1. Introduction
Infragravity (IG) waves are ocean surface waves of low frequencies, typically between 0.004 and 0.04 Hz, that distinct from the wind waves or swells of frequencies between 0.04 and 1 Hz (Bertin et al. 2018). IG wave plays an important role in many coastal processes, such as resonance in harbors (Miles 1974; Bowers 1977; Okihiro et al. 1993; Maa et al. 2010; Thotagamuwage and Pattiaratchi 2014a; Gao et al. 2019; Gao et al. 2020), morphological evolution (Roelvink et al. 2009; Mendes et al. 2020; Ruffini et al. 2020), and hydrodynamics in surf zones (Guedes et al. 2013; Padilla and Alsina 2018; Contardo et al. 2019) and on coral-reef flats (Nwogu and Demirbilek 2010; Pomeroy et al. 2012; Liu and Li 2018). The generation, evolution, and dissipation of IG waves in intermediate and shallow water have therefore been the focus of numerous studies in the past decades (Longuet-Higgins and Stewart 1962, hereinafter LHS62; Symonds et al. 1982; Herbers et al. 1994; Sheremet et al. 2002; Janssen et al. 2003; Battjes et al. 2004; Henderson et al. 2006; Thomson et al. 2006; Baldock 2012; De Bakker et al. 2015; Li et al. 2020). One of the main generation mechanisms of IG waves is the nonlinear wave group forcing, and the relevant analytical frameworks are well established in intermediate water where the resonance between forcing and IG waves is generally weak (Bowers 1992; Janssen et al. 2003; Zou 2011, hereinafter referred to as ZOU11). However, in shallow water where strong resonance occurs, the nonlinear wave group forcing becomes more significant, but currently there is a lack of analytical models. This limits the physical understanding of many important behaviors of IG waves in shallow water, such as IG waves shoaling on nearshore beaches and the recently reported offshore shoal-induced amplification of IG waves (Paniagua-Arroyave et al. 2019; Li et al. 2020).
For nonbreaking waves over a slowly varying topography (|β_{1}| ≪ 1), the existing theoretical works on group-forced IG waves (edge wave excluded) may be categorized into two types. The first type focuses on waves over sloping beaches, where the resonance intensity between group-forcing and IG waves increases from μ = O(1) in intermediate depth to μ ≪ 1 in shallow water. Accordingly, the relative importance between effects of depth variation and local resonance may evolve from off-resonance [β_{1}μ^{−1} = O(β_{1})] to near-resonance [β_{1}μ^{−1} = O(1)] or even greater. In the off-resonant condition, the effect of depth variation has been incorporated analytically as perturbations to the LHS62 solution (Bowers 1992; Van Leeuwen 1992; Janssen et al. 2003). It was found that the topography-induced perturbation at the leading order O(β_{1}) is in quadrature with the LHS62 solution, and therefore causes the phase shift of bound IG waves. However, in the full-resonant condition in shallow water, the resonance is greatly enhanced (μ → 0) so that the LHS62 solution may diverge and the perturbation method fails. In this situation, by numerically solving the governing equation accurate to O(β_{1}), Janssen et al. (2003) concluded that it is not appropriate to employ the asymptotic shallow water limit of the LHS62 solution (~h^{−2.5}) as the shoaling trend.
The second type focuses on the group-forced IG waves over a finite topography in intermediate depth where concomitant free IG waves are scattered from the topography. With μ = O(1) at this region, the second type problem is often the off-resonant case. Starting with a one-dimensional topography in deep water, Molin (1982) reported the free IG waves induced by a bichromatic wave group for the first time. Later on, Mei and Benmoussa (1984) investigated obliquely incident bichromatic waves on two-dimensional topographies with shore parallel contours in intermediate depths. They found that the free IG wave is scattered in a direction different from the incident wave direction. The governing equation of group-forced IG waves additional to the LHS62 solution over variable depth was derived and numerically solved. Liu (1989) corrected the boundary conditions of Mei and Benmoussa (1984) and obtained different numerical results. ZOU11 investigated the IG waves generated by a fully modulated bichromatic wave group propagating over a 1D topography using a multiscale Wentzel–Kramers–Brillouin (WKB) expansion method. Assuming
As illustrated in Fig. 1, as wave groups propagate toward the shoreline from deep to shallow water, they would pass through several zones with different resonant intensities. Apart from the shallow water over a sloping beach close to the shoreline, strong resonance [β_{1}μ^{−1} = O(1)] may also occur at an offshore topography with relatively shallow water depth on the top and could generate remarkable IG waves. For example, we propose the strong resonance at the crest of a large-scale submarine shoal is responsible for the considerably amplified IG waves reported by several observational and numerical studies listed in Table 1. However, the existing off-resonant IG wave solutions cannot explain this phenomenon since it is a single-valued function of wave radiation stress, depth, and bottom slope. Therefore, it predicts the same order of magnitude of group-forced IG waves on both sides of the topography.
IG wave amplification induced by an offshore shoal and the resonance parameters
The objective of the present work is to derive theoretical solutions of group-forced IG waves under off- and nearly-resonant conditions, from intermediate to shallow water depth. The off-resonant solution provides new insight for the relative importance of different generation mechanisms of the nonequilibrium bound IG waves in intermediate depths, while the nearly-resonant solution predicts the shallow water shoaling rate of bound IG waves on a plane sloping beach to be ~h^{−1} for the first time. More importantly, it indicates that in shallow water where strong resonance occurs, the group-forced IG wave is not locally determined but relies on its spatial evolution history. We will also be able to theoretically demonstrate for the first time that on a symmetric underwater shoal topography, the presence of strong resonance near the shoal crest at low tide is responsible for the irreversible asymmetric spatial evolution of group-forced IG waves. The governing equation of the IG wave response to group-forcing is derived in section 2. Analytical solutions for off- and nearly-resonant conditions are obtained and discussed in section 3. Boundary condition of topography-scattered free IG wave is derived in section 4. In section 5, examples of bichromatic and irregular waves on a plane beach are presented. In section 6, group-forced IG waves over a symmetric topography are investigated. Conclusions and main findings are summarized in section 7.
2. Governing equations
3. Analytical solutions
a. Bichromatic waves
Assuming that the topography varies slowly at the scale of group lengths, i.e., |β_{1}| ≪ 1, and
1) Off-resonant solution
2) Nearly-resonant solution
Note that the phase of F_{H}(X) is given by
b. Irregular waves
4. Boundary condition
5. Theoretical results
a. Bichromatic waves over a sloping bottom
1) Offshore ramps
Evolution of the IG waves induced by a bichromatic wave group propagating over downward and upward sloping offshore ramps (Figs. 3a,b) in the intermediate depth range k_{∞}h ∈ [1, 2] (k_{∞} = ω^{2}/g) are given in Figs. 3c and 3d. The predictions of the present OLS [Eqs. (12) and (17)] compare well with the ZOU11 solution.
Figure 3c shows that on the downward sloping ramp, compared with OLS, the NRS [Eqs. (19) and (21)] overestimates the amplitude of additional bound IG wave, while the predicted overall amplitude by OLS and NRS is similar to each other. This is because with increasing depth the scattered free IG wave becomes the predominant component of the additional IG wave due to its much lower shoaling rate, and the predicted amplitude of scattered free waves by NRS is close to that by OLS. On the upward sloping ramp (Fig. 3d), however, the predicted amplitudes of the additional bound IG wave by OLS and NRS are nearly the same. The additional group-forced IG wave becomes dominant over the scattered free wave with decreasing depth due to its greater shoaling rate. As a result, predictions of amplitudes of total additional IG wave by OLS and NRS are similar.
2) Plane sloping beach
For wave groups over a plane sloping beach, predictions of the analytical solutions are validated against the results of the Van Noorloos (2003) experiment, which generated high-quality datasets of IG waves induced by bichromatic waves on a sloping beach with an offshore depth of 0.7 m and bottom slope of 1/35. In addition to the OLS and NRS, the exact solution of the governing Eq. (10) was also numerically calculated without decomposition into forced and free components. As shown in Fig. 4, predictions of the exact solution and NRS are in good agreement with the experimental results. Performance of the OLS is promoted with increasing wave group frequency, i.e., weakening resonant intensity.
To illustrate the evolution of group-forced IG waves through zones of different resonance intensities, examples of bichromatic waves on three sloping beaches (h_{x} = −1/20, −1/35, −1/100) are demonstrated in Fig. 5. The physical parameters of the second case (h_{x} = −1/35) were chosen to be the same as those of the A-4 case in Fig. 4.
For the nearly- to fully-resonant cases of h_{x} = −1/20 and −1/35 [β_{1}μ^{−1} = O(1)]), (Figs. 5a,b), compared with the exact solution predictions, both the amplitude (Figs. 5a,b) and phase lag (Figs. 5d,e) of group-forced IG waves are overestimated by the OLS. The NRS agrees well with the exact solution except for slight overestimations of the phase lag in the vicinity of the breakpoint (kh ≈ 0.5) where the assumption |β_{1}| ≪ 1 is becoming invalid with decreasing depth. In the case of h_{x} = −1/100, the resonance intensity is generally weak (|β_{1}μ^{−1}| ∈ [0.06,1.2]) and the system is mainly off-resonant in the shoaling zone. Accordingly, the OLS agrees with the exact solution in terms of predictions of the amplitude and phase lag in kh ≥ 0.8 (Figs. 5c,f). Meanwhile, predictions of the NRS are nearly the same as the exact solution, suggesting that the NRS might be practically effective over the whole beach. Furthermore, amplitudes of topography-induced additional bound IG wave of two parts [section 3a(1)] are calculated based on the OLS. Amplitudes of additional bound IG waves induced by the enhanced group-forcing (blue line) and by the varying IG wave amplitude with decreasing water depth (red line) are demonstrated in Fig. 5c. Consistent with the theoretical calculations displayed in Fig. 2, the results confirm the dominance of the latter type.
The local shoaling rates of IG waves,
b. Irregular waves on a plane sloping beach
For irregular waves propagating over a plane beach of bottom slope h_{x} = −1/80, comparison among predictions of the exact solution, NRS, and the nonhydrostatic wave model SWASH (Zijlema et al. 2011) is given in Fig. 6. The SWASH model is essentially a Reynolds-averaged Navier–Stokes (RANS) equation solver capable of describing the processes of wave motion with strong nonlinearity (Kirby 2017). The model has been extensively validated against both laboratory (De Bakker et al. 2016) and field data (Rijnsdorp et al. 2015) of nearshore wave evolution with strong nonlinearity. Detailed descriptions of the model can be found in Rijnsdorp et al. (2014) and Smit et al. (2014).
All parameters are the same as the A3 case in the numerical simulation by De Bakker et al. (2016), which has been validated against the laboratory experiment of Ruessink et al. (2013). Irregular waves described by JONSWAP spectrum (Hasselmann et al. 1973) were generated with the significant wave height H_{s} = 0.1 m, the peak period T_{p} = 2.25 s, and peak enhancement factor γ = 20. Time series of the surface elevation predicted by the SWASH model at the toe of the slope was bandpass filtered through the frequency range 0.5ω_{p} ≤ ω ≤ 3ω_{p}, and then transformed into the single-side complex Fourier amplitudes. Then the spectrum of wave radiation stress was calculated according to Eqs. (23) and (24), and employed as the input for the theoretical solution Eq. (25). Note that since the boundary condition Eq. (29) is derived based on the off-resonant assumption (|β_{1}μ^{−1}| = O(β_{1}) ≪ 1), it does not apply to the wave frequency components close to zero frequency. Therefore, we applied the boundary condition only to the wave frequency components that satisfy |β_{1}μ^{−1}| ≤ 0.1 at the toe of the bottom slope, while the response function of the rest of the lower-frequency wave spectral components was calculated using the LHS62 solution neglecting the scattering effect of free IG waves.
Figure 6a shows the spatial evolution of normalized integrated IG wave height over the frequency band 0 ≤ ω ≤ 0.5ω_{p}. The NRS and exact solution predictions are nearly the same, and both agree with those predicted by the SWASH model reasonably well in the shoaling zone. In addition, the predicted phase lag by exact solution is in excellent agreement with that by the SWASH model (Fig. 6b), whereas the NRS gradually overpredicts the phase lag with decreasing depth. The local shoaling rate of the significant wave height predicted by the SWASH model approximately reaches ~h^{−1} near the breakpoint, while the exact solution and NRS predict greater value (Fig. 6c), possibly due to the fact that the narrow-band assumption may become less effective as the bandwidth increases when waves propagate shoreward.
Figure 7a shows the variance density spectrum of group-forced IG waves as a function of depth and frequency by the SWASH model in the shoaling zone using a horizontal resolution of 1 m. The spectrum was generated using the surface elevation of the incident IG waves of 1 h duration, sampled at 10 Hz, and smoothed with a moving window of 50 bins width. Figure 7b shows the comparison between the predicted variance density spectrum by the theoretical model OLS, NRS, and SWASH at the wave breaking depth (k_{p}h = 0.51). Both the NRS and exact solution well captured the shape of the spectrum (Fig. 7b) prior to wave breaking, but overestimated the IG wave energy near the breaking depth as discussed in Fig. 6 earlier. Figure 7b also shows that at extremely low frequencies (0 ≤ ω/ω_{p} ≤ 0.1), the NRS predicts lower wave energy than SWASH while the exact solution agrees reasonably well with SWASH, indicating that the second-order terms of
Figure 7c indicates that the local shoaling rate of IG component is greater for higher frequencies, i.e., lower-resonance intensities. In the band of 0.1 ≤ ω/ω_{p} ≤ 0.5, the single-peaked pattern of the spatial evolution of local shoaling rate can be recognized as in Fig. 6, and for components in lower-resonance intensities the peak is greater and the depth for its occurrence is deeper (Fig. 7c). With decreasing depth, the local shoaling rates in this frequency band eventually approaches the theoretical shallow limit of NRS ~h^{−1}. At extremely low frequencies (0 ≤ ω/ω_{p} ≤ 0.1), the local shoaling rates are evidently lower (Figs. 7c,d). The predicted local shoaling rate by the exact solution decays with decreasing frequency in the same trend as the result of SWASH model, while that predicted by the NRS decays faster. The aforementioned significant second-order effect of
6. Discussions on group-forced infragravity waves over a symmetric topography
To theoretically diagnose the mechanisms for the shoal-induced amplification of IG waves mentioned in the Introduction (Table 1), the behavior of IG waves in the area of near-resonance on a symmetric topography, where h(x) = h(L − x), with L the finite horizontal length of the topography, is analyzed. The topography is assumed to be smooth, so no scattering of free IG wave due to discontinuity in bottom slope is considered except at the offshore edge.
Examples of bichromatic waves propagating across a shoal-mimic topography (Fig. 8a) with different tide levels are given in Fig. 8. For k_{∞}h_{0} = 2 the system is off-resonant over the entire region, the OLS agrees with the exact solution, while the NRS overpredicts the amplitude of the additional bound IG wave (Fig. 8b) and phase lag (Fig. 8e). In this situation, the scattered free wave is of the same order as the additional group-forced wave, therefore the behavior of the total additional wave deviates from that of the additional forced IG wave. Consequently, the predictions of exact solution and OLS of the spatial evolution of additional IG wave only show rough symmetric conjugacy.
For k_{∞}h_{0} = 1, nearly- to fully-resonant area emerges near the crest of the shoal. In this case, the OLS remains in agreement with the exact solution in the range 0 ≤ x/L ≤ 0.29, while the NRS agrees with the exact solution reasonably well over the entire domain (Figs. 8c,f). Additional bound IG wave is considerably amplified over the shoal according to the predictions of both the NRS and exact solution. As the tidal level decreases to k_{∞}h_{0} = 0.5, the prediction by the NRS is almost exactly the same as that by the exact solution, while the OLS deviates from the exact solution significantly (Figs. 8d,g). It is shown that with increasing resonant intensity the maximum phase lag occurs closer to the crest of the shoal where resonant intensity attains maximum (Figs. 8e–g). Since the additional forced IG wave dominates over the scattered free wave in these two strong resonant cases (Figs. 8c,d), the nonequilibrium response contributes to the total response at the leading order
Theoretically, for the more general cases other than a single shoal, the presence of the nearly-resonant area does not always rule out the possibility for the reversible spatial variation of the group-forced IG waves to occur. The reversible condition Eq. (31) suggests that in the nearly-resonant area, the group-forced IG wave would start to evolve reversely from the location where the depth starts to vary symmetrically, and meanwhile the group forcing and group-forced IG waves are nearly in phase/antiphase with each other. Regarding the bathymetry of a single shoal, however, this condition is expected to be unlikely to be satisfied so that the amplification of IG waves should always occur.
For the wave group with significant directional spreading in the field condition, group-forced IG waves propagating in a direction different from the main direction of primary waves may be induced. Future works of extending the present solution to accommodate the 2D wave field are needed to theoretically model both the frequency and directional spreading of group-forced IG waves.
7. Conclusions
The spatial evolution of infragravity (IG) waves induced by wave group over topography in a wide range of resonance intensity and water depth under nonbreaking conditions has been investigated with a newly defined complex response function. The analytical solution for the governing equation at the first order of the relative bottom slope is proposed and validated against the laboratory, theoretical and numerical results. In intermediate water, the off-resonance solution is consistent with the previous second-order nonlinear theory of ZOU11 when the higher-order terms therein are negligible. In shallow water, a novel nearly-resonance solution is obtained and is the first of its kind according to the authors’ knowledge.
The nonequilibrium bound IG wave in quadrature with the wave group forcing and the equilibrium component on flat bottoms is generated by the off-resonant [β_{1}μ^{−1} = O(β_{1})] wave groups in the intermediate depth as indicated by the off-resonant linear solution [Eqs. (12) and (17)]. At the leading order, it is proportional to the normalized bottom slope β_{1}, therefore, secondary compared with the equilibrium component at O(1). It is jointly induced by the spatial variation of magnitude and transmitting speed of group-forcing and the spatial variation of the bound IG wave amplitude [cf. Eq. (15)]. The latter is dominant over the former, and therefore, is the primary contributor to the spatially varying phase coupling between bound IG waves and group-forcing and the energy transfer from primary waves to IG waves.
Opposite to the aforementioned off-resonant case, for the nearly-resonant [β_{1}μ^{−1} = O(1)] wave groups in shallow water, the nearly-resonant solution [NRS, Eqs. (19) and (21)] indicates that both the nonequilibrium and equilibrium group-forced IG waves are of the leading order
The solutions proposed also enable a theoretical investigation of the spatial evolution of group-forced IG waves over a symmetric shoal in intermediate and shallow water to reveal the influences of the strong resonance near the topography crest. When the resonance is weak across the whole topography at high tide, the solution indicates that the spatial evolution of the IG wave amplitude on the shoreward slope of the shoal is roughly the mirror image of that of its counterpart on the seaward slope. As the water level drops sufficiently at low tide, however, strong resonance begins to occur near the crest of the shoal, the group-forced IG wave dominates over the topography-scattered freely propagating IG wave, with the maximum phase lag relative to the equilibrium response occurring near the crest. The spatial distribution of the phase lag becomes increasingly symmetric and remains positive with decreasing water level, which leads to the persistent nonlinear energy transfer from primary waves to IG waves, and therefore, significant amplification of IG wave amplitude across the shoal.
According to the NRS [Eqs. (19) and (21)], the irreversible amplification in the strong resonance area would occur when the group-forced IG wave and group-forcing are not in phase/antiphase at the shoal crest. This finding provides the theoretical explanation for the previously reported IG wave amplification induced by a shoal in field (Paniagua-Arroyave et al. 2019) and numerical models (Li et al. 2020), or a trapezoid low-crest man-made structure such as breakwater (Peng et al. 2009; Zou and Peng 2011). Generally, the NRS suggests that in the nearly-resonant area, the group-forced IG wave would start to evolve reversely from any location where the depth starts to vary symmetrically and meanwhile the forcing and IG wave are in phase/antiphase with each other, regardless of the history of the spatial evolution of these waves.
Acknowledgments
This research work is financed by the National Natural Science Foundation of China (Grant 51779170). The fourth author was supported by NERC Grant NE/E0002129/1.
Data availability statement
The input files and outputs of the numerical wave model SWASH presented in section 5b and the MATLAB codes of the exact solution, off-resonant linear solution, and nearly-resonant solution are openly available on Zenodo.org (10.5281/zenodo.3951926).
APPENDIX A
Spatial Variation of Response Function
APPENDIX B
Integrated Biphase from the Response Function
APPENDIX C
Notation
η^{(1)} | Primary wave surface elevation |
η^{(2)}, η | IG wave surface elevation |
A | Modulated amplitude of primary wave |
a_{1}, a_{2} | Real amplitudes of bichromatic waves |
k | Central wavenumber of bichromatic waves |
ω | Central radian frequency of bichromatic waves |
θ | Phase of carrier wave |
Δθ | Phase of wave group |
Δk | Wavenumber of wave group |
Δω | Radian frequency of wave group |
φ_{g} | Spatial phase of wave group |
g | Gravitational acceleration |
c_{g} | Wave group speed |
c | Wave speed |
S | Wave radiation stress |
Slowly varying part of wave radiation stress | |
Single-side complex amplitude of slowly varying wave radiation stress | |
Single-side complex amplitude of IG waves | |
Single-side complex amplitude of group-forced IG waves | |
Single-side complex amplitude of freely propagating IG waves | |
Δφ | Phase lag of IG waves relative to the equilibrium bound IG wave described by LHS62 |
F | Group-forced IG wave response to radiation stress for bichromatic waves, |
φ_{F} | Phase of group-forced IG wave response function F |
F_{n} | Group-forced IG wave response of the nth frequency component to radiation stress for irregular waves |
x | Horizontal coordinate, positive onshore |
z | Vertical coordinate, positive upward |
t | Time |
X | Slowly varying horizontal coordinate at group scale, dX = Δkdx |
h | Water depth |
h_{x} | Bottom slope |
h_{xx} | Bottom curvature |
β_{1} | Relative bottom slope, β_{1} = h_{x}/(Δkh) |
β_{2} | Relative bottom curvature, β_{2} = h_{xx}/(Δk^{2}h) |
Λ_{1} | First-order normalized derivative of the amplitude of radiation stress with respect to depth, |
Λ_{2} | Second-order normalized derivative of the amplitude of radiation stress with respect to depth, |
δ | First-order normalized derivative of the speed of radiation stress with respect to depth, |
μ | Degree of departure from resonance, |
ϕ_{0} | ϕ at the offshore incident edge x_{0} of the topography, ϕ is an arbitrary quantity |
F_{H} | Homogeneous solution to the truncated governing equation of the response function F |
θ_{H} | Phase of F_{H} |
ZOU11 | Second-order analytical solution of group-forced IG waves of Zou (2011) |
ES | Exact solution of the group-forced IG wave response |
OLS | Off-resonant [β_{1}μ^{−1} = O(1)] linear solution of the group-forced IG wave |
NRS | Nearly-resonant [ |
F_{LHS62} | Bound IG wave response corresponding to the LHS62 solution, F_{LHS62} = μ^{−1} |
F_{Slope} | Nonequilibrium additional bound IG wave response induced by the variable depth at O(β_{1}) in off-resonant condition, F_{Slope} = F_{I} + F_{II} + F_{III} |
F_{I} | Additional bound IG wave response induced by variable group-forcing |
F_{II} | Additional bound IG wave response induced by variable amplitude of equilibrium bound IG wave described by LHS62 |
F_{III} | Additional subharmonic response induced by variable amplitude of IG wave described by F_{H} |
ω_{p} | Peak radian frequency |
k_{p} | Peak wavenumber |
m_{c} | Rank of the cutoff frequency between IG waves and primary waves |
Single-side complex amplitude of the mth frequency component of primary wave | |
H_{IG} | Significant wave height of IG waves |
Δφ_{IG} | Integrated phase lag of IG waves with respect to group-forcing in addition to π for irregular waves |
k_{∞} | Wavenumber in deep water, k_{∞} = ω^{2}/g |
L | Horizontal length of finite topographies |
F_{J}(x) | Symmetrically conjugate part of the nearly-resonant solution of F(x) on symmetric topographies, |
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