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,
4,
5,
6,
7,
8,
9,
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,
12,
13,
14]. Extended literature considers the importance of wave-induced morphodynamic changes [
15,
16,
17] and their impact on environment also from an ecological and socio-economic point of view [
18,
19,
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,
22,
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,
25,
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,
30,
31,
32,
33] or intertidal estuaries [
34,
35,
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,
40,
41,
42,
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,
49,
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 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:
Section 2 presents an overview of previous studies of the wave growth in finite depth generation processes; in
Section 3 the numerical procedure is described and the modified growth curves are shown;
Section 4 reports the application of the curves to Lake Neusiedl, Austria; and
Section 5 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.
In particular, the variables
ε =
g2E/(
U10)
4 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:
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, both expressed in dimensionless form as respectively
δ =
gd/
(U10)
2 and
χ =
gx/(
U10)
2:
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 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:
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
Table 1.
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 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.
The water depth is assigned as uniform over the whole grid, but it varies from a minimum value of 0.1 m to a maximum of 4 m in order to reproduce different conditions of flat bathymetries in the simulated scenarios. This approach is similar to the one followed by Graber and Madsen [
41], although with shallower depths, coherently with the experimental values used to develop the forecasting relationships discussed above. The wind direction is kept constant and aligned with the longitudinal dimension of the grid, while the wind speed is uniform over the domain but variable in the range 6–14 m/s. These assumptions are consistent with the hypotheses on which the forecasting curves are based since the latter require wind with constant speed and direction over the generation area. Moreover, these hypotheses are coherent with what really happens in shallow and confined basins. In these contexts, there is a not so pronounced variability in the wind characteristics during the generation process since it is completed in a shorter time compared to the analogous case of deep water. Moreover, the resulting values of the dimensionless water depth
δ are consistent with those investigated both in Lake Okeechobee and Lake George.
With the aim of evaluating the effects on the wave field due to different bottom friction conditions, the relative dissipation source term is taken according to the formulations of Madsen et al. [
40]. This means that the friction factor
fw entering the equation to compute the spectral wave attenuation is not constant but depends, under the hypothesis of rough turbulent wave motion, on the relative roughness. This is defined as
A/
KN, where
A =
UwT/
2 is one half of the horizontal orbital excursion,
Uw is the maximum bottom orbital velocity,
T is the wave period, and
KN is the Nikuradse equivalent bed roughness.
KN is a parameter to be assigned as input to the model and it is generally taken as a function of the median diameter if the bed is composed of coarse grains, while for fine cohesive grains or mud-beds, it can be set equal to a few millimeters [
54,
55]. An appropriate range of
KN values has been established so as to consider different configurations and granulometric compositions of the bed, as actually happens in shallow coastal regions. The minimum value is 0.0005 m and it is representative of a very flat bed, corresponding approximately to a friction factor of 0.01 [
11]. The highest value is 0.05 m, set as default by SWAN, and it can be related to an irregular rough bottom with ripples. SWAN has been run in stationary mode since there is no interest in following the temporal evolution of the generation process, and overall 625 simulations have been carried out.
Figure 1 shows the arrangement of all the obtained points in terms of dimensionless variables with respect to the semi-empirical limits having the coefficients reported in
Table 1.
As can be observed, the lines tend to define an upper and a lower bound for all the points corresponding to the wave energy and the wave peak frequency νp = 1/Tp, respectively. This result confirms that the forecasting curves (4) and (5) identify the asymptotic conditions that the generated wave field can reach for the assigned water depth and wind speed.
The points have been grouped according to the different bottom roughness height, with the aim of verifying whether they tend to align and, in this manner, define a particular wave field related to the bed condition. Even though there is some data scattering which in any case reduces as the roughness increases, a trend can be recognized as depicted in
Figure 2. The dimensionless wave energy is represented as a function of the dimensionless depth for the
KN values set to compute spectral bottom friction dissipations. The red dashed lines provide the regression of the points, and the resulting coefficients
A and
B entering in Equation (4) are shown in the boxes.
Parameter
B defines the angular coefficient of each line and it does not change significantly between the wave field generated on different bottom roughness conditions. In particular, it remains fairly constant and equal to the value of 1.3 determined by Young and Verhagen [
26], thus confirming the agreement with their results and the experimental data collected in Lake George. On the contrary, parameter
A decreases progressively with the increase of
KN. This means that the position of the lines in the graph and therefore the asymptotic limit that the wave energy can reach, is lowered. The maximum value of
A corresponding to the minimum of
KN is the same as that proposed by Young and Verhagen [
26], and this further confirms the condition of the smooth flat bed of Lake George.
Very similar considerations can also be made on the trend of the dimensionless peak frequency reported in
Figure 3.
In fact, parameter
D maintains an almost constant value even if slightly lower than that of −0.375 originally proposed by Bretschneider [
58] and subsequently confirmed by Young and Verhagen [
26]. On the other hand, there is a dependence of the coefficient
C on the bottom roughness which causes the lines to be raised in the graph with increasing
KN.
In this sense, the angular coefficients of the regression lines can be kept fixed in all the cases analyzed and equal to 1.3 for B and −0.40 for D, while the variability of the remaining parameters A and C with the bed roughness KN cannot be ignored. In order to obtain the functions that associate these coefficients to KN, the estimated values for each class of roughness have been plotted and a regression by means of the least squares method has been performed.
This analysis has led to define two polynomial curves, that can be observed in
Figure 4, having the following equations respectively:
Equations (6) and (7) allow for the determination of the characteristics of the locally generated wave motion on a given depth and for an assigned wind speed, taking into account different roughness heights and therefore relative dissipation conditions of the bottom. A summary of the revised coefficients is reported in
Table 2 for a more immediate comparison with those listed in
Table 1.
With the purpose of better understanding the differences that this approach involves with respect to the assumption of a constant friction factor, a comparative analysis based on the wave characteristics instead of on the related dimensionless variables is preferred. In fact, this choice makes it possible to analyze the trend with the depth of the wave characteristics, i.e., the significant wave height and the peak period, and to highlight where the roughness contribution is more decisive. The wind speed U10 equal to 10 m/s is taken as an example, since it is an average value in the investigated 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 morphologically stable flat bed, for which the friction factor can be assumed as constant and equal to the commonly adopted value of 0.01 [
11]. On the contrary,
KN = 0.05 m can identify a very rough bottom due to the presence of ripples, vegetation, or different grain sizes. The growth curves of Bretschneider [
58] and Young and Verhagen [
26] are reported together with the numerically simulated points for a comparison. Both the significant wave heights, depicted in
Figure 5a1–c1, and the peak periods in
Figure 5a2–c2, show a great reduction if
KN increases.
In particular, the wave height is lowered by 40% compared to the value predicted by assuming a not very dissipative bed. The new forecasting curve of wave energy, having coefficient
A and
B as reported in
Table 2, is almost superimposed on those obtained by Bretschneider [
58] and Young and Verhagen [
26] when
KN is very low, with a maximum percentage difference of 10% and 5% 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. This outcome has been clearly pointed out by the experimental investigations in Lake George. Also in this case, Equation (5) with
C and
D taken according
Table 2, interprets the trend of the period for the lowest value of
KN well, and gives decreasing values for rougher bottoms. These results agree well with the numerical evidence performed by Graber and Madsen [
41] for various conditions of bottom roughness, which clearly demonstrates the trend of decreasing energy and peak periods for rougher seabeds.
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
Table 2, leading to a new set of relations.
The curves depicted as examples in
Figure 5 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 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].