Water Wave Field Research Articles

Exploring the Diversity of Kink Solitons in (3+1)-Dimensional Wazwaz–Benjamin–Bona–Mahony Equation

The Wazwaz–Benjamin–Bona–Mahony (WBBM) equation is a well-known regularized long-wave model that examines the propagation kinematics of water waves. The current work employs an effective approach, called the Riccati Modified Extended Simple Equation Method (RMESEM), to effectively and precisely derive the propagating soliton solutions to the (3+1)-dimensional WBBM equation. By using this upgraded approach, we are able to find a greater diversity of families of propagating soliton solutions for the WBBM model in the form of exponential, rational, hyperbolic, periodic, and rational hyperbolic functions. To further graphically represent the propagating behavior of acquired solitons, we additionally provide 3D, 2D, and contour graphics which clearly demonstrate the presence of kink solitons, including solitary kink, anti-kink, twinning kink, bright kink, bifurcated kink, lump-like kink, and other multiple kinks in the realm of WBBM. Furthermore, by producing new and precise propagating soliton solutions, our RMESEM demonstrates its significance in revealing important details about the model behavior and provides indications regarding possible applications in the field of water waves.

Nonisospectral water wave field: Fast and adaptive modal identification and prediction via reduced-order nonlinear solutions.

Real-world water wave fields exhibit significant nonlinear and nonisospectral characteristics, making it challenging to predict their evolution by relying solely on numerical simulation or exact solutions using integrable system theory. Hence, this paper introduces a fast and adaptive method of modal identification and prediction in nonisospectral water wave fields using the reduced-order nonlinear solution (RONS) scheme. Specifically, we discuss the coarse graining and mode extraction of wave field snapshots from the data-driven and physics-driven perspectives and utilize the RONS method for principle modal prediction of nonisospectral water wave fields. This is achieved by investigating the standard and nonisospectral Gardner system describing nonlinear water waves as a demonstration. Through detailed comparison and analysis, the fundamental solitary behaviors and dispersive effects in the Gardner system are discussed. Subsequently, a neighbor approximation is developed that combines the essences of symbolic precomputation and numerical computation in the RONS procedure, which exploits the locality of nonlinear interactions in water wave fields.

A new probability density function for the surface elevation in irregular seas

To date, the predominant means for computing the probability density function (p.d.f.) for the free surface elevation of a nonlinear, irregular water wave field, free of assumptions involving narrow-bandedness and small directionality, is the approximate Gram–Charlier series solution of Longuet-Higgins (J. Fluid Mech., vol. 17, 1963, pp. 459–480, hereafter LH63). In this paper we re-visit the derivation of this p.d.f. to second order in the wave steepness, utilizing both moment and cumulant generating functions. We show that LH63's approximate solution based on the cumulant generating function, in fact, matches that derived from the moment generating function. Moreover, through a change of variables coupled with complex analysis, it is shown that the approximation employed by LH63 is unnecessary, and the second-order p.d.f. stemming from the cumulant generating function can be represented exactly in terms of the Airy function. The new second-order p.d.f. predicts increased probability of extreme positive surface elevations typical of e.g. rogue waves, relative to both second- and third-order solutions of LH63. This heavy positive tail is inherent, and is explained through comparison of the asymptotic limits of the p.d.f.s for large surface elevations. A semi-theoretical method is also proposed for remedying non-physical spurious oscillations that arise in the negative tail, based on the envelope of the Airy function with negative arguments. This modified negative tail is valid for irregular wave fields having skewness less than or equal to 0.2. The new p.d.f.s are compared against those based on data sets generated from second-order irregular wave theory as well as a fully nonlinear, spectrally accurate numerical wave model. Good accuracy is collectively demonstrated for directionally spread irregular seas in both finite and infinite water depths for a range of directional spreading.

Riemann-Hilbert problem associated with an integrable generalized mixed nonlinear Schrödinger equation

ABSTRACT The generalized mixed nonlinear Schrödinger (GMNLS) equation is a completely integrable model under certain parameter conditions, which has important applications in nonlinear optics, weak nonlinear dispersive water wave and quantum field theory. In this work, we use the unified transformation method to construct a Riemann-Hilbert problem and discuss the initial-boundary value problems of the GMNLS equation on the half-line. Furthermore, we also obtain that some spectral functions are not independent of each other, but meet a paramount global relation.

Identifying Combination of Dark–Bright Binary–Soliton and Binary–Periodic Waves for a New Two-Mode Model Derived from the (2 + 1)-Dimensional Nizhnik–Novikov–Veselov Equation

In this paper, we construct a new two-mode model derived from the (2+1)-dimensional Nizhnik–Novikov–Veselov (TMNNV) equation. We generalize the concept of Korsunsky to accommodate the derivation of higher-dimensional two-mode equations. Since the TMNNV is presented here, for the first time, we identify some of its solutions by means of two recent and effective schemes. As a result, the Kudryashov-expansion method exports a combination of bright–dark binary solitons, which simulate many applications in optics, photons, and plasma. The modified rational sine and cosine functions export binary–periodic waves that arise in the field of surface water waves. Moreover, by using 2D and 3D graphs, some physical properties of the TMNNV were investigated by means of studying the effect of the following parameters of the model: nonlinearity, dispersion, and phase–velocity. Finally, we checked the validity of the obtained solutions by verifying the correctness of the original governing model.

New symmetric bidirectional progressive surface-wave solutions to a generalized fourth-order nonlinear partial differential equation involving second-order time-derivative

New nonlinear fourth-order integrable equation of second-order in time has been established by Wazwaz via collaborating some recursion operators and the sense of Boussinesq equation. The proposed new equation has several real-life applications in the field of water-waves, such as gravity-capillary waves, waves in shallow water, and ocean engineering. The aim of the current study is threefold. First, we show that the propagation of Wazwaz equation is simultaneously moving symmetric bidirectional waves. Second, we extract new novel solutions such as bidirectional lumps and bidirectional kinks, singular-kinks, and periodic-kinks. Finally, by using graphical analysis, we show the physical structures of the recovery solutions and study the significance of the model’s parameters.

Experimental Investigation of Air Bubble Curtain Effects on Water Wave Field

This paper studies the effects of an air bubble curtain on surface water waves. Water particle velocities and free surface elevations were measured simultaneously at two cross-shore locations downstream of the air bubble curtain. Measurements were carried out for regular waves using different air bubble curtain configurations. Free surface elevations were measured using resistive gauges and the instantaneous velocities were acquired using an Acoustic Doppler Velocimeter (ADV). The characteristics of the free surface elevation time series, velocity field and turbulence are analyzed and discussed. The free surface elevation was found to be attenuated by the air bubble curtains. The phase averaged velocity profiles also depict the effect of the air bubbles in the flow field by generating milder longitudinal velocities (u) and by increasing the transverse component of the velocity (w). The increase in the turbulence intensity and the different energy spectrum produced by the air bubble curtain is also observed. The experimental results indicate that the thickness of the air bubble curtain and the total air flow rate affects the wave field.

On the Alber equation for shoaling water waves

The problem of unidirectional shoaling of a water-wave field with a narrow energy spectrum is treated by using a new Alber equation. The stability of the linear stationary solution to small non-stationary disturbances is analysed; and numerical solutions for its subsequent long-distance evolution are presented. The results quantify the physics which causes the gradual decay in the probability of freak-wave occurrence, when moving from deep to shallow coastal waters.

On the statistical properties of inertia and drag forces in nonlinear multi-directional irregular water waves

Abstract

Attenuating surface gravity waves with mechanical metamaterials

Metamaterials and photonic/phononic crystals have been successfully developed in recent years to achieve advanced wave manipulation and control, both in electromagnetism and mechanics. However, the underlying concepts are yet to be fully applied to the field of fluid dynamics and water waves. Here, we present an example of the interaction of surface gravity waves with a mechanical metamaterial, i.e., periodic underwater oscillating resonators. In particular, we study a device composed of an array of periodic submerged harmonic oscillators whose objective is to absorb wave energy and dissipate it inside the fluid in the form of heat. The study is performed using a state-of-the-art direct numerical simulation of the Navier–Stokes equation in its two-dimensional form with free boundary and moving bodies. We use a volume of fluid interface technique for tracking the surface and an immersed boundary method for the fluid–structure interaction. We first study the interaction of a monochromatic wave with a single oscillator and then add up to four resonators coupled only fluid-mechanically. We study the efficiency of the device in terms of the total energy dissipation and find that by adding resonators, the dissipation increases in a nontrivial way. As expected, a large energy attenuation is achieved when the wave and resonators are characterized by similar frequencies. As the number of resonators is increased, the range of attenuated frequencies also increases. The concept and results presented herein are of relevance for coastal protection applications.

Observations, modeling, and inter-comparison of waves from deep to intermediate waters in the East Mediterranean basin

Wind waves are of great importance in various fields of oceanography and engineering. Although they propagate along entire seas and oceans, due to the complexity and high costs in setting deep water measurements, they are mostly measured only in nearshore regions. The connection between deep sea wave conditions and nearshore ones is commonly done using wave propagation models. Nevertheless, due to the scarcity of deep water measurements, the validity of this process is seldomly tested. This work presents newly acquired data for investigating wave evolution from the deep sea far beyond the continental shelf to intermediate coastal waters. The deep water wave measurements were obtained using an upward-looking Acoustic Doppler Current Profiler (ADCP) mounted on a submerged buoy (35 m depth) in the East Mediterranean 50 km offshore Israel (bottom depth of 1500 m). To the best of our knowledge, it is the first time that such a deep sea measurement is carried out in the region. Another ADCP was deployed on the sea bed (26 m depth) 2 km offshore. During this campaign, a substantial storm (Hm0=6 m) was recorded simultaneously by both ADCPs. A wave evolution model accounting for triad and quartet interactions was employed for connecting the two measurements, in order to explain the differences between the two measurements with numerical estimations. A new methodology was developed for resolving spatial variability in the deep sea conditions by assimilating the deep water data into the output of an operational wave model for the Mediterranean Sea. This allowed forming appropriate boundary conditions for propagation of the wave field from deep to coastal waters. Model results show an importance of accounting for inhomogeneity of the wave field in deep water. Nevertheless, the measurements indicate that one cannot expect to fully resolve the exact wave spectra only by applying wave modeling. This emphasizes the importance of conducting such deep wave measurements.

Small water channel network for designing wave fields in shallow water

A small water channel network is proposed for designing shallow water fields, and the network is applied to attain water wave cloaking. The design formula is derived from an analogy between waves in a water channel and in an electric circuit; an approach of a transmission line metamaterial is extended to water waves and the water channel is used as an alternative of the transmission line. The size of the water channel is sufficiently smaller than the wavelength and a number of the channels are periodically connected as a network. This small water channel network makes artificial wave fields, and it works for a wide band of frequencies. First, we make an isotropic network equivalent to a shallow water space with constant depth in order to validate the proposed design method. It shows no wave reflection at the interface due to impedance matching. After that, the proposed theory is applied to designing an anisotropic small water channel network for demonstrating shallow water cloaking. A cylinder is cloaked from waves by the network surrounding the cylinder. Both cases are confirmed with numerical computations by solving the boundary-value problem based on linear potential theory.

Material Transport under a Wave Train in Interaction with Constant Wind: A Eulerian RANS Approach Combined with a Lagrangian Particle Dispersion Model

The interaction of a developed train of gravity deep water waves with suddenly applied winds is investigated in this manuscript. The direction of the wind is the same as that of the wave train (i.e., following) and its imposed surface shear stress is constant and steady. The focus of this study is on a micro-scale water wave field where the time scale is on the order of ten wave periods and the length scale is on the order of ten wave lengths. Accurate 2D Reynolds-averaged Navier–Stokes (RANS) multi-phase simulations of Navier–Stokes equations are performed in a Eulerian framework to capture the flow features, induced by the wave field and the surface wind. The sufficiently large spatial domain in the horizontal direction, combined with the sufficiently long simulation time, permits the development of surface currents and the consequent formation of a near-surface shear layer. The interaction of surface currents with the wave orbital velocity field results in the generation of spilling breaking waves. Downstream of the domain, vertical turbulent structures are observed as the result of such breaking waves. Lagrangian particle tracking is performed, using the RANS simulation velocity and eddy diffusivity data. A second order random acceleration particle tracking method is applied with the vitally important spatial gradients of the eddy diffusivity (Journal of Marine Science and Engineering, 2018, 6, 10.3390/jmse6010007.) also included in calculations. The spatial gradients of the eddy diffusivity were proved to be a key factor in material transport simulations. Our particle tracking results exhibit strong vertical mixing downstream of the domain and by means of visualizing the spiral trajectory of neutrally buoyant particles. Such enhanced vertical mixing (caused by horizontal winds) is the result of the strong near-surface advection (induced by currents) and the turbulence (induced by breaking waves). The objectives of this paper are twofold. Firstly, a numerical approach to simulate wind breaking waves is proposed based on: using K − ϵ RANS model to capture turbulence features, employing the Volume of Fluid Method (VOF) to model the free surface flow, and applying a constant shear stress body force at the interfacial cells to simulate the wind force. Such treatment of the winds eliminates the need for fully resolving the air phase. The computed eddy viscosity profiles are in good agreement with the experimental profiles reported in the literature ( ν t = − κ u * z , Journal of Physical Oceanography, 1977, 7, pp. 248–255; Journal of Physical Oceanography, 1984, 14, pp. 855–863). Secondly, effects of the horizontally applied wind on the vertical mixing and eddy viscosity profiles on the water column are studied. It is observed that, away from the surface and outside the shear layer, the negative horizontal gradient of eddy diffusivity (induced by the dampening effect of breaking surface waves), combined with the downwards advection velocities (induced by breaking waves), results in an enhanced vertical mixing and reduced horizontal drift of transported material.

Oil Droplet Transport under Non-Breaking Waves: An Eulerian RANS Approach Combined with a Lagrangian Particle Dispersion Model

Oil droplet transport under a non-breaking deep water wave field is investigated herein using Computational Fluid dynamics (CFD). The Reynolds-averaged Navier–Stokes (RANS) equations were solved to simulate regular waves in the absence of wind stress, and the resulting water velocities agreed with Stokes theory for waves. The RANS velocity field was then used to predict the transport of buoyant particles representing oil droplets under the effect of non-locally generated turbulence. The RANS eddy viscosity exhibited an increase with depth until reaching a maximum at approximately a wave height below the mean water level. This was followed by a gradual decrease with depth. The impact of the turbulence was modeled using the local value of eddy diffusivity in a random walk framework with the added effects of the gradient of eddy diffusivity. The vertical gradient of eddy viscosity increased the residence time of droplets in the water column region of high diffusivity; neglecting the gradient of eddy diffusivity resulted in a deviation of the oil plume centroid by more than a half a wave height after 10 wave periods.

On three-dimensional Gerstner-like equatorial water waves.

This paper reviews some recent mathematical research activity in the field of nonlinear geophysical water waves. In particular, we survey a number of exact Gerstner-like solutions which have been derived to model various geophysical oceanic waves, and wave-current interactions, in the equatorial region. These solutions are nonlinear, three-dimensional and explicit in terms of Lagrangian variables.This article is part of the theme issue 'Nonlinear water waves'.

Surface wave predictions in weakly nonlinear directional seas

We have employed laboratory and numerical experiments in order to investigate propagation of waves in both long and short-crested wave fields in deep water. For long-crested waves with steepness, ϵ=kcac=0.1 (a fairly extreme case), reliable prediction can be performed with the modified nonlinear Schrödinger equation up to about 40 characteristic wavelengths. For short-crested waves the accuracy of prediction is strongly reduced with increasing directional spread.

Time-reversal of nonlinear waves: Applicability and limitations

Time-reversal (TR) refocusing of waves is one of fundamental principles in wave physics. Using the TR approach, "Time-reversal mirrors" can physically create a time-reversed wave that exactly refocus back, in space and time, to its original source regardless of the complexity of the medium as if time were going backwards. Lately, laboratory experiments proved that this approach can be applied not only in acoustics and electromagnetism but also in the field of linear and nonlinear water waves. Studying the range of validity and limitations of the TR approach may determine and quantify its range of applicability in hydrodynamics. In this context, we report a numerical study of hydrodynamic TR using a uni-directional numerical wave tank, implemented by the nonlinear high-order spectral method, known to accurately model the physical processes at play, beyond physical laboratory restrictions. The applicability of the TR approach is assessed over a variety of hydrodynamic localized and pulsating structures' configurations, pointing out the importance of high-order dispersive and particularly nonlinear effects in the refocusing of hydrodynamic stationary envelope solitons and breathers. We expect that the results may motivate similar experiments in other nonlinear dispersive media and encourage several applications with particular emphasis on the field of ocean engineering.

Fourth-order split-step pseudo-spectral method for the modified nonlinear Schrödinger equation

In the field of water waves, the modified nonlinear Schrödinger (mNLS) equation which models the wave propagation in water is numerically solved by using the split-step pseudo-spectral method. In the present paper, the fourth-order split-step pseudo-spectral method is introduced with better numerical results. The proposed method is based on a split-step method which decomposes the original equation into two parts, a linear problem and a nonlinear problem. In order to demonstrate the high accuracy and capability of the newly proposed method, a simple problem of periodic waves is presented to compare the traditional first-order method with the fourth-order method in terms of the conservation error and computational cost. Meanwhile, another numerical experiment concerning the Peregrine breather solution of the nonlinear Schrödinger (NLS) equation is presented by using the fourth-order split-step pseudo-spectral method. It is found that the fourth-order scheme can provide higher conservative accuracy and is computationally more efficient compared with the first-order scheme.

Reduced-order precursors of rare events in unidirectional nonlinear water waves

We consider the problem of short-term prediction of rare, extreme water waves in irregular unidirectional fields, a critical topic for ocean structures and naval operations. One possible mechanism for the occurrence of such rare, unusually intense waves is nonlinear wave focusing. Recent results have demonstrated that random localizations of energy, induced by the linear dispersive mixing of different harmonics, can grow significantly due to modulation instability. Here we show how the interplay between (i) modulation instability properties of localized wave groups and (ii) statistical properties of wave groups that follow a given spectrum defines a critical length scale associated with the formation of extreme events. The energy that is locally concentrated over this length scale acts as the ‘trigger’ of nonlinear focusing for wave groups and the formation of subsequent rare events. We use this property to develop inexpensive, short-term predictors of large water waves, circumventing the need for solving the governing equations. Specifically, we show that by merely tracking the energy of the wave field over the critical length scale allows for the robust, inexpensive prediction of the location of intense waves with a prediction window of 25 wave periods. We demonstrate our results in numerical experiments of unidirectional water wave fields described by the modified nonlinear Schrödinger equation. The presented approach introduces a new paradigm for understanding and predicting intermittent and localized events in dynamical systems characterized by uncertainty and potentially strong nonlinear mechanisms.

Green’s function retrieval in a field of random water waves

It has recently been demonstrated using a variety of wave types that cross-correlating time series of apparently random waves measured at two locations yields an estimate of the Green’s function that describes the wave field generated at one of those locations and measured at the other. This procedure can be described as random wave interferometry. In this paper random surface gravity wave interferometry is described theoretically, demonstrated using numerical simulations, and investigated experimentally using both wavetank measurements and ocean wave measurements. Simulations and wavetank measurements are in good agreement with theoretical predictions, but the ocean-measurement-based cross correlations do not yield the predicted structure. Possible explanations are discussed.