water

Article Wave Forecasting in Shallow Water: A New Set of Growth Curves Depending on Bed Roughness

Sara Pascolo *, Marco Petti and Silvia Bosa Dipartimento Politecnico di Ingegneria e Architettura, University of Udine, 33100 Udine, Italy; [email protected] (M.P.); [email protected] (S.B.) * Correspondence: [email protected]; Tel.: +39-0432-558713



Received: 15 October 2019; Accepted: 1 November 2019; Published: 5 November 2019


Abstract: Forecasting relationships have been recognized as an important tool to be applied together, or not, with complete numerical modelling in order to reconstruct the wave field in coastal areas properly when the available wave data is limited. In recent years, the literature has offered several comprehensive sets of field experiments investigating the form of the asymptotic, depth-limited wind waves. This has made it possible to reformulate the original deep water equations, taking into account the effects of water depth, if wind waves are locally generated in shallow and confined basins. The present paper is an initial attempt to further contribute to the shallow water forecasting curves which are currently available, also considering the role on the wave generation of a variable equivalent bottom roughness. This can offer the possibility of applying shallow growth curves to a broad variety of contexts, for which bed composition and forms can be different. Simple numerical tests have been conducted to reproduce the fully developed conditions of wave motion with variable roughness values. To validate the new set of equations, they have been applied to a real shallow lake for which both experimental and numerical wave data is available. The comparison of the obtained results is very encouraging in proceeding with this approach.

Keywords: wind wave growth; finite depth; bottom friction; shallow lake

1. Introduction Wind waves govern the nearshore dynamics in combination with tidal and current flows [1,2]. They are really important phenomena in the sediment resuspension mechanism of coastal lagoons, sheltered estuarine basins, and shallow lakes [3–10]. In these contexts, where depth limits both current velocities and the relative bottom shear stresses, locally generated waves trigger the main morphological processes [11–14]. Extended literature considers the importance of wave-induced morphodynamic changes [15–17] and their impact on environment also from an ecological and socio-economic point of view [18–20]. In this sense, the comprehension and evaluation of wind wave dynamics on finite depth represent a key point for the correct management of important eco-systems, such as estuarine and lagoon mudflats, salt marshes, and shallow lakes. Experimental wave measurements are still essential to investigate hydrodynamic and erosion-accretion processes on shallow depths [8,16,21–23], but they are quite rare. The available data has revealed that the growth of wind waves, locally generated on finite depth, is greatly affected by energy dissipations due to the interaction with bottom [24–26]. Both wave heights and periods are limited by water depth, and although very small compared to deep water cases, they are able to induce high bed shear stresses, and to resuspend sediments from the seabed [11,16,27,28]. In absence or lack of field data, numerical modelling is undoubtedly one of the most complete approaches for forecasting waves and their hydrodynamic effects. For this reason, it has been widely adopted over recent years in several coastal applications, many of which are in lagoons [27,29–33] or

Water 2019, 11, 2313; doi:10.3390/w11112313 www.mdpi.com/journal/water Water 2019, 11, 2313 2 of 20 intertidal estuaries [34–36], and in similar shallow water contexts such as lakes [14]. To this purpose, nearshore wind wave spectral models have extended the hindcasting procedure to finite-depth water, taking into account all the source terms linked to bottom dissipations [37,38]. Spectral wave attenuation by bottom friction has been thoroughly examined, both from an analytical and numerical point of view, in order to further understand the influence on the evolution of the spectrum in depth-limited generation cases [39–43]. The main results concern the spectral peak frequency which, with the same wind, remains higher than in deep water, and an upper bound which limits the energy growth at lower frequencies. This means that both the wave peak period and the significant wave height are confined to lower values depending on depth [44,45]. Moreover, bottom friction is an important mechanism in controlling the spectral shape in shallow water, which counteracts the Hasselmann nonlinear energy transfer [46] toward lower frequencies. In this sense, spectral energy is strictly related to depth and bottom roughness and it reaches an equilibrium in the generation process when the bottom friction dissipation approximately balances the nonlinear wave-wave interactions [41]. The existence of an asymptotic limit to wave growth in finite-depths has already been found by Bretschneider by means of field investigations in Lake Okeechobee, USA [25]. Starting from these measurements, a forecasting set of equations has been developed in order to determine the growth of waves generated by winds blowing over relatively shallow water [24,47]. This semi-empirical approach was subsequently improved, based on further and more extensive experimental wave data carried out by Young and Verhagen in the shallow Lake George, Australia [26,48–50]. Forecasting equations represent a valuable aid to preliminary engineering design and for use in the validation of comprehensive spectral models. In particular, the equations provide results quickly, given their simplicity and because they do not require calibration steps. For these reasons, despite the fact that growth curves refer to the nominally ideal conditions of constant wind blowing over a body of uniform finite depth, their usefulness is widely accepted. Many authors have tried out the semi-empirical formulas to reproduce the wave field in shallow lakes and inside coastal lagoons, where tidal flats are characterized by a near horizontal topography. The results have been used both independently [13,18,51] and in combination with numerical modelling [11,52,53]. However, despite these studies, the role of bottom roughness still remains unclear, even if it has been recognized as an important factor in the evolution of the energy spectrum during the generation process in shallow depths [41], since the friction dissipations balance the energy source terms. The current forecasting equations do not depend on the bed topographic features as might be expected, at least in a fully developed state, and do not take into account that the wave field is limited not only by depth but also by bottom friction. Bretschneider [25] has determined the growth curves by adopting a low friction factor, which can be generally representative of very smooth, homogenous and flat beds [11]. Also, the bottom of Lake George is characterized by cohesive mud, where ripples do not develop, and this would explain why the roughness does not directly enter the growth curves by Young and Verhagen [26]. However, it has been proven that the friction factor can considerably increase in the presence of a more irregular bottom, in terms of bedforms and composition [41,54,55]. Moreover, a greater roughness can strongly affect the locally generated wave motion, as underlined in a recent study [11], leading to a significant reduction of the wave height and period and in particular, of the wave bed shear stress. The present paper is a first attempt to reformulate the growth curves taking into account the correlation between the wave field and the bed roughness in the generation process of wind waves in shallow water, starting from the asymptotic state of the fully developed condition. A new set of equations is proposed, consistent with the previous ones, but with the advantage of being expressed in a parametric form that varies according to different bottom states. The study has been developed numerically, by means of the open source spectral model SWAN (Simulating WAves Nearshore) [38], applied to an idealized test condition. The results have been compared to the currently available forecasting equations and arranged on the basis of different roughness values. Finally, the curves Water 2019, 11, 2313 3 of 20 have been validated through an application to a real shallow lake and the obtained wave heights and periods have been compared to both experimental and numerical data. The arrangement of the paper is as follows: Section2 presents an overview of previous studies of the wave growth in finite depth generation processes; in Section3 the numerical procedure is described and the modified growth curves are shown; Section4 reports the application of the curves to Lake Neusiedl, Austria; and Section5 discusses the obtained results.

2. Growth of Depth-Limited Wind Waves The acquired data in Lake Okeechobee represents one of the first important experiments concerning the generation process in shallow and confined basins. Bretschneider and Reid [24] have suggested a simplified numerical procedure for determining the change in wave height for wind waves travelling on a flat bed, taking bottom friction and percolation into account. The best agreement between the numerical method and the wave data, collected both in Lake Okeechobee and in the shallow regions of the Gulf of Mexico, was obtained when a bottom friction factor fw of 0.01 was selected [56]. Ijima and Tang [47] have also combined these results with available deep water and fetch-limited data to develop a set of dimensionless curves which predict the main wave parameters as a function of depth. 2 4 In particular, the variables ε = g E/(U10) and υ = U10/(Tpg), which are related respectively to the total wave energy E and the peak period Tp, are expressed in the general form:

( " #)α1 B1 ε = C1 tanhA1 tanh , (1) tanhA1

  α2 B2 ν = C2 tanhA2 tanh (2) tanhA2 where U10 is the wind speed measured at a reference height of 10 m and g is the gravity acceleration. The coefficients Ai and Bi, with i = 1,2, are polynomial functions of the water depth d and the fetch x, 2 2 both expressed in dimensionless form as respectively δ = gd/(U10) and χ = gx/(U10) :

ai2 bi2 Ai = ai1δ Bi = bi1χ , (3) where aij, bij, with j = 1,2, Ci and αi are constant. The Lake Okeechobee study, within the limits given by the instruments and recording techniques of that time, was more focused on determining the asymptotic values to wind wave growth on shallow depths, rather than on investigating the fetch-evolution. In fact, Bretschneider included the effects of fetch by assuming a deep water growth rate modified by bottom friction. This growth rate asymptotically approached the limits of the fully developed condition empirically evaluated. Also, Vincent and Hughes [44] considered the case of wave generation controlled only by depth, thus excluding dependence on fetch. They defined simplified equations for wave height and period, based on theories about spectral shape and assumptions of nondispersive waves on shallow depth. These equations represent the upperbounds on the experimental data of Bretschneider [25] and Bouws et al. [45]. The role of bottom friction still remains unclear and the authors assumed that its effects were implicitly included by the expression assumed for spectral shape. Graber and Madsen [41] analyzed the importance of the friction factor in determining the growth of waves for arbitrary water depths, with a minimum of 10 m. They applied a parametric model to idealized tests of fetch-limited generation over a horizontal flat bed. A steady, uniform wind was assumed, while the bottom friction was made as varying in order to reproduce different bottom sediments and conditions. Generally, the friction factor depends on the wave characteristics near the bottom [11,57] and on the fluid-sediment interaction; in particular, this mechanism can cause a strong increase in the friction factor when ripples start to form [41,48,54,55]. The values considered by Graber and Madsen [41] are constant in the domain and range from 0 to 0.1 in the performed test cases. For fw = 0.01, analogous to the Bretschneider value [56] and representing a flat immobile bed, the Water 2019, 11, 2313 4 of 20 friction begins to show its influence in shallow water. For the assigned wind speed of 20 m/s, this is the minimum bottom friction necessary to balance the non-linear interactions in the migration of the peak frequency toward lower values. The spectral shape is really dominated by bottom friction only when the friction factor is at least equal to 0.03, representative of a minimum mobile flat bed with silt-sized sediments. A further increase in the friction factor in the presence of coarser grain sizes results in strongly attenuated wave conditions. This study confirms that the wave generation in shallow waters cannot disregard the bottom composition and therefore its roughness. Extensive measurements of wind wave spectra, wind speed, and wind direction were acquired by Young and Verhagen [26] along the main axis of the elongated and shallow Lake George in Australia. This is probably one of the best set of observations of wave growth in finite depths and it has provided the possibility of defining the evolution of waves along the fetch. Furthermore, this study has led to a reformulation of the previous set of forecasting equations, both in the fully developed condition and with varying fetch. The bed of Lake George is quite uniform and relatively horizontal with an approximate water depth of 2 m. This feature makes it close to the ideal test condition of Graber and Madsen [41] although with shallower water. Approximately 65,000 points were used to define, through image techniques, the asymptotic relations between the wave parameters and the depth, in the following dimensionless form:

ε = AδB, (4)

υ = CδD. (5) where A, B, C, and D are constants to be determined. Equations (4) and (5) represent the limits reached in the fully developed generation process and fetch-independent condition. This means that they provide the upper and the lower limit respectively of the wave energy, therefore of wave height, and of the wave peak frequency. Subsequent new wave recordings in Lake George have allowed the shallow depth growth curves to be further revised [49,50]. The experiments were conducted when Lake George was at a very low level with the water depth ranging between 1.15 m and 0.40 m, at the measurement site. Young and Babanin [49] proposed a slight modification to the power law (4) as derived by Young and Verhagen [26]. Breugem and Holthuijsen [50] found a North–South stratification in the data that Young and Verhagen [26] ignored, but, for fully developed conditions, their results still agree well with those of the original authors. The constant values of the coefficients entering in the limit Equations (4) and (5), according to the various authors cited above, are summarized in Table1.

Table 1. Constant values of the coefficients entering in the limit Equations (4) and (5), according to different authors.

Authors ABCD Bretschneider [58] 1.4 10 3 1.5 0.16 0.375 × − − Vincent and Hughes [44] 2.7 10 3 1.5 0.14 0.5 × − − Young and Verhagen [26] 1.06 10 3 1.3 0.20 0.375 × − − Young and Babanin [49] 1.0 10 3 1.2 - - × −

In the present study these coefficients are modified in order to make them variable, depending on the equivalent roughness height of the bottom.

3. Numerical Procedure and Analyses The finite-depth wind wave generation process has been performed numerically to reproduce the distribution of wave heights and peak periods for different water depths and bed roughness conditions. The spectral model used is the 2D open source finite difference model SWAN [38], that solves the energy density balance equation taking into account all physical processes: the positive energy input by wind, the dissipations by whitecapping, bottom friction, and depth-induced wave breaking, and the Water 2019, 11, 2313 5 of 20 energy transfer by quadruplet wave-wave interactions. The domain is a regular computational grid, analogous to a previous study by Pascolo et al. [11] but with a fetch of 200 km along the prevailing axis to ensure fully developed wave conditions. The spatial discretization of 100 m is used both in x- and in y-direction. Water 2019, 11, x FOR PEER REVIEW 5 of 20 The water depth is assigned as uniform over the whole grid, but it varies from a minimum value of 0.1computational m to a maximum grid, analogous of 4 m in to order a previous to reproduce study by di Pascolofferent et conditions al. [11] but ofwith flat a bathymetriesfetch of 200 km in the simulatedalong scenarios.the prevailing This axis approach to ensure is fully similar developed to the onewave followed conditions. by GraberThe spatial and discretization Madsen [41 ],of although 100 withm shallower is used both depths, in x- and coherently in y-direction. with the experimental values used to develop the forecasting relationshipsThe water discussed depth above. is assigned The as wind uniform direction over the is whole kept constant grid, but andit varies aligned from witha minimum the longitudinal value dimensionof 0.1 m of to the a maximum grid, while of 4 the m in wind order speed to reproduce is uniform different over conditions the domain of flat but bathymetries variable in in the the range simulated scenarios. This approach is similar to the one followed by Graber and Madsen [41], 6–14 m/s. These assumptions are consistent with the hypotheses on which the forecasting curves although with shallower depths, coherently with the experimental values used to develop the are based since the latter require wind with constant speed and direction over the generation area. forecasting relationships discussed above. The wind direction is kept constant and aligned with the Moreover,longitudinal these hypothesesdimension of are thecoherent grid, while with the whatwind speed really is happens uniform inover shallow the domain and confinedbut variable basins. in In thesethe contexts, range 6–14 there m/s. is a These not so assumpti pronouncedons are variability consistent in with the the wind hypotheses characteristics on which during the forecasting the generation processcurves since are it based is completed since the in latter a shorter require time wind compared with constant to the speed analogous and direction case of over deep the water. generation Moreover, the resultingarea. Moreover, values ofthese the hypotheses dimensionless are coherent water depth with whatδ are really consistent happens with in thoseshallow investigated and confined both in Lakebasins. Okeechobee In these and contexts, Lake there George. is a not so pronounced variability in the wind characteristics during Withthe generation the aim ofprocess evaluating since it the is completed effects on in the a sh waveorter field time duecompared to diff toerent the analogous bottom friction case of conditions, deep the relativewater. Moreover, dissipation the source resulting term values is taken of the according dimensionless to the water formulations depth δ are of consistent Madsen etwith al. those [40]. This investigated both in Lake Okeechobee and Lake George. means that the friction factor fw entering the equation to compute the spectral wave attenuation is With the aim of evaluating the effects on the wave field due to different bottom friction not constant but depends, under the hypothesis of rough turbulent wave motion, on the relative conditions, the relative dissipation source term is taken according to the formulations of Madsen et roughness. This is defined as A/K , where A = U T/2 is one half of the horizontal orbital excursion, al. [40]. This means that the frictionN factor fw enteringw the equation to compute the spectral wave Uw isattenuation the maximum is not bottom constant orbital but depends, velocity, underT is thethe wavehypothesis period, of rough and K turbulentN is the Nikuradse wave motion, equivalent on bed roughness.the relative roughness.KN is a parameter This is defined to be as assigned A/KN, where as input A = U towT the/2 is model one half and of the it is horizontal generally orbital taken as a functionexcursion, of the U medianw is the maximum diameter ifbottom the bed orbital is composed velocity, T of is coarsethe wave grains, period, while and forKN is fine the cohesiveNikuradse grains or mud-beds,equivalent it bed can roughness. be set equal KN is to aa parameter few millimeters to be assigned [54,55 as]. Aninput appropriate to the model range and it of is KgenerallyN values has beentaken established as a function so asto ofconsider the median diff diametererent configurations if the bed is composed and granulometric of coarse grains, compositions while for of fine the bed, as actuallycohesive happens grains or in mud-beds, shallow coastal it can be regions. set equal The to a minimum few millimeters value [54,55]. is 0.0005 An m appropriate and it is representative range of KN values has been established so as to consider different configurations and granulometric of a very flat bed, corresponding approximately to a friction factor of 0.01 [11]. The highest value is compositions of the bed, as actually happens in shallow coastal regions. The minimum value is 0.0005 0.05 m, set as default by SWAN, and it can be related to an irregular rough bottom with ripples. SWAN m and it is representative of a very flat bed, corresponding approximately to a friction factor of 0.01 has been[11]. The run highest in stationary value is mode 0.05 m, since set as there default is noby interestSWAN, and in following it can be related the temporal to an irregular evolution rough of the generationbottom process,with ripples. and SWAN overall has 625 been simulations run in stationary have been mode carried since out.there is no interest in following Figurethe temporal1 shows evolution the arrangement of the genera oftion all process, the obtained and overall points 625 insimu termslations of have dimensionless been carried variables out. with respectFigure to the1 shows semi-empirical the arrangement limits of having all the theobtain coeedffi pointscients in reported terms of indimensionless Table1. variables with respect to the semi-empirical limits having the coefficients reported in Table 1.

(a) (b)

FigureFigure 1. Scatter 1. Scatter points points numerically numerically obtained obtained with generation generation process process in inshallow shallow depths depths in terms in terms of of dimensionlessdimensionless variables: variables: (a )(a the) the wave wave energy, energy, andand ( (bb)) the the peak peak frequency frequency as functions as functions of water of water depth. depth. The lines represent the semi-empirical Equations (4) and (5) according to the authors listed in Table The lines represent the semi-empirical Equations (4) and (5) according to the authors listed in Table1. 1.

Water 2019, 11, 2313 6 of 20

As can be observed, the lines tend to define an upper and a lower bound for all the points Water 2019, 11, x FOR PEER REVIEW 6 of 20 corresponding to the wave energy and the wave peak frequency νp = 1/Tp, respectively. This result confirms thatAs can the be forecasting observed, curvesthe lines (4) tend and to (5)define identify an upper the asymptotic and a lower conditions bound for all that the the points generated wavecorresponding field can reach to forthe thewave assigned energy and water the depth wave peak and windfrequency speed. νp = 1/Tp, respectively. This result Theconfirms points that have the forecasting been grouped curves according (4) and (5) to iden thetify di fftheerent asymptotic bottom conditions roughness that height, the generated with the aim of verifyingwave field whether can reach they for tendthe assigned to align water and, depth in this and manner, wind speed. define a particular wave field related The points have been grouped according to the different bottom roughness height, with the aim to the bed condition. Even though there is some data scattering which in any case reduces as the of verifying whether they tend to align and, in this manner, define a particular wave field related to roughnessthe bed increases, condition. a trend Even canthough be recognized there is some as depicteddata scattering in Figure which2. Thein any dimensionless case reduces wave as the energy is representedroughness as increases, a function a trend of the can dimensionless be recognized depth as depicted for the KinN Figurevalues 2. set The to dimensionless compute spectral wave bottom frictionenergy dissipations. is represented The as red a function dashed of lines the providedimensionless the regression depth for the of K theN values points, set and to compute the resulting coefficientsspectralA bottomand B frictionentering dissipations. in Equation The (4) red are dash showned lines in provide the boxes. the regression of the points, and the resulting coefficients A and B entering in Equation (4) are shown in the boxes.

FigureFigure 2. Dimensionless 2. Dimensionless wave wave energy energy grouped grouped by by di differentfferent bottom bottom roughnessroughness values: values: (a()a K) NK =N 0.0005= 0.0005 m; m; (b) KN = 0.0001 m; (c) KN = 0.005 m; (d) KN = 0.015 m; (e) KN = 0.05 m. Red dashed lines are regression (b) KN = 0.0001 m; (c) KN = 0.005 m; (d) KN = 0.015 m; (e) KN = 0.05 m. Red dashed lines are regression 2 curvescurves obtained obtained by meansby means of of the the least least squares squares method.method. Correlation Correlation coefficients coefficients R ofR each2 of linear each linear regression are indicated on each plot. The relative coefficients A and B entering in Equation (4) are regression are indicated on each plot. The relative coefficients A and B entering in Equation (4) are shown in the boxes. shown in the boxes.

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WaterParameter 2019, 11, x BFORdefines PEER REVIEW the angular coefficient of each line and it does not change significantly7 of 20 between the wave field generated on different bottom roughness conditions. In particular, it remains Parameter B defines the angular coefficient of each line and it does not change significantly fairly constant and equal to the value of 1.3 determined by Young and Verhagen [26], thus confirming between the wave field generated on different bottom roughness conditions. In particular, it remains the agreement with their results and the experimental data collected in Lake George. On the contrary, fairly constant and equal to the value of 1.3 determined by Young and Verhagen [26], thus confirming parameterthe agreementA decreases with their progressively results and the with experimental the increase data of collectedKN. This in Lake means George. that On the the position contrary, of the linesparameter in the graph A decreases and therefore progressively the asymptotic with the increase limit that of K theN. This wave means energy that can the position reach, is of lowered. the lines The maximumin the graph value and of A thereforecorresponding the asymptotic to the minimum limit that of theKN waveis the energy same as can that reach, proposed is lowered. by Young The and Verhagenmaximum [26], value and thisof A further corresponding confirms to thethe conditionminimum ofof theKN is smooth the same flat as bed that of proposed Lake George. by Young andVery Verhagen similar [26], considerations and this further can confirms also be made the condition on the trend of the ofsmooth the dimensionless flat bed of Lake peak George. frequency reportedVery in Figure similar3. considerations can also be made on the trend of the dimensionless peak frequency reported in Figure 3.

Figure 3. Dimensionless wave peak frequency grouped by different bottom roughness values:Figure(a) 3.K NDimensionless= 0.0005 m; (waveb) K Npeak= 0.0001 frequency m; groupe (c) KN d= by0.005 different m; ( dbottom) KN = roughness0.015 m; values:(e) KN (a=) 0.05KN m. Red= dashed 0.0005 m; lines (b) K areN =regression 0.0001 m; (c curves) KN = 0.005 obtained m; (d)by KN means= 0.015 ofm; the(e) K leastN = 0.05 squares m. Red method. dashed lines Correlation are coeffiregressioncients R 2curvesof each obtained linear regressionby means of are the indicatedleast squares on method. each plot. Correlation The relative coefficients coefficients R2 of eachC and D enteringlinear in regression Equation are (5) indicated are shown on ea inch the plot. boxes. The relative coefficients C and D entering in Equation (5) are shown in the boxes.

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Water 2019, 11, x FOR PEER REVIEW 8 of 20 In fact, parameter D maintains an almost constant value even if slightly lower than that of 0.375 − originallyIn fact, proposed parameter by D Bretschneider maintains an [ 58almost] and constant subsequently value confirmedeven if slightly by Young lower and than Verhagen that of –0.375 [26]. Onoriginally the other proposed hand, there by Bretschneider is a dependence [58] ofand the subsequently coefficient C confirmedon the bottom by Young roughness and Verhagen which causes [26]. theOn linesthe other to be hand, raised there in the is grapha dependence with increasing of the coefficientKN. C on the bottom roughness which causes the linesIn this to be sense, raised the in angular the graph coe withfficients increasing of the regressionKN. lines can be kept fixed in all the cases analyzedIn this and sense, equal the to 1.3angular for B andcoefficients0.40 for ofD the, while regression the variability lines can of be the kept remaining fixed in parameters all the casesA − andanalyzedC with and the equal bed roughnessto 1.3 for BK andN cannot –0.40 befor ignored. D, whileIn the order variability to obtain of the the remaining functions thatparameters associate A theseand C coe withfficients the bed to roughnessKN, the estimated KN cannot values be ignored. for each In class order of to roughness obtain the have functions been plottedthat associate and a regressionthese coefficients by means to ofKN the, the least estimated squares values method for has each been class performed. of roughness have been plotted and a regressionThis analysis by means has of led the to least define squares two polynomial method has curves, been performed. that can be observed in Figure4, having the followingThis analysis equations has led respectively: to define two polynomial curves, that can be observed in Figure 4, having the following equations respectively: 0.205 A = 0.0002 KN− , (6) = −0.205 A 0.0002 K N , (6) 0.061 C = 0.307 KN . (7) = 0.061 C 0.307 K N . (7)

(a) (b)

FigureFigure 4.4. Values of: ( (aa)) the the coefficient coefficient AA enteringentering in in Equation Equation (4) (4) and and (b ()b the) the coefficient coefficient C Centeringentering in inEquation Equation (5), (5), as asa afunction function of of the the equivalent equivalent bottom bottom roughness. roughness. The The dotted dotted lines lines are thethe relativerelative regressionregression curvescurves byby meansmeans ofof thethe leastleast squaressquares method. method.

EquationsEquations (6)(6) andand (7)(7) allowallow forfor thethe determinationdetermination ofof thethe characteristicscharacteristics ofof thethe locallylocally generatedgenerated wavewave motionmotion on on a givena given depth depth and forand an for assigned an assigned wind speed, wind takingspeed, into taking account into di accountfferent roughness different heightsroughness and heights therefore and relative therefore dissipation relative dissipatio conditionsn conditions of the bottom. of the Abottom. summary A summary of the revised of the coerevisedfficients coefficients is reported is reported in Table 2in for Table a more 2 for immediate a more immediate comparison comparison with those with listed those in Tablelisted1 in. Table 1. Table 2. Revised coefficients entering in the limit Equations (4) and (5) as functions of KN.

Table 2. Revised coefficients entering in the limit Equations (4) and (5) as functions of KN. ABCD 0.0002 K 0.205A 1.3B 0.307C K 0.061 D 0.40 N− N − 0.0002 KN−0.205 1.3 0.307 KN0.061 −0.40

WithWith thethe purpose purpose of of better better understanding understanding the dithefferences differences that thisthat approach this approach involves involves with respect with torespect the assumption to the assumption of a constant of a friction constant factor, friction a comparative factor, a comparative analysis based analysis on the wavebased characteristics on the wave insteadcharacteristics of on the instead related of dimensionlesson the related dimensionless variables is preferred. variables In is fact, preferred. this choice In fact, makes this choice it possible makes to analyzeit possible the to trend analyze with the the trend depth with of the the wave depth characteristics, of the wave i.e.,characteristics, the significant i.e. wavethe significant height and wave the peakheight period, and the and peak to highlight period, and where to highlight the roughness where contribution the roughness is more contribution decisive. is The more wind decisive. speed UThe10 equalwind tospeed 10 m U/s10is equal taken to as 10 an m/s example, is taken since as an it example, is an average since valueit is an in average the investigated value in the speed investigated range. speed range. Figure 5 presents the distribution at varying water depths of the wave fields for KN values increased progressively by an order of magnitude from the lowest value of 0.0005 m to the highest one and equal to 0.05 m. The former can be representative, as mentioned above, of a smooth and

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Figure5 presents the distribution at varying water depths of the wave fields for KN values increased progressively by an order of magnitude from the lowest value of 0.0005 m to the highest one and equal to 0.05 m. The former can be representative, as mentioned above, of a smooth and Water 2019, 11, x FOR PEER REVIEW 9 of 20 morphologically stable flat bed, for which the friction factor can be assumed as constant and equal to themorphologically commonly adopted stable flat value bed, of for 0.01 which [11 the]. On friction the contrary,factor can KbeN assumed= 0.05 m as can constant identify and aequal very to rough bottomthe duecommonly to the adopted presence value of ripples,of 0.01 [11]. vegetation, On the contrary, or different KN = 0.05 grain m sizes.can identify The growtha very rough curves of Bretschneiderbottom due [58 to] andthe presence Young and of ripples, Verhagen vegetation, [26] are reportedor different together grain sizes. with theThe numericallygrowth curves simulated of pointsBretschneider for a comparison. [58] and Both Young the and significant Verhagen wave [26] heights, are reported depicted together in Figure with5 a1–c1,the numerically and the peak simulated points for a comparison. Both the significant wave heights, depicted in Figure 5(a1–c1), periods in Figure5a2–c2, show a great reduction if KN increases. and the peak periods in Figure 5(a2–c2), show a great reduction if KN increases.

FigureFigure 5. Distribution 5. Distribution at at varying varying waterwater depths depths of of the the wave wave field field generated generated by a wind by a speed wind U speed10 of 10U 10 of 10 mm/s/s for for di differentfferent KN values:values: (a1 (a1–c1–)c1 the) the significant significant wave wave height height and (a2 and–c2) (thea2– wavec2) the peak wave period. peak The period. The redred and black black dashed dashed lines lines refer refer to the to thegrowth growth curves curves predicted predicted by Bretschneider by Bretschneider [58] and Young [58] and and Young and Verhagen [26] [26 ]respectively. respectively. The The points points are arethe re thesults results obtained obtained by means by meansof the spectral of the spectralmodel. model.

In particular,In particular, the the wave wave height height is is lowered lowered by 40% compared compared to tothe the value value predicted predicted by assuming by assuming a nota verynot very dissipative dissipative bed. bed. TheThe new forecastin forecastingg curve curve of wave of wave energy, energy, having having coefficient coeffi Acient and BA asand B reported in Table 2, is almost superimposed on those obtained by Bretschneider [58] and Young and as reported in Table2, is almost superimposed on those obtained by Bretschneider [ 58] and Young Verhagen [26] when KN is very low, with a maximum percentage difference of 10% and 5% and Verhagen [26] when K is very low, with a maximum percentage difference of 10% and 5% respectively. This confirmsN the more general validity of the approach. The difference between the curves of Bretschneider [58] and Young and Verhagen [26] is much more important if the peak period

Water 2019, 11, 2313 10 of 20 respectively. This confirms the more general validity of the approach. The difference between the curves of Bretschneider [58] and Young and Verhagen [26] is much more important if the peak period is considered.Water 2019, 11 This, x FOR outcome PEER REVIEW has been clearly pointed out by the experimental investigations10 of in 20 Lake George. Also in this case, Equation (5) with C and D taken according Table2, interprets the trend of the periodisfor considered. the lowest This value outcome of K Nhaswell, been and clearly gives pointed decreasing out by values the experimental for rougher investigations bottoms. These in Lake results agreeGeorge. well with Also the in numerical this case, Equation evidence (5) performed with C andby D taken Graber according and Madsen Table 2, [41 interprets] for various the trend conditions of of bottomthe period roughness, for the which lowestclearly value of demonstrates KN well, and gives the trend decreasing of decreasing values for energy rougher and bottoms. peak periods These for results agree well with the numerical evidence performed by Graber and Madsen [41] for various rougher seabeds. conditions of bottom roughness, which clearly demonstrates the trend of decreasing energy and peak 4. Applicationperiods for to rougher Lake Neusiedlseabeds. A4. real Application lake has to been Lake examined Neusiedl to test the applicability of the forecasting curves for fully developed wave conditionsA real onlake shallow has been depths, examined with to the test new the coe appfficientslicabilityA andof theC proposedforecasting by curves Equations for fully (6) and (7). Experimentaldeveloped wave observations conditions ofon bothshallo windw depths, and waveswith the are new available coefficients together A and with C proposed the results by of a completeEquations bidimensional (6) and (7). numerical Experimental application observations performed of both wind by meansand waves of a are spectral available model together [53]. with the results of a complete bidimensional numerical application performed by means of a spectral 4.1. Studymodel Site [53]. and Data Setup

Homorodi4.1. Study Site et al.and [ 53Data] have Setup applied SWAN to Lake Neusiedl, sited between Austria and Hungary, to improve the modelling of locally generated wind waves in very shallow and confined basins. As depictedHomorodi in et Figure al. [53]6, have the Austrianapplied SWAN portion to Lake of Lake Neusiedl, Neusiedl sited hasbetween a quite Austria uniform and Hungary, bathymetry to improve the modelling of locally generated wind waves in very shallow and confined basins. with small depths of about 1.0–1.2 m. This particular aspect makes the context very similar to the ideal As depicted in Figure 6, the Austrian portion of Lake Neusiedl has a quite uniform bathymetry conditionwith usedsmall indepths the present of about paper 1.0–1.2 to m. numerically This particular analyze aspect and makes determine the context the growth very similar of wind to the waves in shallowideal condition water. Furthermore, used in the present the extension paper to of numerically the lake, which analyze stretches and determine toward the North–North growth of wind East, is characterizedwaves in by,shallow at least water. for Furthermore, some wind directionsthe extensio andn of speeds,the lake, suwhichfficiently stretches long toward fetch North–North so that the wave motionEast, can is be characterized considered by, as at completely least for some developed. wind directions and speeds, sufficiently long fetch so that the wave motion can be considered as completely developed.

(a) (b)

FigureFigure 6. Lake 6. Lake Neusiedl Neusiedl depicted depicted (a )(a in) in the the European European context,context, and and (b (b) )with with the the contour contour of bathymetry of bathymetry of theof Austrian the Austrian portion. portion. The The dashed dashed white white lines lines define define thethe sectors of of the the blowing blowing wind wind data data collected collected by theby anemometer the anemometer placed placed in thein the measurement measurement site site shownshown by the the arrow. arrow. The The sectors sectors are arenumbered numbered as as indicatedindicated by the by circles. the circles.

In theirIn study,their study, Homorodi Homorodi et al. et [53 al.] calibrated[53] calibrated and and validated validated the the numerical numerical approach approach with with the the wave data collectedwave data during collected two during experimental two experimental campaigns camp conductedaigns conducted respectively respectively in Julyin July and and October October 2005 2005 near Illmitz, Austria. Each measurement was made for a consecutive period of about one week.

Water 2019, 11, 2313 11 of 20 near Illmitz, Austria. Each measurement was made for a consecutive period of about one week. The most important difference between the two stages was the direction of the dominant winds: during the first period of July it was North-West, while North and North-East during the second one. To take into account the different wave exposure and to select the conditions which lead to a fully developed wave motion, the registered wind directions provided by Homorodi et al. [53] have been grouped into six sectors as indicated in Figure6b. A mean depth has been assigned to each sector as an average value weighted on the sector area, while the fetch has been estimated along its mean direction. In Table3a summary of these parameters is reported.

Table 3. Sectors of blowing winds: the second column reports the directions that limit each sector; the mean depth is calculated as an average value weighted on the sector area; the fetch is taken along the mean direction of the sector.

Sector Directions Mean Depth (m) Fetch (Km)

0 290◦ N–310◦ N 0.82 2.5 1 310◦ N–330◦ N 0.88 3 2 330◦ N–345◦ N 0.84 5 3 345◦ N–360◦ N 0.79 9 4 0◦ N–12◦ N 0.91 15 5 12◦ N–24◦ N 0.90 18

In the campaigns a standard wave pressure gauge was used to reconstruct the wave spectra and the relative wave parameters, in particular the significant wave height Hs and the mean wave period T01 based on the zero-order and first-order moments of the spectra. Homorodi et al. [53] compared the numerical results to both the available wave data and that estimated through the empirical growth curves of Bretschneider. The complete relations (1) and (2), that consider fetch limitation, have been adopted with the coefficients entering in Equation (3) equal to:

3 a = 0.53 a = 0.75 b = 5.65 10− b = 0.5, (8) 11 12 11 × 12 2 a = 0.833 a = 0.375 b = 3.79 10− b = 0.33. (9) 21 22 21 × 22 As the aim of this study is to apply the asymptotic form for the depth-limited Equations (4) and (5), it has been convenient to identify those records which approximately conform to this limit. These relations simplify the previous and more complete ones (1) and (2), which tend to the asymptotic form when the argument of the hyperbolic tangent depending both on fetch and depth is such as to make it close to unity. In this sense, cases for which

" b # b11χ 12 tanh a 0.8 (10) tanh(a11δ 12 ) ≥ have been selected, in analogy to Young and Babanin [49] who considered, for a similar purpose, a band of points within 20% of the maximum values. This approach allows us to identify the fully developed wave conditions taking into account both fetch length and depth for an assigned wind speed. Figures7 and8 show the measured wind data during the periods of October and July 2005 respectively. The period of October was used by Homorodi et al. [53] to calibrate SWAN and the period of July to validate the numerical approach. The blowing directions are depicted in terms of the sectors defined above, with the representative mean fetch reported in the second panel according to values in Table3. The colored bands highlight the selected records of wind speed, depicted in the third panel, for which Equation (10) is satisfied. As can be seen, a longer fetch is a sufficient requirement for low wind speeds, but for moderate or stronger winds even a shorter fetch can still guarantee fully developed wave motion. This outcome agrees with the recent studies of Petti et al. [33] and Pascolo et al. [11], Water 2019, 11, 2313 12 of 20 who have found that the maximum limit of the wave growth in shallow depth is already reached on WatershortWater 20192019 distances,, 1111,, xx FORFOR of PEER aPEER few REVIEWREVIEW km, as also happens in the present case study. 1212 ofof 2020

FigureFigure 7. 7. Wind Wind data data measured measured during during the the campaign campaign of of October October 2005 2005 [53]: [[53]:53]: ( ( aa)) sectors sectors of of blowing blowing winds; winds; ((bb)) thethe relativerelative meanmean fetchfetch accordingaccording toto thethe valuesvalues inin TableTable3 3;;(3; ((cc))) thethethe windwindwind speeds.speeds. The The colored colored bands bands limitedlimited bybyby the thethe dashed dasheddashed lines lineslines correspond correspondcorrespond to thetoto the conditionsthe coconditionsnditions of fully ofof fullyfully developed developeddeveloped wave wave motionwave motionmotion which which satisfywhich satisfyrelationsatisfy relationrelation (10). (10).(10).

FigureFigure 8.8. WindWind data data measured measured during during the the campaign campaign of of July July 2005 2005 [53]: [[53]:53]: ( (aa))) sectors sectorssectors of ofof blowing blowingblowing winds; winds;winds; ((bb)) thethe relativerelative meanmean fetchfetch accordingaccording toto thethe valuesvalues inin TableTable3 3;;(3; ((cc))) thethethe windwindwind speeds.speeds. The The colored colored bands bands limitedlimited bybyby the thethe dashed dasheddashed lines lineslines correspond correspondcorrespond to thetoto the conditionsthe coconditionsnditions of fully ofof fullyfully developed developeddeveloped wave wave motionwave motionmotion which which satisfywhich satisfyrelationsatisfy relationrelation (10). (10).(10). To simulate bottom friction dissipations, Homorodi et al. [53] chose the formulation of TheThe coloredcolored bandsbands highlighthighlight thethe selectedselected recordsrecords ofof windwind speed,speed, depicteddepicted inin thethe thirdthird panel,panel, forfor Madsen et al. [40], setting a K value of 0.001 m which produced the best agreement with the whichwhich EquationEquation (10)(10) isis satisfied.satisfied.N AsAs cancan bebe seen,seen, aa longer longer fetchfetch isis aa sufficientsufficient requirementrequirement forfor lowlow windwind experimental data. No further details on the composition and morphology of the bed are provided, and speeds,speeds, butbut forfor moderatemoderate oror strongerstronger windswinds eveneven aa shortershorter fetchfetch cancan stillstill guaranteeguarantee fullyfully developeddeveloped therefore the same bottom roughness is herein taken to compute the coefficients of the depth-limited wavewave motion.motion. ThisThis outcomeoutcome agreesagrees withwith thethe recentrecent studiesstudies ofof PettiPetti etet al.al. [33][33] andand PascoloPascolo etet al.al. [11],[11], growth curves. The mean depth evaluated for each sector, together with the wind speed U , have been whowho havehave foundfound thatthat thethe maximummaximum limitlimit ofof thethe wavewave growthgrowth inin shallowshallow depthdepth isis alreadyalready10 reachedreached onon used to calculate the dimensionless variable δ entering in Equations (4) and (5). Since the measured shortshort distancesdistances ofof aa fewfew km,km, asas alsoalso happenshappens inin thethe presentpresent casecase study.study. wave period is given as T , a reduction coefficient is necessary to make a comparison with the peak ToTo simulatesimulate bottombottom frictionfriction01 dissipations,dissipations, HomorodiHomorodi etet al.al. [53][53] chosechose thethe formulationformulation ofof MadsenMadsen period obtained by the semi-empirical relations possible. The results derived in terms of T01 and Tp etet al.al. [40],[40], settingsetting aa KKNN valuevalue ofof 0.0010.001 mm whichwhich producedproduced thethe bebestst agreementagreement withwith thethe experimentalexperimental data.data. NoNo furtherfurther detailsdetails onon thethe compositioncomposition andand momorphologyrphology ofof thethe bedbed areare provided,provided, andand thereforetherefore thethe samesame bottombottom roughnessroughness isis hereinherein takentaken toto computecompute thethe coefficientscoefficients ofof thethe depth-limiteddepth-limited growthgrowth curves.curves. TheThe meanmean depthdepth evaluatedevaluated forfor eacheach sector,sector, togethertogether withwith thethe windwind speedspeed UU1010,, havehave beenbeen usedused toto calculatecalculate thethe dimensionlessdimensionless variablevariable δδ enteringentering inin EquationsEquations (4)(4) andand (5).(5). SinceSince thethe measuredmeasured wavewave periodperiod isis givengiven asas TT0101,, aa reductionreduction coefficientcoefficient isis necessarynecessary toto makemake aa comparisoncomparison withwith thethe peakpeak periodperiod

Water 2019, 11, 2313 13 of 20 WaterWater 20192019,, 1111,, xx FORFOR PEERPEER REVIEWREVIEW 1313 ofof 2020 fromobtainedobtained the numerical byby thethe semi-empiricalsemi-empirical simulations relations describedrelations possible.possible. in Section TheThe3 resultsresultshave been derivedderived arranged inin termsterms in ofof order TT0101 andand to determine TTpp fromfrom thethe an averagenumericalnumerical value simulationssimulations of the ratio descdescT01ribedribed/Tp. This inin SectionSection coeffi cient 33 havehave has beenbeen turned arrangedarranged out to inin be orderorder equal toto to determinedetermine 0.78. anan averageaverage valuevalue ofof thethe ratioratio TT0101//TTpp.. ThisThis coefficientcoefficient hashas turnedturned outout toto bebe equalequal toto 0.78.0.78. 4.2. Results 4.2. Results 4.2.The Results results in terms of significant wave height and mean period are plotted in Figures9 and 10, respectively,TheThe resultsresults for the inin periodtermsterms ofof of significantsignificant October and wavewave July. heightheight andand meanmean periodperiod areare plottedplotted inin FiguresFigures 99 andand 10,10, respectively,respectively, forfor thethe periodperiod ofof OctoberOctober andand July.July.

FigureFigureFigure 9. 9.9.October OctoberOctober 2005: 2005:2005: ( (a(aa))) sectors sectorssectors ofof blowing blowing windswinds withwith with thethe the sese selectedlectedlected coloredcolored colored recordsrecords records thatthat that correspondcorrespond correspond toto fullyto fullyfully developed developeddeveloped wave wavewave conditions; conditions;conditions; ( (b(bb))) significantsignificantsignificant wave wave heights;heights; ((c (cc)) ) meanmean mean wavewave wave periods.periods. periods. AsterisksAsterisks Asterisks indicate measured data; the bold blue line indicates the results given by the growth curve with KN of indicateindicate measured measured data; data; the the bold bold blue blue lineline indicatesindicates the results results given given by by the the growth growth curve curve with with KNK ofN of 0.0010.0010.001 m, m,m, the thethe dashed dasheddashed red redred line lineline the thethe forecasting forecastingforecasting valuesvalues obtained obtained withwith with thethe the BretschneiderBretschneider Bretschneider formula,formula, formula, andand and thethe the blackblackblack dotted dotteddotted line lineline those thosethose obtained obtainedobtained with withwith Young YoungYoung and andand VerhagenVerhagenVerhagen formula.formula.

FigureFigureFigure 10. 10.10.July JulyJuly 2005: 2005:2005: ( a ((a)a)) sectors sectorssectors of ofof blowing blowingblowing windswinds with with thethe the selectedselected selected coloredcolored colored recordsrecords records thatthat that correspondcorrespond correspond toto fullyto fullyfully developed developeddeveloped wave wavewave conditions; conditions;conditions; ( (b(bb))) significantsignificantsignificant wave wave heights;heights; ((c (cc)) ) meanmean mean wavewave wave periods.periods. periods. AsterisksAsterisks Asterisks indicate measured data; the bold blue line indicates the results given by the growth curve with KN of indicateindicate measured measured data; data; the the bold bold blue blue lineline indicatesindicates the results results given given by by the the growth growth curve curve with with KNK ofN of 0.0010.0010.001 m, m,m, the thethe dashed dasheddashed red redred line lineline the thethe forecasting forecastingforecasting valuesvalues obtained obtained withwith with thethe the BretschneiderBretschneider Bretschneider formula,formula, formula, andand and thethe the blackblackblack dotted dotteddotted line lineline those thosethose obtained obtainedobtained with withwith Young YoungYoung and andand VerhagenVerhagenVerhagen formula.formula.

Water 2019, 11, 2313 14 of 20

Water 2019, 11, x FOR PEER REVIEW 14 of 20 In the graphs, the values provided by the new depth-limited growth curves, defined in the present study,In are the compared graphs, the to thevalues measured provided data, by the the simulation new depth-limited results of Homorodi growth curves, et al. [ 53defined] obtained in the by presentmeans of study, a complete are compared 2D application, to the andmeasured those computeddata, the simulation according toresults Bretschneider of Homorodi [58] and et al. Young [53] obtainedand Verhagen by means [26] forecastingof a complete formulations. 2D application, and those computed according to Bretschneider [58] and YoungFocusing and on Verhagen the wave heights[26] forecasting registered formulations. in the period of October, the first three days of observation, fromFocusing the 14th to on 17th, the show wave a substantialheights registered agreement in betweenthe period the measuredof October, values the andfirst thosethree evaluated days of observation,through the newfrom forecastingthe 14th to curves.17th, show On a the subs contrary,tantial agreement for the following between days, the measured there is a values systematic and thoseover-estimation evaluated ofthrough the wave the height new forecasting also through curves. the complete On the contrary, 2D modelling. for the This following discrepancy days, wasthere not is acommented systematic on over-estimation by Homorodi et of al. the [53 ],wave and it height could alsoalso be through attributable the tocomplete an instrumental 2D modelling. error. Minor This discrepancydifferences are was found not betweencommented the measuredon by Homorodi and calculated et al. [53], wave and periods it could except also forbe theattributable highest values to an instrumentalrecorded between error. the Minor 16th differences and 18th of are October. found Thisbetween discrepancy the measured can be and attributed calculated to lowwave frequency periods exceptcomponents for the in highest the measured values recorded energy density between spectra, the 16th as and reported 18th of by October. Homorodi This et discrepancy al. [53] at 12:00 can onbe attributedthe 17th of to October. low frequency These components components can in be the related measured to bounded energy long density waves spectra, which are as not reported present by in Homorodithe modelled et energyal. [53] densityat 12:00 spectra on the and17th that of Oc leadtober. to an These overestimation components of thecan meanbe related period, to thisbounded being long waves which are not present in the modelled energy density spectra and that lead to an calculated through the moments m0 and m1. overestimationThe case for of Julythe mean is diff period,erent, forthis which being bothcalculated the wave through heights the moments and periods m0 and forecasted m1. by the growthThe curves case for are July very is close different, to those for measured. which both the wave heights and periods forecasted by the growthTo analyzecurves are the very comparison close to better,those measured. the selected periods which correspond to the colored bands, and thereforeTo analyze to a fully the developed comparison wave better, condition, the selected have been periods grouped. which The correspond correlation to graphsthe colored between bands, the andmeasured therefore wave to data a fully and thosedeveloped calculated wave have condition, been plotted, have been as proposed grouped. in FigureThe correlation 11 for the graphs period betweenof October. the measured wave data and those calculated have been plotted, as proposed in Figure 11 for the period of October.

(a1) (b1) (c1) (d1)

(a2) (b2) (c2) (d2)

Figure 11.11. CorrelationCorrelation betweenbetween the the fully fully developed developed wave wave data data measured measured in October in October 2005 and2005 that and given that N givenby: (a1-2 by:) the(a1-2 growth) the growth curve with curveKN with= 0.001 K m; = 0.001 (b1-2 )m; Bretschneider (b1-2) Bretschneider [58] formulation; [58] formulation; (c1-2) Young (c1-2 and) VerhagenYoung and [26 Verhagen] formulation; [26] (d1-2formulation;) Homorodi (d1-2 et al.) Homorodi [53] bidimensional et al. [53] (2D) bidimensional spectral application. (2D) spectral application. For the quantitative evaluation of the results, the bias parameter (BIAS), root mean square errorFor (RMSE the quantitative), scatter index evaluation (SI), andof the correlation results, the coebiasffi parametercient (R2) ( haveBIAS), been root mean computed square as error the (followingRMSE), scatter expressions: index (SI), and correlation coefficient (R2) have been computed as the following expressions:

N 1 N σ = ()− = 1 ()− 2 RMSE 2 = xy BIAS xi yi , RMSE xi yi , SI = ⋅100 , R , N  N  N σ σ (11) i=1 i=1 1 x y  yi N i=1

Water 2019, 11, 2313 15 of 20

tuv XN XN 1 1 2 RMSE 2 σxy BIAS = (xi yi), RMSE = (xi yi) , SI = 100, R = , (11) N − N − N · σxσy i=1 i=1 1 P Water 2019, 11, x FOR PEER REVIEW N yi 15 of 20 i=1 where xi is the value predicted by the model or the forecasting curves and yi is the measured value, where xi is the value predicted by the model or the forecasting curves and yi is the measured value, σxy σxy the covariance, σx and σy the relative standard deviation. The obtained coefficients are reported in the covariance, σx and σy the relative standard deviation. The obtained coefficients are reported in Table4 4..

Table 4. CoefficientsCoefficients related related to wave to wave heights heights and andperiods periods measured measured and computed and computed referring referring to the Octoberto the October period. BIAS period., biasBIAS parameter;, bias parameter; RMSE, rootRMSE mean, square root mean error; square SI, scatter error; index;SI, R scatter2,correlation index; coefficient.R2,correlation coefficient.

Statistical KN = 0.001 m Bretschneider [58] Young andYoung Verhagen and [26 ] HomorodiHomorodi et al. et [53 al.] Statistical KN = 0.001 m Bretschneider [58] Coefficients Verhagen [26] [53] coefficients Hs T01 Hs T01 Hs T01 Hs T01 Hs T01 Hs T01 Hs T01 Hs T01 BIAS 0.034 m 0.140 s 0.047 m 0.286 s 0.058 m 0.069 s 0.014 m 0.062 s − − − BIASRMS 0.0440.034 m m 0.204− s0.140 0.058s m 0.047 m 0.312 s 0.286 0.065s 0.058 m m 0.147−0.069 s s 0.035 0.014 m m 0.116−0.062 s s RMSSI 29.5670.044 m 13.738 0.204 s 38.661 0.058 m 21.010 0.312 s 43.486 0.065 m 9.911 0.147 s 23.3050.035 m 7.788 0.116 s 2 RSI 0.88129.567 0.81713.738 0.86138.661 0.837 21.010 0.88143.486 0.8379.911 0.87823.305 0.906 7.788 R2 0.881 0.817 0.861 0.837 0.881 0.837 0.878 0.906 This has also been done for the July period: the correlation graphs are depicted in Figure 12 and the relativeThis has coe alsofficients been done reported for the in Table July 5period:. the correlation graphs are depicted in Figure 12 and the relative coefficients reported in Table 5.

(a1) (b1) (c1) (d1)

(a2) (b2) (c2) (d2)

Figure 12. CorrelationCorrelation between the the fully developed wave wave data data measured measured in July 2005 and that given by: ( (a1-2a1-2)) the the growth growth curve curve with with KN == 0.0010.001 m; m; (b1-2 (b1-2) )Bretschneider Bretschneider [58] [58] formulation; formulation; ( (c1-2c1-2)) Young Young and and Verhagen [[26]26] formulation;formulation; ((d1-2d1-2)) HomorodiHomorodi etet al.al. [[53]53] bidimensionalbidimensional (2D)(2D) spectralspectral application.application.

Table 5. Coefficients related to wave heights and periods measured and computed referring to the Table 5. Coefficients related to wave heights and periods measured and computed referring to the July period. July period.

Statistical KN = 0.001 m Bretschneider [58] Young and Verhagen [26] Homorodi et al. [53] Young and Homorodi et al. [53] StatisticalCoefficients HKN = 0.001T m H BretschneiderT [58] H T H T s 01 s 01 s Verhagen01 [26] s 01 coefficientsBIAS 0.005 m 0.079 s 0.009 m 0.350 s 0.019 m 0.006 s 0.028 m 0.009 s − Hs − T01 Hs T01 Hs − T01 − Hs − T01 RMS 0.027 m 0.158 s 0.034 m 0.365 s 0.030 m 0.121 s 0.036 m 0.083 s BIAS SI −14.2500.005 m 11.014−0.079 s 18.068 0.009 m 25.470 0.350 s 16.070 0.019 m 8.418−0.006 s 18.982−0.028 m 5.772 −0.009 s R2 0.941 0.847 0.933 0.857 0.941 0.857 0.938 0.930 RMS 0.027 m 0.158 s 0.034 m 0.365 s 0.030 m 0.121 s 0.036 m 0.083 s SI 14.250 11.014 18.068 25.470 16.070 8.418 18.982 5.772 R2 0.941 0.847 0.933 0.857 0.941 0.857 0.938 0.930

The correlation graphs confirm that the available forecasting curves of Bretschneider [58] and Young and Verhagen [26] actually provide an upper limit to wave growth, especially for wave

Water 2019, 11, 2313 16 of 20

The correlation graphs confirm that the available forecasting curves of Bretschneider [58] and Young and Verhagen [26] actually provide an upper limit to wave growth, especially for wave heights. The new curves with assigned equivalent bed roughness provide an estimate that is halfway between the aforementioned curves and the results of the 2D numerical simulation, and this is clearly highlighted by the comparison of the relative coefficients. The application of SWAN undoubtedly gives the values closest to those measured, but it requires a more detailed and complete approach. In this sense, the proposed study has produced the expected results, given that the forecasting curves arise from the need to estimate the characteristics of wave motion through a certainly simplified but more rapid methodology.

5. Discussion and Conclusions Numerical modelling is certainly a very valid tool to investigate wave and current hydrodynamics, especially where the complexity of the phenomena is such that it cannot be described by more simplified approaches. However, the required computational effort can be significant and a comparison with experimental and analytical results is always needed to calibrate and validate models, as in the study of Homorodi et al. described above [53]. In the absence of measured data, especially of wave motion, it is often necessary to proceed with a wave prediction system. The forecasting growth curves offer the possibility to estimate the wave characteristics more rapidly than a complete numerical approach, but still providing reliable values. In fact, the SMB (Sverdrup–Munk–Bretschneider) method [58] is widely recognized to be the most convenient and robust approach to use for computing wave heights and periods in deep water when a limited amount of data and time are available. Similarly, in shallow water contexts, the curves of Bretschneider [58] and Young and Verhagen [26] have become a widespread key reference for many applications. The domains are above all coastal areas, lagoons or lakes characterized by quite uniform bottoms and shallow depths. The interaction with the bottom plays a crucial role in determining the wave components, but there is still an ongoing debate as to how the roughness can be involved in the forecasting equations. This study suggests the possibility of making the equations dependent on the bed roughness, while remaining consistent with those of Bretschneider [58] and Young and Verhagen [26]. At the same time, some important intuitions on the friction influence, as for example the outcomes determined in the numerical field by Graber and Madsen [41], are considered. This first step has been carried out by starting from the simpler asymptotic form of the growth curves, corresponding to a fully developed wave condition, which is therefore not affected by the fetch limitation. The proposed approach is able to adequately address the question of estimating the main wave characteristics if the wind speed, the depth, and the equivalent bottom roughness are assigned. The coefficients entering the forecasting Equations (4) and (5) have been reformulated as a function of the KN value, as indicated in Table2, leading to a new set of relations. The curves depicted as examples in Figure5 seem to confirm the validity of the method. In fact, the lowest value of KN, i.e., 0.0005 m, returns the limits of Young and Verhagen [26] exactly both for the wave heights and peak periods. This value is representative of a friction factor equal to 0.01, for which bottom resistance is beginning to show its influence, as already found by Graber and Madsen [41]. Moreover, the same value is also compatible with both that considered by Bretschneider [58] and the description of the bottom conditions of Lake George, very smooth and with no bedforms, where Young and Verhagen [26] carried out their important experimental measurements. On the contrary, the increase in roughness, and therefore of the friction factor, causes the substantial reduction of Hs and Tp. The new curves also show the good adaptability to scatter points determined numerically with SWAN, further validating the results. The application of the new equations system to the real case study satisfies the expectations for a better estimate of the wave field generated in shallow and confined basins, compared to Bretschneider [58] and Young and Verhagen [26]. Furthermore, the forecasted values of wave heights Water 2019, 11, 2313 17 of 20 and periods, taking into account an equivalent roughness of 0.001 m, are very close to those computed by means of a complete 2D numerical modelling [53]. The domain is very similar to the test case involved in the preliminary numerical investigations and also to Lake George, with a very small depth of about 1.0 m. The quantitative analyses of the results have been performed by selecting the conditions of wind speed and depth which ensure a fully developed wave field. The criterion established and formalized in Equation (10) has highlighted that in depth-limited conditions, the wave motion becomes independent from the fetch length over shorter distances of some km, if the wind speed is greater than about 5 m/s. This outcome agrees with the Young and Babanin study [49], which collected fully developed conditions over a maximum fetch length of about 7 km for a water depth ranging from 0.4 m to 1.5 m and wind speeds of no less than 6 m/s. Petti et al. [33] also found that the maximum wave heights and periods are reached over a distance of 3–4 km, inside the microtidal basin of a shallow lagoon. In this case, the bed roughness can be greater than 0.001 m, since the composition of the sediments can vary and ripples or irregularities can arise, leading to an increase of the friction factor. The differences between the values derived from the Young and Verhagen formulation and those computed by the new set of equations are not so important with KN values lower than 0.001 m, also given the very shallow water. However, these initial results encourage future developments of this study, providing for the application of the new curves to coastal or lagoon contexts, with more irregular seabeds, and the necessary comparison with observed wave data. A further improvement to the method presented deals with considering the cases of fetch-limited wave generation in shallow water, which requires the reformulation of the general expressions of the wave growth as originally defined by Bretschneider [58] and revised by Young and Verhagen [26].

Author Contributions: M.P., S.P., and S.B. contributed equally to this work. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Fredsøe, J.; Deigaard, R. Mechanics of Coastal Sediment Transport; Advanced Series on Ocean Engineering; World Scientific: Singapore, 1992; Volume 3, p. 369. 2. Nielsen, P. Coastal and Estuarine Processes; Advanced Series on Ocean Engineering; World Scientific: Singapore, 2009; Volume 29, p. 345. 3. Fagherazzi, S.; Palermo, C.; Rulli, M.C.; Carniello, L.; Defina, A. Wind waves in shallow microtidal basins and the dynamic equilibrium of tidal flats. J. Geophys. Res. 2007, 112, F02024. [CrossRef] 4. Le Hir, P.; Roberts, W.; Cazaillet, O.; Christie, M.; Bassoullet, P.; Bacher, C. Characterization of intertidal flat hydrodynamics. Cont. Shelf Res. 2000, 20, 1433–1459. [CrossRef] 5. Friedrichs, C.L. Tidal flat morphodynamics: A synthesis. In Treatise on Estuarine and Coastal Science; Estuarine and Coastal Geology and Geomorphology; Hansom, J.D., Fleming, B.W., Eds.; Elsevier: Amsterdam, The Netherlands, 2011; Volume 3, pp. 137–170. 6. Green, M.O. Very small waves and associated sediment resuspension on an estuarine intertidal flat. Estuar. Coast. Shelf Sci. 2011, 93, 449–459. [CrossRef] 7. Green, M.O.; Coco, G. Review of wave-driven sediment resuspension and transport in estuaries. Rev. Geophys. 2014, 52, 77–117. [CrossRef] 8. Shi, B.; Cooper, J.R.; Li, J.; Yang, Y.; Yang, S.L.; Luo, F.; Yu, Z.; Wang, Y.P. Hydrodynamics, erosion and accretion of intertidal mudflats in extremely shallow waters. J. Hydrol. 2019, 573, 31–39. [CrossRef] 9. Sheng, Y.P.; Lick, W. The transport and resuspension of sediments in a shallow lake. J. Geophys. Res. 1979, 84, 1809–1826. [CrossRef] 10. Carper, G.L.; Bachmann, R.W. Wind Resuspension of Sediments in a Prairie Lake. Can. J. Fish. Aquat. Sci. 1984, 41, 1763–1767. [CrossRef] 11. Pascolo, S.; Petti, M.; Bosa, S. On the Wave Bottom Shear Stress in Shallow Depths: The Role of Wave Period and Bed Roughness. Water 2018, 10, 1348. [CrossRef] Water 2019, 11, 2313 18 of 20

12. Petti, M.; Pascolo, S.; Bosa, S.; Bezzi, A.; Fontolan, G. Tidal Flats Morphodynamics: A new Conceptual Model to Predict Their Evolution over a Medium-Long Period. Water 2019, 11, 1176. [CrossRef] 13. Chao, X.; Jia, Y.; Shields, F.D.; Wang, S.S.Y.; Cooper, C.M. Three-dimensional numerical modeling of cohesive sediment transport and wind wave impact in a shallow oxbow lake. Adv. Water Resour. 2008, 31, 1004–1014. [CrossRef] 14. Jin, K.R.; Wang, K.H. Wind generated waves in Lake Okeechobee. JAWRA J. Am. Water Resour. Assoc. 2007, 34, 1099–1108. [CrossRef] 15. Carniello, L.; Defina, A.; D’Alpaos, L. Morphological evolution of the Venice lagoon: Evidence from the past and trend for the future. J. Geophys. Res. 2009, 114, F04002. [CrossRef] 16. Shi, B.W.; Yang, S.L.; Wang, Y.P.; Bouma, T.J.; Zhu, Q. Relating accretion and erosion at an exposed tidal wetland to the bottom shear stress of combined current–wave action. Geomorphology 2012, 138, 380–389. [CrossRef] 17. Zhu, Q.; van Prooijen, B.C.; Wang, Z.B.; Yang, S.L. Bed-level changes on intertidal wetland in response to waves and tides: A case study from the Yangtze River Delta. Mar. Geol. 2017, 385, 160–172. [CrossRef] 18. Bachmann, R.W.; Hoyer, M.V.; Canfield, D.E., Jr. The Potential for Wave Disturbance in Shallow Florida Lakes. Lake Reserv. Manag. 2000, 16, 281–291. [CrossRef] 19. Kirby, R. Practical implications of tidal flat shape. Cont. Shelf Res. 2000, 20, 1061–1077. [CrossRef] 20. Callaghan, D.P.; Bouma, T.J.; Klaassen, P.; van der Wal, D.; Stive, M.J.F.; Herman, P.M.J. Hydrodynamic forcing on salt-marsh development: Distinguishing the relative importance of waves and tidal flows. Estuar. Coast. Shelf Sci. 2010, 89, 73–88. [CrossRef] 21. Hu, Z.; Wang, Z.B.; Zitman, T.J.; Stive, M.J.F.; Bouma, T.J. Predicting long-term and short-term tidal flat morphodynamics using a dynamic equilibrium theory. J. Geophys. Res. Earth Surf. 2015, 120, 1803–1823. [CrossRef] 22. Shi, B.; Cooper, J.R.; Pratolongo, P.D.; Gao, S.; Bouma, T.J.; Li, G.; Li, C.; Yang, S.L.; Wang, Y.P. Erosion and accretion on a mudflat: The importance of very shallow-water effects. J. Geophys. Res. Ocean. 2017, 122, 9476–9499. [CrossRef] 23. Jalil, A.; Li, Y.; Zhang, K.; Gao, X.; Wang, W.; Sarwar Khan, H.O.; Pan, B.; Ali, S.; Acharya, K. Wind-induced hydrodynamic changes impact on sediment resuspension for large, shallow Lake Taihu, China. Int. J. Sediment Res. 2019, 34, 205–215. [CrossRef] 24. Bretschneider, C.L.; Reid, R.O. Change in Wave Height Due to Bottom Friction, Percolation and Refraction; Tech. Memo No. 45; Beach Erosion Board Engineer Research and Development Center (U.S.): Boston, MA, USA, 1954. 25. Bretschneider, C.L. Generation of Wind Waves in Shallow Water; Beach Erosion Board; Tech. Memo No. 51; Beach Erosion Board Engineer Research and Development Center (U.S.): Boston, MA, USA, 1954. 26. Young, I.R.; Verhagen, L.A. The growth of fetch limited waves in water of finite depth. 1. Total energy and peak frequency. Coast. Eng. 1996, 29, 47–78. [CrossRef] 27. Umgiesser, G.; Sclavo, M.; Carniel, S.; Bergamasco, A. Exploring the bottom shear stress variability in the Venice Lagoon. J. Mar. Syst. 2004, 51, 161–178. [CrossRef] 28. Carniello, L.; Silvestri, S.; Marani, M.; D’Alpaos, A.; Volpe, V.; Defina, A. Sediment dynamics in shallow tidal basins: In situ observations, satellite retrievals, and numerical modeling in the Venice Lagoon. J. Geophys. Res. Earth Surf. 2014, 119, 802–815. [CrossRef] 29. Cappucci, S.; Amos, C.L.; Hosoe, T.; Umgiesser, G. SLIM: A numerical model to evaluate the factors controlling the evolution of intertidal mudflats in Venice Lagoon, Italy. J. Mar. Syst. 2004, 51, 257–280. [CrossRef] 30. Fagherazzi, S.; Wiberg, P.L. Importance of wind conditions, fetch, and water levels on wave-generated shear stresses in shallow intertidal basins. J. Geophys. Res. 2009, 114, F03022. [CrossRef] 31. Carniello, L.; Defina, A.; D’Alpaos, L. Modeling sand-mud transport induced by tidal currents and wind waves in shallow microtidal basins: Application to the Venice Lagoon (Italy). Estuar. Coast. Shelf Sci. 2012, 102, 105–115. [CrossRef] 32. Zhou, Z.; Coco, G.; van der Wegen, M.; Gong, Z.; Zhang, C.; Townend, I. Modeling sorting dynamics of cohesive and non-cohesive sediments on intertidal flats under the effect of tides and wind waves. Cont. Shelf Res. 2015, 104, 76–91. [CrossRef] Water 2019, 11, 2313 19 of 20

33. Petti, M.; Bosa, S.; Pascolo, S. Lagoon Sediment Dynamics: A Coupled Model to Study a Medium-Term Silting of Tidal Channels. Water 2018, 10, 569. [CrossRef] 34. Gorman, R.M.; Neilson, C.G. Modelling shallow water wave generation and transformation in an intertidal estuary. Coast. Eng. 1999, 36, 197–217. [CrossRef] 35. Petti, M.; Bosa, S.; Pascolo, S.; Uliana, E. Marano and Grado Lagoon: Narrowing of the Lignano Inlet. IOP Conf. Ser. Mater. Sci. Eng. 2019, 603, 032066. [CrossRef] 36. Petti, M.; Pascolo, S.; Bosa, S.; Uliana, E.; Faggiani, M. Sea defences design in the vicinity of a river mouth: The case study of Lignano Riviera and Pineta. IOP Conf. Ser. Mater. Sci. Eng. 2019, 603, 032067. [CrossRef] 37. Holthuijsen, L.H.; Booij, N.; Herbers, T.H.C. A prediction model for stationary, short-crested waves in shallow water with ambient currents. Coast. Eng. 1989, 13, 23–54. [CrossRef] 38. Booij, N.; Ris, R.C.; Holthuijsen, L.H. A third-generation wave model for coastal regions, Part I, Model description and validation. J. Geophys. Res. 1999, 104, 7649–7666. [CrossRef] 39. Shemdin, O.H.; Hsiao, S.V.; Carlson, H.E.; Hasselmann, K.; Schulze, K. Mechanisms of wave transformation in finite-depth water. J. Geophys. Res. 1980, 85, 5012–5018. [CrossRef] 40. Madsen, O.S.; Poon, Y.-K.; Graber, H.C. Spectral wave attenuation by bottom friction: Theory. In Proceedings of the 21th International Conference Coastal Engineering, ASCE, Costa del Sol-Malaga, Spain, 20–25 June 1988; pp. 492–504. 41. Graber, H.C.; Madsen, O.S. A Finite-Depth Wind-Wave Model. Part I: Model Description. J. Phys. Oceanogr. 1988, 18, 1465–1483. [CrossRef] 42. Young, I.R.; Verhagen, L.A. The growth of fetch limited waves in water of finite depth. Part 2. Spectral evolution. Coast. Eng. 1996, 29, 79–99. [CrossRef] 43. Zijlema, M.; van Vledder, G.P.; Holthuijsen, L.H. Bottom friction and wind drag for wave models. Coast. Eng. 2012, 65, 19–26. [CrossRef] 44. Vincent, C.L.; Hughes, S.A. Wind Wave Growth in Shallow Water. J. Waterw. Port Coast. Ocean Eng. 1985, 111, 765–770. [CrossRef] 45. Bouws, E.; Günther, H.; Rosenthal, W.; Vincent, C.L. Similarity of the wind wave spectrum in finite depth water: 1. Spectral form. J. Geophys. Res. 1985, 90, 975–986. [CrossRef] 46. Hasselmann, K. On the non-linear energy transfer in a gravity-wave spectrum Part 1. General theory. J. Fluid Mech. 1962, 12, 481–500. [CrossRef] 47. Ijima, T.; Tang, F.L.W. Numerical Calculation of Wind Waves in Shallow Water. Coast. Eng. 1966, 38–49. [CrossRef] 48. Young, I.R.; Banner, M.L.; Donelan, M.A.; McCormick, C.; Babanin, A.V.; Melville, W.K.; Veron, F. An Integrated System for the Study of Wind-Wave Source Terms in Finite-Depth Water. J. Atmos. Ocean. Technol. 2005, 22, 814–831. [CrossRef] 49. Young, I.R.; Babanin, A.V. The form of the asymptotic depth-limited wind wave frequency spectrum. J. Geophys. Res. 2006, 111, C06031. [CrossRef] 50. Breugem, W.A.; Holthuijsen, L.H. Generalized Shallow Water Wave Growth from Lake George. J. Waterw. PortCoast. Ocean Eng. 2007, 133, 173–182. [CrossRef] 51. Mariotti, G.; Fagherazzi, S. A two-point dynamic model for the coupled evolution of channels and tidal flats. J. Geophys. Res. Earth Surf. 2013, 118, 1387–1399. [CrossRef] 52. Mariotti, G.; Fagherazzi, S. Wind waves on a mudflat: The influence of fetch and depth on bed shear stresses. Cont. Shelf Res. 2013, 60, S99–S110. [CrossRef] 53. Homorodi, K.; Józsa, J.; Krámer, T. On the 2D modelling aspects of wind-induced waves in shallow, fetch-limited lakes. Period. Polytech. Civ. Eng. 2012, 56, 127–140. [CrossRef] 54. Soulsby, R.L. Dynamics of Marine Sands: A Manual for Practical Applications; Thomas Telford Publications: London, UK, 1997; 249p. 55. Whitehouse, R.J.S.; Soulsby, R.L.; Roberts, W.; Mitchener, H.J. Dynamics of Estuarine Muds; Technical Report; Thomas Telford: London, UK, 2000. 56. Bretschneider, C.L. Revisions in wave forecasting: Deep and shallow water. Coast. Eng. Proc. 1957, 1, 30–67. [CrossRef] Water 2019, 11, 2313 20 of 20

57. Pascolo, S.; Petti, M.; Bosa, S. Wave–Current Interaction: A 2DH Model for Turbulent Jet and Bottom-Friction Dissipation. Water 2018, 10, 392. [CrossRef] 58. CERC (U.S. Army Coastal Engineering Research Center). Shore Protection Manual; U.S. Army Coastal Engineering Research Center: Washington, DC, USA, 1973; Volume 1.

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