Deep-water Waves Research Articles

Overtopping Metrics and Coastal Safety: A Case of Study from the Catalan Coast

Design criteria for coastal defenses exposed to wave overtopping are usually assessed by mean overtopping discharges and maximum individual overtopping volumes. However, it is often difficult to give clear and precise limits of tolerable overtopping for all kinds of layouts. A few studies analyzed the relationship between wave overtopping flows and hazard levels for people on sea dikes, confirming that one single value of admissible mean discharge or individual overtopping volume is not a sufficient indicator of the hazard, but detailed characterization of flow velocities and depths is required. This work presents the results of an experimental campaign aiming at analyzing the validity of the safety limits and design criteria for overtopping discharge applied to an urbanized stretch of the Catalan coast, exposed to significant overtopping events every stormy season. The work compares different safety criteria for pedestrians. The results prove that the safety of pedestrians on a sea dike can be still guaranteed, even for overtopping volumes larger than 1,000 L/m. Sea storms characterized by deep-water wave height between 3.6 and 4.5 m lead to overtopping flow depth values larger than 1 m and flow velocities up to 20 m/s. However, pedestrian hazard is proved to be linked to the combination of overtopping flow velocity and flow depth rather than to single maximum values of one of these parameters. The use of stability curves to assess people’s stability under overtopping waves is therefore advised.

Three-dimensional sound scattering from transversely symmetric surface waves in deep and shallow water using the equivalent source method.

This paper proposes a propagation model to calculate the three-dimensional (3-D) sound scattering from transversely symmetric sea surface waves in both deep and shallow water using the equivalent source method (ESM). The 3-D sound field is calculated by integrating an assembly of two-dimensional (2-D) transformed fields with different out-of-plane wavenumbers through a cosine transform. Each 2-D solution is calculated using the ESM incorporating a complex image method that can efficiently and accurately solve the 2-D water/seabed Green's function. The oscillatory cosine integral is accurately calculated using a segmented integral scheme requiring relatively few 2-D solutions, which can be further improved through the use of parallel computation. The model is validated by comparison with a 3-D Helmholtz-Kirchhoff method for deep water and a finite element method for a shallow water wedge with both a fluid and an elastic seabed. The model is as accurate as the finite element approach but more numerically efficient, which enables Monte Carlo simulations to be performed for random rough surfaces in order to study the scattering effects at a reasonable computational cost. Also, 3-D pulse propagation in the shallow water wedge is demonstrated to understand the out-of-plane scattering effects further.

Lie Symmetry Analysis on Benjamin-Ono Equation

Benjamin-Ono Equation are significantly important in describes the one-dimensional internal waves in deep water. Because of its significance, Lie symmetry reduction were chosen to reduce the equation and hence solve the equation. Lie symmetry analysis is one of the powerful methods to solve partial differential equation. Due to its effectiveness, this method is widely applied in solving equation in various field. In this paper, calculation of symmetry of the equation was first present, followed by reduction of the equation. The equation are reduced from non-linear partial differential equation (PDE) to ordinary differential equation (ODE) and hence analytic solution of the partial differential equation was obtained by solving the reduced ODE.

Beach morphodynamic characteristics and classifications on the straight coastal sectors in the west Guangdong

Currently beach morphodynamic classification is the most important foundation to conduct associated coastal geomorphological studies. This paper carried out beach morphodynamic classifications for 12 straight beaches on headland-bay coasts based on field survey and evaluated the applicability of the most widely used dimensional fall velocity parameter (Ω) and relative tidal range parameter (RTR). One reflective, five intermediate and six non-barred dissipative beaches were visually classified and sand size seemed to be a key factor to differentiate these beaches. The studied beaches were in relatively low wave energy environments (Hs < 1 m) and the absolute deep-water wave energy level of P0 = 3 KWm−1 was supposed to a critical threshold to characterize the applicability of the Ω and RTR parameters. These two morphodynamic parameters were applicable for the beaches with P0 &gt; 3 KWm−1 and MSR &lt; 2 m. It was found that the model of the traditional winter-and-summer profiles was not applicable in the study area in despite of distinct wave seasonality. The studied beaches were more possible to hover around a limited range due to relatively low background wave environments and variability without considering typhoon impacts, which needs further research on actual breaker wave conditions and beach morphodynamic type responses to typhoon events.

Numerical simulations of collinear finite amplitude steady-state resonant waves in deep water

In this work, numerical simulations of steady-state resonant wave system have been conducted by high-order spectral method (HOS) to study its evolution mechanism in deep water. Convergent high-order series solutions of steady-state resonant waves are first obtained by homotopy analysis method (HAM). Theoretical solutions together with disturbances of different orders of magnitude are then served as the initial solutions in HOS. It is found that as more accurate wave components are generated at the initial stage of the numerical simulation, the time that amplitude of the two largest wave components keeps unchanged increases. Steady-state resonant waves with time-independent spectra can be obtained if sufficient number of wave components are generated at the initial stage. At the end of simulation, additional new wave components that have not been considered at the initial stage appear in the spectra due to four-wave resonant interactions. For steady-state resonant waves with random disturbances, the numerical simulations confirm again the existence of steady-state resonant waves. Besides, energy transfer among different components is more remarkable for steady-state resonant waves with small disturbances before the wave breaking.

Effects of coherent dynamics of stochastic deep-water waves.

A method of windowed spatiotemporal spectral filtering is proposed to segregate different nonlinear wave components and to calculate the surface of free waves. The dynamic kurtosis (i.e., produced by the free wave component) is shown able to contribute essentially to the abnormally large values of the surface displacement kurtosis, according to the direct numerical simulations of realistic sea waves. In this situation the free wave stochastic dynamics is strongly non-Gaussian, and the kinetic equation for sea surface waves fails. Traces of coherent wave patterns are found in the Fourier transform of the directional irregular sea waves; they may form "jets" in the Fourier domain which strongly violate the classic dispersion relation.

An experimental comparison of the velocities and energies of focused spilling waves in deep water

It is important to achieve a better understanding of the wave energy variations occurring as a steep wave evolves towards breaking. Laboratory experiments of focused waves are performed to investigate the kinematics of primary breaking waves. When a primary wave evolves towards breaking, the increase in the maximum kinetic energy is stronger than the increase in the maximum potential energy. Furthermore, the accuracy of the average energy of the wave train analyzed by the wave spectrum is discussed. At measuring locations far from the breaking onset point, the average energy of the wave train analyzed by the wave spectrum is slightly larger (0.6–2.5%) than the average energy calculated by the measured free-surface elevation and fluid velocities. In contrast, at measuring locations close to the breaking point, the nonlinearity of the wave train is strong and the average energy analyzed by the wave spectrum underestimates the measured wave energy. The error ranges of the total energy loss rate of deep-water wave breaking calculated by wave spectrum analysis are 1.5–3.79% and ±1.9% for measuring locations close to and far from the breaking point, respectively. Although the absolute value of the error is not prominent, the error range is not negligible relative to the energy loss rate of a single spilling wave. Finally, a wave index ϵ ∗ (fs/fp) is proposed to reveal the characteristics of the wave energy variations during the incipient breaking stage under different wave conditions.

Theory of deep-water surface gravity waves derived from a Lagrangian

Theory of deep-water surface gravity waves derived from a Lagrangian

Assessment of Shallow Water Random Wave-Induced Scour at the Trunk Section of Breakwaters using Deep Water Wind and Wave Conditions

Myrhaug, D. and Ong, M.C., 2020. Assessment of shallow water random wave-induced scour at the trunk section of breakwaters using deep water wind and wave conditions. In: Malvarez, G. and Navas, F. (eds.), Global Coastal Issues of 2020. Journal of Coastal Research, Special Issue No. 95, pp. 289–293. Coconut Creek (Florida), ISSN 0749-0208This article provides a simple analytical method giving estimates of random wave-induced scour at the trunk section of vertical-wall and rubble-mound breakwaters in shallow water from deep water wind and wave conditions. Results are exemplified by using a Pierson-Moskowitz model wave spectrum for deep water wind waves with the mean wind speed at the 10 m elevation above the sea surface as the parameter. The significant value of the scour depth within a sea state of random waves is provided and an example typical for field conditions is given. The method should serve as a useful tool for assessing shallow water random wave-induced scour based on input from deep water wind and wave conditions.

Tropical Cyclone Induced Wave Setup around New Caledonia during Cyclone COOK (2017)

Jullien, S.; Aucan, J.; Lefevre, J.; Peltier, A., and Menkes, C.E., 2020. Tropical cyclone induced wave setup around New Caledonia during cyclone COOK (2017). In: Malvarez, G. and Navas, F. (eds.), Global Coastal Issues of 2020. Journal of Coastal Research, Special Issue No. 95, pp. 1454-1459. Coconut Creek (Florida), ISSN 0749-0208.Tropical cyclone (TC)-induced coastal flooding arises from a combination of atmospheric surge, and energetic wave-induced setup. TC-generated waves in deep water considerably evolve in shallow water before reaching the coast. Islands surrounded by a barrier reef and lagoon are a particular case of steep and irregular bathymetry over which waves undergo various transformations, and induce complex wave-current interactions. The present study investigates the TC-induced coastal impact on the reef environment of New Caledonia in the tropical southwest Pacific. An unstructured wave-current model simulation encompassing New Caledonia and Vanuatu is used to describe the complete TC-induced waves impact, from wave generation to propagation and transformation in the coastal area. Comparison with in-situ measurements are performed during cyclone COOK in 2017. The results show that coastlines with a double protection (barrier and fringing reefs) are relatively sheltered from the waves induced by TCs (significant wave height is 3 m at the coast vs. 13 m offshore) while shores fronting the reef passes are more severely exposed with waves reaching 9 m close to the coast. Setups and currents that are associated to wave breaking on the barrier reef induce strong lagoon currents and localized sea level anomalies that does not necessarily reach the coast. In more sheltered bays, wave breaking at the entrance of the bay acts against outgoing currents. This phenomenon leads to an additional elevation of the mean water level in the bay.

New solutions of the C.S.Y. equation reveal increases in freak wave occurrence

In this article we study the time evolution of broad banded, random inhomogeneous fields of deep water waves. Our study is based on solutions of the equation derived by Crawford, Saffman and Yuen in 1980, (Crawford et al., 1980). Our main result is that there is a significant increase in the probability of freak wave occurrence than that predicted from the Rayleigh distribution. This result follows from the investigation of three related aspects. First, we study the instability of JONSWAP spectra to inhomogeneous disturbances whereby establishing a wider instability region than that predicted by Alber’s equation. Second, we study the long time evolution of such instabilities. We observe that, during the evolution, the variance of the free surface elevation and thus, the energy in the wave field, localizes in regions of space and time. Last, we compute the probabilities of encountering freak waves and compare it with predictions obtained from Alber’s equation and the Rayleigh distribution.

Soliton Turbulence in Approximate and Exact Models for Deep Water Waves

We investigate and compare soliton turbulence appearing as a result of modulational instability of the homogeneous wave train in three nonlinear models for surface gravity waves: the nonlinear Schrödinger equation, the super compact Zakharov equation, and the fully nonlinear equations written in conformal variables. We show that even at a low level of energy and average wave steepness, the wave dynamics in the nonlinear Schrödinger equation fundamentally differ from the dynamics in more accurate models. We study energy losses of wind waves due to their breaking for large values of total energy in the super compact Zakharov equation and in the exact equations and show that in both models, the wave system loses 50% of energy very slowly, during few days.

Multiple Soliton Interactions on the Surface of Deep Water

The paper presents the long-time dynamics with multiple collisions of breathers in the super compact Zakharov equation for unidirectional deep water waves. Solutions in the form of breathers were found numerically by the Petviashvili method. In the terms of envelope and the assumption of the narrow spectral width the super compact equation turns into the well known exact integrable model—nonlinear Schrödinger equation, and the breather solution in this case turns into envelope soliton. The results of numerical simulations show that two main scenarios of long-time dynamics occur during numerous collisions of breathers. In the first case, one of the breathers regularly takes a number of particles from the other one at each collision and in the second one a structure resembling the bi-soliton solution of nonlinear Schrödinger equation arises during the collision. Despite these scenarios, it is shown that after numerous collisions the only one breather having initially a larger number of particles remains.

Wave–vortex interactions and effective mean forces: three basic problems

Three examples of wave–vortex interaction are studied, in analytically tractable weak refraction regimes with attention to the mean recoil forces, local and remote, that are associated with refractive changes in wave pseudomomentum fluxes. Wave-induced mean forces of this kind can be persistent, with cumulative effects, even in the absence of wave dissipation. In each example, a single wavetrain propagates past a single vortex. In the first two examples, in a two-dimensional, non-rotating acoustic or shallow-water setting, the focus is on whether or not the wavetrain overlaps the vortex core. In the overlapping case, the recoil has a local contribution given by the Craik–Leibovich force on the vortex core, the vector product of Stokes drift and mean vorticity. (For a quantum vortex this contribution is called the Iordanskii force arising from the Aharonov–Bohm effect on a phonon current.) However, in all except one special limiting case there are additional “remote” contributions, mediated by Stokes-drift-induced return flows that can intersect the vortex core well away from locations where the waves are refracted. The third example is a non-overlapping, remote-recoil-only example in a rapidly rotating frame, in which the waves are deep-water gravity waves and the mean flow obeys shallow-water quasigeostrophic dynamics. Contrary to what might at first be thought, the Ursell “anti-Stokes flow” induced by the rotation – an Eulerian-mean flow tending to cancel the Stokes drift – fails to suppress remote recoil. There are nontrivial open questions about extending these results to regimes of stronger refraction, especially regarding the scope of the “pseudomomentum rule” for the wave-induced recoil forces.

Practical flow-representations for arbitrary singularity-distributions in ship and offshore hydrodynamics, with applications to steady ship waves and wave diffraction-radiation by offshore structures

Diffraction-radiation of regular waves by offshore structures and flows around ships advancing in calm water or in regular waves are commonly analyzed via potential-flow methods based on the Green functions that satisfy the corresponding free-surface boundary conditions. This realistic, practical, and widely-used approach requires evaluation of free-surface flows due to arbitrary (notably constant, linear or quadratic) distributions of singularities (sources and dipoles) over (flat or curved) panels of various shapes (notably rectangles and triangles). Indeed, reliable, efficient and practical methods to evaluate the flows due to arbitrary singularity-distributions over hull-surface panels is a crucial core-element of the Green-function method in ship and offshore hydrodynamics. This core-issue is the object of the study. Free-surface flows due to singularity-distributions over panels (and related ‘influence coefficients’) are ordinarily evaluated via a two-step procedure that involves evaluation of a Green function G and its gradient ∇G (a Fourier integration) and subsequent integration of G and ∇G over a panel (a space integration). This common approach involves notorious analytical and numerical complexities related to the complicated singularities of the Green functions in ship and offshore hydrodynamics. An alternative general approach, applicable to generic dispersion relations and arbitrary distributions of singularities, is expounded. This alternative approach is based on a Fourier–Kochin representation of free-surface effects, in which the space integration over hull-panels is performed first and the Fourier integration is performed subsequently. Thus, the Green function and its gradient are not evaluated in this approach. Indeed, this usual first step is bypassed, and the flow due to a singularity distribution is evaluated directly. A major advantage of the Fourier–Kochin method is that the panel-integration is a trivial task as it merely involves integration of an exponential–trigonometric function. A crucial element of the approach expounded in the study is a general analytical decomposition of free-surface effects into waves and a local flow. The waves in this fundamental flow decomposition are expressed as single integrals along the dispersion curves defined by the dispersion relation in the Fourier plane, and the local flow is given by a double Fourier integral that has a smooth integrand dominant within a compact region of the Fourier plane. This analytical flow representation does not involve approximations, i.e. is mathematically exact, as is verified via numerical applications for two main classes of flows in ship and offshore hydrodynamics: ships advancing in calm water, and diffraction-radiation of regular waves by offshore structures. These applications also demonstrate that the general approach expounded in the study provides a practical, remarkably simple, basis that is well suited for accurate and efficient evaluation of flows due to arbitrary singularity-distributions. Indeed, the approach yields a smooth wave and local-flow decomposition that avoids the complexities related to the evaluation and subsequent hull-panel integration of the singular wave and local-flow components in the classical Green functions of ship and offshore hydrodynamics. The general approach expounded in the study is applicable to other classes of flows, notably wave diffraction-radiation by ships advancing through waves in deep water and by offshore structures in finite water-depth, and these important particular applications will be considered in sequels to the study.

Evolution characteristics and quantization of wave period variation for breaking waves

The wave parameters (the wave height and period) are important environmental factors in the ocean engineering design. General numerical wave models, such as SWAN and WAVEWATCH, are widely employed to simulate the wave conditions. However, significant differences were observed between the measurement and the simulation for both the wave height and period, which asks for wave model improvements. The differences are mainly due to the uncertainty of parameterizing various physical processes, including the wave breaking. The energy transfer and loss during the wave breaking involves an important physical mechanism, and the energy dissipation and the period changes are not well studied. This paper studies the deep and shallow water wave breaking using the wave focusing and the slope platform random wave experiments. The characteristics of the wave periods under different conditions are studied in detail, including the period variation. The results show that the periods change during the wave propagation and breaking processes. The energy transfer caused by the strongly nonlinear interaction between the wave components, as well as the energy loss caused by the wave breaking, are the primary causes. The corresponding relationships are established by fitting the data. For the deep water breaking waves induced by the wave focusing, the spectrally averaged period (SAP) increases, and a positive correlation between the rate of change and the wave steepness is found. In the shallow water, the nonlinear interactions are stronger than in the deep water, the wave periods are significantly reduced, and a negative correlation between the rate of change and a nonlinear parameter is found. The inherent mechanism of the period variation is analyzed based on the energy spectrum distribution variations. The contributions of the nonlinear interactions and the wave breaking to the SAP evolution are discussed.

Nonlinear amplification of ocean waves in straits

Using the exact Hasselmann equation, we study wind-driven deep-water ocean waves in a strait with the wind directed orthogonally to the shore. The strait has “dissipative” shores with no reflection from the shorelines. We show that the evolution of wave turbulence can be divided into two different regimes in time. During the first regime, waves propagate with the wind, and the wind-driven sea can be described by self-similar solutions of the Hasselmann equation. The second regime starts after a sufficiently significant accumulation of wave energy at the downwind boundary. From this instant, an ensemble of waves propagating against the wind starts to form. Moreover, waves orthogonal to the wind arise and propagate along the strait. The wave system eventually reaches an asymptotic stationary state in which two types of wave motion coexist: an ensemble of self-similar waves propagating with the wind and quasimonochromatic waves propagating almost orthogonally to the wind direction and tending to slant against the wind at the angle of 15° with respect to the shore of turbulence origination. These “secondary waves” arise only as a result of an intensive nonlinear wave interaction. The total wave energy exceeds its expected value approximately by a factor of two compared with the energy calculated in the absence of shores. We expect that this amplification increases substantially in the presence of reflective shores. We propose calling this “secondary” laser-like mechanism “nonlinear ocean wave amplification” (abbreviated NOWA).

From modulational instability to focusing dam breaks in water waves

We report water wave experiments performed in a long tank where we consider the evolution of nonlinear deep-water surface gravity waves with the envelope in the form of a large-scale rectangular barrier. Our experiments reveal that, for a range of initial parameters, the nonlinear wave packet is not disintegrated by the Benjamin-Feir instability but exhibits a specific, strongly nonlinear modulation, which propagates from the edges of the wavepacket towards the center with finite speed. Using numerical tools of nonlinear spectral analysis of experimental data we identify the observed envelope wave structures with focusing dispersive dam break flows, a peculiar type of dispersive shock waves recently described in the framework of the semi-classical limit of the 1D focusing nonlinear Schrodinger equation (1D-NLSE). Our experimental results are shown to be in a good quantitative agreement with the predictions of the semi-classical 1D-NLSE theory. This is the first observation of the persisting dispersive shock wave dynamics in a modulationally unstable water wave system.

Rip currents in the non-tidal surf zone with sandbars: numerical analysis versus field measurements

Rip currents, which are local seaward-directed jets with their mean velocity exceeding 0.5 m/s, have been a subject of many studies since the 1940s. They are an important part of the nearshore current system and in specific hydro- and litomorphological conditions can cause changes in the local bathymetry. Thus, a detailed analysis of the characteristics of this phenomenon is crucial both to public safety and hydroengineering. The main purpose of this research is to determine the wave conditions of a multi-bar non-tidal coastal zone environment in which rip currents can occur. In this study, we focus on a multi-bar non-tidal coastal zone environment located in the Southern Baltic Sea, where rip current driving forces are mostly reduced to the wind and wind-induced waves. This is one of very few comprehensive approaches to exploring the possibility of rip currents occurrence in such environmental conditions. During two field expeditions, there were carried out in situ measurements exploiting two GPS drifters. The results indicate the formation of irregular non-longshore flows (related to rip currents) in the studied area. To answer the question under what conditions the formation of rip currents takes place, an extended modelling experiment was performed. Deep-water wave conditions typical of the studied area were chosen due to bouy measurements. The total of 589 combinations of the significant wave height, the mean period and wave direction values were examined as test cases. The coastal flow in the area and tracks of virtual drifters were simulated by XBeach numerical model for all test cases. As a result, 589 nearshore currents fields were generated and two scenarios were indicated: a regular circulation (dominated by the longshore current) which is typical of this area (547 cases), and flows with rip current features (42 cases). This reflects the results of the field measurements carried out. It can be concluded that the wave direction is a dominating factor in the formation of rip currents. Namely the flows of this type may occur in the area of interest when the direction of a deep water wave is almost perpendicular to the shore. Such situations occur rarely. They cover about 7% of the days of the year. Thus, rip currents do not appear to be a significant factor in the reconstruction of the sea bottom in the studied area.

Time domain simulation of focused waves by High-Level Irrotational Green–Naghdi equations and Harmonic Polynomial Cell method

Wave focusing is one of the main mechanisms that can generate freak waves in deep water. The strongly nonlinear High-Level Irrotational Green–Naghdi (HLIGN) equations and the Harmonic Polynomial Cell (HPC) method are used here to simulate the focused waves. Numerical simulations are performed in intermediate water depth as well as in deep water. In intermediate water, the HLIGN free surface and wave kinematics are then compared with the well-accepted laboratory data and the results of the HPC method. The effect of water depth on focused waves is also carried out. It is demonstrated that both the HLIGN equations and the HPC method can accurately simulate the surface elevation and particle velocity of focused waves compared with the experiments. The velocity distribution is distinct for the focused waves with the same crest elevations but in different water depths. The HLIGN equations are considered to be an accurate and efficient wave model to investigate focused waves in the intermediate and deep water cases.