Nonlinear Random Waves Research Articles

Nonlinear Wave Kinematics near the Ocean Surface

AbstractEstimation of second-order, near-surface wave kinematics is important for interpretation of ocean surface remote sensing and surface-following instruments, determining loading on offshore structures, and understanding of upper-ocean transport processes. Unfortunately, conventional wave theories based on Stokes-type expansions do not consider fluid motions at levels above the unperturbed fluid level. The usual practice of extrapolating the fluid kinematics from the unperturbed free surface to higher points in the fluid is generally reasonable for narrowband waves, but for broadband ocean waves this results in dramatic (and nonphysical) overestimation of surface velocities. Consequently, practical approximations for random waves are at best empirical and are often only loosely constrained by physical principles. In the present work, the authors formulate the governing equations for water waves in an incompressible and inviscid fluid, using a boundary-fitted coordinate system (i.e., sigma or s coordinates) to derive expressions for near-surface kinematics in nonlinear random waves from first principles. Comparison to a numerical model valid for highly nonlinear waves shows that the new results 1) are consistent with second-order Stokes theory, 2) are similar to extrapolation methods in narrowband waves, and 3) greatly improve estimates of surface kinematics in random seas.

On quantitative errors of two simplified unsteady models for simulating unidirectional nonlinear random waves on large scale in deep sea

To investigate nonlinear random wave dynamics or statistics, direct phase-resolved numerical simulation of nonlinear random waves in deep sea on large-spatial and long-temporal scales are often performed by using simplified numerical models, such as those based on the Nonlinear Schrödinger Equation (NLSE). They are efficient and can give sufficiently acceptable results in many cases but they are derived by assuming narrow bandwidth and small steepness. So far, there has been no formula to precisely predict the quantitative errors of such simplified models. This paper will present such formulas for estimating the errors of enhanced NLSE based on the Fourier transform and quasi spectral boundary integral method when they are applied to simulate ocean waves on large-spatial and long-temporal scales (about 128 peak wavelengths and 1000 peak periods). These formulas are derived by fitting the errors of the simplified models, which are estimated by comparing their wave elevations with these obtained by using a fully nonlinear model for simulating the cases with initial conditions defined by two commonly used ocean wave spectra with a wide range of parameters. Based on them, the suitable regions for the simplified models to be used are shown.

Nonlinear random optical waves: Integrable turbulence, rogue waves and intermittency

We examine the general question of statistical changes experienced by ensembles of nonlinear random waves propagating in systems ruled by integrable equations. In our study that enters within the framework of integrable turbulence, we specifically focus on optical fiber systems accurately described by the integrable one-dimensional nonlinear Schrödinger equation. We consider random complex fields having a Gaussian statistics and an infinite extension at initial stage. We use numerical simulations with periodic boundary conditions and optical fiber experiments to investigate spectral and statistical changes experienced by nonlinear waves in focusing and in defocusing propagation regimes. As a result of nonlinear propagation, the power spectrum of the random wave broadens and takes exponential wings both in focusing and in defocusing regimes. Heavy-tailed deviations from Gaussian statistics are observed in focusing regime while low-tailed deviations from Gaussian statistics are observed in defocusing regime. After some transient evolution, the wave system is found to exhibit a statistically stationary state in which neither the probability density function of the wave field nor the spectrum changes with the evolution variable. Separating fluctuations of small scale from fluctuations of large scale both in focusing and defocusing regimes, we reveal the phenomenon of intermittency; i.e., small scales are characterized by large heavy-tailed deviations from Gaussian statistics, while the large ones are almost Gaussian.

Modelling of the temporal and spatial evolutions of weakly nonlinear random directional waves with the modified nonlinear Schrödinger equations

The temporal and spatial evolutions of nonlinear wave group with an initial Gaussian envelope are theoretically studied under the governing of MNLS equations, demonstrating that the temporal and spatial versions of numerical model are not always consistent in the whole evolution process, particularly in the presence of strong nonlinearity. Moreover, a large set of numerical simulations, performed respectively by these two versions of numerical model, are systematically compared to mechanically generated waves with different initial directional spreading and Benjamin–Feir Index, mainly focusing on the evolution properties of surface elevations such as the coefficients of skewness and kurtosis, the probability density function, and the maximal surface elevation. On the whole, it can be argued that the statistical properties of both numerically simulated wave fields are basically consistent with the laboratory observations.

Drag forces on aquatic plants in nonlinear random waves plus current

Plant-flow interactions are characterised by an assemblage of processes acting at different temporal and spatial scales. In order to mathematically characterise these interactions, such processes have to be parameterised given some simplifications. Typically, drag coefficients are derived from experiments to characterise the plant reconfiguration and wave energy dissipation processes. By reviewing the different plant drag coefficients CD valid in oscillatory flows, this study first highlights the lack of normalisation of the different existing CD formulations and identifies possibilities for a standardisation of the formulations for oscillatory and steady flows. Then, by taking into account the wave crest height distribution of a sea state condition, this study further develops a stochastic method to compute the expected wave induced forces on a plant in linear/nonlinear random waves plus current based on two different CD formulations for waves alone and waves plus current. This method improves the characterisation of the stochastic plant–flow interactions by allowing the calculation of expected values under different random wave plus currents conditions. Results are compared to a classic deterministic approach and some differences are identified, calling for further investigations against experimental datasets. Based on the appropriate CD formulations, this study finally revealed that wave nonlinearities have a significant effect on expected wave forces for a higher wave activity, and that in presence of an increasing current, the effect of wave nonlinearities decreases while the expected wave forces increase.

Suspended sediments due to random waves including effects of second order wave asymmetry and boundary layer streaming

The paper provides a practical stochastic method by which the suspended sediment concentration due to long-crested (2D) and short-crested (3D) nonlinear random waves can be calculated. The approach is based on assuming the waves to be a stationary narrow-band random process, and by using the parameterized formulas valid for regular waves presented in Soulsby, R.L., 1997. Dynamics of Marine Sands. Thomas Telford, London. The Forristall, G.Z., 2000. Journal of Physical Oceanography. 30, 1931–1943. wave crest height distribution representing both 2D and 3D nonlinear random waves is also adopted. The model covers sediment suspension over rippled beds and for sheet flow. Comparisons are made with random wave data from Thorne, P.D., Williams, J.J., Davies, A.G., 2002. Journal of Geophysical Research. 107 (C8), 3178 for flow over rippled beds. An example for sheet flow using data typical to field conditions is also included to illustrate the approach.

Equivalent design wave approach for structural analysis of floating pendulum wave energy converter

Offshore structures are installed in a variety of water depths, under harsh environments, for long stretches of time, and waves are the most interrelated factor directly affecting their structural integrity. Thus, such structures should be designed to satisfy the demands of wave characteristics. In the preliminary structural design phase, a method for predicting non-linear random wave characteristics and extreme wave conditions must be determined. A structural response is not a linear value corresponding to maximum wave height, due to the related periods; that is, certain waves of less than maximum height may have greater effects on a structure depending on the wave period. This study focuses on selecting design conditions that generate a significant hydrodynamic load on a floating pendulum wave energy converter (FPWEC). For this purpose, it finds an equivalent regular wave height capable of generating the same hydrodynamic load, critical wave period and wave direction by conducting a response analysis. Generally, the subsequent equivalent design wave conditions are used to investigate an FPWEC's global strength. This study also applies a global structural analysis for the FPWEC using the determined equivalent design wave. The obtained results should be useful for understanding the global structural responses of FPWECs.

Intermittency in integrable turbulence.

We examine the statistical properties of nonlinear random waves that are ruled by the one-dimensional defocusing and integrable nonlinear Schrödinger equation. Using fast detection techniques in an optical fiber experiment, we observe that the probability density function of light fluctuations is characterized by tails that are lower than those predicted by a Gaussian distribution. Moreover, by applying a bandpass frequency optical filter, we reveal the phenomenon of intermittency; i.e., small scales are characterized by large heavy-tailed deviations from Gaussian statistics, while the large ones are almost Gaussian. These phenomena are very well described by numerical simulations of the one-dimensional nonlinear Schrödinger equation.

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.

On the Propagation of Weakly Nonlinear Random Dispersive Waves

We study several basic dispersive models with random periodic initial data such that the different Fourier modes are independent random variables. Motivated by the vast physics literature on related topics, we then study how much the Fourier modes of the solution at later times remain decorrelated. Our results are sensitive to the resonances associated with the dispersive relation and to the particular choice of the initial data.

Burial and scour of truncated cones due to long-crested and short-crested nonlinear random waves

This paper provides a practical stochastic method by which the burial and scour depths of truncated cones exposed to long-crested (2D) and short-crested (3D) nonlinear random waves can be derived. The approach is based on assuming the waves to be a stationary narrow-band random process, adopting the Forristall (2000) wave crest height distribution representing both 2D and 3D nonlinear random waves. Moreover, the formulas for the burial and the scour depths for regular waves presented by Catano-Lopera et al. (2011) for truncated cones are used. An example of calculation is also presented.

Simulation of irregular waves over submerged obstacle on a NURBS potential numerical wave tank

In this paper, a fully non-linear three-dimensional Numerical Wave Tank (NWT) is developed for studying propagation and scattering of non-linear random sea wave over bottom submerged bars. The simulation of fully non-linear free-surface is based on Non-Uniform Rational B-Spline formulation (NURBS) as a novel approach and Mixed Eulerian-Lagrangian method (MEL). High-order boundary integral equation is used to solve the Laplace equation in the Eulerian frame. To update the free-surface, time marching approach including material node method and fourth order Runge-Kutta time integration scheme is used. To obtain appropriate numerical solutions for wave propagation problem, damping zone is set at the downstream. Also, the NURBS approximation is employed to evaluate the velocity of the free-surface particles. Propagation of regular and irregular waves in a NWT is investigated and compared with the available experimental and numerical data. Transmission of the random sea wave over submerged bars is also compared with the experimental and prior numerical studies.

反射波を伴う非線形性不規則波の水位・水面勾配の結合確率分布に関する研究

反射波を伴う非線形性不規則波の水位・水面勾配の結合確率分布に関する研究

Large-time evolution of statistical moments of wind–wave fields

AbstractWe study the long-term evolution of weakly nonlinear random gravity water wave fields developing with and without wind forcing. The focus of the work is on deriving, from first principles, the evolution of the departure of the field statistics from Gaussianity. Higher-order statistical moments of elevation (skewness and kurtosis) are used as a measure of this departure. Non-Gaussianity of a weakly nonlinear random wave field has two components. The first is due to nonlinear wave–wave interactions. We refer to this component as ‘dynamic’, since it is linked to wave field evolution. The other component is due to bound harmonics. It is non-zero for every wave field with finite amplitude, contributes both to skewness and kurtosis of gravity water waves and can be determined entirely from the instantaneous spectrum of surface elevation. The key result of the work, supported both by direct numerical simulation (DNS) and by the analysis of simulated and experimental (JONSWAP) spectra, is that in generic situations of a broadband random wave field the dynamic contribution to kurtosis is small in absolute value, and negligibly small compared with the bound harmonics component. Therefore, the latter dominates, and both skewness and kurtosis can be obtained directly from the instantaneous wave spectra. Thus, the departure of evolving wave fields from Gaussianity can be obtained from evolving wave spectra, complementing the capability of forecasting spectra and capitalizing on the existing methodology. We find that both skewness and kurtosis are significant for typical oceanic waves; the non-zero positive kurtosis implies a tangible increase of freak wave probability. For random wave fields generated by steady or slowly varying wind and for swell the derived large-time asymptotics of skewness and kurtosis predict power law decay of the moments. The exponents of these laws are determined by the degree of homogeneity of the interaction coefficients. For all self-similar regimes the kurtosis decays twice as fast as the skewness. These formulae complement the known large-time asymptotics for spectral evolution prescribed by the Hasselmann equation. The results are verified by the DNS of random wave fields based on the Zakharov equation. The predicted asymptotic behaviour is shown to be very robust: it holds both for steady and gusty winds.

Effects of Sand-Clay Mixtures on Scour Around Vertical Piles Due to Long-Crested and Short-Crested Nonlinear Random Waves

This note provides a practical stochastic method by which the effects of sand-clay mixtures on the maximum equilibrium scour depth around vertical piles exposed to long-crested (2D) and short-crested (3D) nonlinear random waves can be derived. This is made by using the regular wave formulas for scour depth for sand-clay mixtures by Dey et al. (2011, Scour at Vertical Piles in Sand-Clay Mixtures Under Waves,” J. Waterway, Port, Coastal, Ocean Eng., 137(6), pp. 324–331) and the stochastic method presented by Myrhaug and Ong (2013, “Scour Around Vertical Pile Foundations for Offshore Wind Turbines Due to Long-Crested and Short-Crested Nonlinear Random Waves,” ASME J. Offshore Mech. Arctic Eng., 135(1), p. 011103). Thus the present results are supplementary to those presented in the latter reference. An example calculation is provided.

Scour Around Vertical Pile Foundations for Offshore Wind Turbines Due to Long-Crested and Short-Crested Nonlinear Random Waves

This paper provides a practical stochastic method by which the maximum equilibrium scour depth around vertical piles exposed to long-crested (2D) and short-crested (3D) nonlinear random waves can be derived. The approach is based on assuming the waves to be a stationary narrow-band random process, adopting the Forristall wave crest height distribution (Forristall, 2000, “Wave Crest Distributions: Observations and Second-Order Theory,” J. Phys. Oceanogr., 30, pp. 1931–1943) representing both 2D and 3D nonlinear random waves, and using the regular wave formulas for scour depth by Sumer et al. (1992, “Scour Around Vertical Pile in Waves,” J. Waterway, Port, Coastal, Ocean Eng., 114(5), pp. 599–641). An example calculation is provided. Tentative approaches to related random wave-induced scour cases are also suggested.

Addendum to “Bottom friction and erosion beneath long-crested and short-crested nonlinear random waves”: Relevance for plug flow formation

The purpose of this note is to point out the relevance and applicability of the results in Myrhaug and Holmedal (2011) for plug flow formation beneath 2D and 3D nonlinear random waves.

非線形不規則波の水位・水面勾配の同時計測およびその結合確率分布に関する研究

非線形不規則波の水位・水面勾配の同時計測およびその結合確率分布に関する研究

Evolution of basic equations for nearshore wave field

In this paper, a systematic, overall view of theories for periodic waves of permanent form, such as Stokes and cnoidal waves, is described first with their validity ranges. To deal with random waves, a method for estimating directional spectra is given. Then, various wave equations are introduced according to the assumptions included in their derivations. The mild-slope equation is derived for combined refraction and diffraction of linear periodic waves. Various parabolic approximations and time-dependent forms are proposed to include randomness and nonlinearity of waves as well as to simplify numerical calculation. Boussinesq equations are the equations developed for calculating nonlinear wave transformations in shallow water. Nonlinear mild-slope equations are derived as a set of wave equations to predict transformation of nonlinear random waves in the nearshore region. Finally, wave equations are classified systematically for a clear theoretical understanding and appropriate selection for specific applications.

Scour around spherical bodies due to long-crested and short-crested nonlinear random waves

This paper provides a practical stochastic method by which the maximum equilibrium scour depth around spherical bodies exposed to long-crested (2D) and short-crested (3D) nonlinear random waves can be derived. The approach is based on assuming the waves to be a stationary narrow-band random process, adopting the Forristall (2000) wave crest height distribution representing both 2D and 3D nonlinear random waves, and using the regular wave formulas for scour and self-burial depths by Truelsen et al. (2005). An example calculation is provided.