Deep-water Waves Research Articles

Harmonic stability of standing water waves

A numerical method is developed to study the stability of standing water waves and other time-periodic solutions of the free-surface Euler equations using Floquet theory. A Fourier truncation of the monodromy operator is computed by solving the linearized Euler equations about the standing wave with initial conditions ranging over all Fourier modes up to a given wave number. The eigenvalues of the truncated monodromy operator are computed and ordered by the mean wave number of the corresponding eigenfunctions, which we introduce as a method of retaining only accurately computed Floquet multipliers. The mean wave number matches up with analytical results for the zero-amplitude standing wave and is helpful in identifying which Floquet multipliers collide and leave the unit circle to form unstable eigenmodes or rejoin the unit circle to regain stability. For standing waves in deep water, most waves with crest acceleration below $A_c=0.889$ are found to be linearly stable to harmonic perturbations; however, we find several bubbles of instability at lower values of $A_c$ that have not been reported previously in the literature. We also study the stability of several new or recently discovered time-periodic gravity-capillary or gravity waves in deep or shallow water, finding several examples of large-amplitude waves that are stable to harmonic perturbations and others that are not. A new method of matching the Floquet multipliers of two nearby standing waves by solving a linear assignment problem is also proposed to track individual eigenvalues via homotopy from the zero-amplitude state to large-amplitude standing waves.

Breaking waves in deep water: measurements and modeling of energy dissipation

In the presence of strong winds, ocean surface waves dissipate significant amounts of energy by breaking. Here, breaking rates and wave-following turbulent dissipation rate measurements are compared with numerical WAVEWATCH III estimates of bulk energy dissipation rate. At high winds, the measurements suggest that turbulent dissipation becomes saturated; however, the modeled bulk dissipation continues to increase as a cubic function of wind speed. Similarly, the mean square slope (i.e., the steepness) of the measured waves becomes saturated, while the modeled mean squared slope grows linearly with wind speed. Only a weak relation is observed between breaker fraction and wind speed, possibly because these metrics do not capture the scale (e.g., crest length) of the breakers. Finally, the model skill for basic parameters such as significant wave height is shown to be sensitive to the dissipation rate, indicating that the model skill may be compromised under energetic conditions.

New Analytical Solutions for Time Fractional Benjamin-Ono Equation Arising Internal Waves in Deep Water

In this article, the author sets up the abundant traveling wave solutions for time fractional Benjamin-Ono equation which was introduced to describe internal waves in stratified fluids by using Jacobi elliptic function expansion method. The traveling wave solutions are expressed in terms of the hyperbolic functions, the trigonometric functions and the rational functions. It can be seen that the obtained results are found to be important for the statement of some physical demonstrations of problems in mathematical physics and ocean engineering. In ocean engineering Benjamin-Ono equations are generally used in computer simulation for the water waves in deep water and open seas.

Experimental study of wave propagation characteristics on a simplified coral reef

The wave transformation over slope topography is widely studied, but most of studies are for common coastal slopes. This paper presents an experimental study of 2-D regular and irregular wave transformations on a simplified coral reef with a steep slope of 1:5 and a horizontal reef flat, focusing on the characteristics of the waves that break on the reef flat. The analyzed results show that the estimates of the wave breaking made by using the well-known previous formulae do not agree completely with the experimental results. When the waves break on a reef flat, the relative breaking wave height (H/d)b is related to the incident deep-water wave steepness and the relative water depth db/L0. Hence, a new criterion for breaking waves on a reef flat is proposed. Furthermore, in view of the fact that the local Ursell number is commonly used to parameterize the wave nonlinearity, the relationships among the skewness, the asymmetry, and the local Ursell number are also presented. Experimental data confirm that when the Ursell number is greater than 30, the absolute values of the skewness and the asymmetry on a reef flat are greater than those on a steep slope.

Forecasting Sudden High Waves on the Coasts of East/Japan Sea

Oh, J.; Kim, I.-C.; Suh, K.-D., and Kim, S., 2019. Forecasting sudden high waves on the coasts of East/Japan Sea. In: Lee, J.L.; Yoon, J.-S.; Cho, W.C.; Muin, M., and Lee, J. (eds.), The 3rd International Water Safety Symposium. Journal of Coastal Research, Special Issue No. 91, pp. 96-100. Coconut Creek (Florida), ISSN 0749-0208.During the recent decade, marine accidents and property damage due to high waves have frequently occurred on the coasts of East/Japan Sea, especially during the winter season. One of the major characteristics of the waves is its suddenness associated with rapid development of high waves from a calm state of sea. In 2016, Kim, Oh, and Suh named these waves ‘sudden high waves’ and proposed a criterion for sudden high waves by comparing the wave measurement data and the record of marine accidents and property damage on the east coast of the Korean Peninsula. Their intensity parameter of sudden high waves was proposed in terms of local wave height and wavelength. The criterion for sudden high waves is proposed as To enhance its generality, in this study, a similar analysis is made in terms of deepwater wave height and wavelength. It is also shown that the parameter can be forecasted if accurate wave forecast data is available. To show this, the spatio-temporal variation of the parameter is calculated using the wave hindcast data, and the wave measurement data and the record of damage are examined during the period when the parameter exceeds the criterion on the coasts.

Characteristics of ocean waves off Fortaleza, CE, Brazil, extracted from 1-year deep-water measured data

One year of deep-water wave hourly records collected by a buoy deployed off Ceara state, northeastern Brazilian coast, were used to estimate some regional wave climate characteristics and annual cycles. Waves generated by four wind sources were identified. Southeastern direction showed to be the most effective, especially in the period from June to November due to southeast trade winds system intensification. Several dispersive arrivals of north Atlantic swell were recorded, and their generation zones were identified, which allows determining the region of main sources of these northern waves. Analysis of sea storms showed that although most of them had southeast direction, they could also come from the north. These northern storms sometimes are highly energetic, leading to damage in coastal structures. Only 37 records in the analyzed dataset around 0.4% of the total contained rogue waves. Although the number of occurrences of rogue waves was small, all of them were higher than 2.5 m.

Survey of the near wake of an axial-flow hydrokinetic turbine in the presence of waves

Flow field results are presented for the near-wake of an axial-flow hydrokinetic turbine in the presence of surface gravity waves. The turbine is a 1/25 scale, 0.8 m diameter, two bladed turbine based on the U.S. Department of Energy's Reference Model 1 tidal current turbine. Measurements were obtained in the large towing tank facility at the U.S. Naval Academy with the turbine towed at a constant carriage speed and a tip speed ratio selected to provide maximum power. The turbine has been shown to be nearly scale independent for these conditions. The selected wave form was intended to represent oceanic swell encountered off the U.S. eastern seaboard. The resulting model wave is a deep water wave, in terms of relative depth, traveling with the “current”, in the opposite direction of the towing carriage. Velocity measurements were obtained using a submersible, planar particle image velocimetry (PIV) system at streamwise distances of up to two diameters downstream of the rotor plane. PIV ensembles were obtained for phase locked conditions with the reference blade at the horizontal position. Phase averaged results for no-wave and wave conditions are presented for comparison showing further expansion of the wake and shear layer in the presence of waves as compared to the no-wave case. When ensembles are selectively sampled on the wave phase, a high degree of coherency is shown to remain and the wake width is shown to undulate with the passing of the wave, with the vertical displacement range on the same order as that of a particle under similar conditions. The impact of waves on turbine tip vortex helical structure is also examined. Waves are shown to change the location of adjacent helices. In the streamwise direction, this changes the pitch of the helix, shown in previous studies to affect the downstream wake recovery distance. In the vertical direction, depending on the wave-induced flow field at the time they were created, vortices are forced outward, into the mean flow or inward, into the wake core, potentially enhancing kinetic energy transport and accelerating the re-energization process.

On pressure forcing techniques for numerical modelling of unidirectional nonlinear water waves

A method of incorporating pressure forcing into a nonlinear potential flow wave model is presented. A semi-analytical pseudo-spectral method is used to calculate dynamic response of a water body exposed to evolving local pressure distribution. Surface slope coherent and slope proportional pressure functions are directly applied through a pressure term appearing in the dynamic free-surface boundary condition of a formulated initial boundary-value problem. First, a monochromatic pressure distribution is used to generate steady regular waves of permanent form. The pressure-induced wave motion exhibits stable harmonic structure for deepwater, transitional water and shallow water waves. In the next step, a more complex pressure system is used to initiate multi-component wave propagation. It is demonstrated that the proposed method provides well-posed initial conditions for studying various water wave scenarios within a framework of nonlinear potential flow solutions.

The Effect of Random Wind Forcing in the Nonlinear Schrödinger Equation

The influence of a strong and gusty wind field on ocean waves is investigated. How the random wind affects solitary waves is analyzed in order to obtain insights about wave generation by randomly time varying wind forcing. Using the Euler equations of fluid dynamics and the method of multiple scales, a random nonlinear Schrödinger equation and a random modified nonlinear Schrödinger equation are obtained for randomly wind forced nonlinear deep water waves. Miles theory is used for modeling the pressure variation at the wave surface resulting from the wind velocity field. The nonlinear Schrödinger equation and the modified nonlinear Schrödinger equation are computed using a relaxation pseudo spectral scheme. The results show that the influence of gusty wind on solitary waves leads to a randomly increasing ocean wave envelope. However, in a laboratory setup with much smaller wave amplitudes and higher wave frequencies, the influence of water viscosity is much higher. This leads to fluctuating solutions, which are sensitive to wind forcing.

Calculation of Phase Velocities in the Field of Sea Surface Waves

Specific features of phase velocities of sea waves due to the angular distribution of wave energy and the second harmonic of a dominant wave with a velocity differing from the phase velocity of a free wave at the same frequency have been analyzed. The presence of wave components with different velocity projections on a given direction or different propagation velocities leads to a deviation in the estimates of phase velocities from the values resulting from a variance relationship for gravitational waves in deep water. It has been shown that the phase spectrum in field of sea waves varies nonlinearly with increasing distance between the relevant points. As a consequence, the estimates for phase velocity are functions of distance. The deviations of the estimates for phase velocity from the linear theory have been shown to depend on the stage of wave field development.

Phase field lattice Boltzmann model for air-water two phase flows

Two phase flows occur in different forms with liquid and gas in general, among which, the interaction of the flow of air and water is a common scenario. However, modeling the two phase flow still remains a challenge due to the large density ratio between them and different space-time scales involved in the flow regimes. In the present work, the lattice Boltzmann (LB) model capable of handling large density ratio (1000) and high Reynolds number (104) simultaneously is proposed. The present model consists of two sets of LB equations, one for the flow field in terms of normalized pressure-velocity formulation and the other for the solution of the conservative Allen-Cahn equation to capture the interface. The numerical tests such as stationary drop, bubble coalescence, and capillary wave decay have been performed, and the results exhibit excellent mass conservation property. The capability of the present model to handle complex scenarios has been tested through test cases, for example, rise of an air bubble, splash of a water droplet on a wet bed, and Rayleigh-Taylor instability. In all test cases, the simulation results agree well with the available reference data. Finally, as an application of the present model, the breaking of a deep water wave with high Reynolds number (104) is simulated. The plunging breaker with wave overturning and the generation of secondary jet and splashes are well described by the present LB model. The evolution of wave energy dissipation during and after breaking is in agreement with the reference data.

Focusing deep-water surface gravity wave packets: wave breaking criterion in a simplified model

Geometric, kinematic and dynamic properties of focusing deep-water surface gravity wave packets are examined in a simplified model with the intent of deriving a wave breaking threshold parameter. The model is based on the spatial modified nonlinear Schrödinger equation of Dysthe (Proc. R. Soc. Lond. A, vol. 369 (1736), 1979, pp. 105–114). The evolution of initially narrow-banded and weakly nonlinear chirped Gaussian wave packets are examined, by means of a trial function and a variational procedure, yielding analytic solutions describing the approximate evolution of the packet width, amplitude, asymmetry and phase during focusing. A model for the maximum free surface gradient, as a function of $\unicode[STIX]{x1D716}$ and $\unicode[STIX]{x1D6E5}$, for $\unicode[STIX]{x1D716}$ the linear prediction of the maximum slope at focusing and $\unicode[STIX]{x1D6E5}$ the non-dimensional packet bandwidth, is proposed and numerically examined, indicating a quasi-self-similarity of these focusing events. The equations of motion for the fully nonlinear potential flow equations are then integrated to further investigate these predictions. It is found that a model of this form can characterize the bulk partitioning of $\unicode[STIX]{x1D716}-\unicode[STIX]{x1D6E5}$ phase space, between non-breaking and breaking waves, serving as a breaking criterion. Application of this result to better understanding air–sea interaction processes is discussed.

Numerical Simulations of Modulated Waves in a Higher-Order Dysthe Equation

The nonlinear stage of the modulational (Benjamin–Feir) instability of unidirectional deep-water surface gravity waves is simulated numerically by the fifth-order nonlinear envelope equations. The conditions of steep and breaking waves are concerned. The results are compared with the solution of the full potential Euler equations and with the lower-order envelope models (the 3-order nonlinear Schrodinger equation and the standard 4-order Dysthe equations). The generalized Dysthe model is shown to exhibit the tendency to re-stabilization of steep waves with respect to long perturbations.

Assessing Morphologic Controls on Atoll Island Alongshore Sediment Transport Gradients Due to Future Sea-Level Rise

Atoll islands’ alongshore sediment transport gradients depend on how island and reef morphology affect incident wave energy. It is unclear, though, how potential atoll morphologic configurations influence shoreline erosion and/or accretion patterns, and how these relationships will respond to future sea-level rise (SLR). Schematic atoll models with varying morphologies were used to evaluate the relative control of individual morphological parameters on alongshore transport gradients. Incident wave transformations were simulated using a physics-based numerical model and alongshore erosion and accretion was calculated using empirical formulae. The magnitude of the transport gradients increased with SLR: initial erosion or accretion patterns intensified. Modeled morphologic parameters that significantly influenced alongshore transport were the atoll diameter, reef flat width, reef flat depth, and island width. Modeled atolls with comparably small diameters, narrow and deep reef flats with narrow islands displayed greater magnitudes of erosion and/or accretion, especially with SLR. Windward island shorelines are projected to accrete toward the island’s longitudinal ends and lagoon due to SLR, whereas leeward islands erode along lagoon shorelines and extend toward the island ends. Oblique island, oriented parallel to the incident deepwater wave direction, shorelines are forecast to build out leeward along the reef rim and toward the lagoon while eroding along regions exposed to direct wave attack. These findings make it possible to evaluate the relative risk of alongshore erosion/accretion on atolls due to SLR in a rapid, first-order analysis.

Storm Waves at the Shoreline: When and Where Are Infragravity Waves Important?

Infragravity waves (frequency, f = 0.005–0.05 Hz) are known to dominate hydrodynamic and sediment transport processes close to the shoreline on low-sloping sandy beaches, especially when incident waves are large. However, in storm wave conditions, how their importance varies on different beach types, and with different mixes of swell and wind-waves is largely unknown. Here, a new dataset, comprising shoreline video observations from five contrasting sites (one low-sloping sandy beach, two steep gravel beaches, and two compound/mixed sand and gravel beaches), under storm wave conditions (deep water wave height, H0 up to 6.6 m, and peak period, Tp up to 18.2 s), was used to assess: how the importance and dominance of infragravity waves varies at the shoreline? In this previously unstudied combination of wave and morphological conditions, significant infragravity swash heights (Sig) at the shoreline in excess of 0.5 m were consistently observed on all five contrasting beaches. The largest infragravity swash heights were observed on a steep gravel beach, followed by the low-sloping sandy beach, and lowest on the compound/mixed sites. Due to contrasting short wave breaking and dissipation processes, infragravity frequencies were observed to be most dominant over gravity frequencies on the low-sloping sandy beach, occasionally dominant on the gravel beaches, and rarely dominant on the compound/mixed beaches. Existing empirical predictive relationships were shown to parameterize Sig skillfully on the sand and gravel beaches separately. Deep water wave power was found to accurately predict Sig on both the sand and gravel beaches, demonstrating that, under storm wave conditions, the wave heights and periods are the main drivers of infragravity oscillations at the shoreline, with the beach morphology playing a secondary role. The exception to this was the compound/mixed beach sites where shoreline infragravity energy remained low.

Revisiting infinite mapping layer for open domain problems

We revisit and investigate the application of an infinite mapping layer in the context of different physical fields. This layer remaps the infinite physical domain to a finite extension covering the boundary of the computational domain. Thus, infinite domain problems are solved efficiently by this advanced finite element method. Compared to infinite elements, this method has major practical advantages since it uses standard basis functions. However, the solution at the outer boundary of the infinite layer must be bounded. In principle, this method can be applied to arbitrarily shaped mapping layers.The presented method is verified against three physical field problems and their analytic solution. Firstly, the computation of the electrostatic potential in an unbounded region is investigated. Secondly, a benchmark problem of mechanical engineering demonstrates the method for continuum mechanics. Finally, eigenmodes of deep water waves are solved accurately. For these three cases, we assess three different mapping functions of tangent, exponential, and rational type and comment on their suitability for different physical field problems. The method guarantees high efficiency and accuracy, as shown by the application examples.

Shoreline evolution near river mouth: Case study of Petrace River (Calabria, Italy)

An analysis and understanding of coastline variability and coastal erosion–accretion trends is important to enable scientists and local decision-makers to make appropriate decisions concerning regulatory and coastal planning.The paper describes the shoreline changes which have occurred near the mouth of the Petrace River over the last 60 years, using end point rate (EPR) and net shoreline movement (NSM) statistics. The area extends for about 5 km from the groin near the port of Gioia Tauro on the North side to the port of Palmi on the South side, and which for this study has been divided into 14 transepts orthogonal to the coast with an average spacing of 350 m. The diachronic analysis of the shoreline was carried out through the comparison of cartography data, which consists of aerophotogrammetry (by CASMEZ of 1954 and IGM of 1985, provided by the Calabria Basin Authority and previously georeferenced), orthophotos of 1996, 2006 and 2012 taken from the Open Data section of the National Geoportal and satellite imagery of 2017 provided by Google Earth Pro.The paper also analyzed the main factors influencing the coastal dynamics, in particular the deep-water wave climate, the longshore sediment transport, and the river sediment contribution, produced by variations of hydraulic structures, rainfall regime, land cover and soil erosion by water (WSE). Cross-analysis of these factors in the case study showed that the rainfall regime had a considerable influence on these changes. This result is consistent with the hydrological characteristics of most of the Calabrian rivers and in general of many Mediterranean rivers in which WSE plays an important role. The methodology for analyzing shoreline changes and related correlations with the factors involved described in the paper is therefore applicable to many geographical and hydrological contexts which are similar to the Petrace River mouth.

Simulation of unidirectional propagating wave trains in deep water using a fully non-hydrostatic model

A new fully non-hydrostatic model is developed and then applied to simulate the evolution of unidirectional propagating waves in intermediate and deep water. The ability and efficiency of the model are examined using experimental data. Then, the evolutions of initially uniform and perturbed wave trains are simulated. Frequency downshift phenomena occur during the wave evolution. For a small initial steepness, there are approximately two incidents of partial recurrence. However, with increasing initial wave steepness, the energy transfers from the lower sideband to the carried wave decreases, and a permanent frequency downshift is more likely to occur. For the initially imposed wave trains, if the wave steepness is lower than 0.11, an approximate Fermi-Pasta-Ulam (FPU) phenomenon occurs. For wave trains with a steepness of approximately 0.13, a partial recurrence still can appear. Recurrence is not observed in larger wave trains. Additionally, the kinematics of extreme waves formed during the wave evolution are also investigated, and the non-dimensional horizontal velocity profiles under the extreme crests are converged and can be well modelled by the theoretically exponential curve. In addition, the crest speeds of the steep waves are less than the linear wave speeds. It is found that larger waves have slower crest speeds.

The influence of shelf bathymetry and beach topography on extreme total water levels: Linking large-scale changes of the wave climate to local coastal hazards

Total water levels (TWLs) at the coast are driven by a combination of deterministic (e.g., tides) and stochastic (e.g., waves, storm surge, and sea level anomalies) processes. The contribution of each process to TWLs varies depending on regional differences in climate and framework geology, as well as local-scale variations in beach morphology, coastal orientation, and shelf bathymetry. Large-scale changes to the climate altering the frequency, direction, and intensity of storms, may therefore propagate to the nearshore differently, amplifying or suppressing local coastal hazards and changing the exposure of coastal communities to extreme TWLs. This study investigates the hydrodynamic and geomorphologic factors controlling local TWLs along high-energy United States coastlines where wave-influences dominate TWLs. Three study sites in the states of Washington, Oregon, and California are chosen to explore how regional and local differences in beach topography and wave transformation over shelf bathymetry drives variations in the magnitude and impacts of extreme TWLs. Results indicate that TWLs are most influenced by wave transformation processes in locations with steep beach slopes and complex offshore bathymetry, while beach topography influences the severity of coastal impacts. Once the relative morphologic controls on TWLs are better understood, hypothetical future climate scenarios are explored to assess how changes to the average deepwater wave climate (height, period, and direction) may alter local TWLs when compared to estimates of likely sea level rise and future coastal management strategies. Changes to the wave climate are found to be as detrimental to the coastline as sea level rise in some locations, where small variations of the TWL drive large, nonlinear changes in hours of impact to the backshore beach. Overall, this study develops an approach for quantifying the range of hydrodynamic and morphologic controls on the magnitude of TWLs which will ultimately better prepare coastal communities for uncertain changes to the global climate.

Lagrangian Transport by Nonbreaking and Breaking Deep-Water Waves at the Ocean Surface

AbstractUsing direct numerical simulations (DNS), Deike et al. found that the wave-breaking-induced mass transport, or drift, at the surface for a single breaking wave scales linearly with the slope of a focusing wave packet, and may be up to an order of magnitude larger than the prediction of the classical Stokes drift. This model for the drift due to an individual breaking wave, together with the statistics of wave breaking measured in the field, are used to compute the Lagrangian drift of breaking waves in the ocean. It is found that breaking may contribute up to an additional 30% to the predicted values of the classical Stokes drift of the wave field for the field experiments considered here, which have wind speeds ranging from 1.6 to 16 m s−1, significant wave heights in the range of 0.7–4.7 m, and wave ages (defined here as cm/u*, for the spectrally weighted phase velocity cm and the wind friction velocity u*) ranging from 16 to 150. The drift induced by wave breaking becomes increasingly more important with increasing wind friction velocity and increasing significant wave height.