NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-24-157-2017Subharmonic resonant excitation of edge waves by breaking surface wavesAbchaNizarnizar.abcha@unicaen.frZhangTongleiEzerskyAlexanderPelinovskyEfimhttps://orcid.org/0000-0002-5092-0302DidenkulovaIrahttps://orcid.org/0000-0003-0913-9167Morphodynamique Continentale et Côtière UMR6143, CNRS,
Normandie Univ, UNICAEN, 14000 Caen, FranceNizhny Novgorod State Technical University n.a. R.E. Alekseev, 24
Minin Str., Nizhny Novgorod 603950, RussiaInstitute of Applied Physics, 46 Uljanov Str., Nizhny Novgorod 603950,
RussiaNational Research University, Higher School of Economics, Nizhny
Novgorod 603950, RussiaMarine Systems Institute at Tallinn University of Technology,
Akadeemia tee 15A, 12618 Tallinn, EstoniaNizar Abcha (nizar.abcha@unicaen.fr)28March201724215716524October20162November201619February201727February2017This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/24/157/2017/npg-24-157-2017.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/24/157/2017/npg-24-157-2017.pdf
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.
Introduction
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
Ω/2 in a thin horizontal layer of viscous fluid placed on a
horizontal plate oscillating vertically with a frequency of Ω (Faraday,
1831). After Faraday, such parametric excitation of waves was observed in
hydrodynamics (Douady, 1990; Cerda and Tirapegui, 1998), plasma physics
(Okutani, 1967; Kato et al., 1965), chemically active media
(Fermandez-Garcia et al., 2008), and other systems. Such parametric excitation also
occurs in the ocean. Surface waves approaching the shore from the open sea
with a frequency of Ω can excite the so-called edge waves with a
frequency of Ω/2. Edge waves propagate along the coastline with their
amplitudes decreasing in the offshore direction (Ursell, 1952; Grimshaw, 1974;
Guza and Davis, 1974; Evans and McIver, 1984; Johnson, 2005, 2007). Interest
in parametrically excited edge waves is related to their ability to
significantly affect morphological characteristics of sea coasts. Edge waves
may contain enough energy to be responsible for beach erosion. They may also
focus, forming a freak wave (Pelinovsky et al., 2010). Sometimes, edge waves
are also associated with beach cusp formation (Guza and Imman, 1975; Komar,
1998; Masselink, 1999; Dodd et al., 2008; Coco and Murray, 2007).
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.
Theoretical model
Let us start from the non-breaking scenario, when long waves propagate over
some changing bottom geometry, h=h(x). In this
case, they can be described by 2-D nonlinear shallow water equations:
∂u∂t+u∂u∂x+v∂u∂y+g∂η∂x=0,∂u∂t+u∂v∂x+v∂v∂y+g∂η∂y=0,∂η∂t+∂∂xuh+η+∂∂yvh+η=0,
where (u,v) are the two components of the depth-averaged
horizontal velocity, η=η(x,y,t) is the free surface displacement, and g is the
gravity acceleration. In a linear approximation, the system Eqs. (1)–(3) can be
transformed into a 2-D wave equation:
∂2η∂t2-g∂∂xh(x)∂η∂x+h(x)∂2η∂y2=0.
Note that Eq. (4) describes both surface waves propagating
perpendicular to the shore and generated edge waves. For edge waves, we
assume that they propagate along the shore and consider a linear change of
the bottom slope h(x)=βx= tan αx. In this case, an elementary
solution of Eq. (4) has the following form:
η=bcos(Ωnt-ky)⋅e-kxLn(x),Ωn=(2n+1)βgk,n=0,1,2,…
where Ln are the Laguerre polynomials, b is a wave
amplitude, k is a wave number along the propagation direction,
Ω is a wave frequency, and n is the number of the mode.
By using two edge waves propagating in opposite directions, it is also
possible to compose a solution corresponding to a standing edge wave:
η=bcos(Ωn,mt)sin(kmy)Ln(x),km=π(1+2m)/L,Ωn,m=(2n+1)βgkm,m=0,1,2,…
Here, we used the boundary conditions v(x,y,t)=0 at
y=±L/2, where L is a channel width.
For surface waves propagating perpendicular to the shore, Eq. (4)
transforms into a 1-D wave equation,
∂2η∂t2-g∂∂xh(x)∂η∂x=0,
and has a solution:
ηx,t=a0J04ω3xgβcosωt,
where J0 is the Bessel function of the first kind,
ω is a frequency, and a0 is an amplitude of
the generated surface waves.
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 Ω≈ω/2. It is the so-called
parametric subharmonic resonance. In this case, we can write down the
equation for slowly varying wave amplitude b of the excited edge
waves with frequency Ω (Rabinovich et al., 2000):
∂b∂t=-γb+μb∗+iΔb+(iσ-ρ)bb2.
Here, γ represents an exponential decay of edge waves due to
the viscous dissipation, Δ=Ω-ω/2 is a
detuning between frequencies of edge waves and the external parametric
forcing, σ is a nonlinear frequency shift, ρ
is a nonlinear damping coefficient, and b* is a complex conjugate. This
equation was initially obtained for Faraday ripples excited by a homogeneous
oscillating field. For edge waves excited by surface waves propagating
perpendicular to the shore, an expression for a coefficient μ has been obtained in previous studies (Akylas, 1983; Minzoni and Whitham, 1977; Yang, 1995):
μ=a0ω34gβ2S(β).
Here, S is a coefficient depending on a bottom slope
α. For small slopes α, S≈6.7×10-2. The nonlinear frequency shift σ
has been calculated in Minzoni and Whitham (1977). The nonlinear damping
coefficient ρ has been discussed in Yang (1995).
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).
Experimental set-up
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 α to the
horizontal bottom so that tan α=β= 0.20; the water depth in the flume, h, has been kept at 0.25 m
(see Fig. 1). As one can see from Fig. 1, in this geometric configuration
only a small part of the flume can be used for experiments. Three resistance
probes – P1, P2, P3 (see Fig. 1) – have been used to measure the water surface
displacement.
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 µm have
been added into the water.
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 µm. Using a vertical light sheet (photodiode 532 nm with a
cylindrical lens), it has been possible to visualize the cross section of the
water in the x-z plane. The size of the visualization domain is 40 cm × 30 cm.
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 f= 1.08 Hz,
aL= 0.66 cm: (a) the
full time series recorded by probes P2 and P3; (b) zoom of the time series
recorded during the time interval 50 s < t < 55 s, and (c) during the time
interval 85 s < t < 90 s.
Data processing and results
The subharmonic instability described above is investigated in the flume for
different values of (aL, f), where aL
is an amplitude of surface waves in the vicinity of the wavemaker,
aL≈a0, and f is the
frequency of the wavemaker. In order to understand whether the instability
really occurs, we analyse the signals from probes P2 and P3. Before each
experiment, we wait for 5–10 min to let all the perturbation
in the flume decay and let the wavemaker work in calm water conditions.
Power spectrum frequency: (a) in the absence of breaking
waves, the first peak indicates the edge wave frequency, while the second
peak indicates the surface elevation frequency; (b) in the presence of breaking
waves, the peak for the edge wave frequency is suppressed.
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 < t < 95 s and 85 s < t < 90 s
is given in Fig. 2b and c, respectively. The power frequency spectra
for two surface wave regimes (with and without wave breaking) are shown in
Fig. 3. The first spectrum (Fig. 3a) is the fast Fourier transform (FFT) of the signal shown in
Fig. 2a. This is a spectrum in the absence of wave breaking, where the first
peak indicates the edge wave frequency and the second peak indicates the
surface elevation frequency. The second frequency spectrum (Fig. 3b) is
plotted in the presence of breaking waves and indicates the suppression of the
peak for the edge wave frequency.
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 π.
These two criteria (period doubling and a phase shift equal to π) are
used to identify parametric instability. To confirm an appearance of edge
waves as a result of subharmonic instability, we analyse the water level
oscillations. It is found that subharmonic oscillations represent the mode,
where maxima of horizontal displacement (antinodes) occur near the lateral
walls of the flume, while its zeroes (nodes) are observed in the middle of
the flume. This mode is a superposition of two edge waves propagating in
opposite directions. A spatial period of these edge waves is 2 times larger
than the width of the flume. Snapshots of water surface over the time
interval equal to half of the edge wave period are shown in Fig. 4.
Snapshots of water surface over the time interval equal to
half of the edge wave period (approximately 1 s),
f= 1.06 Hz, and
aL= 1.3 cm.
Subharmonic instability starts with an exponential growth of an infinitely
small perturbation. To describe the instability in the system, partitioning of
a (aL, f) plane into different stability regions is
performed. Results of this analysis are demonstrated in Fig. 5.
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 (aL, f)
above this curve. If the amplitude of surface waves decreases from a finite
value above Curve 1, generation of edge waves is observed in a small region
(3) between Curves 1 and 2 (see triangles in Fig. 5). When we start from
the regime without edge wave generation (points below Curve 2) and increase
the amplitude of surface waves, instability will occur above Curve 1. This
type of instability is called subcritical instability.
The partition of a plane (aL, f) into regions with
different regimes shown in Fig. 5 corresponds to two qualitatively different
conditions of wave excitation schematically shown by boxes (I) and (II). In
Region I, surface waves excited by the wavemaker and propagating to the
shore undergo a plunging wave breaking. In Region II, waves do not break.
Image processing of the high-speed camera data shows that such excitation
occurs only when the wave breaking parameter Br > 0.9. Under the wave breaking parameter, we mean Br =Umax2/ gR,
where Umax is the maximal flow velocity and R is the maximal wave run-up height on the shore (Didenkulova, 2009).
Partition of a
(aL, f)
plane into different stability regions of the system; circles correspond to
a parametric instability, diamonds correspond to stability regimes, and
triangles indicate the regime of subcritical instability.
(a) Dependence of the exponential index of parametric
instability γ on the
surface wave amplitude aL, shown by
the black dots, and (b) dependence of the kinetic turbulent energy
components on the surface wave amplitude
aL;
Vx is shown by blue diamonds, while
Vy is shown by black squares. Solid
lines represent a fit to the experimental data.
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
aL are shown in Fig. 6a and b. The dependence of the
exponential index γ on the amplitude of surface
waves aL is found by processing corresponding time series
similar to those shown in Fig. 2a. For this, we select time intervals where
the edge wave amplitude grows and calculate γ by
exponential approximation of the time-dependent amplitude.
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 x= 0.9 m from the shoreline. At this point, it is
possible to neglect the turbulence caused by the near-bottom oscillating
boundary layer and detect the wave breaking turbulence.
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 f/2, f,
3f/2, 2f, 5f/2, and 3f. It is shown that the
intensity of turbulence grows sufficiently if the amplitude of surface waves
aL is larger than 0.8 cm (see Fig. 6b).
Discussion
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
Ωn. The frequency of the zero edge wave mode Ω0 is as
follows:
Ω0=βgπL=3.41rads-1,f0≈0.54Hz.
To estimate the dissipation rate of edge waves, we study the time evolution
of the edge wave amplitude after stopping the parametric excitation. Edge
waves decay exponentially and in this way we measure the decay rate
γ, which is estimated as γ= 0.1 s-1. For the resonance condition Δ= 0, parametric
instability occurs when the wave amplitude exceeds the critical wave
amplitude a0:
a0=γ4gβ2ω3S(β)≈0.76cm.
The theoretical value of the parametric instability threshold is calculated
using the free surface displacement. To compare experimental and theoretical
values of the threshold, we need to measure the surface wave amplitude at
x= 0. As it has been noted in several studies (see, for example,
Denissenko et al., 2011), this value can be measured indirectly. We find it
using the visualization of the flow in the middle of the flume by the laser
sheet at a time preceding the development of the edge wave parametric
instability (see Fig. 7).
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 %.
(a) Time series measured by P1 with
aL= 1 cm,
f= 1.06 Hz; (b) non-dimensional wave
amplitude and phase obtained by the Hilbert transformation; (c) power
spectrum of the signal (a) in log–log scale.
Dependence of wave-propagating energy (f= 1.06 Hz) E2 (energy at shore) on E1 (energy near the
wavemaker) for different amplitudes of excitation aL.
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 ηm cos(ωt+Φ), where ηm is a slow varying
amplitude and Φ is a slow varying phase. To
extract the amplitude and the phase of the signal, the Hilbert
transformation is used:
η^(t)=1πPV∫-∞+∞η(t,τ)t-τdτ=ηmsin(ωt+Φ),
where PV denotes the principal value of the integral. It is also
possible to determine the wave amplitude and phase:
η(t)=Rea(t)exp(iωt,a(t)=aeiΦ,
where
a=η2+η^2,Φ=arctan(η^/η)-ωt.
Extracted amplitudes and phases for the time series measured in the presence of
the surface wave breaking are shown in Fig. 9. The time series itself is
given in Fig. 9a, while the extracted amplitudes and phases are shown in
Fig. 9b. The root mean square of phase and amplitude fluctuations for the
intensive wave breaking (a= 1.4 cm) is
Φ2≈0.1,(a-a)2a≈0.1.
It is also possible to estimate the influence of chaotic phases and
amplitudes on the parametric wave excitation. It has been revealed that
chaotic phases decrease the effective amplitude of the external force
(Petrelis et al., 2005). Suppose that the wave breaking leads to the
Gaussian noise; then, the corresponding decrease in the external forcing may
be estimated as (Petrelis et al., 2005)
e-(Φ2/2≈0.995.
This small decrease in the effective external forcing cannot explain
suppression of the parametric excitation during the wave breaking regime;
therefore, the influence of the turbulence seems to be more important.
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 νed. We measure experimentally some components of the
kinematic turbulent energy at the edge wave background (see Fig. 6b).
According to our measurements, the most important components of shear
stresses are related to the longitudinal component of turbulent fluctuations
Vx (see Fig. 6b).
The eddy viscosity νed is proportional to the turbulent
energy. For the wave breaking case, one can consider νed to be
proportional to a2 (see Fig. 6b). In this case, the
exponential decay γ in Eq. (9) has the following form:
γ=γ0+γ1a2, where γ0 is the exponential
decay of edge waves in the absence of wave breaking, and γ1 is responsible for energy dissipation due to the eddy viscosity.
Since the external forcing μ grows linearly with the surface
wave amplitude and the dissipation is proportional to the amplitude squared,
the parametric instability is suppressed for large surface wave amplitudes.
We observe this effect in our experiment under the surface wave breaking
regime.
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:
E=ρg2Cgr∫(η-<η¯>)2dt,
where Cgr=dωdK is the group velocity of the harmonic
component corresponding to the peak frequency f; g is
acceleration of gravity; ρ is water density; and η and
<η¯> are free surface displacement and mean water level,
respectively.
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
f= 1.06 Hz. We observe that evolution of dependence follows a law
of power. An energy dissipation of the order of 25 % occurs in the absence of the
edge waves. These losses are caused by viscous dissipation and contact line
damping. In the presence of edge waves, the energy dissipation can reach
50 %.
Conclusions
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.
Acknowledgements
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
References
Akylas, T. R.: Large-scale modulations of edge waves, J. Fluid Mech., 132,
197–208, 1983.
Blondeaux, P. and Vittori, G.: The nonlinear excitation of synchronous edge
waves by a monochromatic wave normally approaching a plane beach, J. Fluid
Mech., 301, 251–268, 1995.
Buchan, S. J. and Pritchard, W. G.: Experimental observations of edge
waves, J. Fluid Mech., 288, 1–35, 1995.
Cerda, E. A. and Tirapegui, E. L.: Faraday's instability in viscous fluid,
J. Fluid Mech., 368, 195–228, 1998.
Coco, G. and Murray, B. A.: Pattern in the sand: form forcing templates
to self-organization, Geomorphology, 91, 271–290, 2007.Denissenko, P., Didenkulova, I., Pelinovsky, E., and Pearson, J.: Influence of the nonlinearity on
statistical characteristics of long wave runup, Nonlin. Processes Geophys., 18, 967–975, 10.5194/npg-18-967-2011, 2011.
Didenkulova, I.: New trends in the analytical theory of long sea wave runup,
in: Applied Wave Mathematics: Selected Topics in Solids, Fluids, and
Mathematical Methods, Springer, 265–296, 2009.
Dodd, N., Stoker, A. M., Calvete, D., and Sriariyawat,A.: On beach cusp
formation, J. Fluid Mech., 597, 145–169, 2008.
Douady, S.: Experimental study of the Faraday instability, J. Fluid Mech.,
221, 383–409, 1990.
Dubinina, V. A., Kurkin, A. A., Pelinovsky, E. N., and Poloukhina, O. E.: Weakly
nonlinear periodic Stokes edge waves, Izvestiya, Atmos. Ocean. Phys., 40, 464–469, 2004.
Evans, D. V. and McIver, P.: Edge waves over a shelf: full linear theory, J.
Fluid Mech., 142, 79–95, 1984.
Ezersky, A. B. and Matusov, P. A.: Time-space chaos of capillary waves
parametrically excited by noise, Radiophys. Quantum El., 37,
828–836, 1994.Ezersky, A., Abcha, N., and Pelinovsky, E.: Physical simulation of resonant wave run-up on a beach,
Nonlin. Processes Geophys., 20, 35–40, 10.5194/npg-20-35-2013, 2013.
Faraday, M.: On the forms and states of fluids on vibrating elastic
surfaces, Philos. T. R. Soc. Lond. 52, 319–340, 1831.Fernández-García G., Roncaglia D. I., Pérez-Villar V.
Muñuzuri A. P, and Pérez-Muñuzuri V.: Chemical-wave dynamics in
a vertically oscillating fluid layer, Phys. Rev. E, 7, 026204, 10.1103/PhysRevE.77.026204, 2008.
Galletta, V., and Vittori, G.: Nonlinear effects on edge wave development,
Eur. J. Mech. B-Fluid. 23, 861–878, 2004.
Goring, D. and Nikora, V.: Despiking acoustic Doppler velocimeter, J.
Hydraul. Eng., 128, 117–126, 2002.
Grimshaw, R.: Edge waves: a long wave theory for oceans of finite depth, J.
Fluid Mech., 62, 775–791, 1974.
Guza, R. T. and Davis, R. E.: Excitation of edge waves bywaves incident on a
beach, J. Geophys. Res., 79, 1285–1291, 1974.
Guza, R. T. and Inman, D. L.: Edge waves and beach cusps, J. Geophys. Res.,
80, 2997–3012, 1975.
Huntley, D. A. and Bowen, A. J.: Beach cusps and edge waves, Proc 16th
Conf. Coastal, 1378–1393, 1980.
Johnson, R. S.: Some contributions to the theory of edge waves, J. Fluid
Mech., 524, 81–97, 2005
Johnson, R. S.: Edge waves: theories past and present, Philos. T. R. Soc.
A, 365, 2359–2376, 2007.
Kato, K., Yoseli, M., and Kiyama, S.: Excitation of plasma oscillation by
parametric resonance, J. Phys. Soc. Jpn., 20, 2097–2098, 1965.
Komar, P.: Beach Processes and Sedimentation, Prentice Hall, New York, 1998.
Masselink, G.: Alongshore variation in beach cusp morphology in a coastal
embayment, Earth Surf. Proc. Land., 24, 335–347, 1999.
Minzoni, A. A. and Whitham, G. B.: On the excitation of edge waves on
beaches, J. Fluid Mech., 79, 273–287, 1977.
Nikora, V. and Goring, D.: ADV measurements of turbulence: Can we improve
their interpretation?, J. Hydraul. Eng., 124, 630–634, 1998.
Okutani, J.: Excitation of plasma oscillations by parametric resonance, J.
Phys. Soc. Jpn., 23, 110–113, 1967.
Pelinovsky, E., Polukhina, O., and Kurkin, A.: Rogue edge waves in the
ocean, Eur. Phys. J.-Spec. Top., 185, 35–44, 2010.Petrelis, F., Aumaıtre, S., and Fauve, S.: Effect of Phase Noise on
Parametric Instabilities, Phys. Rev. Lett. 94, 070603, 10.1103/PhysRevLett.94.070603, 2005.
Rabinovich, M. I., Ezersky, A. B., and Weidman, P. D.: The Dynamics of
Pattern,
World Sci., Singapore, 2000.
Ursell, F.: Edge waves on a sloping beach, P. R. Soc. A, 214, 79–97,
1952.Yang, J.: The stability and nonlinear evolution of edge wave, Stud.
Appl. Math., 95, 229–246, 1995.
Yeh, H.: Nonlinear progressive edge waves: their instability and evolution,
J. Fluid Mech. 152, 479–499, 1985.