Occurrence probability of a freak wave in a nonlinear wave field

The Edgeworth’s form of a cumulative expansion of the probability density function (PDF) of surface elevation expands the maximum wave height distribution to predict the occurrence probability of freak waves. This study investigated the enhancement of the occurrence probability of freak waves due to the fourth order moment of surface elevation, kurtosis, change and found that the nonlinear effects on the occurrence probability of a freak wave linearly depends on kurtosis for a small number of waves N=250. The statistical theory was compared with field data, and freak waves sometimes appear when not expected by the Rayleigh theory, but they were predicted by the proposed theory.     

Comparison of linear and nonlinear extreme wave statistics

An extremely large (“freak”) wave is a typical though rare phenomenon observed in the sea. Special theories (for example, the modulation instability theory) were developed to explain mechanics and appearance of freak waves as a result of nonlinear wave-wave interactions. In this paper, it is demonstrated that the freak wave appearance can be also explained by superposition of linear modes with the realistic spectrum. The integral probability of trough-to-crest waves is calculated by two methods: the first one is based on the results of the numerical simulation of a wave field evolution performed with one-dimensional and two-dimensional nonlinear models. The second method is based on calculation of the same probability over the ensembles of wave fields constructed as a superposition of linear waves with random phases and the spectrum similar to that used in the nonlinear simulations. It is shown that the integral probabilities for nonlinear and linear cases are of the same order of values     

A note on nearshore wave features: Implications for wave generation

This paper analyses 10 years of wave data from the Mediterranean Spanish (Catalan) coast considering the mean wave climate and storm events from the standpoint of wind-wave momentum transfer and wave prediction. The data, registered by a buoy at about 12 km from the coastline, revealed two main groups of wave storms, with NW and E directions. NW storms correspond to a fetch-limited situation since the intense wind blows from land. Low-pressure centres located over the Mediterranean Sea produce easterly storms. Near the coast the eastern winds from the sea are replaced by NW winds coming from meteorological patterns over northern Spain and south-western France. Wave storms are classified and studied to obtain their main features (including spectral width, wave length, wave age and bimodality) and discussed in terms of wind-wave momentum transfer for operational wave predictions. Observations show a complex coastal wave climate. Fetch-limited storms presented smaller spectral widths while varying wind situations presented larger widths due to the presence of bimodal spectra. These wave features are highly relevant for wind–ocean momentum transfer and, thus, for current and wave predictions. The spectral width proved to be a good indicator of sea complexity and is thus applicable for improved wind drag estimations. A new drag coefficient formulation is proposed, based on existing wind dependent drag expressions, but including also spectral wave properties (a spectral width parameter) that highlights the characteristics of wind-wave generation under pre-existing swell. Such a formulation, once properly validated with field observations, is expected to improve wind-wave predictions.     

A note on wave action conservation in a dissipative current wave motion

We consider steady, slowly varying water waves propagating on a steady current over a gently sloping bed, so-called current depth refraction. All expressions are correct to second order in wave amplitude. Formulating the energy equation for the fluctuating motion in terms of wave action (wave energy divided by intrinsic angular frequency) results in an expression, where the dissipative term is strikingly similar to wave action itself. It is simply the ‘extra’ dissipation (per unit area) caused by the fluctuating motion (i.e. total dissipation minus the effect of current acting on total mean bed shear stress) divided by the intrinsic angular frequency. We call it ‘wave action dissipation’. An inconsistency in Phillips' (1977) book is pointed out. A new formula for the calculation of wave amplitudes along rays is set forth.     

A note on significant wave height

In this note conservative bounds for significant crest height and amplitude obtained from the crossing intensity of a sea are presented. For Gaussian models of a sea level, the Rayleigh approximation for the distributions of amplitude and crest height is proved to provide conservative values for the expected significant wave characteristics. The results are illustrated by examples in which both Gaussian and non-Gaussian models for a sea are considered.     

Effects of wave breaking on wave statistics for deep-water random wave train

The experimental studies of the breaking effects on wave statistics for deep-water random waves are presented. It is especially focused on the behavior of kurtosis of surface elevations due to wave breaking. Wave breaking suppresses the maximum limit of kurtosis of the surface elevation, although skewness depends on characteristic wave steepness. The mean instantaneous wave steepness of breaking waves defined using the zero-down-crossing method was much lower than expected from the Stokes waves.     

Extreme value statistics for wave run-up on a natural beach

Statistics of wave run-up maxima have been calculated for 149 35-minutes data runs from a natural beach. During the experiment incident wave height varied from 0.4 to 4.0 m, incident wave period from 6 to 16 s, and beach face slope from 0.07 to 0.20. Four extreme statistics were calculated; the maximum run-up height during each run, the 2% exceedence level of shoreline elevation, the 2% exceedence height for individual run-up peaks, and the 2% exceedence level for swash height as determined by the zero-upcrossing method. These statistics were best parameterized when normalized by the incident significant wave height and plotted against the Iribarren number, ξ = β/(H/L₀)^1/2. The swash data (with set-up removed) showed less scatter than total run-up (with set-up included). For Iribarren number greater than 1.5 the run-up was dominated by the incident frequencies, for lower Iribarren number longer period motions dominated the swash. A reasonable value of wave steepness for a fully developed storm sea is 0.025 so that a storm Iribarren number can be estimated as 6.3 times the beach slope. Using this and an offshore design wave height, the included graphs may provide guidance in determining a design run-up height.     

弱抽运下光学参量过程中压缩真空场的光子统计性质

在理论和实验上研究了光学参量振荡中产生的弱压缩真空场的光子统计行为. 弱压缩真空具有强烈的光子聚束效应,这种比热光场更强的聚束效应在量子光学和量子测量中具有重要的应用. 利用远离阈值的光学参量振荡(optical parametric oscillator, OPO )过程,在实验上产生了该弱压缩真空输出,运转波长在铯原子线附近. 通过Hanbury-Brown-Twiss(HBT)测量了OPO输出光场的二阶关联函数,实验结果与理论分析基本一致.     

A numerical study of nonlinear wave run-up on a vertical plate

A finite difference model based on a recently derived highly-accurate Boussinesq-type formulation is presented. Up to the third-order space derivatives in terms of the velocity variables are retained, and the horizontal velocity variables are re-formulated in terms of a velocity potential. This decreases the total number of unknowns in two horizontal dimensions from seven to five, simplifying the implementation, and leading to increased computational efficiency. Analysis of the embedded properties demonstrates that the resulting model has applications with errors of 2 to 3% for (wavenumber times depth) kh ≤ 10 in terms of dispersion and kh ≤ 4 in terms of internal kinematics. The stability and accuracy of the discrete linearised systems are also analysed for both potential and velocity formulations and the advantages and disadvantages of each are discussed. The velocity potential model is then used to study physically demanding problems involving highly nonlinear wave run-up on a bottom-mounted (surface-piercing) plate. New cases involving oblique incidence are considered. In all cases, comparisons with recent physical experiments demonstrate good quantitative accuracy, even in the most demanding cases, where the local wave steepness can exceed (waveheight divided by wavelength) H / L = 0.20. The velocity potential model is additionally shown to have numerical advantages when dealing with wave–structure interactions, requiring less smoothing around exterior structural corners.     

关于风浪预报公式和风区宽度的注记

从定常波的风浪波高随风距成长的预报公式出发 ,在一定条件下 ,导出各波向上风浪波高分量与该波向上的风距之间的关系 ,以便在不规则风区上推算风浪波高和平均波向 ;同时 ,对风区宽度作了定量化的分析。     

A discrete correction scheme for envelope approach of wave group statistics

The existence of empty envelope excursions (EEE) brings error to the envelope approach of wave group statistics, which identifies wave group by envelope upcrossing of a critical level. A group number correction scheme is suggested in this paper to exclude EEE from wave group statistics. To this end, the Ditlevson and Lindgren [J. Sound Vib. 122 (1988) 571] theory about the fraction of empty excursion envelopes (FEEE) is examined to see if it fits for ocean waves. The sea waves are simulated with Monte Carlo method and with P-M and JONSWAP spectrums. The values of FEEE of the simulated waves are investigated and compared with the theory of Ditlevson and Lindgren. The comparison shows that, at the second-order approximation, theoretical predictions of FEEE are close to those derived from simulations. This approximate analytical expression of FEEE is then employed to form a group number correction scheme. Comparisons between numerical and theoretical results of wave group properties show that this correction scheme is quite effective.     

A note on wave energy dissipation over steep beaches

This paper revisits the derivation of the parametric surf zone model proposed by Baldock et al. [Baldock, T. E., Holmes, P., Bunker, S. & Van Weert, P. 1998 Cross-shore hydrodynamics within an unsaturated surf zone. Coast. Eng. 34, 173–196.]. We show that a consistent use of the proposed Rayleigh distribution for surf zone wave heights results in modification of the expressions for the bulk dissipation rate and enhanced dissipation levels on steep beaches and over-saturated surf zone conditions. As a consequence, the modification proposed herein renders the model robust even on steep beaches where it could otherwise develop a shoreline singularity.     

On an improvement of a nonlinear iterative scheme for nonlinear wave profile prediction

The authors of the present paper present an iterative scheme to calculate the nonlinear wave profiles [Jang, T.S., Kwon, S.H., 2005. Application of nonlinear iteration scheme to the nonlinear water wave problem: Stokian wave. Ocean Engineering, in press]. The nonlinear operator was constructed from the dynamic boundary condition of the free surface. The initial input of the iterative process was linear potential. The linear dispersion relation was utilized. The authors of the present paper suggest an improved scheme in terms of accuracy and speed of convergence by utilizing the nonlinear dispersion relation. The existence and uniqueness of the improved scheme are illustrated in this paper. The calculation results together with Fast Fourier transform revealed that the improved scheme made it possible to predict higher-order nonlinear characteristics of the Stokes’ wave.     

A nonlinear wave profile correction of the diffraction of a wave by a long breakwater: Fixed point approach

The authors of the present paper have suggested an iterative scheme to calculate the nonlinear wave profiles [Jang and Kwon, 2005. Application of nonlinear iteration scheme to the nonlinear water wave problem: Stokes wave. Ocean Engineering 32, 1862–1872]. The scheme was shown to be good for estimating nonlinear wave profiles. In the study, the iterative scheme is applied to the wave-diffraction problem by a long breakwater to calculate a diffracted wave by the breakwater. The iterative solution of diffraction was compared with the linear solution of Sommerfeld, 1896. [Mathematische Theoried der Diffraction. Mathematical Annals 47, 317–374]. For a small wave slope, the two solutions were in good agreement. However, the scheme enabled us to observe the nonlinear behaviors of a beating phenomenon and of wave profile such as Stokes’ wave for a relatively large wave slope: as the wave slope becomes larger, we can examine the nonlinear wave characteristics of the actual shapes of waves, i.e., the crests are steeper and the troughs are flatter.     

A conservative nonlinear filter for the high-frequency range of wind wave spectra

Filtering of the high-frequency part of a wind wave spectrum may be useful in a numerical wind wave model for various reasons. First, it can be used to augment (or be part of) a parameterization of the resonant nonlinear interactions, that are essential to third-generation wind wave models. Second, when combined with a dynamic time stepping scheme for source term integration, it may result in smoother (and hence faster) wave model integration. In this study, such a filter is proposed, based on the traditional Discrete Interaction Approximation (DIA) for the resonant four-wave nonlinear interactions. This filter retains all conservative properties of the interactions. For small time steps and/or smooth spectra, it is formulated as a traditional source term. For larger time steps and/or non-smooth spectra it is formulated as a filter. This formulation guarantees stability of the filter itself and will enhance overall computational stability in a full wave model. The stability properties of this filter are illustrated using traditional wave growth computations. Examples are given where the filter improves model economy, and where it is shown to remove spurious high-frequency noise from a wave model.     

A note on the theoretical comparison of wave staffs and wave rider buoys in steep gravity waves

The sea surface displacements measured by a wave staff, fixed in horizontal position, and a wave rider buoy, which moves with the water particles, are compared. Second-order theory shows that, to this approximation, the second harmonic in the surface displacement is not measured by the wave rider. Here a simple approximation to maximum amplitude gravity waves is considered; then a freely floating wave rider gives an approximately sinusoidal record at a lower frequency than the wave, while a tethered wave rider gives results depending on the mooring system. The horizontal motion of a tethered buoy still affects the measurements, particularly of the second and higher harmonics.     

Effect of sampling rate on wave height statistics

This paper examines the effects of digital processing, and therefore discretisation or sampling, of sea surface elevations that are, in reality, continuous. Using random linear wave theory, probability distributions for the measured (as opposed to actual) wave amplitudes and heights have been obtained which are conditional on the sampling rate. It is shown that with low sampling rates there are significant departures from the usual Rayleigh distribution. Rates of 1 Hz or less may lead to significant underestimation of the probability of very large waves. An analysis of full-scale measurements obtained from a platform in the North Sea supports these results.     

A note on the derivation of wave action balance equation in frequency space

In this paper the wave action balance equation in terms of frequency-direction spectrum is derived.A theoretical formulation is presented to generate an invariant frequency space to replace the varying wavenumber space through a Jacobian transformation in the wave action balance equation.The physical properties of the Jacobian incorporating the effects of water depths are discussed.The results provide a theoretical basis of wave action balance equations and ensure that the wave balance equations used in the SWAN or other numerical models are correct.It should be noted that the Jacobian is omitted in the wave action balance equations which are identical to a conventional action balance equation.     

A note on estimation of the Jacobian elliptic parameter in cnoidal wave theory

A new and simple calculating technique for the Jacobian elliptic parameter is presented in this study. The technique is very useful in generating a train of cnoidal waves in both laboratory and numerical wave tanks. Upon specification of water depth, the wave height and either the wave period or the wavelength, the proposed technique uses the Newton–Raphson method to estimate the Jacobian elliptic parameter directly, without trial and error procedures or look-up in tables. It is shown that the technique provides equally accurate results as the ad hoc methods previously used.