Statistical distribution of nonlinear random water wave surface elevation

This study deals with the development of statistical modeling for water wave surface elevation by using a method that combines a dynamic solution with random process statistics. Ocean wave data taken from four NOAA (National Oceanic and Atmospheric Administration) buoys moored in the northeast Pacific were used to validate the model. The results indicated that the nonlinear probability density distribution of ocean wave surface elevation derived from the model described the measurements much better than Gaussian distribution and Longuet-Higgins distribution.     

Theoretical and experimental investigation on nonlinear interaction among wave groups

Wave group is important in ocean wave theory and applications. In the past, nonlinear interaction among wave groups has been studied on the basis of the nonlinear Schr?dinger equation. Using this theoretical approach, we found that the nonlinear interaction among wave groups causes asymmetry in the shape of the wave envelope (steeper in the front of the curve of the envelope). An important consequence of this asymmetry is that the highest wave in a wave group appears one individual wave length ahead of the center of the wave group. Further results show that the degree of envelope asymmetry increases with increasing spectral width and the wave steepness. This theoretical analysis has been supplemented by a systematic experimental study of wind waves. Laboratory and some open sea wave data were analyzed. The results show that the shape of the wind wave envelope of wind waves has the same asymmetry predicted by the theoretical approach. The observed degree of deformation of the envelope also increases with increasing spectral width and the wave steepness as predicted by theory. These conclusions have important ramifications for practical applications of ocean wave theory.     

A new probability distribution function of wave steepness

Wave steepness is an important characteristic describing the severity of sea state in ocean engineering. In the existing theoretical and experimental studies, wave steepness is often substituted by some related quantities. In this paper, a new probability density function (pdf) of steepness, which is a pdf of the steepness in its original definition, is obtained for narrowband Gaussian processes. The drawback inherent in the previous theoretical pdfs of steepness, that is, the probability density at zero steepness is nonzero, has been eliminated. Laboratory experiments were conducted in a wind-wave flume to measure the wave steepness distribu- tion. Comparisons among laboratory measurements and some theoretical pdfs of steepness show that the new pdf generally fits the data better than the one proposed by Zheng et al. (1999).     

CALCULATIONS OF WAVE BREAKING PROBABILITY AND DISTRIBUTION OF BREAKING WAVE HEIGHTS IN DEEP WATER

lareODUcrI0NThephenomenonofbreakingwavesintheoasnoocurswheneveramomentahiyhighcrestmecheSanunstablecondition.It0ccursintendtimhy,andtheoccurrencefre-qUencydePendsonthescastate.ManystudAshaddricaniedoutontheoccurmcefre-qUencyofbrmkingwavesindeepweter.0chiandTsai(l983),LongueHiggins(l975)andVanD0mandPamn(l975)nadstheptalictonofbrmkingwavesoccurrenceindeePwater,usingabaskingcriterionforindividualwavesbasedonthewavesmpness.ThecriterioncanbeexpmeedintennSofthewaVehdghtHandperiodTasH>PgT'(…     

Directional Spectrum of Wind Waves: Part Ⅰ- Model and Derivation

1　Introduction　Numerousinvestigationsondeepwaterwindwavespectrumhavebeen performed (Phillips ,195 8;Bretschneider,195 9;PiersonandMoscowitz ,196 4;Hasselmannetal.,1973;Donelanetal.,1985 ;Ban ner ,1990 ;Wenetal.,1999) .Ondimensionalground ,Phillips (195 8)suggestedthattheequilibriumfre quencyspectrumofwindwavesfordeepwatershouldbe proportionaltoω- 5,andthecorrespondingwavenumberspectrumshouldbe proportionaltok- 4,whereωistheangularfrequencyandkisthewavenumber.Forfully developedwindwaves…     

A relationship between wave steepness and wave age for wind waves in deep water

Studying the relationship between wave steepness and wave age is important for describing wind wave growth with energy balance equation of significant waves. After invoking the dispersion rela- tion of surface gravity wave in deep water, a new relationship between wave steepness and wave age is revealed based on the “3/2-power law” (Toba, 1972), in which wave steepness is a function of wave age with a drag coefficient as a parameter. With a given wave age, a larger drag coefficient would lead to larger wave steepness. This could be interpreted as the result of interaction between wind and waves. Comparing with previous relationships, the newly proposed one is more consistent with observational data in field and laboratory.     

Wave extraction with portable high-frequency surface wave radar OSMAR-S

High frequency surface wave radar (HFSWR) has now gained more and more attention in real-time monitoring of sea surface states such as current, waves and wind. Normally a small-aperture antenna array is preferred to a large-aperture one due to the easiness and low cost to set up. However, the large beam-width and the corresponding incorrect division of the first- and second-order Doppler spectral regions often lead to big errors in wave height and period estimations. Therefore, for the HFSWR with a compact cross-loop/monopole antenna (CMA), a new algorithm involving improved beam-forming (BF) and spectral division techniques is proposed. On one hand, the cross-spectrum of the output sequence by the conventional beam-forming (CBF) with all the three elements and the output with only the two loops is used in place of the CMA output self-spectrum to achieve a decreased beam-width; on the other hand, the better null seeking process is included to improve the division accuracy of the first- and second-order regions. The algorithm is used to reprocess the data collected by the portable HFSWR OSMAR-S during the Sailing Competition of the 16th Asian Games held in Shanwei in November 2010, and the improvements of both the correlation coefficients and root-mean-square (RMS) errors between the wave height and period estimations and in situ buoy measurements are obvious. The algorithm has greatly enhanced the capabilities of OSMAR-S in wave measurements.     

Comparison of retrieving methods of ocean wave periods from satellite altimeter with buoy measurements

For validating the results of retrieved mean wave period, four empirical algorithms established previously are introduced. Based on the data of over five years derived from TOPEX satellite altimeter for the entire East China Sea, ocean wave periods were calculated and statistical comparison among them was performed. The retrieved mean wave period obtained with our new distribution parameters showed better agreement with the wave period TB measured by buoy than that calculated by other three algorithms. The difference between the mean values of and that of TB is 0.16 s and the RMSE (root mean square error) of is the lowest value (0.48).     

Wave steepness retrieved from scatterometer data in a genetic algorithm

Wave steepness is an important characteristic of a high sea state, and is widely applied on wave propagations at ports, ships, offshore platforms, and CO2 circulation in the ocean. Obtaining wave steepness is a difficult task that depends heavily on theoretical research on wavelength distribution and direct observations. Development of remote-sensing techniques provides new opportunities to study wave steepness. At present, two formulas are proposed to estimate wave steepness from QuikSCAT and ERS-1/2 scatterometer data. We found that wave steepness retrieving is not affected by radar band, and polarization method, and that relationship of wave steepness with radar backscattering cross section is similar to that with wind. Therefore, we adopted and modified a genetic algorithm for relating wave steepness with radar backscattering cross section. Results show that the root-mean-square error of the wave steepness retrieved is 0.005 in two cases from ERS-1/2 scatterometer data and from QuikSCAT scatterometer data.     

Directional spectrum of wind waves: Part I—model and derivation

A new model on the directional spectrum of wind waves for deep water is proposed based on the statistics of wind waves. This model contains three parameters: the wave age, the inverse spectral bandwidth and the local spectral-peak angular frequency. The inverse spectral bandwidth is a robust parameter for describing the spectral steepness of wind waves. Using the inverse spectral bandwidth parameter, the proposed model can well describe various observations obtained from the open ocean and laboratory tank.     

Directional Spectrum of Wind Waves: Part Ⅱ- Comparison and Confirmation

1　Introduction1.1　Proposed ModelonDirectionalFrequencySpec trum　ThisisthePartⅡofthetwo papersetondirection alspectraofwindwaves.Anewmodelonthedirec tionalspectrumofwindwavesfordeepwaterispro posedbasedonthestatisticsofwindwavesinthePartⅠ .Substituting (4 5 )ofPartⅠinto (4 0 )andaddingapeak enhancementitemγΓ,weobtainS(ω ,) =0 .0 0 93αaαwD( ,k) ωpω2 ξ- 4g2ω5×exp - 2 ξ+14[bωp +(1-b)ωp]4ω4 γΓ. (1)Here ,αaisthewaveagefactorofspectralcoefficientdefinedbyαa = ω0 .55…     

Relationship between wave steepness and wave age in the course of wind wave growth

It is traditionally assumed that the relationship between wave steepness and wave age is independent of the wind wave growth state. In fact, the traditional relationship can not describe the whole course of wind wave growth. This paper assumes that the relationship between wave steepness and wave age changes with the variety of dimensionless fetch. Based on the relationship proposed by Hou and Wen (1990), a new relationship in the course of wind wave growth is revealed. Comparisons between the present study and other previous relationships show that this new relationship explains better the observations than the other existing relationships.In the case of small fetch, wave age value increases more quickly than other models while it is in opposition to that in the case of large fetch. The result in present paper can clearly reflect the whole course of wind wave growth, it is an improvement for traditional results.     

Crest-height distribution of nonlinear random waves

The wave crest is an important factor for the design of both fixed and floating marine structures. Wave crest height is a dominant parameter in assessing the likelihood of wave-in-deck impact and resultant severe damage. Many empirical and theoretical distribution functions for wave crest heights have been proposed, but there is a lack of agreement between them. It is of significance to develop a better new nonlinear wave crest height distribution model. The progress in the research of wave crest heights is reviewed in this paper. Based on Stokes’ wave theory, an approximate nonlinear wave crest-height distribution formula with simple parameters is derived. Two sets of measured data are presented and compared with various theoretical distributions of wave crests obtained from nonlinear wave models and analysis of the comparison is given in detail. The new crest-height distribution model agrees well with observations. Also, the new theoretical distribution is more accurate than the other methods cited in this paper and has a greater range of applications.     

A background error covariance model of significant wave height employing Monte Carlo simulation

The quality of background error statistics is one of the key components for successful assimilation of observations in a numerical model.The background error covariance(BEC) of ocean waves is generally estimated under an assumption that it is stationary over a period of time and uniform over a domain.However,error statistics are in fact functions of the physical processes governing the meteorological situation and vary with the wave condition.In this paper,we simulated the BEC of the significant wave height(SWH) employing Monte Carlo methods.An interesting result is that the BEC varies consistently with the mean wave direction(MWD).In the model domain,the BEC of the SWH decreases significantly when the MWD changes abruptly.A new BEC model of the SWH based on the correlation between the BEC and MWD was then developed.A case study of regional data assimilation was performed,where the SWH observations of buoy 22001 were used to assess the SWH hindcast.The results show that the new BEC model benefits wave prediction and allows reasonable approximations of anisotropy and inhomogeneous errors.     

THE WAVE GROUPING EFFECT ON WAVE FORCES ON A VERTICAL BREAKWATER

Linear wave theory and Longuet-Higgins and Steward's (1964) group-induced second-order longweve (GSLW) theory ware used in this study on the grouping effect on wave forces acting on a verticalbreakwater. The calculated variance of total wave pressure on the vertical breakwater was closer tothe measured value if the wave grouping effect was considered.     

STUDY ON METHOD OF NUMERICAL SIMULATION OF NONLINEAR RANDOM WAVES

This new method can simulate the nonlinear random wavcs processes by computer if the higherorder moments of the probability distribution of the sea surface elevation reflecting the nonlinearity ofthe sea wave are given. Compared with other methods, this method has greater accuracy andflexibility, wider application and faster simulation. Statistical analysis of the sea surface elevationdistribution of the simulated wave process showed obviously the Gram-Charlier series can be used to depictthe distribution of the sea surface elevation.     

STATISTICAL PROPERTIES OF HIGH-FREQUENCY INTERNAL WAVES IN QINGDAO OFFSHORE AREA OF THE YELLOW SEA

Densely-sampled thermistor chain data obtained from a shallow-water acoustics experiment in the Yellow Sea off the coast of Qingdao were analyzed to examine the statistical properties of the 6 to 520 cpd frequency band internal waves observed. The negative skewness coefficients and the greaterthan-3 kurtosis coefficients indicated non-Gaussianity of the internal waves. The probability distributions were negatively skewed and abnormally high peaks. Nonlinear properties, as exemplified by the asymmetric waveshapes of the internal waves in the offshore area are described quantitatively.     

A mechanism of the effect of non-uniform current on the spectrum of short wind waves

A mechanism is suggested in this paper concerning the effect of non-uniform current on the spectrum of short wind waves. According to this mechanism, a non-uniform current brings changes to the breaking criteria of short wind waves through modulating the surface drift, and hence enhances or weakens wave breaking. Some modification is proposed to the source term, which represents the spectral rate of wave energy dissipation due to wave breaking so that the source term can incorporate this mechanism. In order to illustrate whether this mechanism is significant, a real case is studied, in which the wind waves propagate on a tidal current flowing over the sea bottom covered with sand waves. Finally, the effect of the new mechanism on the equilibrium spectrum of small scale gravity waves is discussed. Numerical estimates suggest that, for water depths less than 50 m and wavelengths less than 1 m, this current field may result in distinct spatial variations of the wave breaking criteria, the spectral rate of wave energy dissipation and the equilibrium spectrum of short gravity waves.     

Probability Distribution of the Hull Motion and Mooring Line Tension of Two Floating Systems

The statistical and distribution characteristics of the responses of a floater and its mooring lines are essential in designing floating/mooring systems.In general,the dynamic responses of offshore structures obey a Gaussian distribution,assuming that the structural system,and sea loads are linear or weakly nonlinear.However,mooring systems and wave loads are considerably nonlinear,and the dynamic responses of hull/mooring systems are non-Gaussian.In this study,the dynamic responses of two types of floaters,semi-submersible and spar platforms,and their mooring lines are computed using coupled dynamic analysis in the time domain.Herein,the statistical characteristics and distributions of the hull motion and mooring line tension are discussed and compared.The statistical distributions of the dynamic responses have strong non-Gaussianity and are unreasonably fitted by a Gaussian distribution for the two floating and mooring systems.Then,the effects of water depth,wave parameters,and low-frequency and wave-frequency components on the non-Gaussianity of the hull motion,and mooring line tension are investigated and discussed.A comparison of the statistical distributions of the responses with various probability density functions,including the Gamma,Gaussian,General Extreme Value,Weibull,and Gaussian Mixture Model(GMM)distributions,shows that the GMM distribution is better than the others for characterizing the statistical distributions of the hull motion,and mooring line tension responses.Furthermore,the GMM distribution has the best accuracy of response prediction.     

INTERNAL WAVE EIGENMODES AND THEIR WEAK NONLINEAR RESONANT INTERACTION IN A NONLINEARLY STRATIFIED FLUID- A PRELIMINARY THEORETICAL AND EXPERIMENTAL STUDY

A preliminary theoretical and experimental study was conducted on internal wave modes and their weak nonlinear resonant interaction in a nonlinearly stratified fluid . An asymptotical solution of the modes and a dispersion relation of internal waves in a stratified fluid with density profile similar to that in our experiment were obtained theoretically . The resonant-interaction mechanism to 2nd order approximation is also discussed . The resonant interaction of the 3rd and 4th mode internal waves excited by the unstable 1st mode wave is analyzed on the basis of data obtained by conductivity probes. The resonant-interaction condition, , is examined . It is shown that the resonant instability increases with pycnocline thickness and wave maker driving frequency .