Nonlinear Wave Model Research Articles

Pulsed Bessel-Gauss beams: a depleted wave model for type II second-harmonic generation.

In this work, a three-dimensional and time-dependent nonlinear wave model to describe the generation of pulsed Bessel-Gauss second-harmonic waves (SHWs) is presented. Three coupled equations, two for ordinary and extraordinary fundamental waves and one for extraordinary SHWs, describing type II second-harmonic generation (SHG) in a KTiOPO4 (KTP) crystal were solved by considering the depletion of fundamental waves (FWs). The results examined the validity of nondepleted wave approximation against the energy of pulses, beam spot size, and interaction length. It was shown that for pulses with spot sizes of ωf=80 μm and energy of 0.8j, the nonlinear interaction was accomplished over a distance of ∼5 mm. Therefore, for KTP crystals with lengths longer than 5mm, the nondepleted wave approximation can no longer be valid. To be valid, the crystal must be shorter than the interaction length, i.e., 5mm.

Soft shore protection methods: The use of advanced numerical models in the evaluation of beach nourishment

Beach nourishment is one of the worldwide most common soft shore protection methods. However, the design of these projects is usually based on empirical equations and rules, leaving large margins of error regarding their expected efficiency. In the present work, an advanced wave and sediment transport numerical model is developed and tested in the evaluation of beach nourishment. Non-linear wave transformation in the surf and swash zone is computed by a non-linear breaking wave model based on the higher order Boussinesq equations, for breaking and non-breaking waves. The new Camenen and Larson (2007), transport rate formula for non-cohesive sediments (involving unsteady aspects of the sand transport phenomenon) is adopted for estimating the sheet flow sediment transport rates, as well as the bed load and suspended load over ripples. Suspended sediment transport rate is incorporated by solving the 2DH depth-integrated transport equation. Model results are compared with experimental data of both profile (cross-shore) and planform morphology evolution; the agreement between the two is considered to be quite satisfactory.

Discussion of “A 2DH nonlinear Boussinesq-type wave model of improved dispersion, shoaling, and wave generation characteristics” by Chondros and Memos

The procedure proposed by Chondros and Memos [Chondros and Memos, 2014, A 2DH nonlinear Boussinesq-type wave model of improved dispersion, shoaling, and wave generation characteristics, Coast. Eng., 91: 99–122] for improving dispersion, shoaling and wave generation characteristics of a 2DH Boussinesq-type wave model is discussed. We realize that the rationale behind utilizing variable coefficients in Boussinesq equations is not valid for natural waves and the technique of using surface elevation to obtain wave number in numerical simulation is inaccurate. In addition, the numerical examples provided in the paper are also not convincing.

Magnetization boundary conditions at a ferromagnetic interface of finite thickness

We develop a systematic approach to derive boundary conditions at an interface between two ferromagnetic materials in the continuous medium approximation. The approach treats the interface as a two-sublattice material, although the final equations connect magnetizations outside of the interface and therefore do not explicitly depend on its structure. Instead, the boundary conditions are defined in terms of some average properties of the interface, which may also have a finite thickness. In addition to the interface anisotropy and symmetric exchange coupling, this approach allows us to take into account coupling resulting from inversion symmetry breaking in the vicinity of the interface, such as the Dzyaloshinskii-Moriya antisymmetric exchange interaction. In the case of negligible interface anisotropy and Dzyaloshinskii-Moriya exchange parameters, the derived boundary conditions represent a generalization of those proposed earlier by Barnaś and Mills and are therefore named ‘generalized Barnaś–Mills boundary conditions’. We demonstrate how one could use the boundary conditions to extract parameters of the interface via fitting of appropriate experimental data. The developed theory could be applied to modeling of both linear and non-linear spin waves, including exchange, dipole-exchange, magnetostatic, and retarded modes, as well as to calculations of non-uniform equilibrium micromagnetic configurations near the interface, with a direct impact on the research in magnonics and micromagnetism.

Non-linear plane waves in materials having hexagonal internal structure

Three different continuum limits for modeling non-linear plane waves in two-dimensional hexagonal lattice are obtained. New coupled non-linear continuum equations are obtained to study the interaction of a macro-strain wave and the waves caused by variations in an internal structure. New analytical solutions are obtained to describe localized non-linear strain waves. It is shown that the solutions are different from those of the 1D lattice model due to the inclusion of non-neighboring interactions in a lattice.

Modelling of the spatial evolution of extreme laboratory wave Heights with the nonlinear Schrödinger and Dysthe equations

The spatial evolution of nonlinear wave groups with the same initial Gaussian envelope are theoretically studied under different kind of governing equations, confirming that the higher order nonlinear Schrödinger equation formulation of Dysthe is capable of representing the asymmetric features of propagating wave packets which have narrow spectral bandwidth in deep water. Moreover, laboratory experiments run in an offshore wave basin are systematically compared with a large set of numerical simulations, respectively performed by using the third-order and fourth-order nonlinear wave models, mainly focusing on the aspects of higher order statistical moments such as the coefficients of skewness and kurtosis of the surface elevation, and the exceedance distributions of wave heights in different random sea states. The modulational instability, resulting from a quasi-resonant four wave interaction in the unidirectional sea state, can be indicated by the coefficient of kurtosis, which has shown an appreciable correlation with the abnormal wave density from a statistical point of view. Compared with Dysthe simulation, the NLS model presents a more intensive energy focusing, particularly at the intermediate stage of evolution process in the presence of strong nonlinearity.

A 2DH nonlinear Boussinesq-type wave model of improved dispersion, shoaling, and wave generation characteristics

A modified Boussinesq-type model is derived to account for the propagation of either regular or irregular waves in two horizontal dimensions. An improvement of the dispersion and shoaling characteristics of the model is obtained by optimizing the coefficients of each term in the momentum equation, expanding in this way its applicability in very deep waters and thus overcoming a shortcoming of most models of the same type. The values of the coefficients are obtained by an inverse method in such a way as to satisfy exactly the dispersion relation in terms of both first and second-order analyses matching in parallel the associated shoaling gradient. Furthermore a physically more sound way to approach the evaluation of wave number in irregular wave fields is proposed. A modification of the wave generator boundary condition is also introduced in order to correctly simulate the phase celerity of each input wave component. The modified model is applied to simulate the propagation of breaking and non-breaking, regular and irregular, long and short crested waves in both one and two horizontal dimensions, in a variety of bottom profiles, such as of constant depth, mild slope, and in the presence of submerged obstacles. The simulations are compared with experimental data and analytical results, indicating very good agreement in most cases.

Calculating crest statistics of shallow water nonlinear waves based on standard spectra and measured data at the Poseidon platform

This paper introduces a new approach for the calculation of the wave crest distribution of shallow water nonlinear waves by utilizing a Transformed Rayleigh method where the second order nonlinear wave model is incorporated. The proposed Transformed Rayleigh method is based on a deterministic and time instantaneous functional transformation. It is very efficient and accurate and can be used for engineering purposes. Meanwhile, the correction for the effect of bottom on the wave spectrum, a procedure that is frequently overlooked by some researchers, has been integrated into the Transformed Rayleigh method. The proposed new approach has been first applied for calculating the wave crest height exceedance probabilities of sea states with standard JONSWAP spectra corresponding to different water depths, and the calculation results have been favorably validated against Monte Carlo simulation results. Meanwhile, the wave crest height exceedance probabilities of these sea states obtained from using the Jahns and Wheeler finite depth wave crest height distribution model have also been included for comparison purpose. It is found that in all cases the Transformed Rayleigh method can offer better predictions than the Jahns and Wheeler model. The Transformed Rayleigh method is then applied to calculate the wave crest height exceedance probabilities of a combined sea state with a bimodal Torsethaugen spectrum corresponding to a water depth of 25m, and its accuracy and efficiency are again favorably validated by using Monte Carlo simulations. Finally, the Transformed Rayleigh method is applied to predict the wave crest height exceedance probabilities of a sea state with the surface elevation data measured at the Poseidon platform in the Japan Sea, and its accuracy and efficiency have been once again convincingly substantiated.

Nonlinear Waves and Dispersive Equations

Nonlinear dispersive equations are models for nonlinear waves in a wide range of physical contexts. Mathematically they display an interplay between linear dispersion and nonlinear interactions, which can result in a wide range of outcomes from finite time blow-up to scattering. They are linked to many areas of mathematics and physics, ranging from integrable systems and harmonic analysis to fluid dynamics and general relativity. The conference did focus on the analytic aspects and PDE aspects.

Finite-depth effects on solitary waves in a floating ice sheet

A theoretical and numerical study of two-dimensional nonlinear flexural-gravity waves propagating at the surface of an ideal fluid of finite depth, covered by a thin ice sheet, is presented. The ice-sheet model is based on the special Cosserat theory of hyperelastic shells satisfying Kirchhoff׳s hypothesis, which yields a conservative and nonlinear expression for the bending force. From a Hamiltonian reformulation of the governing equations, two weakly nonlinear wave models are derived: a 5th-order Korteweg–de Vries equation in the long-wave regime and a cubic nonlinear Schrödinger equation in the modulational regime. Solitary wave solutions of these models and their stability are analysed. In particular, there is a critical depth below which the nonlinear Schrödinger equation is of focusing type and thus admits stable soliton solutions. These weakly nonlinear results are validated by comparison with direct numerical simulations of the full governing equations. It is observed numerically that small- to large-amplitude solitary waves of depression are stable. Overturning waves of depression are also found for low wave speeds and sufficiently large depth. However, solitary waves of elevation seem to be unstable in all cases.

Interaction of surface waves at very close wavenumbers

We show that interaction of two monochromatic waves at the water surface enters a different dynamic regime if their wavenumbers become very close. The study is conducted by means of a fully nonlinear wave model. In the course of evolution of the two waves, downshifting of the initial wave energy and growth of the first mode occur depending on wave steepness and dk/k. Behaviour of these features changes if dk/k < 0.0025: both downshifting and growth rate become independent of dk/k, accompanied by rapid transfer of wave energy to large scales.

Interface depth used in a two-layer model of nonlinear internal waves

A two-layer model includes three parameters: interface depth h 1, upper layer density $$\rho_{1}$$ , and lower layer density $$\rho_{2}$$ . Many theoretical and laboratorial studies of internal waves, as well as most numerical models, are based on the two-layer assumption. However, these three parameters cannot be directly measured because a pycnocline in the real ocean has finite thickness, and the densities in both the mixed layer and the deep ocean are not constant. In the present study, seven different methods are used to determine the interface depth of the two-layer model and compared with the depth of maximum vertical displacement: the depth of maximum buoyancy frequency (Ν max), the depth where the first mode eigenfunction has its maximum (Φmax), the depth where the lowest mode temperature empirical orthogonal function has its maximum, the depth where either the two-layer Korteweg–de Vries (KdV) or Benjamin–Ono equation has closest coefficients with their continuously stratified counterparts, and the same KdV approach with stratification replaced by two idealized distributions. The multi-ship measurement conducted near the Luzon Strait is used for deep ocean comparison, and two measurements conducted in the east of Dongsha Atoll are used for shallow water comparison. The results show that in the deep ocean, the KdV approach with idealized type I stratification gives the interface closest to the depth of maximum vertical displacement. In shallow waters, the KdV approach agrees with the measurement best.

Evaluation of nearshore wave models in steep reef environments

To provide coastal engineers and scientists with a quantitative evaluation of nearshore numerical wave models in reef environments, we review and compare three common- ly used models with detailed laboratory observations. These models are the following: (1) SWASH (Simulating WAves till SHore) (Zijlema et al. 2011), a phase-resolving nonlinear shallow-water wave model with added nonhydrostatic terms; (2) SWAN (Simulating WAve Nearshore) (Booij et al. 1999), a phase-averaged spectral wave model; and (3) XBeach (Roelvink et al. 2009), a coupled phase-averaged spectral wave model (applied to modeling sea-swell waves) and a nonlinear shallow-water model (applied to modeling infragravity waves). A quantitative assessment was made of each model's ability to predict sea-swell (SS) wave height, infragravity (IG) wave height, wave spectra, and wave setup (η) at fivelocationsacross the laboratory fringingreefprofile of Demirbilek et al. (2007). Simulations were performed with the "recommended"empirical coefficients as documented for each model, and then the key wave-breaking parameter for each model (α in SWASH and γ in both SWAN and XBeach) was optimized to most accurately reproduce the observations. SWASH, SWAN, and XBeach were found to be capable of predicting SS wave height variations across the steep fringing reef profile with reasonable accuracy using the default coeffi- cients. Nevertheless, tuning of the key wave-breaking parameter improved the accuracy of each model's predictions. SWASH and XBeach were also able to predict IG wave height and spectral transformation. Although SWAN was capable of modeling the SS wave height, in its current form, it was not capable of modeling the spectral transformation into lower frequencies, as evident in the underprediction of the low- frequency waves.

Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model

The aim of this work is the quantification and prediction of rare events characterized by extreme intensity in nonlinear waves with broad spectra. We consider a one-dimensional nonlinear model with deep-water waves dispersion relation, the Majda–McLaughlin–Tabak (MMT) model, in a dynamical regime that is characterized by a broadband spectrum and strong nonlinear energy transfers during the development of intermittent events with finite-lifetime. To understand the energy transfers that occur during the development of an extreme event we perform a spatially localized analysis of the energy distribution along different wavenumbers by means of the Gabor transform. A statistical analysis of the Gabor coefficients reveals (i) the low-dimensionality of the intermittent structures, (ii) the interplay between non-Gaussian statistical properties and nonlinear energy transfers between modes, as well as (iii) the critical scales (or critical Gabor coefficients) where a critical amount of energy can trigger the formation of an extreme event. We analyze the unstable character of these special localized modes directly through the system equation and show that these intermittent events are due to the interplay of the system nonlinearity, the wave dispersion, and the wave dissipation which mimics wave breaking. These localized instabilities are triggered by random localizations of energy in space, created by the dispersive propagation of low-amplitude waves with random phase. Based on these properties, we design low-dimensional functionals of these Gabor coefficients that allow for the prediction of the extreme event well before the nonlinear interactions begin to occur.

The “Carey Number” continued—Mechanisms governing the horizontal array coherence length in shallow water acoustics

The use of simplified “feature models” (geometric idealizations of specific, isolated ocean features) for coastal oceanographic features can allow one to calculate acoustically useful quantities approximately and even generate analytic forms for them. Feature models for coastal fronts, eddies, internal tides, linear and nonlinear internal waves, and spice are presented, and the scattering of sound from these objects is calculated. This allows one to estimate the useful quantity Lcoh, the horizontal coherence length that represents a physical limit for array signal processing. Continuing work on calculations of Lcoh, including source/receiver motion, will be presented. [Work sponsored by the Office of Naval Research.]

Influence of second-order random wave kinematics on the design loads of offshore wind turbine support structures

The impact of wave model nonlinearities on the design loads of wind turbine monopile foundations is delineated based on a second-order nonlinear random wave model that satisfies the boundary conditions at the free surface and by including the effects of convective acceleration in the inertial loads. The second-order nonlinear water kinematics is developed based on a Gram Charlier series expansion using the first four stochastic moments of the wave process. The wave surface velocities and accelerations are expressed using a Taylor series expansion about the mean sea level, which satisfies to the second-order, the unsteady Bernoulli equation and normal flow condition at the free surface. The operating design loads on the monopile are computed using fully coupled and uncoupled methods. The computation of the mud level loads based on the inclusion of nonlinear wave surface kinematics is compared with those obtained when using linear waves and Wheeler stretching. The effect of the spatial derivatives of the wave velocity on the wave surface kinematics is quantified and shown to determine the wave spectral cut-off frequency limit. The spatial derivatives of wave velocity also participate in the expression for the wave convective acceleration, whose effect is demonstrated on the inertial loads on the foundation in the presence of ocean currents. The effect of nonlinear water kinematics on the monopile design load reveals the large frequency bandwidth of wave structure interaction, but the phase differences between the hydrodynamic loads with the rotor loads tend to lower the probability of joint simultaneous extreme peaks in hydrodynamics and rotor loads.

Extended models of non-linear waves in liquid with gas bubbles

In this work we generalize the models for non-linear waves in a gas–liquid mixture taking into account an interphase heat transfer, a surface tension and a weak liquid compressibility simultaneously at the derivation of the equations for non-linear waves. We also take into consideration high order terms with respect to the small parameter. Two new non-linear differential equations are derived for long weakly non-linear waves in a liquid with gas bubbles by the reductive perturbation method considering both high order terms with respect to the small parameter and the above-mentioned physical properties. One of these equations is the perturbation of the Burgers equation and corresponds to main influence of dissipation on non-linear waves propagation. The other equation is the perturbation of the Burgers–Korteweg–de Vries equation and corresponds to main influence of dispersion on non-linear waves propagation.

Local-in-Space Criteria for Blowup in Shallow Water and Dispersive Rod Equations

We unify a few of the best known results on wave breaking for the Camassa--Holm equation (by R. Camassa, A. Constantin, J. Escher, L. Holm, J. Hyman and others) in a single theorem: a sufficient condition for the breakdown is that $u_0'+|u_0|$ is strictly negative in at least one point $x_0$ of the real line. Such blowup criterion looks more natural than the previous ones, as the condition on the initial data is purely local in the space variable. Our method relies on the introduction of two families of Lyapunov functions. Contrary to McKean's necessary and sufficient condition for blowup, our approach applies to other equations that are not integrable: we illustrate this fact by establishing new local-in-space blowup criteria for an equation modeling nonlinear dispersive waves in elastic rods.

An improved wave-vector frequency-domain method for nonlinear wave modeling

In this paper, a recently developed wave-vector frequency-domain method for nonlinear wave modeling is improved and verified by numerical simulations and underwater experiments. Higher order numeric schemes are proposed that significantly increase the modeling accuracy, thereby allowing for a larger step size and shorter computation time. The improved algorithms replace the left-point Riemann sum in the original algorithm by the trapezoidal or Simpson's integration. Plane waves and a phased array were first studied to numerically validate the model. It is shown that the left-point Riemann sum, trapezoidal, and Simpson's integration have first-, second-, and third-order global accuracy, respectively. A highly focused therapeutic transducer was then used for experimental verifications. Short high-intensity pulses were generated. 2-D scans were conducted at a prefocal plane, which were later used as the input to the numerical model to predict the acoustic field at other planes. Good agreement is observed between simulations and experiments.

非線形長波モデルと流体粒子法による津波シミュレータの開発

We develop a tsunami simulator integrating a 3-D fluid simulation technology that runs on large-scale parallel computers using smoothed-particle hydrodynamics (SPH) method, together with a 2-D tsunami propagation simulation technique using a nonlinear shallow water wave model. We use the 2-D simulation to calculate tsunami propagation of scale of about 1000km from epicenter to near shore. The 3-D SPH method can be used to calculate the force that a tsunami can exert on a building, and to simulate flooding patterns in urban area of at most km scale. By applying the processing power of computers to the technologies resulting from this research, we seek to contribute to improved disaster preparedness and disaster mitigation through a better understanding of tsunami's mechanisms.