Parametric excitation of edge waves with a frequency 2 times less than the frequency of surface waves propagating perpendicular to the inclined bottom is investigated in laboratory experiments. The domain of instability on the plane of surface wave parameters (amplitude–frequency) is found. The subcritical instability is observed in the system of parametrically excited edge waves. It is shown that breaking of surface waves initiates turbulent effects and can suppress the parametric generation of edge waves.

The study of parametric excitation of waves with half of their external
frequency has a long history. The first papers on this subject were published by
M. Faraday, who described excitation of capillary ripples with a frequency of

Analytical solutions for edge waves excited by non-breaking surface waves have been obtained in prior studies (Akylas, 1983; Minzoni and Whitham, 1977; Yeh, 1985; Yang, 1995; Blondeaux and Vittori, 1995; Galletta and Vittori, 2004; Dubinina et al., 2004). The correlation between characteristics of edge waves and spectra of surface waves approaching the shore is studied in situ (Huntley and Bowen, 1978). This kind of study is hard for analysis and interpretation of the results due to the irregularity of the coastline and the complex spectra of the approaching surface waves.

Laboratory experiments on parametric excitation of edge waves are described in Buchan and Pritchard (1995). The main advantage of such experiments is the freedom to define the bottom geometry and spectrum of the approaching surface waves. However, none of the studies mentioned above considered wave breaking, whereas in natural conditions surface waves often break while propagating towards the coastline. Thus, the influence of wave breaking on a parametric instability still remains an open question. In the present paper, we concentrate on the influence of wave breaking on characteristics of parametrically excited edge waves.

The paper is organized as follows. In Sect. 1, we focus on the theoretical description of the problem by providing the nonlinear equation for parametric excitation of edge waves. Section 2 is devoted to the experimental set-up, while Sect. 3 presents the results of measurements. In Sect. 4, we discuss the experimental data with respect to their theoretical interpretation. The main results are summarized in the conclusion.

Let us start from the non-breaking scenario, when long waves propagate over
some changing bottom geometry,

By using two edge waves propagating in opposite directions, it is also
possible to compose a solution corresponding to a standing edge wave:

In the linear approximation waves, Eqs. (6) and (8) are independent. If nonlinear
effects are taken into consideration (Eqs. 1–3), coupling between
the two types of waves takes place. In the first approximation of
nonlinearity, surface waves described by Eq. (8) can generate edge waves
described by Eq. (6) if

The experimental set-up: resistance probes: vertical (P1) and horizontal (P2, P3), a high-speed video camera (2), a wavemaker of a piston type (3), an inclined bottom (4), and the acoustic Doppler velocimeter (ADV).

The experiments have been performed in the wave flume of the Laboratory of
Continental Coastal Morphodynamics of University of Caen Normandy, France. This
flume has a length of 18 m and width of 0.5 m. The flume is equipped with
a piston type of wavemaker controlled by the computer. For construction of
an inclined bottom slope, a PVC plate of 0.01 m thickness has been used.
The plate has been placed at an angle

The first of these, the immobile probe P1, has been placed at a distance of
1 cm from the wavemaker, while probes P2 and P3 have been glued to the
inclined plate. The latter two probes placed along the bottom slope allow us
to measure wave run-up and run-down. In addition, the run-up height can be
identified by image processing from the high-speed camera operating with a
frame rate of 100 Hz (see Fig. 1). The wavemaker oscillating with a given
frequency and amplitude allows us to excite the targeted mode described by
Eq. (8). The wavemaker can work in two regimes. The first regime
controls the amplitude of the wavemaker displacement, while the second one
controls the amplitude of the force applied to the wavemaker. In both
regimes, it is not possible to control the free surface displacement.
Therefore, to study the surface wave characteristics, simultaneous
measurements of a free surface displacement near the wavemaker and the
shoreline have been carried out. For velocity fields (all three components
of the flow velocity), the acoustic Doppler velocimeter (ADV) has been used.
The quality of the signal registered by ADV strongly depends on the
concentration of particles in the liquid. Therefore, in order to get a
better signal, some small particles with a diameter of 10

For visualization of a free surface displacement in the breaking zone by the
high-speed camera, the water has also been seeded with sand particles of
10

Our excitation frequency range was chosen following our published study about the physical simulation of resonant wave run-up on a beach (see Ezersky et al., 2013). In this study, we describe edge waves excited by the third resonant mode of the system.

Example of wave instability developing from a natural
perturbation with

The subharmonic instability described above is investigated in the flume for
different values of (

Power spectrum frequency:

An example of the signals from P2 and P3 is shown in Fig. 2a, whereas a more
detailed zoom of the time series for intervals 50 s <

It can be seen that in the beginning of the record the waves have the same
frequency and phase as the wavemaker (Fig. 2b). However, after instability
arises (Fig. 2c), the amplitude of generated edge wave increases and the
period doubles compared to the period of surface waves. The phase shift
between the signals recorded by probes P2 and P3 is approximately

Snapshots of water surface over the time interval equal to
half of the edge wave period (approximately 1 s),

Subharmonic instability starts with an exponential growth of an infinitely
small perturbation. To describe the instability in the system, partitioning of
a (

Instability occurs if the frequency of surface waves is close to a double
frequency of edge waves. Curve 1 represents a border of the supercritical
instability regime which occurs for points (

The partition of a plane (

Partition of a
(

It is found that while surface wave breaking leads to the appearance of the hydrodynamic turbulence, turbulence itself leads to a decrease in the amplitude of excited edge waves and suppression of subharmonic generation for large-amplitude surface waves.

Dependences of the increment of edge wave instability and intensity of
turbulent velocity fluctuation on the amplitude of surface waves

Parameters of the turbulence are measured by ADV in the middle of the
experimental flume, 0.04 m below the free surface (0.14 m from the bottom),
at a distance of

Visualization of the free surface displacement: 1 indicates the water surface; 2 indicates the inclined bottom; max and min correspond to the maximum and minimum values of the free surface displacement.

Here, we should specify some difficulties related to the characteristic
features of ADV signals. The recorded ADV signals contain the so-called
spikes, which are filtered using the MATLAB algorithm (Nikora and Goring,
1998; Goring and Nikora, 2002). Another problem occurs due to the complex
structure of the velocity field in the breaking zone, which represents a
mixture of turbulence and velocities caused by both surface and edge waves.
In this case, the impact of surface and edge wave components is removed by
filtering harmonics with frequencies

Thus, the range of parameters corresponding to the parametric excitation of
edge waves is found experimentally. Now, using the theoretical Eq. (10),
we can estimate the threshold of parametric excitation of edge waves. For
this, we need to find the eigenfrequencies of edge waves in the flume

Note that while the parametric instability threshold is determined, there was no surface wave breaking, which corresponds to Region II in Fig. 5.

Comparison of experimental and theoretical values of the instability threshold: triangles correspond to the theoretical formula; diamonds represent experimental data.

Figure 7 shows what occurs before the development of the parametric instability, when amplitudes of edge waves are zero. To estimate the surface wave amplitude, the measured crest-to-trough wave height (Fig. 7) is divided by 2. Comparison of the experimental and theoretical values of the instability threshold is shown in Fig. 8. One can see from Fig. 8 that theoretical values are larger than experimental ones by approximately 30 %.

Dependence of wave-propagating energy (

Note that even when the surface wave breaking takes place, the parametric excitation of edge waves still occurs. However, the parametric excitation is suppressed for large amplitudes of surface waves. The reason for this could be the following. The wave breaking results in the irregularity of the surface wave field: amplitudes and phases of the waves vary chaotically. Evidently, wave breaking also leads to the appearance of small-scale turbulence in the near-shore zone. Below, we discuss the impact of these two physical mechanisms on the suppression of the parametric instability.

The parametric wave excitation by the irregular oscillating field has been studied in Ezersky and Matusov (1994) and Nikora et al. (2005). It was shown that chaotic amplitudes and phases of the external wave field lead to an increase in the threshold of parametric excitation and decrease in the amplitude of parametrically excited oscillations.

Let us check whether these results can explain the decrease in the edge wave
amplitude in the presence of the wave breaking. For this, we calculate amplitudes
and phases of surface waves. After narrow-band filtering generated by the
wavemaker, surface waves may be described as

Wave breaking generates turbulence, and the intensity of turbulent velocity
fluctuations grows with the surface wave amplitude. On the other hand,
turbulence leads to the appearance of turbulent shear stresses and eddy
viscosity

The eddy viscosity

Since the external forcing

From the measurements, to calculate the energy dissipation rate, we have
study the energy of wave propagating in the flume. We compare wave energy
near the wavemaker with wave energy at shore. The wave energy (energy on a
unit length in the direction transversal to the direction of wave
propagation) is estimated as follows:

The typical dependence of E2 (energy at shore) on E1 (energy near the wavemaker)
is shown in Fig. 10 for different amplitudes of excitation for

The parametric edge wave excitation is studied for different regimes of surface wave propagation. We have found that for parametrically excited edge waves there is a region of subcritical instability, which is manifested by the hysteresis: different regimes of edge wave excitation are observed in the case of decrease or increase in the surface wave amplitude. Note that subcritical instability was not observed in Buchan and Pritchard (1995), though their experimental conditions were very close to those in our experiment.

The increase in the surface wave amplitude leads to the appearance of wave breaking. The wave breaking regime itself does not prevent parametric excitation of edge waves; only the developed wave breaking can suppress parametric excitation of edge waves. We compare the two possible mechanisms of the parametric instability suppression: (i) phase irregularity of the external forcing and (ii) generation of the hydrodynamic turbulence. We have found that the most probable mechanism responsible for the increase of the parametric instability threshold and suppression of parametric excitation of edge waves is the hydrodynamic turbulence which appears as a result of wave breaking.

The data used by this study are experimental. The data are freely available but not otherwise published in any publicly accessible database. The experimental data can nonetheless be provided upon request via email to the corresponding author, Nizar Abcha (nizar.abcha@unicaen.fr).

The authors declare that they have no conflict of interest.

This work is dedicated to Alexander Ezersky, who was the key author and the main driver of this study. Last summer, he sadly passed away after a long-lasting fight with cancer, leaving the manuscript unfinished. Until his last days, he tried to dedicate his time to work, including the results presented here. Therefore, it is important for us to conclude his work in memory of a dear friend and colleague.

The present study was supported by Russian Presidential grants MD-6373.2016.5 and NS-6637.2016.5. Ira Didenkulova and Efim Pelinovsky also thank the visitor programme of the University of Caen Normandy, which allowed this fruitful collaboration. Edited by: R. Grimshaw Reviewed by: two anonymous referees