Relative Magnitude of Infragravity Waves at Coastal Dikes with Shallow Foreshores: A Prediction Tool

Introduction

Methods

Physical Model Description

The physical modeling was performed at Flanders Hydraulics Research in a wave flume 70-m long, 4-m wide, and 1.45-m deep (Altomare et al. 2016). The experiments simulated the transformation of moderately steep spilling waves, with deep-water wave steepness, s₀ ≤ 0.26 and ξ₀ < 0.5 (Table 1), over a smooth 1:50 sloping foreshore backed by a 1:2 sloping dike (Fig. 1) with varying offshore significant wave heights (H_m0,deep), spectral peak periods (T_p), and initial still-water depths (h_toe). The flume, equipped with second-order wave generation, produced irregular waves that corresponded to a JONSWAP-type spectrum with a peak enhancement factor of 3.3 and a duration equal to 1,000 times T_p (∼1,000 waves). The variations of water–surface elevations were measured using 10 resistance-type gauges, all synchronously sampling at 50 Hz (Fig. 1). Considering a geometric scaling of 1:25, conditions at the dike toe were measured 1 m (prototype scale) seaward of the dike base (at gauge 10). Table 1 provides a summary of the test conditions.

Test No.	Model scale (1:25)			Prototype scale			ξ0	Htoe/Hm0,deep	flow (Hz)	βb	${\tilde{H}}_{\mathrm{IG}}$
	Hm0,deep (m)	Tp(s)	htoe (m)	Hm0,deep (m)	Tp (s)	htoe (m)
1	0.067	2.30	0.030	1.675	11.50	0.757	0.22	0.452	0.06	0.50	1.05
2	0.057	2.53	0.001	1.425	12.65	0.025	0.26	0.018	0.06	0.53	1.43
3	0.085	2.32	0.001	2.125	11.60	0.020	0.20	0.009	0.05	0.47	1.74
4	0.100	2.32	0.001	2.500	11.60	0.025	0.18	0.010	0.05	0.48	1.82
5	0.121	2.30	0.001	3.025	11.50	0.025	0.17	0.008	0.05	0.42	2.01
6	0.068	2.28	0.050	1.700	11.40	1.250	0.22	0.735	0.07	0.41	0.85
7	0.059	2.48	0.050	1.475	12.40	1.262	0.26	0.856	0.06	0.48	0.70
8	0.065	2.28	0.050	1.625	11.40	1.250	0.22	0.769	0.06	0.48	0.82
9	0.114	2.31	0.050	2.850	11.55	1.250	0.17	0.439	0.05	0.41	1.20
10	0.115	2.53	0.050	2.875	12.65	1.250	0.19	0.435	0.05	0.42	1.12
11	0.060	2.29	0.031	1.500	11.45	0.787	0.23	0.525	0.06	0.49	0.96
12	0.052	2.48	0.002	1.300	12.40	0.050	0.27	0.038	0.06	0.55	1.39

Numerical Model Description

Governing Equations

For this study, we applied the open-source XBeach numerical model. The nonhydrostatic mode (XB-NH) resolves both infragravity and sea–swell wave motions and is thus referred to as phase-resolving. It computes depth-averaged flow due to waves and currents using the nonlinear shallow water equations, including the effect of vegetation. The governing equations, in one-dimensional form, follow

$${\displaystyle \frac{\mathrm{\partial }\eta }{\mathrm{\partial }t}+{\displaystyle \frac{\mathrm{\partial }uh}{\mathrm{\partial }x}=0}}$$

(2)

$${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}-v_h{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}=-{\displaystyle \frac{1}{\rho }{\displaystyle \frac{\mathrm{\partial }(\overline{q}+\rho g\eta )}{\mathrm{\partial }x}-c_f{\displaystyle \frac{u|u|}{h}+{\displaystyle \frac{F_v}{\rho h}}}}}}}}$$

(3)

where x and t = horizontal spatial and temporal coordinates, respectively; η = free surface elevation; u = depth-averaged cross-shore velocity; v_h = horizontal eddy viscosity (Smagorinsky 1963); h = water depth; ρ = density of water; c_f = bed friction factor; and F_v = depth-averaged vegetation force (van Rooijen et al. 2016). The depth-averaged dynamic (nonhydrostatic) pressure normalized by the density, $\overline{q}$, is computed from the mean of the dynamic pressure at the surface (q_s, assumed to be zero) and the bed (q_b, by assuming a linear change over depth). In order to compute the dynamic pressure at the bed, the contributions of advective and diffusive terms to the vertical momentum balance are considered negligible:

$${\displaystyle \frac{\mathrm{\partial }w}{\mathrm{\partial }t}+{\displaystyle \frac{\mathrm{\partial }q}{\mathrm{\partial }z}=0}}$$

(4)

where w = vertical velocity; and z = vertical coordinate. The vertical velocity at the bed is set by the kinematic boundary condition

$$w_b=u{\displaystyle \frac{\mathrm{\partial }(\eta -h)}{\mathrm{\partial }x}}$$

(5)

and the vertical momentum balance at the surface follows

$${\displaystyle \frac{\mathrm{\partial }{w}_s}{\mathrm{\partial }t}=2{\displaystyle \frac{q_b}{h}-{\displaystyle \frac{\mathrm{\partial }{w}_b}{\mathrm{\partial }t}}}}$$

(6)

The dynamic pressure at the bed is then solved by combining Eq. (6) and the local continuity equation:

$${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+{\displaystyle \frac{w_s-w_b}{h}=0}}$$

(7)

In the present study, we obtain c_f from the Manning's roughness coefficient (n) as (Roelvink et al. 2015)

$$c_f={\displaystyle \frac{n^{2}g}{\sqrt[3]{h}}}$$

(8)

F_v is obtained by integrating the drag force per unit height (F_D) over the vegetation height (a_h) (Dalrymple et al. 1984):

$$F_D={\displaystyle \frac{1}{2}\rho C_D{b}_v{N}_{v}u|u|}$$

(9)

where C_D = drag coefficient; b_v = vegetation stem diameter; and N_v = vegetation density. Though not assessed as part of the present study, the performance of XBeach in simulating the effects of vegetation has been previously validated with data from two physical experiments, showing high skill (van Rooijen et al. 2016). Note that a recent study using SWASH reveals that the effects of porosity and inertia of vegetation can be important for wave propagation in dense vegetation fields (Suzuki et al. 2019). However, those effects were not taken into consideration in this study since the vegetation considered here is not so dense to have a significant influence on the results.

To control the computed location and magnitude of depth-limited wave breaking, a hydrostatic front approximation is applied. The pressure distribution under breaking waves is therefore considered to be hydrostatic once the local surface steepness exceeds a maximum value (Smit et al. 2010). This criterion is controlled by the user-specified maxbrsteep parameter.

Model Setup

The numerical simulations were first configured using a 1D approach to represent actual flume conditions. In the current study, we apply the configuration of Lashley et al. (2018) that obtained reasonably accurate results when applied to similar shallow environments (fringing reefs). As such, the maximum breaking wave steepness (maxbrsteep) was set to 0.5, compared with its default value of 0.6; and n was set to 0.012 s/m^1/3 to represent a relatively smooth flume bottom (Suzuki et al. 2017). The cross-shore grid spacing (Δx) applied varied from a maximum of L₀/100 (offshore) to a minimum of 1 m (dike toe). This choice of varying Δx minimized computation time while ensuring that the waves were accurately resolved. No further tuning of the model was done as part of this study.

For validation, the model was forced at its boundary with time series of velocities derived from the paddle motion observed during the physical experiment, which corresponded to a JONSWAP-type spectrum and included the generation of second-order bound waves. Likewise, the extent of numerical model domain corresponded to that of the physical flume (Fig. 1).

While assessing the influence of the varied environmental parameters (described in “Environmental Parameters for Numerical Simulations”), the model domain was allowed to vary depending on the water depth and foreshore slope (Fig. 2). These simulations were forced with irregular waves generated using parametric (JONSWAP-type) spectra with a peak enhancement factor of 3.3. For the numerical simulations where directional spreading was considered, a 2DH approach was adopted with cyclic lateral boundaries. This approach assumed alongshore uniformity and used an alongshore grid spacing, Δy = 2Δx, which was found to be optimal considering both accuracy and computational demand. All other model parameters were set to match the 1D simulations.

It should be noted that the numerical model, like the physical experiments, assumes bound IG waves at its boundary and does not consider potentially free IG waves arriving at the boundary from distant sources. Likewise, neither the physical experiment nor the 1D numerical simulations contain (2D) trapped IG waves, which might have an effect in the field. They do, however, consider the generation of free IG waves resulting from the shoaling and breaking of the SS-wave groups within the model domain.

Validation Metrics

In order to assess the performance of XB-NH, we applied the following objective functions: Scatter Index (SCI), as a normalized measure of error [Eq. (10)]; and Relative Bias (Rel.bias), as an indicator of prediction bias [Eq. (11)]:

$$\mathrm{SC}{\mathrm{I}}_{\mathrm{\Psi }}={\displaystyle \frac{\sqrt{{\displaystyle \frac{1}{n}{{\sum }_{i=1}^N({\mathrm{\Psi }}_{\mathrm{XB}}^i-{\mathrm{\Psi }}_{\mathrm{Obs}}^i)}^2}}}{{\displaystyle \frac{1}{n}{\sum }_{i=1}^N{\mathrm{\Psi }}_{\mathrm{Obs}}^i}}}$$

(10)

and

$$\mathrm{Rel}.\mathrm{bia}{\mathrm{s}}_{\mathrm{\Psi }}={\displaystyle \frac{{\sum }_{i=1}^N({\mathrm{\Psi }}_{\mathrm{XB}}^i-{\mathrm{\Psi }}_{\mathrm{Obs}}^i)}{{\sum }_{i=1}^N{\mathrm{\Psi }}_{\mathrm{Obs}}^i}}$$

(11)

where Ψ = the parameter being evaluated, in a sample size N; and subscripts XB and Obs = XBeach predictions and observations during the physical experiment, respectively. Lower SCI values (<0.15) indicate accurate model predictions (Roelvink et al. 2018), while a positive or negative Rel.bias estimate indicates a systematic over- or under-prediction, correspondingly.

Results and Discussions

XBeach Validation

Before using XB-NH to generate the synthetic dataset, we first verify that it accurately simulates the hydrodynamics of shallow foreshore environments by comparing it to the observations of the physical experiment. The mean water levels $(\overline{\eta })$ predicted by XB-NH compare well with those observed during the physical experiment (Fig. 3), with minor SCI (0.069) and Rel.bias (0.053) error values. In particular, the location and magnitude of wave-induced setup (the rise in $\overline{\eta }$), shoreward of the breakpoint, where h_b = H_m0/Ω, is well represented [Fig. 4(a)]. Here, the ratio of breaking waves to local water depth (Ω) is taken as 0.5; this value was found to best represent the onset of wave breaking [Fig. 4(b)].

Likewise, Fig. 4(b) shows good agreement between the modeled and observed H_m0,IG and H_m0,SS. This cross-shore profile shows the growth of H_m0,SS during shoaling (seaward of the breakpoint) and its subsequent dissipation by depth-induced breaking. Conversely, H_m0,IG is enhanced during shoaling but continues to grow as SS waves break, resulting in IG-wave dominance at the dike toe. This is further seen in the transformation of wave spectra; Fig. 5 shows that XB-NH accurately captures the shift in spectral density, C_ηη(f) from SS to IG frequencies. This apparent transfer of energy to lower frequencies is due to the dissipation of SS waves by depth-induced breaking and the concurrent enhancement of IG waves. It should be noted that the results here differ from those in Lashley et al. (2019b), as the observed surface elevation time series are used here as model input and not parametric spectra, for increased accuracy.

Considering the validation metrics, both H_m0,IG and H_m0,SS show little bias (Rel.bias values of 0.021 and −0.01, respectively); while H_m0,IG predictions, though still considered accurate [SCI = 0.111, Fig. 6(a)], show more scatter than predictions of H_m0,SS [SCI = 0.04, Fig. 6(b)]. That said, note that this difference in scatter is related to the low values of H_m0,IG observed. The SCI is defined as the root-mean-square error (RMSE) between the model and observations, normalized by the mean of the observations [Eq. (10)]. Both predictions of H_m0,IG and H_m0,SS have similar RMSE values: 0.002 and 0.003 m, respectively; however, the lower mean of the observed H_m0,IG results in a larger SCI value. Nevertheless, both SCI values are considered to be well within acceptable values: SCI < 0.15 or within 15% error (Roelvink et al. 2018).

To investigate the sensitivity of the predicted $\overline{\eta }$, H_m0,IG, and H_m0,SS to the random wave components imposed at the model boundary, each XBeach simulation was run ten times with a new random-wave time series. However, the effects were found to be negligible (<3% variation in H_m0,IG and <1% in both $\overline{\eta }$ and H_m0,SS).

Reflection of Infragravity Waves at the Dike

Fig. 4(c) shows the cross-shore evolution of the total incoming and outgoing H_m0,IG for Test No. 11. The incoming IG waves gain energy over the shallow foreshore as SS waves shoal (x > 28 m) and break (x ≥ 41.5 m), before finally experiencing near-complete reflection at the dike [x = 46 m, Fig. 4(c)]. The superposition of the incoming and the reflected outgoing IG signals results in a maximum total signal at the dike [Fig. 4(b)]. Following reflection, the outgoing waves experience inverse shoaling, also referred to as deshoaling (Battjes et al. 2004), where they reduce in amplitude with increasing water depth, that is as the wave celerity increases [Fig. 4(c)].

To investigate the presence of a standing wave, we analyze the evolution of wave spectral density at each cross-shore location, resulting in the spatial plot shown in Fig. 7. This was done first considering only the incoming signal [Fig. 7(a)], obtained by running the simulation without the dike, to exclude the effects of reflection, and then for the total signal where reflection from the dike is included [Fig. 7(b)]. A pattern of curved spectral ridges and troughs is clearly visible at low frequencies in Fig. 7(b) but absent in Fig. 7(a). These locations of minimum and maximum C_ηη(f) correspond to nodes and antinodes in a standing wave (Buckley et al. 2018; Klopman and van der Meer 1999; Symonds et al. 1982). This is further corroborated using Eq. (21), as the predicted f_node line corresponds well with the minima in C_ηη(f).

Infragravity Wave-Generation Mechanism

The high spatial resolution of the numerical model allows the presentation of the sequence of cross-correlations (R_ηA) between the modeled local η^IG(t) and A(t) offshore (x = 0 m), at each cross-shore location, as a continuous spatial plot with ridges of positive and negative correlations (Fig. 8). The slight negative correlation seen at zero time lag (τ = 0 s) corresponds to the trough of a bound IG wave that is out of phase with the incident wave groups (Longuet-Higgins and Stewart 1962).

At x ∼ 10 m, both this negative signal and a bar of positive correlation, which represents the crest of the bound wave, becomes more visible. Shoreward of this point (x > 10 m), the positive/negative form of the cross-correlation becomes stronger; this is consistent with an incident bound wave developing an asymmetric form during shoaling (Baldock and Huntley 2002; List 1992). Concurrently, the lag between the wave group, travelling with velocity (c_g), based on linear wave theory [black dashed line in Fig. 8(a and b)], and the trough of the bound wave increases. This phase shift allows the transfer of energy between the wave group and the bound wave, resulting in IG wave growth (van Dongeren and Svendsen 1997).

Shoreward of the breakpoint (x_b = 41.5 m), the ridge of positive correlation is strongly enhanced, suggesting the contribution of an incoming breakpoint-generated IG wave. The now-enhanced IG wave is reflected at the dike (x = 46 m, τ ∼ 23 s) and propagates to the offshore boundary (x = 0 m, τ ∼ 53 s) as a free long wave; that is, a wave propagating with celerity equal to $\sqrt{gh}$ [dashed black-and-white line in Fig. 8(a and b)].

Theoretically, breakpoint forcing would result in both incoming and outgoing IG waves propagating away from the breakpoint. However, this is not immediately evident in Fig. 8(a), as the outgoing signal is dominated by that reflected at the dike. To investigate this further, the simulation was rerun with the dike removed and the landward boundary set to absorbing–generating (weakly reflective). In the absence of a reflective boundary, a negatively correlated ridge may be seen extending from the breakpoint (x_b = 41.5 m, τ ∼ 30 s) to offshore (x = 0 m, τ ∼ 49 s) [Fig. 8(b)]. This negative cross-correlation corresponds to a seaward propagating free IG-wave generated by SS-wave breaking (Baldock 2006; Contardo and Symonds 2013; Pomeroy et al. 2012). Thus, the incoming IG waves shoreward of the breakpoint [x > 41.5 m, Fig. 8(a)] are the combined result of enhanced bound waves and the shoreward propagating component of breakpoint forced waves. Accordingly, the IG waves propagating seaward (outgoing) are the combined result of the wave reflected at the dike and the wave radiated seaward directly from the breakpoint.

This finding is further supported by the normalized bed-slope parameter analysis where β_b ranged from 0.41 to 0.55 for the 12 physical model tests (Table 1). These values are over the threshold for bound-wave shoaling as the main generation mechanism (β_b ≤ 0.3) but still under the threshold for breakpoint forcing to dominate (β_b ≥ 1). This suggests that both generation mechanisms do indeed contribute to H_m0,IG,toe.

Influence of Environmental Parameters

Deep-Water Significant Wave Height and Peak Period

Given that H_m0,deep and T_p are known to be correlated, we jointly assess their influence on nearshore wave conditions. Fig. 9 shows the variation in H_m0,IG,toe, H_m0,SS,toe, and ${\tilde{H}}_{\mathrm{IG}}$ with changes in H_m0,deep and T_P at the offshore boundary, while all other parameters were held constant: h_toe = 1 m, σ = 0°, cot α_fore = 50, W_veg = 0 m, n = 0.012 s/m^1/3, and cot α_dike = ∞. Both H_m0,IG,toe [Fig. 9(a)] and H_m0,SS,toe [Fig. 9(b)] increase linearly with increasing H_m0,deep and, to a much lesser extent, T_p, made evident by the minimal scatter displayed. Likewise, Fig. 9(c) displays a clear positive relationship between H_m0,deep and ${\tilde{H}}_{\mathrm{IG}}$, with minor scatter associated with variations in T_p. This relationship is well described (coefficient of determination, R² = 0.94) by the following expression, obtained using the linear least-squares method:

$${\tilde{H}}_{\mathrm{IG}}=a\cdot H_{m0,\mathrm{deep}}^{0.5}$$

(25)

where the fitted coefficient a = 0.358 (0.344, 0.371) m⁻¹ for the previous reference parameter values; numbers in parentheses indicate the lower and upper 95% confidence bounds, respectively.

The strong dependence of both H_m0,IG,toe and H_m0,SS,toe on H_m0,deep indicates an unsaturated surf zone, typical of steeper foreshore slopes (Power et al. 2010; Stockdon et al. 2006), where the inner surf zone wave heights are equally dependent on the local water depth (h_toe) and the offshore wave height (H_m0,deep). This dependence also justifies the relationship obtained between ${\tilde{H}}_{\mathrm{IG}}$ and H_m0,deep [Eq. (25)]; that is, ${\tilde{H}}_{\mathrm{IG}}$ exhibits a negative exponential (or negative power) relationship with h_toe/H_m0,deep, as shown in Lashley et al. (2019b), and hence a positive power relationship with H_m0,deep/h_toe. This ratio of offshore wave height to local water depth has been found to accurately describe both the intensity (Hofland et al. 2017; Lashley et al. 2018) and location of wave breaking (Yao et al. 2013), which in turn determine the magnitude of nearshore waves. Thus, for the constant water depth simulated here, (h_toe = 1 m), the power law holds. The influence of variations in h_toe is assessed, separately, in “Initial Water Depth at the Dike Toe.”

We also assessed the influence of additional parameters that combine H_m0,deep and T_p: s₀ [Fig. 10(a)]; $\sqrt{H_{m0,\mathrm{deep}}{L}_0}$ [Fig. 10(b)], as recommended by Stockdon et al. (2006), and $H_{m0,\mathrm{deep}}^2{T}_p$ [Fig. 10(c)], as proposed by Inch et al. (2017). However, while these parameters may describe the individual components (H_m0,IG,toe and H_m0,SS,toe) well, as shown in their respective studies, they yielded lower R² values here, compared with Eq. (25). This suggests that T_p has only a minor influence on ${\tilde{H}}_{\mathrm{IG}}$.

Deep-Water Directional Spreading

Considering wave flume (σ = 0°), typical swell (σ = 10°), and typical wind–sea (σ = 24°) conditions, a total of 72 estimates of H_m0,IG,toe and H_m0,SS,toe were obtained. Both parameters show negative relationships with σ (Fig. 11). However, H_m0,IG,toe experiences 2.6 times more dissipation (on average) than does H_m0,SS,toe for the same increase in σ, suggesting that wave energy at IG frequencies is significantly more sensitive to variations in σ. A change in σ from wave flume conditions to a typical wind–sea results in a 36% reduction in H_m0,IG,toe compared with an 16% reduction in H_m0,SS,toe. This finding emphasizes the need to consider directional spreading in the offshore forcing, as the assumption of unidirectionality leads to unrealistically high estimates of H_m0,IG,toe (van Dongeren et al. 2003). As H_m0,IG,toe is more sensitive to changes in σ than H_m0,SS,toe, ${\overline{\gamma }}_{\sigma }$ decreases linearly with increasing σ (Fig. 12):

$${\overline{\gamma }}_{\sigma }=1-0.01\cdot \sigma$$

(26)

where ${\overline{\gamma }}_{\sigma }=1$ corresponds to the reference case.

Initial Water Depth at the Dike Toe

Each of the ten values of h_toe simulated (Table 2) was exposed to the 24 different combinations of H_m0,deep and T_p, resulting in 240 estimates of H_m0,IG,toe [Fig. 13(a)] and H_m0,SS,toe [Fig. 13(b)] for analysis. In addition to the influence of h_toe, the combined influence, if any, of H_m0,deep or T_p may be observed in the colormaps presented in Fig. 13.

Both H_m0,IG,toe and H_m0,SS,toe appear to decrease as the initial water depth becomes shallower (Fig. 13); however, this reduction is more notable in H_m0,SS,toe, where a clear linear trend is observed. The points in Fig. 13(b) which appear to deviate from the general trend, at h_toe = 1.5 m and h_toe = 3.5 m, correspond to simulations with H_m0,deep = 1 m and H_m0,deep = 3 m, respectively. In these simulations, the relative water depth, h_toe/H_m0,deep > 1.0 and H_m0,SS,toe is not yet depth limited.

To quantify the influence of h_toe, we analyze the average response of ${\tilde{H}}_{\mathrm{IG}}$ to variations in h_toe, relative to that obtained for the reference case [${\overline{\gamma }}_h$, Eq. (24)], where h_toe = 1 m. Fig. 14 shows that ${\overline{\gamma }}_h$, and thus ${\tilde{H}}_{\mathrm{IG}}$, on average, increases exponentially with decreasing water depth. This observation is in agreement with the findings of Hofland et al. (2017), which showed that the spectral wave period (T_m−1,0) at the dike toe, which also represents the relative contribution of lower-frequency (IG) waves, increased exponentially with shallower water depths. By fitting a curve to the data, we obtain the following expression for the influence of h_toe:

$${\overline{\gamma }}_h=1.04\cdot e^{-1.4\cdot h_{\mathrm{toe}}}+0.9\cdot e^{-0.19\cdot h_{\mathrm{toe}}}$$

(27)

where ${\overline{\gamma }}_h=1$ corresponds to the reference case.

Influence of Foreshore Slope

H_m0,IG,toe initially increases as cot α_fore increases, reaching a maximum value at cot α_fore = 50 [Fig. 15(a)]. This is due to the combination of both the bound-IG wave shoaling and breakpoint-generation mechanisms (0.3 < β_b < 1), as described in “Infragravity Wave-Generation Mechanism.” However, as the foreshore becomes milder, as cot α_fore increases further, H_m0,IG,toe decreases exponentially (note the log scale of the x-axis). On the other hand, H_m0,SS,toe shows a consistent negative exponential relationship with cot α_fore [Fig. 15(b)]. As SS waves begin to break further offshore with increasing cot α_fore, the surf zone becomes wider and results in increased dissipation.

These trends lend support to the arguments of Lara et al. (2011) and Baldock (2012) that suggest that the bound IG wave may not be released during SS-wave breaking, but remains locked and therefore dissipates in the surf zone concurrently with the SS waves that force it. This dissipation is less evident for steeper slopes [Figs. 16(a and b)], as the bound wave may indeed experience some dissipation, but breakpoint forcing is strong inside the relatively narrow surf zone, thus leading to higher values of H_m0,IG,toe. However, as the foreshore becomes milder (cot α_fore > 100; β_b < 0.3) and the surf zone wider, breakpoint forcing becomes weak, since its forcing (breaking) region becomes large in comparison to the wave-group motion (Baldock and Huntley 2002). This absence of breakpoint forcing and the dissipation of the bound IG wave in the surf zone results in lower values of H_m0,IG,toe [Fig. 16(c)].

The mean response of ${\tilde{H}}_{\mathrm{IG}}$ to variations in cot α_fore, relative to the reference case [${\overline{\gamma }}_f$, Eq. (28)], shows three distinct trends (Fig. 17), dependent on the characteristics of wave breaking (represented here by cot α_fore and the relative water depth, h_toe/H_m0,deep). For steeper slopes (cot α_fore < 100), ${\overline{\gamma }}_f$ increases logarithmically with increasing values of cot α_fore (red line in Fig. 17). This growth continues for milder slopes (cot α_fore > 100) when h_toe/H_m0,deep ≥ 0.2; however, for lower values of h_toe/H_m0,deep, the surf zone becomes very wide and ${\overline{\gamma }}_f$ decreases (blue line in Fig. 17). The relatively large variation in ${\overline{\gamma }}_f$ (magenta line) for cot α_fore = 500 and cot α_fore = 1,000, shown by the length of the error bars, is due to the increased influence of T_p on the breakpoint location and the overall surf zone width for very mild slopes.

$$\begin{array}{r}{\overline{\gamma }}_f=\{\begin{array}{cc}1.56-3.09\cdot \cot {\alpha }_{\mathrm{fore}}^{-0.44}& \cot {\alpha }_{\mathrm{fore}}\le 100\\ 0.51\cdot \cot {\alpha }_{\mathrm{fore}}^{0.18}& \cot {\alpha }_{\mathrm{fore}}>100,h_{\mathrm{toe}}/H_{m0}\ge 0.2\\ 1.62\cdot \cot {\alpha }_{\mathrm{fore}}^{-0.08}& \cot {\alpha }_{\mathrm{fore}}>100,h_{\mathrm{toe}}/H_{m0}<0.2\end{array}\end{array}$$

(28)

where ${\overline{\gamma }}_f=1$ corresponds to the reference case (cot α_fore = 50).

In addition, we examined how ${\tilde{H}}_{\mathrm{IG}}$ relates to the well-known normalized bed slope parameter (β_b), proposed by Battjes et al. (2004). In Fig. 18(a), ${\tilde{H}}_{\mathrm{IG}}$ shows a negative exponential relationship with β_b; with scatter further highlighting the influence of h_toe/H_m0,deep. By combining the two parameters, the scatter is reduced significantly in Fig. 18(b). Therefore, in addition to describing the IG-wave generation mechanism, β_b also describes – to some extent – the relative magnitude of the IG waves.

Vegetated Cover

H_m0,IG,toe [Fig. 19(a)] and H_m0,SS,toe [Fig. 19(b)] both show negative exponential relationships with W_veg. The presence of vegetation, from W_veg = 0 m to W_veg = 200 m, notably reduces the wave heights at the dike toe; however, the effect is more significant for H_m0,IG,toe than H_m0,SS,toe, with 60% and 46% reduction (on average), respectively. As a larger portion of the water column is under wave motion for longer-period (IG) waves, vegetation is able to more effectively reduce flow velocities. Therefore, submerged vegetation, also referred to as “canopies,” attenuates IG-wave components more than do shorter-period (SS) components, as reported by Lowe et al. (2007) and Koftis et al. (2013).

Though the initial attenuation effect between nonvegetated (W_veg = 0 m) cover and some vegetation (W_veg = 200 m) is striking, a further increase in the width of vegetated cover does not significantly affect either H_m0,IG,toe or H_m0,SS,toe (Fig. 19). A similar response is seen in the average response of ${\tilde{H}}_{\mathrm{IG}}$ to variations in W_veg $({\overline{\gamma }}_v)$, where the inclusion of 400 m of vegetation reduces ${\overline{\gamma }}_v$ by 40%, compared with the nonvegetated (reference) case, while a further increase in W_veg has only a minor effect (Fig. 20). This relationship may be described by

$${\overline{\gamma }}_v=0.94\cdot e^{-W_{\mathrm{veg}}/500}+0.06\cdot e^{W_{\mathrm{veg}}/500}$$

(29)

where ${\overline{\gamma }}_v=1$ corresponds to the reference case (W_veg = 0 m).

Bottom Friction

With respect to bottom friction, we assessed three general bed conditions: smooth concrete (wave flume, n = 0.012 s/m^1/3), mud (n = 0.020 s/m^1/3), and sand (n = 0.03 s/m^1/3). However, both H_m0,IG,toe and H_m0,SS,toe were found to be insensitive to n, experiencing only minor dissipation (<2%) with increasing n (not shown here). This finding agrees with previous studies that found that the dissipation of nearshore waves was governed by depth-induced wave-breaking (Henderson et al. 2006; Thomson et al. 2006; van Dongeren et al. 2007) and attenuation by vegetation (Baron-Hyppolite et al. 2019), but was insensitive to bottom friction. Thus, the influence of n on ${\tilde{H}}_{\mathrm{IG}}$ can be considered negligible.

Dike Slope

The presence of a dike increases H_m0,IG,toe by a factor of 2.1 (on average) compared with the no-dike (cot α_dike = ∞) scenario [Fig. 21(a)]. This is due to the constructive inference of the incoming and reflected IG waves at the dike toe (as described in “Reflection of Infragravity Waves at the Dike”). This occurs for dike slopes as mild as cot α_dike = 10, though a minor reduction in H_m0,IG,toe is seen as the dike slope becomes milder (as cot α_dike increases). H_m0,SS,toe experiences a similar increase in amplitude for cot α_dike = 2 compared with the no-dike scenario [Fig. 21(b)]; however, this increase is only by a factor of 1.3 (on average). This difference in amplification is due to the frequency-dependence of wave reflection – that is, lower-frequency (longer-period) waves experience greater reflection than those at higher frequencies. Furthermore, as the dike slope becomes milder (cot α_dike > 2), the influence of the dike on H_m0,SS,toe quickly becomes negligible [Fig. 21(b)].

As a result, the mean response of ${\tilde{H}}_{\mathrm{IG}}$ to variations in cot α_dike $({\overline{\gamma }}_d)$ shows an initial positive relationship with cot α_dike, where it increases as the dike slope becomes milder, up to cot α_dike = 6, but then decreases slightly for cot α_dike = 10 (Fig. 22). The increased variation in ${\overline{\gamma }}_d$ for larger values of cot α_dike is due to the influence of T_p on wave reflection, where longer-period waves experience greater reflection at the dike, resulting in a larger ${\tilde{H}}_{\mathrm{IG}}$ values.

$${\overline{\gamma }}_d=1.3-0.02\cdot \cot {\alpha }_{\mathrm{dike}}^2+0.24\cdot \cot {\alpha }_{\mathrm{dike}}$$

(30)

where ${\overline{\gamma }}_d=1$ corresponds to the reference case (no-dike scenario).

Predicting Nearshore Infragravity-Wave Dominance

By combining Eqs. (25) to (30), we may estimate ${\tilde{H}}_{\mathrm{IG}}$ based on prior knowledge of offshore forcing (H_m0,deep and σ), foreshore conditions (h_toe, cot α_fore and W_veg), and dike slope (cot α_dike) using:

$${\tilde{H}}_{\mathrm{IG}}=0.36\cdot H_{m0,\mathrm{deep}}^{0.5}\cdot {\overline{\gamma }}_{\sigma }\cdot {\overline{\gamma }}_h\cdot {\overline{\gamma }}_f\cdot {\overline{\gamma }}_v\cdot {\overline{\gamma }}_d$$

(31)

where subscripts σ, h, f, v, and d = initial water depth, directional spreading, foreshore slope, vegetated cover, bottom friction, and dike slope, respectively. To consider only incident wave conditions (no reflection), ${\overline{\gamma }}_d$ takes a value of 1. For conditions where ${\tilde{H}}_{\mathrm{IG}}>1$ (IG-wave dominance), the magnitude of nearshore IG waves exceeds that of SS waves and governs wave-driven processes, such as wave runup and overtopping; thus, under such conditions IG waves should not be neglected.

Fig. 23 presents the combined results of ${\tilde{H}}_{\mathrm{IG}}$ for the physical modeling, observations obtained from 12 simulations and numerical modeling, 672 XBeach simulations. The proposed empirical model [Eq. (31)], developed using XBeach simulations alone, not only describes the entire dataset well (R² = 0.94) but also shows good agreement with the observations made during the physical experiment (SCI = 0.098 and Rel.bias = −0.07). The outliers shown in Fig. 23 are largely associated with simulations where nearshore conditions are not significantly influenced by the water depth (h_toe/H_m0,deep = 1) and ${\tilde{H}}_{\mathrm{IG}}$ values are low (<0.5).

To verify the validity of OAT approach taken here, where the influence of each parameter was assessed independently, a test-case scenario, where multiple parameter values differed from the reference case, was simulated in XB-NH and the results compared with Eq. (31). This test case, with H_m0,deep = 9 m, T_p = 15 s, σ = 0°, h_toe = 0.5 m, cot α_fore = 100, W_veg = 200 m, and cot α_dike = 6, was also well represented by Eq. (31), shown by the blue marker in Fig. 23. This shows that the approach taken here does indeed take into account the interaction between environmental parameters.

Conclusions

Data Availability Statement

Acknowledgments

Notation

A

envelope of the sea and swell waves (m);

a

fitted coefficient in the prediction of ${\tilde{H}}_{\mathrm{IG}}$ (m⁻¹);

a_h

vegetation height (m);

b_v

vegetation stem diameter (m);

C_D

drag coefficient (−);

C_ηη

wave energy density (m²/Hz);

c

wave speed (m/s);

c_f

bed friction factor (−);

c_g

wave-group velocity (m/s);

c_in

incoming wave speed (m/s);

c_out

outgoing wave speed (m/s);

F_D

drag force (N/m);

F_v

depth-averaged vegetation force (N);

f_low

mean frequency of the IG wave at breakpoint (Hz);

f_node

nodal frequency of standing wave (Hz);

f_p

peak frequency (Hz);

${\tilde{H}}_{\mathrm{IG}}$

relative magnitude of the infragravity waves at the dike toe (−);

H_m0

local significant wave height (m);

H_m0,deep

significant wave height offshore in deep water (m);

H_m0,IG

significant wave height in the infragravity frequency band (m);

H_m0,IG,toe

significant wave height in the infragravity frequency band, at the dike toe (m);

H_m0,SS

significant wave height in the sea and swell frequency band (m);

H_m0,SS,toe

significant wave height in the sea and swell frequency band, at the dike toe (m);

h

local water depth (m);

h_b

water depth at breakpoint (m);

h_toe

initial water depth at the dike toe (m);

k

wave number (rad/m);

L

local wavelength (m);

L₀

wavelength in deep water (m);

m

mode of the dike–foreshore system (standing wave) (−);

N_v

vegetation stem density (stems/m²);

n

manning roughness coefficient (s/m^1/3);

$\overline{q}$

depth-averaged dynamic (nonhydrostatic) pressure normalized by the density (−);

Rel.bias

Relative Bias (−);

R_ηA

cross-correlation coefficient (−);

R²

coefficient of determination (−);

s

user-defined directional spreading factor (XBeach) (−);

s₀

deep-water wave steepness (−);

SCI

scatter Index (−);

T_m−1,0

spectral wave period (s);

T_p

peak wave period in deep water (s);

u

depth-averaged cross-shore velocity (m/s);

v_h

horizontal viscosity (m²/s);

W_veg

width of vegetated cover (m);

x

cross-shore location (m);

x_b

cross-shore breakpoint location (m);

x_dike

cross-shore dike location (m);

α_dike

dike slope angle (°);

α_fore

foreshore slope angle (°);

β_b

normalized bed slope (−);

$\overline{\gamma }$

Influence factor (−);

Δx

cross-shore grid spacing (m);

Δy

alongshore grid spacing (m);

η

surface elevation (m);

η^IG

low-pass filtered surface elevation (m);

$\overline{\eta }$

mean water level relative to the dike toe (m);

${\eta }_{\mathrm{in}}^{\mathrm{IG}}$

incoming low-pass filtered surface elevation (m);

${\eta }_{\mathrm{out}}^{\mathrm{IG}}$

outgoing low-pass filtered surface elevation (m);

η^SS

high-pass filtered surface elevation (m);

ξ₀

breaker index (Iribarren number) (−);

ρ

density of water (kg/m³);

σ

directional spreading (°);

σ_A

standard deviation of the wave envelope time series (m);

σ_η

standard deviation of the low-pass filtered surface elevation time series (m);

ω

angular frequency (rad/s); and

Ω

ratio of breaking waves to water depth (−).

References