Elsevier

Coastal Engineering

Volume 167, August 2021, 103917
Coastal Engineering

Dispersive characteristics of non-linear waves propagating and breaking over a mildly sloping laboratory beach

https://doi.org/10.1016/j.coastaleng.2021.103917Get rights and content

Highlights

  • Large deviations of wavenumber and phase velocity spectra from the linear wave dispersion.

  • Deviations observed in both the shoaling region and the surf zone (κph up to 0.7).

  • Bispectral analyses highlight the role of non-linear interactions between triads.

  • Stronger coupling between triads results in greater high-frequency bound energy and larger deviations.

Abstract

The dispersive characteristics of unidirectional irregular waves propagating and breaking over a mildly sloping beach are examined using a highly-resolved laboratory dataset. Cross-spectral analyses are used to determine the cross-shore evolution of (single-valued) dominant wavenumber κ and phase velocity c spectra, and lead to the identification of four different regimes of propagation: I - a linear regime where short waves mostly propagate as free components; II - a shoaling regime where non-linear effects at high harmonics are significant but primary components follow the linear wave dispersion relation; III - a shoaling regime near the mean breaking point location, where amplitude dispersion effects at primary components are important; IV - a surf zone regime, where all components propagate slightly faster than non-dispersive shallow water waves. Bispectral analyses performed onshore of the shoaling region show that the presence of forced energy at high harmonics, which originate from non-linear interactions between triads of frequencies, are responsible for the deviations of wavenumber and phase velocity spectral estimates from predictions by the linear dispersion relation, confirming the findings from previous field-based studies. A Boussinesq approximation of the non-linear energy exchanges between triads is then used to quantify the relative amount of forced energy at high harmonics and explain the differences in dispersion properties observed in the shoaling region between broad and narrow-band spectra. Larger relative amounts of forced energy at high frequencies, which suggest more efficient non-linear energy transfers, are found to be associated with larger deviations of dominant κ and c from predictions by the linear dispersion relation.

Introduction

Wind-generated surface gravity waves (hereafter short waves) are the principal driver of nearshore dynamics. Close to shore, short waves eventually break and through this process, they enhance the vertical and horizontal mixing of the water column (e.g., Ting and Kirby, 1996; Drazen and Melville, 2009; Clark et al., 2012), drive a setup near the shoreline (e.g., Longuet-Higgins and Stewart, 1964; Stive and Wind, 1982) and control the nearshore circulation at various temporal scales (e.g., Svendsen, 1984; Peregrine and Bokhove, 1998; Bühler and Jacobson, 2001; Bonneton et al., 2010; Castelle et al., 2016). At first order, linear wave theory correctly predicts a number of physical processes associated with the propagation of short waves, such as the refraction or shoaling of directionally spread seas (e.g. Longuet-Higgins, 1956; Guza and Thornton, 1980; Elgar et al., 1990, and many others). However, as waves shoal and interact with a sloping, mobile bed, non-linear processes become dominant. These are responsible for the changes observed in wave shape, from nearly symmetric to more (horizontally) skewed in the shoaling region and more (vertically) asymmetric prior to breaking and in the surf zone (e.g. Elgar and Guza, 1985a, b; Doering and Bowen, 1995; Michallet et al., 2011; Rocha et al., 2017). Non-linear effects not only affect the sea surface elevation but also the near-bottom wave orbital velocities and thus play a crucial role in short- and long-term beach morphodynamics (Doering and Bowen, 1986, 1995; Hoefel and Elgar, 2003; Berni et al., 2013; van der Zanden et al., 2017).

Weakly non-linear triad interactions (e.g. Phillips, 1960; Freilich et al., 1984; Elgar and Guza, 1985a) occurring as short waves propagate landward over a sloping bottom are responsible for these changes in the wave field. The interaction of two primary components of frequencies f1 and f2 excite a secondary component f (either sum f1+f2 or difference f1f2), which is bound to the statistically independent primary components. As such, the bound wave component f does not follow the linear wave dispersion relation (Phillips, 1960; Longuet-Higgins and Stewart, 1962; Freilich et al., 1984). As opposed to non-linear resonant interactions between quadruplets in deep water (Hasselmann, 1962), which require very large distances to be effective, non-linear coupling between triads in nearshore areas are non- or near-resonant and can be very efficient in transforming incident wave spectra over just few typical wavelengths (e.g., see Freilich et al., 1984, and the references therein). As both forced (or ‘bound’) and free components of directionally spread seas can co-exist in a wave field, there is no longer a unique relation between a frequency and wavenumber (e.g. Herbers and Guza, 1994). When forced components dominate over a region of the spectrum, large deviations from predictions by the linear wave dispersion relation can be observed in (single-valued) dominant wavenumber and phase velocity spectra (Thornton and Guza, 1982; Freilich et al., 1984; Elgar and Guza, 1985b). In particular near the breaking point or in the surf zone, most wave components of a typical sea-surface spectrum travel at the speed of non-dispersive shallow-water waves (e.g., see Thornton and Guza, 1982; Elgar and Guza, 1985b; Catalán and Haller, 2008; Tissier et al., 2011), which is due to the dominance of amplitude dispersion effects over frequency ones (Herbers et al., 2002). As noted by Laing (1986), the deviations of measured wave phase speed from predictions by the linear dispersion relation, discussed here for nearshore waves, are quite analogous to those observed in growing seas (e.g., see Ramamonjiarisoa and Coantic, 1976; Mitsuyasu et al., 1979; Crawford et al., 1981; Donelan et al., 1985). In such conditions, growing short-wave fields are dominated by modulated trains of finite amplitude waves to which high-frequency components are bound (Lake and Yuen, 1978; Coantic et al., 1981).

In practice, knowledge on the spatial structure of the wave field is generally lacking and the presence of forced energy is therefore difficult to quantify. As forced components at high harmonics are characterized by lower wavenumbers than free components of the same frequency, large errors from depth-inversion algorithms based on the linear wave dispersion relation can be expected in regions where non-linear effects are important (e.g., see Holland, 2001; Brodie et al., 2018). The over-predictions of the dominant wavenumbers at high frequencies also explain the commonly reported ‘blow-up’ when reconstructing the free surface elevation from sub-surface pressure measurements with the linear dispersion relation (Bonneton and Lannes, 2017; Bonneton et al., 2018; Mouragues et al., 2019; Martins et al., 2020b). This is related to the fact that forced high harmonics are much less attenuated across the vertical than free components of the same frequency (e.g. Herbers and Guza, 1991; Herbers et al., 1992). Nonetheless, most field-based studies on non-linear wave transformation in the shoaling region employed sub-surface hydrodynamic data (whether pressure or orbital wave velocity), not corrected or corrected for depth-attenuation using the linear wave dispersion relation. Field-based studies on wave non-linearity also suffer from other limitations such as a poor spatial resolution and the distance over which waves can be studied. In particular, the cross-shore location where non-linear effects at high harmonics become predominant remains largely unknown.

The present paper uses a high-resolution laboratory dataset (GLOBEX, see Ruessink et al., 2013) to study the dispersive properties of irregular waves propagating and breaking over a mildly sloping beach. Besides confirming past findings, the GLOBEX dataset stands out from previously-published field observations for several reasons. The free surface is directly measured with wave gauges and it is highly-resolved in space (several points per wavelength at any stage of propagation). The former aspect removes uncertainties currently existing on energy levels at high harmonics as measured in the field by sub-surface pressure sensors, and where the choice of the surface elevation reconstruction method has a strong influence (Bonneton et al., 2018; Mouragues et al., 2019; Martins et al., 2020b). Furthermore, the experiments considered unidirectional irregular waves, which removes uncertainty about directional effects. In section 2, the high spatial and temporal resolution experimental dataset collected during GLOBEX is briefly presented. Section 3 introduces the cross-spectral and bispectral analysis techniques and describes the weakly non-linear numerical approach employed here for predicting the cross-shore evolution of energy spectra. In Section 4, the cross-spectral analysis is performed on the surface elevation data from adjacent wave gauges to extract dominant wavenumber spectra κ(f), phase velocity spectra c(f) and their evolution across the entire wave flume. From this analysis, we identify four regimes of propagation ranging from a linear up to a surf zone situation, where wavenumber and phase velocity spectra display specific characteristics. In section 5, the bispectral analysis is used to quantify non-linear energy transfers towards harmonics, which play a fundamental role in the patterns observed at high frequencies in wavenumber and phase velocity spectra. The dominant wavenumber is shown to vary with the amount of forced energy at a particular frequency, with larger deviations from the linear wave dispersion expected for higher forced-to-free energy ratios. Finally, section 6 briefly discusses the results and provides the concluding remarks of this study.

Section snippets

Experimental dataset

The Gently sLOping Beach Experiment (GLOBEX) project was performed in a 110-m-long, 1-m-wide, and 1.2-m-high wave flume, located in the Scheldegoot in Delft, the Netherlands (Ruessink et al., 2013). The experiments aimed at collecting high-resolution data of free surface elevation and current velocities in order to study infragravity wave dynamics and short-wave propagation and non-linearities (e.g., see de Bakker et al., 2015; Tissier et al., 2015; Rocha et al., 2017). A combination of 21

Computation of wavenumber and phase velocity spectra

Cross-spectral analysis between adjacent wave gauges is used to compute the dominant wavenumber and phase velocity spectra across the wave flume. As this approach provides phase differences (or delay) between two signals in the frequency domain (e.g., see Ochi, 1998), it has been successfully used in the past to study, in both the laboratory and field, the dispersive properties of ocean waves propagating in deep (e.g., see Ramamonjiarisoa and Coantic, 1976; Mitsuyasu et al., 1979) and

Wavenumber and phase velocity spectra of shoaling and surf zone waves

In this section, we present and describe the main results from the cross-spectral and bispectral analyses. The computation of dominant wavenumber κ and phase velocity c spectra for varying degrees of non-linearity resulted in the identification of four different regimes of propagation, which broadly consist of: a linear regime (stage I), a shoaling regime relatively far from the mean breaking point (stage II), a shoaling regime near the mean breaking point (stage III) and a surf zone regime

Role of non-linear energy transfers on κ and c

Most differences observed in wavenumber and phase velocity spectra between broad and narrow-banded wave conditions concentrate in the shoaling region (stages II and III). In stage II, c values at 2fp and 3fp during A2 lie between c(fp) and the values predicted by linear wave theory while c(fp), c(2fp) and c(3fp) are all equal during A3 (Fig. 5e-f). As these differences between broad versus narrow-band spectra are likely explained by the relative importance of forced energy at those frequencies

Discussion and concluding remarks

Cross-spectral and bispectral analyses were employed on a highly-resolved surface elevation dataset to study the dispersive properties of waves shoaling and breaking over a mildly sloping beach. For all wave tests considered here, four regimes of propagation (I to IV) with specific characteristics in dominant wavenumber and phase velocity spectra could be defined using a local Ursell number. Stage II (Ur~0.3) is particularly interesting as it simultaneously shows significant non-linear effects

CRediT authorship contribution statement

Kévin Martins: Conceptualization, Methodology, Software, Investigation, Writing – original draft, Funding acquisition. Philippe Bonneton: Conceptualization, Investigation, Supervision, Writing – review & editing. Hervé Michallet: Investigation, Writing – review & editing, Funding acquisition.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

. Energy density spectra imposed at a fictive x1 position, with different ratio of forced to free second harmonic amplitude. In the absence of forced components, the forcing corresponds to a JONSWAP spectrum with parameter similar to A3 (see Table 1).

. Wavenumber (a) and phase velocity (b) spectra computed on the synthetic

Acknowledgments

Kévin Martins greatly acknowledges the financial support from the University of Bordeaux, through an International Postdoctoral Grant (Idex, nb. 1024R-5030). The GLOBEX project was supported by the European Community's Seventh Framework Programme through the Hydralab IV project, EC Contract 261520. The GLOBEX data used in this research can be accessed on Zenodo at https://zenodo.org/record/4009405 or from the authors, and can be used under the Creative Commons Attribution 4.0 International

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