Deep-water Waves Research Articles

Lagrangian transport for two-dimensional deep-water surface gravity wave groups

The Lagrangian trajectories of neutrally buoyant particles underneath surface gravity wave groups are dictated by two physical phenomena: the Stokes drift results in a net displacement of particles in the direction of propagation of the group, whereas the Eulerian return flow, as described by the multi-chromatic wave theory of Longuet-Higgins & Stewart (1962 J. Fluid Mech. 13 , 481–504. ( doi:10.1017/S0022112062000877 )), transports such particles in the opposite direction. By pursuing a separation of scales expansion, we develop simple closed-form expressions for the net Lagrangian displacement of particles. By comparing the results from the separation of scales expansion at different orders in bandwidth, we study the effect of frequency dispersion on the local Lagrangian transport, which we show can be ignored for realistic sea states.

Scattering of water waves by vertical porous barriers: An analytical approach

An analytical approach is proposed here to study scattering of deep water waves by a submerged or a surface piercing vertical porous barrier. It involves a connection between two wave potentials of which one is the solution of a boundary value problem associated with wave scattering by the porous barrier and the other is the solution of a complementary type problem where barrier and gap positions are interchanged and solid barrier takes the position of the porous barrier. The connection also involves an auxiliary or a connection wave potential. The potential for the solid barrier problem involves incident wave forcing while the auxiliary potential describes a solid barrier type problem that involves a non-physical forcing. The solution procedure of Ursell (Ursell, 1947) is chosen to solve these boundary value problems explicitly in the case of normal wave incidence as it also determines necessary exact behavior of the potential at the barrier edge. The reflection coefficients are also connected and the reflection amplitudes of the normally incident wave against the vertical porous barriers are obtained analytically. Numerical results for reflection and transmission coefficients are presented.

Topological water wave states in a one-dimensional structure.

Topological concepts have been introduced into electronic, photonic, and phononic systems, but have not been studied in surface-water-wave systems. Here we study a one-dimensional periodic resonant surface-water-wave system and demonstrate its topological transition. By selecting three different water depths, we can construct different types of water waves - shallow, intermediate and deep water waves. The periodic surface-water-wave system consists of an array of cylindrical water tanks connected with narrow water channels. As the width of connecting channel varies, the band diagram undergoes a topological transition which can be further characterized by Zak phase. This topological transition holds true for shallow, intermediate and deep water waves. However, the interface state at the boundary separating two topologically distinct arrays of water tanks can exhibit different bands for shallow, intermediate and deep water waves. Our work studies for the first time topological properties of water wave systems, and paves the way to potential management of water waves.

A forecast model of freak wave generations in deep water

In ocean engineering designs, random extreme events are the most difficultly determined environmental load factors. Appearances of freak waves in the normal sea states are just such events. To reduce their potential destruction of marine structures, a forecast model is studied to predict whether freak waves occur in the wave propagation. In this model, the modified fourth-order nonlinear Schrödinger equation is employed as the deep-water wave model, which controls the evolution of complex wave envelopes. The measured wave train is set as its initial condition which needs to be changed into its corresponding complex wave envelope by use of the Hilbert transform method to be input into the wave model. During the evolution of complex wave envelope, its corresponding wave heights are estimated and compared with the definition of freak waves. If a freak wave is captured, its occurrence position and time are given. Three cases of observed, simulated and laboratory wave trains as initial conditions are performed to predict the generation of freak waves. Results show that measured wave trains can be simply and accurately input into this forecast model through the Hilbert transform method and this model can predict the generation of freak waves within some space and time in its traveling. In addition, larger groups than usual group height or length increase the probability of freak wave generation. This forecast model may provide marine activities with a safety warning in some open seas.

Characterizing JONSWAP rogue waves and their statistics via inverse spectral data

Rogue waves in random sea states modeled by the JONSWAP power spectrum are high amplitude waves arising over non-uniform backgrounds that cannot be viewed as small amplitude modulations of Stokes waves. In the context of Nonlinear Schrödinger (NLS) models for waves in deep water, this poses the challenge of identifying appropriate analytical solutions for JONSWAP rogue waves, investigating possible mechanisms for their formation, and examining the validity of the NLS models in these more realistic settings. In this work we investigate JONSWAP rogue waves using the inverse spectral theory of the periodic NLS equation for moderate values of the period. For typical JONSWAP initial data, numerical experiments show that the developing sea state is well approximated by the first few dominant modes of the nonlinear spectrum and can be described in terms of a 2- or 3-phase periodic NLS solution. As for the case of uniform backgrounds, proximity to instabilities of the underlying 2-phase solution appears to be the main predictor of rogue wave occurrence, suggesting that the modulational instability of 2-phase solutions of the NLS is a main mechanism for rogue wave formation and that heteroclinic orbits of unstable 2-phase solutions are plausible models of JONSWAP rogue waves. To support this claim, we correlate the maximum wave strength as well as the higher statistical moments with elements of the nonlinear spectrum. The result is a diagnostic tool widely applicable to both model or field data for predicting the likelihood of rogue waves. Finally, we examine the validity of NLS models for JONSWAP data, and show that NLS solutions with JONSWAP initial data are described by non-Gaussian statistics, in agreement with the TOPEX field studies of sea surface height variability.

The subtidal morphology of microtidal shore platforms and its implication for wave dynamics on rocky coasts

The subtidal portions of shore platforms are important geomorphic features as they can modify deep-water wave energy before it impacts the intertidal platform edge. In this study an integrated marine and terrestrial aerial LiDAR dataset is used to analyse the morphology of the subtidal portion of shore platforms. Semi-horizontal intertidal platforms on an 85km along stretch of microtidal, open-ocean, rocky coast in Victoria, Australia are investigated and described quantitatively. Three distinct types of subtidal morphology occur; (i) a steep cliff with a mean slope of 8–18°, (ii) a gently sloping ramp with a mean slope of <3°, and (iii) a subtidal terrace/reef. It is inferred that the type of subtidal morphology present on a platform will determine the relative impact of marine and subaerial processes in the intertidal and supratidal zones.

Sediment Transport Beyond the Surf Zone Under Waves and Currents of the Non-Tidal Sea: Lubiatowo (Poland) Case Study

Abstract The paper deals with the sandy coastal zone at Lubiatowo in Poland (the south Baltic Sea). The study comprises experimental and theoretical investigations of hydrodynamic and lithodynamic processes in the coastal region located close to the seaward boundary of the surf zone and beyond the surf zone. The analysis is based on field data collected at the IBW PAN Coastal Research Station in Lubiatowo. The data consist of wind velocity reconstructed from the long-term wave climate, deep-water wave buoy records and sea bottom soil parameters. Nearbed flow velocities induced by waves and currents, as well as bed shear stresses are theoretically modelled for various conditions to determine sediment motion regimes in the considered area. The paper discusses the possibility of occasional intensive sediment transport and the occurrence of distinct sea bed changes at bigger water depths.

Wave observation in the marginal ice zone with the TerraSAR-X satellite

This article investigates the penetration of ocean waves into the marginal ice zone (MIZ), observed by satellite, and likewise provides a basis for the future cross-validation of respective models. To this end, synthetic aperture radar images from the TerraSAR-X satellite (TS-X) and numerical simulations of the European Centre for Medium-Range Weather Forecasts (ECMWF) are used. The focus is an event of swell waves, developed during a storm passage in the Atlantic, penetrating deeply into the MIZ off the coast of Eastern Greenland in February 2013. The TS-X scene which is the basis for this investigation extends from the ice-free open ocean to solid ice. The variation of the peak wavelength is analysed and potential sources of variability are discussed. We find an increase in wavelength which is consistent with the spatial dispersion of deep water waves, even within the ice-covered region.

A kinetic model for particle–surface interaction applied to rain falling on water waves

We present a model for estimating the momentum flux from water drops falling onto a moving free surface. The theory is based on a kinetic approach whereby individual drops are modelled as point particles with mass and velocity, and are described collectively by a distribution function $f(t,\boldsymbol{x},\boldsymbol{v},r)$. We show that the resulting momentum flux can be readily incorporated into free-surface Navier–Stokes and Euler models. As an illustration of this approach we examine the interaction between rainfall and linear deep-water surface waves. This particular application is not fundamentally different from the study of Le Méhauté &amp; Khangaonkar (J. Phys. Oceanogr., vol. 20 (12), 1990, pp. 1805–1812), but our methodology is more general and is novel in its use of a kinetic approach with an all-purpose drop distribution function. The applicability of the model to linear surface waves is found to be valid for surface-wave wavelengths ranging from approximately 3 to 250 m. We further show that rainfall modifies the usual wave dispersion relationship and induces wave amplification, or damping, depending on the rain rate, the rain impact angle and the wavelength of the surface wave. We solve for the amplification and damping rates analytically and show, among other results, that rain falling vertically will always damp the surface waves.

Analytical Approximation in Deep Water Waves

Analytical Approximation in Deep Water Waves

Interaction of surface gravity wave motion with elastic bottom in three-dimensions

A general three-dimensional hydroelastic model is developed to study the effect of elastic bottom on surface gravity wave motion in three-dimensions under the action of uniform compressive force based on linearized theory of water wave in finite water depth. The elastic bottom bed is modeled as a thin plate theory. The progressive wave characteristics in different wave modes are analyzed in both the cases of deep and shallow water waves. Further, the linearized long wave equation under shallow water approximation in a direct manner is derived and compared the results obtained based on the small amplitude wave theory. Three-dimensional Green's function associated with surface gravity wave motion with elastic bed is derived using the fundamental source potentials. Fourier-type expansion formula and corresponding orthogonal mode-coupling relations are derived in finite depth and finite/semi-infinite width in three-dimensions. Utilizing the expansion formula, two classical problems (i) forced motion and (ii) wave reflection by a rigid wall in a channel of finite width and finite/semi-infinite length are illustrated which will play significant role in the analysis of wave-structure interaction problems arising in ocean engineering.The effects of elastic bottom on surface gravity waves are analyzed by presenting several numerical results on reflection wave amplitudes in different cases. Further, the behavior of free motions and oscillations in a basin of finite width and finite/semi-infinite length with elastic bed under the action of uniform compressive force are discussed.

Impacts of enhanced central Pacific ENSO on wave climate and headland-bay beach morphology

Wave climate and Pacific basin coastal behaviour associated with El Niño Southern Oscillation (ENSO) is understood at a reconnaissance level, but the coastal response to different central Pacific (CP) versus eastern Pacific (EP) flavours of ENSO is unknown. We show that CP ENSO events produce different patterns of directional wave power to EP ENSO along the southeast Australian shelf and southwest Pacific region, because of significant variability in trade-wind wave generation. The modulation of the trade wind wave climate during CP ENSO has thus far been neglected in existing coastal process studies. We also show that coastal change between CP and EP ENSO cannot be inferred from shifts in the deepwater wave climate. This is because variability in trade wind wave generation is masked in deepwater by the persistence of high power extra-tropical waves that have reduced impact on nearshore processes due to high wave refraction. Morphodynamic modelling in a headland-bay beach indicates that CP ENSO leads to higher coastal erosion potential and slower post-storm recovery than EP ENSO during an El Niño/La Niña cycle. We show that the alongshore variability in beach morphological type can be used to model the static equilibrium planform response for each ENSO phase. Results indicate that shoreline response to ENSO in most headland-bay beach coasts is not as simple as the existing paradigm that (anti-) clockwise rotation occurs during El Niño (La Niña). Our methods provide a second-order approach to project coastal response and predict the discrete shoreline rotations for ENSO flavours.

Wave Energy as a Control on Dune Development on two Regressive Barriers in Southern Brazil

ABSTRACT Dillenburg, S.; Hesp, P.; Cecilio, R., and Miot da Silva, G., 2016. Wave Energy as a Control on Dune Development on two Regressive Barriers in Southern Brazil. In: Vila-Concejo, A.; Bruce, E.; Kennedy, D.M., and McCarroll, R.J. (eds.), Proceedings of the 14th International Coastal Symposium (Sydney, Australia). Journal of Coastal Research, Special Issue, No. 75, pp. 273–277. Coconut Creek (Florida), ISSN 0749-0208. Two regressive barriers (Cassino and Curumim) were developed during the middle and late Holocene along gentle re-entrants of the Rio Grande do Sul coast, and they have prograded up to 17 km and 5 km, respectively. While Cassino is marked by a surface morphology characterised predominantly by foredune ridges, Curumim displays a surface completely covered by widely spaced ridges, corresponding to phases of transgressive dunefield formation. As wave period and shoreline orientation are very similar for both barriers, and considering that they are subjected to the same deepwater wave clima...

Simulation by discrete mass modeling of offshore wind turbine system with DC link

This paper presents an integrated model for an offshore wind turbine taking into consideration a contribution for the marine wave and wind speed with perturbations influences on the power quality of current injected into the electric grid. The paper deals with the simulation of one floating offshore wind turbine equipped with a permanent magnet synchronous generator, and a two-level converter connected to an onshore electric grid. The use of discrete mass modeling is accessed in order to reveal by computing the total harmonic distortion on how the perturbations of the captured energy are attenuated at the electric grid injection point. Two torque actions are considered for the three-mass modeling, the aerodynamic on the flexible part and on the rigid part of the blades. Also, a torque due to the influence of marine waves in deep water is considered. Proportional integral fractional-order control supports the control strategy. A comparison between the drive train models is presented.

On the distribution of wave height in shallow water

The statistical distribution of the height of sea waves in deep water has been modelled using the Rayleigh (Longuet-Higgins, 1952) and Weibull distributions (Forristall, 1978). Depth-induced wave breaking leading to restriction on the ratio of wave height to water depth requires new parameterisations of these or other distributional forms for shallow water. Glukhovskiy (1966) proposed a Weibull parameterisation accommodating depth-limited breaking, modified by van Vledder (1991). Battjes and Groenendijk (2000) suggested a two-part Weibull–Weibull distribution. Here we propose a two-part Weibull-generalised Pareto model for wave height in shallow water, parameterised empirically in terms of sea state parameters (significant wave height, HS, local wave-number, kL, and water depth, d), using data from both laboratory and field measurements from 4 offshore locations. We are particularly concerned that the model can be applied usefully in a straightforward manner; given three pre-specified universal parameters, the model further requires values for sea state significant wave height and wave number, and water depth so that it can be applied. The model has continuous probability density, smooth cumulative distribution function, incorporates the Miche upper limit for wave heights (Miche, 1944) and adopts HS as the transition wave height from Weibull body to generalised Pareto tail forms. Accordingly, the model is effectively a new form for the breaking wave height distribution. The estimated model provides good predictive performance on laboratory and field data.

Can the Rayleigh distribution be used to determine extreme wave heights in non-breaking swell conditions?

A reliable set of tools for prediction of low-exceedance design waves is of high importance when designing coastal protection structures. The significant wave parameters are typically obtained from buoys or numerical wave propagation models and design values are found by extreme analysis. Statistical wave height distributions are used to transform the significant wave height to lower exceedance wave heights. These extreme single waves will cause the highest loads and wave overtopping volumes on structures and thereby represent the design conditions. An under-prediction of the design maximum wave height causes unsafe designs, while an over-prediction causes too conservative and thus expensive designs. The wave height distribution by Longuet-Higgins (1952) (Rayleigh-distribution) for deep-water non-breaking waves is in the present paper evaluated against data from numerical tests with long period and long-crested swell waves. The numerical model is validated against data from physical model tests. Generally, it is concluded that the Rayleigh-distribution is under-predicting the low-exceedance wave heights in irregular swell waves. This is expected to be caused by wave non-linearity and thus a new modified wave height distribution is suggested, where the shape parameter in the distribution is dependent on the wave non-linearity, represented by the Ursell-number. The new proposed wave height distribution for non-linear and non-breaking waves is highly applicable for practical engineering design of both near-shore and offshore structures under influence of swell-waves.

Breaking-onset, energy and momentum flux in unsteady focused wave packets

Breaking waves on the ocean surface transfer energy and momentum into currents and turbulence. What is less well understood, however, is the associated total loss of wave energy and momentum flux. Further, finding a robust and universal diagnostic parameter that determines the onset of breaking and its strength is still an open question. Derakhti &amp; Kirby (J. Fluid Mech., vol. 761, 2014, pp. 464–506) have recently studied bubble entrainment and turbulence modulation by dispersed bubbles in isolated unsteady breaking waves using large-eddy simulation. In this paper, a new diagnostic parameter ${\it\xi}(t)$ is defined based on that originally proposed by Song &amp; Banner (J. Phys. Oceanogr., vol. 32, 2002, pp. 2541–2558), and it is shown that using a threshold value of ${\it\xi}_{th}=0.05$, the new dynamic criteria is capable of detecting single and multiple breaking events in the considered packets. In addition, the spatial variation of the total energy and momentum flux in intermediate- and deep-water unsteady breaking waves generated by dispersive focusing is investigated. The accuracy of estimating these integral measures based on free surface measurements and using a characteristic wave group velocity is addressed. It is found that the new diagnostic parameter just before breaking, ${\it\xi}_{b}$, has a strong linear correlation with the commonly used breaking strength parameter $b$, suggesting that ${\it\xi}_{b}$ can be used to parameterize the averaged breaking-induced dissipation rate and its associated energy flux loss. It is found that the global wave packet time and length scales based on the spectrally weighted packet frequency proposed by Tian et al. (J. Fluid Mech., vol. 655, 2010, pp. 217–257), are the reasonable estimations of the time and length scales of the carrier wave in the packet close to the focal/break point. A global wave steepness, $S_{s}$, is defined based on these spectrally weighted scales, and its spatial variation across the breaking region is examined. It is shown that the corresponding values of $S_{s}$ far upstream of breaking, $S_{s0}$, have a strong linear correlation with respect to $b$ for the considered focused wave packets. The linear relation, however, cannot provide accurate estimations of $b$ in the range $b&lt;5\times 10^{-3}$. A new scaling law given by $b=0.3(S_{s0}-0.07)^{5/2}$, which is consistent with inertial wave dissipation scaling of Drazen et al. (J. Fluid Mech., vol. 611, 2008, pp. 307–332), is shown to be capable of providing accurate estimates of $b$ in the full range of breaking intensities, where the scatter of data in the new formulation is significantly decreased compared with that proposed by Romero et al. (J. Phys. Oceanogr., vol. 42, 2012, pp. 1421–1444). Furthermore, we examine nonlinear interactions of different components in a focused wave packet, noting interactive effect on a characteristic wave group velocity in both non-breaking and breaking packets. Phase locking between spectral components is observed in the breaking region as well, and subsequently illustrated by calculating the wavelet bispectrum.

Effect of sea level rise on nearshore significant waves and coastal structures

In this paper, the method to assess the impact of sea level rise under regular waves, as proposed by Townend (1994), is extended to irregular waves in order to estimate the changes in nearshore significant waves and the parameters related to hydraulic performance and stability of inclined coastal structures. The relative changes in wavelength, refraction coefficient, shoaling coefficient, and wave height for significant waves are presented as functions of the relative change in water depth. The calculated relative changes in wave characteristics are then used to estimate the effect of sea level rise on coastal structures by calculating the relative changes in wave run-up height, overtopping discharge, crest freeboard, and armor weight of the structures. The relative changes in wave characteristics are expressed as functions of relative water depth and deepwater wave angle or wave steepness. The relative changes in the structure-related parameters are expressed as functions of the relative change in wave height and the wave height and crest freeboard before the sea level rise.

Estimation of mean and extreme waves in the East China Seas

Accurately estimating the mean and extreme wave statistics and better understanding their directional and seasonal variations are of great importance in the planning and designing of ocean and coastal engineering works. Due to the lack of long-term wave measurement data, the analysis of extreme waves is often based on the numerical wave hind-casting results. In this study, the wave climate in the East China Seas (including the Bohai Sea, the Yellow Sea and the East China Sea) for the past 35 years (1979–2013) is hind-casted using a third generation wave model – WAMC4 (Cycle 4 version of WAM model). Two sets of reanalysis wind data from NCEP (National Centers for Environmental Prediction, USA) and ECMWF (European Centre for Medium-range Weather Forecasts) are used to drive the wave model to generate the long-term wave climate. The hind-casted waves are then analysed to study the mean and extreme wave statistics in the study area. The results show that the mean wave heights decrease from south to north and from sea to land in general. The extreme wave heights with return periods of 50 and 100 years in the summer and autumn seasons are significantly higher than those in the other two seasons, mainly due to the effect of typhoon events. The mean wave heights in the winter season have the highest values, mainly due to the effect of winter monsoon winds. The comparison of extreme wave statistics from both wind fields with the field measurements at several nearshore wave observation stations shows that the extreme waves generated by the ECMWF winds are better than those generated by the NCEP winds. The comparison also shows the extreme waves in deep waters are better reproduced than those in shallow waters, which is partly attributed to the limitations of the wave model used. The results presented in this paper provide useful insight into the wave climate in the area of the East China Seas, as well as the effect of wind data resolution on the simulation of long-term waves.

New hydroelastic solitary waves in deep water and their dynamics

A numerical study of fully nonlinear waves propagating through a two-dimensional deep fluid covered by a floating flexible plate is presented. The nonlinear model proposed by Toland (Arch. Rat. Mech. Anal., vol. 289, 2008, pp. 325–362) is used to formulate the pressure exerted by the thin elastic sheet. The symmetric solitary waves previously found by Guyenne &amp; Părău (J. Fluid Mech., vol. 713, 2012, pp. 307–329) and Wang et al. (IMA J. Appl. Maths, vol. 78, 2013, pp. 750–761) are briefly reviewed. A new class of hydroelastic solitary waves which are non-symmetric in the direction of wave propagation is then computed. These asymmetric solitary waves have a multi-packet structure and appear via spontaneous symmetry-breaking bifurcations. We study in detail the stability properties of both symmetric and asymmetric solitary waves subject to longitudinal perturbations. Some moderate-amplitude symmetric solitary waves are found to be stable. A series of numerical experiments are performed to show the non-elastic behaviour of two interacting stable solitary waves. The large response generated by a localised steady pressure distribution moving at a speed slightly below the minimum of the phase speed (called the transcritical regime in the literature) is also examined. The direct numerical simulation of the fully nonlinear equations with a single load reveals that in this range the generated waves are of finite amplitude. This includes a perturbed depression solitary wave, which is qualitatively similar to the large response observed in experiments. The excitations of stable elevation solitary waves are achieved by applying multiple loads moving with a speed in the transcritical regime.