : the JONSWAP expression, the drag law friction model of Collins and the eddy–viscosity model of Madsen. The eddy–viscosity model of Weber was incorporated into the SWAN code. Using Collins’ and Weber’s expressions, the depth- and fetch-limited wave growth laws obtained in the nearly idealized situation of Lake George can be reproduced. The wave model has shown the best performance using the formulation of Weber. This formula has some advantages over the other formulations. The expression is based on theoretical and physical principles. The wave height and the peak frequency obtained from the SWAN runs using Weber’s bottom friction expression are more consistent with the measurements. The formula of Weber should therefore be preferred when modelling waves in very shallow water. q 2001 Elsevier Science B.V. All rights reserved.

Keywords: Waves; SWAN; Wave bottom friction; Lake George

1. Introduction Lake George, AustraliaŽ Young and Verhagen, 1996. Hereafter YV. is as unique as the JONSWAP experi- One of the main problems to advance our knowl- mentŽ. Hasselmann et al., 1973 . The data obtained edge about how to model wind waves in very shal- from the lake in water of limited depth provide a low water is lack of data from measurements. Con- nearly idealized situation to test and analyze several trary to the situation in deep water, the dynamics of of the most widely used bottom friction formula- waves in shallow water areas are dominated by their tions. interaction with the bottom. The growth by wind, There are different mechanisms for wave energy propagation, non-linear interactions, energy decay dissipation at the bottom, such as energy dissipation and possibly the enhancement of whitecapping, are through percolation, friction, motion of a soft muddy all linked to how the waves interact with the bottom. bottom and bottom scattering. The relative strength To this respect, the wave measurements campaign in of those mechanisms depends on the bottom conditions; type of sediment and the presence or absence ) of sand ripples, and on the dimensions of such Corresponding author. Fax: q32-16-321-989. E-mail address: [email protected] ripples. It appears that the bottom friction is the most Ž.R. Padilla-Hernandez´ . important mechanism for energy decay in sandy

0378-3839r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S0378-3839Ž. 01 00010-2 132 R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148 coastal regionsŽ. Shemdin et al., 1978 . The energy propagation velocities in spectral spaceŽ frequency decay by bottom friction has been a subject of and directional space.Ž. . The first term of Eq. 1 investigation and a large number of dissipation mod- represents the local rate of change of action density els for bottom friction have been proposed since the in time. The second and third terms stand for propa- pioneering paper of Putman and JohnsonŽ. 1949 . All gation of action in geographical space. The fourth those models reflect the divergence of opinions on term expresses the shifting of action density in fre- how to model physical mechanisms present in the quency space due to variations in depth and currents. wave–bottom interaction process. One of the recent And the fifth term reproduces depth-induced and formulations proposed to simulate the wave energy current-induced refraction. The source term SŽs dissipation by bottom friction is the eddy–viscosity SŽ..s ,u at the right-hand side of the action balance model of WeberŽ. 1989 . Investigating the effect of equation accounts for the effects of generation, dissi- the bottom friction dissipation on the energy balance pation and nonlinear wave–wave interactions. More using several formulations, Luo and MonbaliuŽ. 1994 explicitly, the source terms in SWAN include wave concluded that there was no evidence to determine energy growth by wind input Sin; wave energy trans- which friction formulation performs best. The work fer due to wave–wave non-linear interactions Snl presented here reflects the search for evidence. Ž.both quadruplets and triads ; decay of wave energy

To reach the objective, the numerical wave model due to whitecapping Sds; bottom friction S bf ; and SWAN was run with the bottom friction source term depth-induced wave breaking Sbk . as ‘unknown’ in order to reproduce the Lake George A detailed description of the SWANŽ. Cycle 2 measurementsŽ. YV in the best possible way. Be- model, the incorporated source terms and the numer- sides the three formulations already present in SWAN ical solution method can be found in RisŽ. 1997 , Ž.Booij et al., 1999; Section 3.2 , also the formulation Holthuijsen et al.Ž. 1999 and Booij et al. Ž. 1999 . for bottom friction formulation by WeberŽ. 1989 was used. To this end, it was introduced in the SWAN model code. Although all of the individual source 3. The models of bottom friction dissipation term formulations are open to discussion, it is assumed that the SWAN model computes the energy 3.1. Dissipation of waÕe energy as a function of balance as a whole correctly. By only analyzing the bottom stress term of dissipation by bottom friction, an attempt is made to select a formulation to be used in depth- Komen et al.Ž. 1994 start with the linearized limited situations. momentum equation for the bottom boundary layer flow which in the case of pure wave motionŽ without 2. The SWAN wave model ambient currents. reads Eu 11Et The SWANŽ Simulation of WAves in Nearshore q = ps Ž.2 Et rrEz areas. model is based on the action balance equation. The equation solved by the SWAN model reads where t is time, z is the vertical coordinate, r is the EN EEE density of the water, u and p the Reynolds-average q q q Ž.cN Ž.cN Ž.cNs horizontal velocity and pressure, respectively, and t Et Ex xyE y Es the turbulent stress in the wave boundary layer. They E Stot obtain an expression for the wave energy dissipation q Ž.cNu s Ž.1 Eus due to bottom friction: where NŽ.s ,u is the wave action density Žs 1 t FŽ..s ,u rs ; F is the wave energy density; t is the S Ž.k sy PU Ž.3 bf g¦;r k time; s is the relative frequency; u is the wave direction; cxy, c , are the propagation velocities in where the bottom friction depends on the known free geographical x-, y-space; and cs and cu are the orbital velocity Ž.Uk of the waves at the bottom and R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148 133 on the unknown turbulent bottom stress Ž.t ; the bottom friction dissipation differ mainly in the ex- subscript k denotes a given wave number. pression given to the dissipation coefficient Cf . An exact solution for t in Eq.Ž.Ž 2 and hence for Below, the expressions that are currently imple-

Sbf Ž.k in Eq. Ž.. 3 does not exist, not even for a mented in the SWAN modelŽ version 40.01, Holthui- simple flow. To overcome the problem, several ap- jsen et al. 1999. as well as Weber’s formulation for proaches have been proposed. Most of the ap- bottom friction dissipation are explained briefly. proaches result in a turbulent shear stress expressed 3.2. Expressions for the dissipation coefficient C as a function of a friction coefficient and of a f free-stream orbital velocityŽ orbital velocity at the 3.2.1. The JONSWAP model top of the boundary layer. . There are two distinct This is the simplest expression for bottom dissipa- formulations for t ; the first is to retain a spectral tion. It was proposed by Hasselmann et al.Ž. 1973 . description. The second is to represent the range of CfJ is assumed to be constant and is given by frequencies by a single frequency, for example, the G peak frequency, resulting in an integral form. C s Ž.8 fJ g Usually, t is expressed in a drag law as ts r << ts r where g is the acceleration due to gravity. From the 1 2 rCDDUU or, alternatively, as 1 2 rC - results of the JONSWAP experiment, they found a Ž.U rmsU, where C D is a drag coefficient, U is the value for G of 0.038 m2 sy3. As long as a suitable wave orbital velocity at the bottom and Urms is the root mean square of the orbital velocity. Taking value for G is chosen, this expression performs well s r << in many different conditions. The value for G can be CfD1 2C U or in the alternative expression C f s r different for swell and for wind sea. Bouws and 1 2CUDrmsŽ. results in KomenŽ. 1983 found that the JONSWAP expression tsr 2 y3 CfU Ž.4 with a value of 0.038 m s for G yielded too low Substitution of Eq.Ž. 4 in the dissipation Eq. Ž. 3 dissipation rates for depth-limited wind sea condi- yields for every wave component with wavenumber tions in the North Sea. They selected a value of 2 y3 k: 0.067 m s in order to obtain a correct equilibrium solution for a steady state. The JONSWAP formula- 1 sy 2 tion is also implemented as the default friction for- SbfŽ.k CU f²: Žk . Ž.5 g mulation in the WAM modelŽ. Komen et al., 1994 .

2 The mean square of the bottom velocity, ²ŽUk . :, 3.2.2. The Madsen model can be associated with the wave component having Madsen et al.Ž. 1988 derived a bottom friction the wavenumber k. Rewriting the expressionŽ. 5 in formulation based on the eddy–viscosity concept, terms of the wave spectrum, one obtains: fw r C s ²:U 2 1 2 Ž.9 k fM '2 S Ž.k sy2CF Ž.k Ž.6 bf f sinh2kh where 1r2 or equivalentlyŽ. as expressed in SWAN model 1r2 2 gk ²:U 2 s FŽ.s ,u ds du Ž.10 HH sinh2kh C s 2 s u sy f s u and f is a non-dimensional friction factor. In the Sbf Ž., 2 F Ž., Ž.7 w g sinh kh SWAN model, the following formulation, based on the work of JonssonŽ. 1966 , for f is used: where Cf is a dissipation coefficient with the dimen- w y1 a sion in m s , and FkŽ.and F Žs ,u .are the en- s b F fw 0.3 for 1.57 ergy–density spectrum in wavenumber-space or in K N frequency-direction space, respectively. The vector 11 aa q s q bb) sŽ.Žs u u . log10m flog 10 for 1.57 k k12, k kcos , ksin is the wavenumber 4''fww4 f KKNN vector with modulus k and direction u, and s is the relative frequency. The different formulations for the Ž.11 134 R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148

sy where mfb0.08, a is a representative near-bot- Solving the Navier–Stokes equations in the turbu- tom excursion amplitude: lent boundary layer and using perturbation theory, 1 Weber derived the following dissipation coefficient. 1 2 s s u s u ) ) ab 2HH 2 FŽ., d d Ž.12 s j q j sinh kh CfW u Ä4Tk Ž.0 Tk Ž.0 .15 Ž. and K is the bottom roughness length. Graber and N CfW depends on the wave spectrum FkŽ., the water MadsenŽ. 1988 implemented the expression of Mad- depth h, and the bottom roughness K N through the sen in a parametric wind sea model for finite water friction velocity u), and on the radian frequency depths. s Ž.v 2ps through j 0.

1 3.2.3. The Collins model q 1 4Ž.y0Nh v 2 4K v Hasselmann and CollinsŽ. 1968 derived a formu- j ss2 Ž.16 lation for the bottom friction dissipation. They re- 0 ž/k u))ž/30k u lated the turbulent bottom stress to the external flow j by means of a quadratic friction law. The dissipation The variable 0 reflects the ratio between the coefficient they derived reads: roughness length and the boundary layer thickness, which scales with u)rv; k is the von Karman´´ UUij Ž.q C s2c d ²:U q Ž.13 constant set equal to 0.4; y0 h is the theoretical f ½5ij ¦;U bottom level and T is defined as k where dij is the Kroneker delta function;²: denotes X X r 1 Ker j qiKei j Ž 22q .1 2 Ž.00 Ž. the ensemble average, U is equal to U12U , T Ž.j sy kj Ž.17 k 002 Ker j qiKei j U12and U are the near bottom orbital velocity Ž.00 Ž. components, and c is a drag coefficient determined experimentally as a function of the bottom rough- Tk is a dimensionless complex function and depends ness. Hasselmann and Collins proposed a value for c on the radian frequency and thus on the wavenumber j q equal to 0.015. through the argument 0. Ker iKei is the zero CollinsŽ. 1972 simplified the expression Ž. 13 for order Kelvin functionŽ Abramowitz and Stegun, the dissipation coefficient by leaving out the depen- 1965. . The prime denotes the derivative with respect j dence on the direction of the wave component and to the argument 0. Details of the derivation of the by using the total wave induced bottom velocity: dissipation coefficient in the eddy–viscosity model Ž. 1 are given in Weber 1989, 1991 . It is of interest to look at the differences and similarities between the s 2 2 CfC 2cU²: Ž.14 different formulations. The expression by Weber will where ²U 2 :Ž.can be computed from Eq. 10 . Ex- be used here as the reference since it models explic- pressionŽ. 14 is the one implemented in the SWAN itly the bottom friction dissipation mechanism. model. The value of the drag coefficient c was set to The JONSWAP friction model does not interpret 0.015. Cavaleri and Malanotte-RizzoliŽ. 1981 imple- bottom dissipation in terms of a physical mechanism mented this friction model in a parametric wave such as percolation, friction or bottom motion. model. Weber’s and Madsen’s formulations differ in the fact that Madsen’s model approximates the random 3.2.4. The Weber eddy–Õiscosity model wave field by an AequivalentB monochromatic wave. Weber’s model for the spectral energy dissipation This approximation is applied at an early stage of due to friction in the turbulent wave boundary layer their calculations. Therefore, Madsen’s expression is is based on the eddy–viscosity concept. In this model, only valid for a narrow, singled-peaked spectrum. In the turbulent shear stress is parameterized in analogy fact, Collins’ drag-law dissipation expression is red- with the viscous stress, with the coefficient of molec- erived. However, in the Madsen formulation the ular viscosity replaced by a turbulent eddy–viscosity friction coefficient fw depends explicitly on the wave coefficient. field and on the roughness length. R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148 135

The formula of Weber is able to compute the The si. The scatter index dissipation rate directly from the bottom roughness rmse length through the stress parameterization. That of- sis Ž.20 fers the possibility to adapt the dissipation rate ac- (XY cording to the changing roughness under different The re. The relative error or index of agreement wave–current regimes. This could be important in some coastal areas. Moreover, the expression of N=rmse2 Weber maintains the spectral description and can be res1y applied to complex situations, where the wave field pe cannot be easily represented by one wave component <

4.3.2. Calibration In order to tune the friction coefficients of every friction model, the third caseŽ. high wind speed was

chosen. The combined scatter indexŽ. sic for Hs and Tp was taken as the cost function value to be minimized. The definition for the sic reads s q r sic siŽ.HsPsi Ž.T 222 Ž. The results using the default value for the friction coefficient in every friction model is taken as a referenceŽ. see Table 1 . An identical definition for the combined relative errorŽ. rec is used, replacing re for si in Eq.Ž. 22 .

The si for Hspand T and the sic are shown in Fig. 2. As can be observed in the figure, the behavior of the si is different for the different models. This Fig. 1. Bathymetry of Lake George and the locations of the eight figure shows how sensitive the wave parameters Ž Hs wave gauges. Depth contour interval is 0.5 m starting from 0. and Tp. are to variations in the friction coefficient in the case of the JONSWAP and the Collins expressions, and to variations in the roughness length, in For the computations, three northerly wind cases the case of the Madsen and the Weber expressions. were selected from the bench mark data set for For the JONSWAP formulation and starting from SWANŽ. SBMSWAN, case F41LAKGR . These cases the default value, it is clear that lowering the value are three typical examples, i.e., a low wind speed of the friction coefficient decreases the sic of Hs and s y1 s T . The si of T reaches a minimum at a value of ŽU106.5 m s.Ž , medium wind speed U 10 10.8 pp y1 s y1 0.030 m2 sy3 for G . For an interval of G-values, the m s.Ž and a high wind speed U10 15.2 m s. . The computations were carried out with SWAN si of Tpsis constant, but the results for H start version 40.01 using the WAM Cycle 3 formulations Ž.Booij et al., 1999 . The wave–wave triad interactions and depth-induced wave breaking were turned Table 1 on with default parameter values. For the bottom Friction coefficient values for the different friction models and the friction, one of the above formulations is used. As in resulting sic RisŽ. 1997 , Station 1 is taken as the up-wave bound- The values in bold are the default and the underlined are the ary in the simulation. This avoids uncertainty in the optimal values for the Lake George case. location of the northern shore because of seasonal Run No. JONSWAP Madsen Collins Weber variation in water depth. Since no directional wave G sic K NNsic c sic K sic spectrum is available at Station 1, the directional wxmm wx mm distribution of the waves is approximated with a 1 0.005 0.105 0.10 0.091 0.0005 0.120 0.01 0.079 cos2 Ž.u directional distribution. For the computa- 2 0.010 0.090 0.25 0.090 0.0025 0.106 0.10 0.074 tions, a directional resolution of 10 0 and a logarith- 3 0.020 0.079 0.50 0.086 0.005 0.093 0.25 0.071 D s 4 0.025 0.074 1.0 0.080 0.010 0.089 0.50 0.070 mic frequency resolution Ž.f 0.1 f between 0.166 5 0.030 0.070 3.0 0.076 0.015 0.081 0.75 0.070 and 2.0 Hz for the low wind case and between 0.125 6 0.034 0.082 5.0 0.082 0.020 0.078 1.0 0.074 and 1.0 Hz for the medium and high wind case are 7 0.038 0.080 10.0 0.093 0.025 0.074 2.5 0.111 chosen. The spatial resolution is 250 m both in x 8 0.045 0.078 20.0 0.172 0.030 0.071 5.0 0.165 and y direction. To account for seasonal variations 9 0.055 0.082 30.0 0.211 0.035 0.077 10.0 0.203 in water level, the water level was increased over the 10 0.067 0.092 50.0 0.214 0.040 0.081 20.0 0.236 q q q 11 0.075 0.096 70.0 0.214 0.045 0.081 40.0 0.241 entire lake with 0.1, 0.3 and 0.27 m for the 12 0.085 0.120 80.0 0.214 0.060 0.085 60.0 0.242 low, medium and high wind case, respectively. R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148 137

Fig. 2. The si for Hspand T , and sic in the function of the friction coefficient value in the Lake George case. The friction coefficient default value is indicated by a vertical line and the optimal valueŽ. the chosen value to run the three cases of Lake George is indicated by a circle.

deteriorating. Nevertheless, lowering the value of G that the reference value for G was taken as 0.067, improves the sic about 23% compared with the which is the appropriate value for depth-limited results using the default value for G Ž.see Table 1 . wind-sea case in the Southern North SeaŽ Bouws and The value retained for G is 0.030. It should be noted Komen, 1983. , and not as 0.038 which corresponds

Fig. 3. Idem Fig. 2 but for rmse. 138 R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148 to swell conditions. The most suitable value for this For the friction model of Weber, lowering the sic case is even smaller than the value for swell condi- leads to improvement for both Hspand T . The tions. The lowest value of the sic here corresponds changes of the si are rather smooth making the with the lowest value of si for Tp. choice of the best value for K N easy. The optimal Using the formulation of Madsen improvement of value for K N in Weber’s formulation was chosen = y4 sic leads to improvement for both Hspand T . That is equal to 7.5 10 m. The improvement for the sic because both parameters Hspand T are underesti- using the optimum value compared to the sic with mated using the reference value. The optimal value the default value is about 71%. = y3 for K N was chosen as 3 10 m. The improve- Comparing the si, rmse and the reŽ Figs. 2, 3 and ment for the sic compared to the sic with the default 4, respectively. , the results using the formulation of value is about 64%. The lowest value of the sic does Weber are more consistent in a statistical sense. The not correspond neither with the lowest value of the si optimal value for K N , according to the sic cost for Hspnor with the lowest value of the si for T . function, gives also the best value for the si, the rmse The sic varies little using the formulation of and the re for Hsp. For T , the magnitude of those Collins for a region of c-values around the reference statistical parameters is quite close to the best values. value. For higher than the reference value of c, the si That is not the case for the other formulations, for Hspimproves but the si for T increases. Going to especially not for the Collins’ and Madsen’s formu- lower values of c, the si of both wave parameters lations. The lowest value for the sicŽ highest value of increases. The value retained for c was taken equal rec. does not correspond neither to the lowest si and to 0.030. According to Table 1 and Fig. 2, the sic has rmseŽ. highest re for Hs nor to the lowest si and the best value but is only 14% different from the sic rmseŽ. highest re for Tp. As can be seen in Figs. 2, 3 with the default c-value. The value retained for c, and 4, the choice of the best value for the friction however, does neither produce the lowest value of coefficient or the roughness length is prescribed the si for Hspnor for T . Looking at the rmseŽ. Fig. 3 mainly by the si of Tpp. T is the parameter that and the recŽ. Fig. 4 , it is not so clear that the wave improves most during the tuning process. Note that model gives the best results for a c-value equal to Ris et al.Ž. 1999 remarked that using the WAM cycle 0.030. 3 formulations, SWAN systematically overestimates

Fig. 4. Idem Fig. 2 but for the re and rec. R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148 139 the significant wave height and underestimates the Comparing it with the value of 0.070, which is mean wave period. approximately the value of the sic using the optimal It should be mentioned that a wave model run was coefficients in the four bottom dissipation expres- made using a roughness length corresponding to the sionsŽ. see Table 1 , the difference is about 70%. grain size at the bottom of the lake which is fine clay Ž.Ris, 1997 of about 1.=10y6 m in diameter. The 4.3.3. Validation model was run using the expressions of Weber and Once the appropriate values for the friction coeffi- Madsen. Comparing the results, using the expression cients and roughness length were chosenŽ see Table = of Weber, between the retained value for K N Ž7.5 1, values underlined. , the other two selected cases 10y4 m. and the roughness length corresponding to for Lake George were run using these values. It is the grain sizeŽ 1.=10y6 m. the sic increases by assumed that the bottom condition did not change. about 30%. The si for Hs increases by 54% and the As can be see in Fig. 5, the significant wave si for Tp decreases by 12%. Using the Madsen height and the peak period are relatively well mod- = y6 expression with K N equal to 1 10 m, the differ- eled by SWAN using either of the four friction ence in the sic between the retained value for K N models, except at the last three stations in case one. Ž3=10y3 m. and the roughness length correspond- At these stationsŽ. from 6 to 8 , the wave parameters ing to the grain sizeŽ 1=10y6 m. is about 38%. The do not show a monotonic behavior. That can possi- si gets worse by 13% for Tpsand by 55% for H .As bly be ascribed to unresolved variation in the wind can be seen from these results, even at very small field. Such variation has been observed by YV. values for K N the dissipation by bottom friction still Those three stations were therefore eliminated from plays a role. the statistical calculations for case one only. A small

A wave model run without bottom friction for the underestimation of Tp can be observed in the medium case of high wind speed was made to see how and low wind cases. important the bottom friction is. The value of the sic Fig. 6 shows the statistical parameters for the not considering the bottom friction is about 0.119. three wind cases for each of the four bottom friction

Fig. 5. SWAN model results for the different bottom friction formulationsŽ. with the optimal value for the friction coefficients and observations of the significant wave heightŽ. left panels and peak period Ž right panels . in nearly ideal generation conditions in Lake George. ObservationsŽ) .Ž.Ž.Ž.Ž. , JONSWAP q , Madsen \ , Collins ` and Weber e formulations. 140 R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148

Fig. 6. rmseŽ.Ž.Ž top panels , si middle panels and re bottom panels . for wave height Ž. left panels and peak period Ž right panels . , for the three cases in Lake George. The AmeanB case represents the mean value for the three cases. JONSWAP Ž.q , Madsen Ž.\ , Collins Ž.` and Weber Ž.e formulations.

dissipation formulations used. The results from the SWAN gives the best approximation for Hs but the SWAN runs are to a certain extent similar. The approximation for Tp is not that good. From results, statistics show that using the formula of Collins, not shown here, using the default value for cs0.015,

Fig. 7. Combined scatter index sicŽ. top panel and the combined relative error rec Ž bottom panel . for the wave height and the peak period. The AmeanB case represents the mean values for the three cases. JONSWAP Ž.q , Madsen Ž.\ , Collins Ž.` and Weber Ž.e formulations. R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148 141

Table 2 Formulas for non-dimensional energy Ž.´ and non-dimensional frequency Ž.n against non-dimensional depth, obtained from the results of SWAN using the different bottom friction models Model JONSWAP Madsen Collins Weber ´s 4.1=10y31.76d 2.9=10y31.66d 1.4=10y31.42d 1.4=10y31.46d ns 0.14dy0.472 0.15dy0.430 0.20dy0.375 0.20dy0.375

CfCgives the worst approximation for H s but the formulations. The peak period in the low wind case best approximation for Tp. Žfor this wind field the bottom friction plays hardly Using the four bottom friction dissipation expres- any role. is very well reproduced using the formula- sions, the largest error for Hs is in the high wind tion of Weber. This is not the case using the other case compared with the other two cases. This can be formulations, as can be seen in Figs. 5 and 6. partially ascribed to a ‘deviation’ of the measure- 4.3.4. Depth-limited waÕe growth ments from the monotonic behavior in the locations In order to refine the expressions of Bretschneider 2 and 7 as can be seen in Fig. 5. Looking at the Ž.1958 for both non-dimensional energy Ž.´ and mean values of rmse, si and re for the three cases non-dimensional peak frequency Ž.n , YV used their Ž.Fig. 6 , the best performance for H corresponds to s full data set of about 65,000 data points. They found the use of the Collins formulation and the best that the asymptotic depth-limited growth can be con- performance for T corresponds to the Weber formu- p sidered dependent on the non-dimensional depth Ž.d lation. Fig. 7 shows the performance of the SWAN only. They found the following limits: using the different bottom friction formulations in s = y3 1.3 the function of the combined statisticsŽ. sic and rec ´ 1.06 10 d Ž.23 ns dy0 .375 of Hspand T . As can be seen, SWAN has the best 0.20 Ž.24 performance using Weber’s formulation followed by The non-dimensional parameters are defined as 24r Collins, and then by the JONSWAP and the Madsen gE U10 for the non-dimensional energy Ž.´ ,

Fig. 8. Comparison of the SWAN results using the different bottom friction formulations and the formula from Young and VerhagenŽ. 1996 for non-dimensional energy ´ Ž.top panel and non-dimensional peak frequency n Žbottom panel . against non-dimensional depth. 142 R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148

Table 3 results were stored for different fetches ranging from Statistics comparing the values obtained from SWAN using the 15 to 15,000 km. The friction coefficients and rough- different friction formulations with the values according to the ness length used have the optimal values retained formulae from YV in the case of depth-limited growth from the tuning runs. The expressions given in Table Model JONSWAP Madsen Collins Weber 2 were computed taking the maximum levels of y y ´ bias 0.002 0.001 0.000 0.000 non-dimensional energy and the minimum non-di- rmse 0.003 0.002 0.000 0.000 si 1.969 1.404 0.368 0.380 mensional peak frequency at every non-dimensional re 0.800 0.816 0.955 0.953 fetch of the different runs. n bias 0.055 0.055 0.000 0.000 Fig. 8 shows the different retained results for ´ rmse 0.055 0.055 0.000 0.000 and n together with the expressionsŽ. 23 and Ž. 24 of si 0.243 0.241 0.000 0.000 YV. Table 3 shows the statistics comparing them re 0.950 0.941 1.000 1.000 with the formulas obtained by YV. Although all of them give a good approximation to the measurements, it is clear that the formulae of Weber and r fUp10g for the non-dimensional peak frequency Collins give the best results. The fit to the n curve of r 2 Ž.n , gd U10 for the non-dimensional depth Ž.d , g YV is perfect. Because of the good approximation of is the gravitational acceleration, E is the total energy all formulations to the formula from YV, Table 3 of the spectrum, U10 is the wind speed measured at a confirms that selecting sic as the parameter to be reference height of 10 m, and d is the water depth. minimized was a good selection. To calculate the depth-limited growth, SWAN Even though YV considered that the asymptotic was run in one-dimensional mode for each of the depth-limited energy growth depends on non-dimen- friction models. Several runs were performed using sional depth only, the numerical results show that the different depths ranging from 2.5 to 20 m. The wind depth-limited growth is a function of the roughness speed was set equal to 20 m sy1 Ž to work in the length as well. If the bottom friction and bed mate- same non-dimensional depth interval as YV. . The rial were not important in fetch- and depth-limited

Fig. 9. Comparison of the SWAN results using the formulation of Weber with different roughness length Ž.K N and the formula from Young and VerhagenŽ. 1996 for non-dimensional energy ´ Žtop panel . and non-dimensional peak frequency n Žbottom panel . against non-dimensional depth. R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148 143

s r 2 conditionsŽ. as assumed by YV , then the value as- and x is the non-dimensional fetch. Ž x gx U10 ., signed to K N should not matter and the results x is the distance and would be expected to be the same. This is not the s 1r m dy0 .375r m case when the wave model is run using a different A2 1.505 Ž.31 roughness length in the friction dissipation formula- B s16.3911r m xy0.27r m Ž.32 tion. To exemplify this, the SWAN model was run 2 for depth-limited wave growth with Weber’s formu- The coefficients n and m control the rate of lation for bottom friction. Fig. 9 shows the results for transition from AdeepB to Adepth limitedB conditions. different values for K N. Going from larger to smaller YV performed a non-linear least squares analysis on values of K N , the ´ increases and the n decreases. their selected data set to determine n and m. Their Hence, one should expect that the curves for ´ analysis yielded n equal to 1.74 and m equal to growth against non-dimensional fetch change as well. y0.37. ExpressionsŽ. 27 and Ž. 28 give a family of This implies that the expressionsŽ. 23 and Ž. 24 should curves, one for each value of d. take into account the bottom roughness. The asymp- The results from SWAN are compared with the totes for ´ and n can be expressed as: equations given by YV. Figs. 10 and 11 show the results from SWAN using the different bottom fric- ´sAd B Ž.25 tion formulations for d equal to 0.10 and 0.50, respectively. The deep water asymptotic forms of nsCd D Ž.26 Eqs.Ž. 27 and Ž. 28 and the same equations but for shallow water Žn equal to 1.74 and m equal to One can see from Fig. 9 that the roughness length y0.37. are also shown. As can be seen from those has more influence on A and C than on B and D. figures, SWAN overestimates the total energy for = y5 Changing K N from 0.10 to 5 10 m, A changes very short x. In particular, energy growth in the around 250%, B changes 8%. C and D change 30% high frequency rangeŽ. very short fetches is usually and 9%, respectively. K N has more impact on the overestimated by SWAN. This overestimation is ob- energy than on the frequency. This suggests that A served systematically. According to Ris et al.Ž. 1999 , and C are functions of K N. the overestimation of energy at short fetches can be ascribed to the linear wave growth term of Cavaleri and Malanotte-RizzoliŽ. 1981 . But in this work, the 4.3.5. Fetch-limited waÕe growth linear growth term was not taken into account. Re- Using their data from Lake George, YV proposed sults from the wave model using the linear wave a generalized form to the shallow water limits for the growth shows no relevant differences with the results ´ growth of non-dimensional energy Ž.and non-di- without the linear growth term. The observed overes- n mensional peak frequency Ž.with non-dimensional timation of energy should be ascribed to another x fetch Ž. reason. The search for such reason is beyond the

n scope of this work. B s = y3 1 The SWAN runs with the formulations of Collins ´ 3.64 10 tanh A1tanhŽ. 27 ½5tanh A1 and Weber for bottom friction dissipation catch the asymptotic levels of ´ and n given by the expres- m sion of YV quite well, better than when using the B2 ns0.133 tanh A tanhŽ. 28 expressions of JONSWAP and Madsen. But as can 2 tanh A ½52 be seen in Fig. 11, the wave model reproduces better the levels of non-dimensional energy when using the where expression of Weber than when using the expression of Collins. A s0.2921r n d 1.3r n 29 1 Ž. To quantify the differences between the model r results and the expression of YV for fetch-limited s = y511 n x r n B1 Ž.4.396 10 Ž.30 growth, the wave model was run for a range of 144 R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148

Fig. 10.Ž. a Non-dimensional energy ´ and Ž. b non-dimensional peak frequency n against non-dimensional fetch x for a non-dimensional depth d of 0.10. The deep water asymptotic form of Eqs.Ž. 20 and Ž. 23 is shown in dashed line. The same equations but for shallow water as found by YV Ž.ns1.74 and msy0.375 is in solid line. SWAN was run using JONSWAP Ž.Ž.Ž.q , Madsen \ , Collins ` and Weber Ž.e formulation. values of d. The four statistical parameters given in Fig. 12 shows the statistical parameters compar- Section 4.2 are computed. ing ´ from the wave model results against the ´

Fig. 11. Idem Fig. 10 but for a non-dimensional depth d of 0.50. SWAN was run using JONSWAP Ž.q , Madsen Ž.` , Collins Ž dotted . and Webers Ž.e formulations. R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148 145

Fig. 12. The bias, rmse, si and re against non-dimensional depth d. The comparison is done for wave growth in fetch-limited conditions between the non-dimensional energy ´ from SWAN with the different bottom friction dissipation formulations and the formula from Young and VerhagenŽ. 1996 . computed from the YV expressionŽŽ.. Eq. 27 for a points at d-values of 0.10 and 0.50, respectively. To range of values of d. Fig. 13 shows the same as Fig. calculate the fetch-limited growth, SWAN was run in 12 but for n. In this way, Figs. 10 and 11 are one-dimensional mode for each of the friction mod- represented statistically in Figs. 12 and 13 as two els. Several runs were performed using a depth equal

Fig. 13. Idem Fig. 12 but for non-dimensional frequency. 146 R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148 to 20 m and wind speed ranging from 10 to 31.3 m is more consistent in shallow water regions. The sy1 Ž to work in the non-dimensional depth range SWAN model was run with the three formulations from 0.1 to 1.0. . The total fetch for every run was originally included plus the eddy–viscosity formula- 15,000 km with a resolution of 5 km. The fetch to tion of Weber. The data of Lake George were used compute the statistical parameters was different for to tune the friction coefficients of every formulation every d, depending on when the energy computed such that the combined scatter index was minimal. from Eq.Ž. 27 becomes constant Ž no changes in the This exercise revealed different levels of difficulty in seventh significant digit. . The fetch range is from tuning the different friction coefficients. 440 km for d equal to 0.1 to 997 km when d is Weber’s model showed the best performance in equal to 1.0. At small d-values, SWAN gives results the cases of depth- and fetch-limited wave growth. In which have almost the same bias and rmse using the case of depth-limited wave growth, the fit of the either of the four bottom friction dissipation expres- calculated curve for the non-dimensional peak fre- sions. From Fig. 12a and b, it is evident that results quency to the one obtained by YV is as good as start diverging going to deeper waters. At higher perfect. For the non-dimensional energy, the statisti- d-values, the use of the expressions of JONSWAP cal values, rmse, si and re, show that the results and Madsen gives the largest bias and rmse. Apply- using Weber’s formulation are superior in approaching the formulations of Collins and Weber, one ing the equations obtained from YV. In the case of obtains smaller bias and rmse, with a preference for fetch-limited wave growth, the formulation of Weber Weber’s expression. Looking at the four statistical showed the best performance in approaching the parametersŽ. Fig. 12 , the results of SWAN using the equations of YV derived from the measurements. formula of Weber approach better the non-dimen- Running the SWAN model using Weber’s formula sional energy ´ computed by Eq.Ž. 27 and its behav- with different roughness length suggests that in the ior is more uniform along the non-dimensional depth equations for depth- and fetch-limited wave growth axis, as can be seen in Fig. 12d. the effect of bottom roughness should be included. Fig. 13 shows the statistics for the non-dimen- Formulations for dissipation by bottom friction, sional frequency n of the SWAN results using the such as the model by Madsen or Weber, which take different bottom friction dissipation formulations. In explicitly physical parameters for bottom roughness contrast with ´, the bias and the rmse of n do not into account, should be preferred in wave modeling have the same value for small d. Contrary to the in shallow water areas. They offer the possibility to results for ´, the bias and the rmse decrease with adapt the dissipation rate according to the changing increasing d. roughness under different wave or wave–current The statistical measures for ´ indicate that using conditions. the expressions of Weber the wave model results Besides showing the best performance, the for- approach quite well the values computed from Eq. mula of Weber has some other advantages. It was, at Ž.27 , better than using the other three bottom friction least for the Lake George case, easier to tune than formulations. With respect to n, the wave results are the other formulations. The tuning parameter, namely, of similar quality when the expressions of Collins the bottom roughness length has a physical meaning. and Weber are used and better than when using one It gives information about the bottom boundary layer, of the other two expressions. through the friction velocity. It retains a spectral One can therefore conclude that in fetch- and description making this formulation more reliable for depth-limited conditions, the computed wave param- a multi-modal wave spectra. It can be extended to eters are more consistent when the bottom friction the combined wave–current situation, important in dissipation expression of Weber is used. situations where the tidal currents play a significant role in the dynamics of the coastal zone. The above conclusions are based on two major 5. Summary and conclusions assumptions. The first assumption states that the The main objective was to investigate and clarify bottom friction dissipation is the only ‘unknown’ which bottom friction formulation performs best or source term and the other source terms are repre- R. Padilla-Hernandez,´ J. MonbaliurCoastal Engineering 43() 2001 131–148 147

s q sented correctly. The second assumption states that sic Combined scatter index ŽŽsi Hs . si- r wx the conclusions would be the same if the bottom Ž..Tp 2– characteristics of the Lake George would, for exam- t Timewx s wx ple, change from fine clay to a sandy bottom. It Tp Wave peak period s would therefore be interesting to redo the above u Reynolds-average velocityw m sy1 x exercise when better formulations for the other source u) Friction velocity at the bottomw m sy1 x r terms and or when similar measurements but in a Uk Orbital velocity at the bottom for a given situation with a different bottom material become wave numberw m sy1 x available. U10 Wind speed at 10 meters above the water levelw m sy1 x r Notation ²U 2 :1 2 Root mean square of the orbital motion at wx w y1 x ab Near bottom excursion-amplitude m the bottom m s c Coefficient in the Collins expression for the x, y Horizontal coordinateswx m dissipation coefficientwx – z Vertical coordinatewx m wx w 2 y3 x fw Non-dimensional friction factor – G Coefficient in CfJ ms w y1 x wx Cf Dissipation coefficient ms d Non-dimensional depth – w y1 x wx CfJ JONSWAP dissipation coefficient m s dij Kronecker delta – w y1 x wx CfM Madsen dissipation coefficient m s ´ Non-dimensional energy – w y1 x wx CfC Collins dissipation coefficient m s u Wave direction rad w y1 x wx CfW Weber dissipation coefficient m s k von Karman constant – wx cxy, c Wave propagation velocities in geographi- n Non-dimensional peak frequency – cal x-, y-spacew m sy1 x r Density of the waterw kg my3 x wx cs , cu Wave propagation velocities in spectral s-, s Relative frequency rad u-spacew sy2 , rad sy1 x t Turbulent stress in the wave boundary layer d Water depthwx m wNmy2 x E Total energy of the wave spectrumw m2 x x Non-dimensional fetchwx – f Frequencywx Hz v Radian frequencyw rad sy1 x F Energy density spectrumw m2 s rady1 x g Acceleration due to gravityw m sy2 x h Total water depthwx m wx Hs Significant wave height m y Acknowledgements k Wavenumber vectorw m 1 x k Wavenumberw my1 x wx R.P.H. gratefully acknowledges financial support K N Roughness length m N Wave action densityw m22 s rady1 x from the Consejo Nacional de Ciencia y Tecnologıa´ y Ž. p Reynolds-average pressurew N m 2 x CONACYT, Mexico´ . We also thank WL Delft re Relative errorwx – Hydraulics for the SBMSWAN and Ian Young, Uni- s q versity of New South Wales, Canberra, Australia, for rec Combined relative error ŽŽre Hs . re- r wx the data of Lake George. Ž..Tp 2– rmse Root mean square errorwx –

Sbf Dissipation of wave energy by bottom fric- tionw m2 s rady1 x References

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