Wave Height Distribution Research Articles

Efficient computational method for joint distributions of heights and periods of nonlinear ocean waves

This paper proposes a novel method for efficient prediction of joint distributions of heights and periods of nonlinear ocean waves. The proposed novel method utilizes a transformed linear simulation which is based on a Hermite transformation model where the transformation is chosen to be a monotonic cubic polynomial, calibrated such that the first four moments of the transformed model match the moments of the true process. This proposed novel method is utilized to predict the joint distributions of wave heights and periods of a sea state with the surface elevation data measured at the Gulfaks C platform in the North Sea, and the novel method's accuracy and efficiency are favorably validated by using comparisons with the results from an empirical joint distribution model, from a linear simulation model and from a second-order nonlinear simulation model.

Characteristics of global waves based on the third-generation wave model SWAN

Long-term wave data at a global scale are crucial for the studying characteristics of global waves, such as spatial distributions and extreme waves. These characteristics play significant roles in the design, construction, installation and operation activities of marine structures. Because of the limited resolution and coverage of instrumental measurements, the global wave hindcast is widely used. To develop this hindcast, the Simulating WAves Nearshore (SWAN) model is forced with the ERA-Interim wind database from 1979 to 2017 with a resolution of 0.25° in both longitude and latitude. The hourly wave parameters, such as the significant wave height and the mean period, are validated against buoy measurements over the Northwest Pacific Ocean, the Northeast Pacific Ocean and the Northwest Atlantic Ocean. Based on the 39-year simulated results, the spatial distributions of mean and maximal significant wave heights are analysed for all 39 years, every decade, 4 seasons, 12 months and special years; extreme waves are studied at the global scale and at every computational point.

Individual wave height and wave crest distributions based on field measurements from the northern North Sea

This paper concerns full-scale wave measurements of individual wave heights and crest heights at one location with water depth 190 m in the northern North Sea from 2004 up to date. The surface elevation was recorded by a Saab WaveRadar REX from one location in the northern North Sea. The measured individual wave heights and crest heights have been compared to well-known theoretical distributions. For wave heights, the Rayleigh distribution is found to form a conservative upper bound, whilst the Forristall distribution is considered to show a good accuracy, though somewhat conservative in the most severe sea states. For crest heights, the Rayleigh distribution is found to be nonconservative, whilst the Forristall distributions for long- and short-crested sea both provide reasonable descriptions. Based on the present study, the Forristall distributions for individual wave height and crest heights in long-crested seas are considered to provide the most appropriate descriptions of these measured wave data.

Long-term distributions of individual wave and crest heights

This paper considers three types of method for calculating return periods of individual wave and crest heights. The methods considered differ in the assumptions made about serial correlation in wave conditions. The long-term distribution of individual waves is formed under the assumption that either (1) individual waves, (2) the maximum wave height in each sea state or (3) the maximum wave height in each storm are independent events. The three types of method are compared using long time series of synthesised storms, where the return periods of individual wave heights are known. The methods which neglect serial correlation in sea states are shown to produce a positive bias in predicted return values of wave heights. The size of the bias is dependent on the shape of the tail of the distribution of storm peak significant wave height, with longer-tailed distributions resulting in larger biases. It is shown that storm-based methods give accurate predictions of return periods of individual wave heights. In particular, a Monte Carlo storm-based method is recommend for calculating return periods of individual wave and crest heights. Of all the models considered, the Monte Carlo method requires the fewest assumptions about the data, the fewest subjective judgements from the user and is simplest to implement.

A generalised equivalent storm model for long-term statistics of ocean waves

To calculate the return periods of individual wave or crest heights, the long-term distribution of sea states must be combined with the short-term distribution of individual wave or crest heights conditional on sea state. This is normally achieved using an equivalent storm model to parameterise the distribution of the maximum wave or crest height in a storm. A new equivalent storm model is introduced that generalises the approach of Tromans and Vanderschuren (1995). The generalised equivalent storm (GES) method is significantly simpler than equivalent storm methods that model the temporal evolution of the significant wave height in a storm. The GES method is applied to long time series of wave buoy measurements for deep and shallow water sites and demonstrated to be more accurate than existing methods at representing the statistical characteristics of measured storms. Return periods of crest heights from the GES method are shown to be more robust to uncertainties in the fitted models of the equivalent storm parameters than estimates from temporal evolution methods such as the equivalent triangular storm and equivalent power storm model.

Experiments on Surface Waves Interacting with Flexible Aquatic Vegetation

Surface wave interaction with aquatic vegetation appears to play a key role in coastal hydro-morpho-dynamics. As an example, the presence of a dense meadow at intermediate water depth is usually associated with a stable and resilient shore. Wave-meadow interactions are investigated here by means of physical modelling, with a focus on wave height distribution and hydrodynamics. The central part of a wave flume is covered by flexible artificial seagrass, composed of polyethylene leaves. This vegetation is tested in both near emergent and submerged conditions. The wave height reduction is evaluated by means of a drag coefficient defined from linear wave theory, which contains all the unknowns of the adopted methodology. The behaviour of such a coefficient is investigated as a function of a wave related Reynolds number. The influence of the flexibility of the leaves is also considered, together with a wave frequency parameter. The results show a complex behaviour with three different trends for near rigid, intermediate or highly flexible leaves. Amplitudes of the orbital velocities are investigated and show a fairly good match with the linear wave theory. On the contrary, the mean velocity along the water column appears to be modified by the seagrass for submerged leaves.

Assessment of statistical parameter uncertainty in the reliability analysis of jacket platforms

Statistical uncertainty is undoubtedly introduced in the estimation of parameters usually involved in the physical or probabilistic models employed for reliability analysis. Provided that such models parameters are uncertain, the probability of failure and the reliability index become uncertain variables as well. In this work, a general formulation is introduced to account for parameter uncertainties in the reliability analysis of jacket platforms under storm conditions. A point estimates method based on the Rosenblatt transformation is used to compute the statistics of the failure probability and the reliability index. It has the advantage that the estimating points and weights are readily defined. The effect of considering parameter uncertainties in the probability distribution of maximum wave height and the environmental load model was assessed. The relative influence of the different sources of parameter uncertainty and the implications of neglecting it were examined. Credible bounds of failure probability and coefficients of variation of the reliability index were evaluated as means of charactering uncertainty in their estimation. A decision problem regarding optimal probabilities of failure for structural design using a risk model is also analyzed, showing that depending on the coefficient of variation of the failure probability greater safety margins may be required.

Numerical prediction of Normal and Extreme Waves at Fukushima Offshore Site

In this study, wave hindcasting was performed at Fukushima floating offshore wind turbine demonstration site by using Wave Watch III, and the simulation results were validated by using measurement data. Following results were obtained. It was found that the use of the computational domain which covers whole the Pacific Ocean improves the prediction accuracy of significant wave height and wave period. By applying correction factors to the extreme value of the significant wave height results in the accurate estimation of the average value and the frequency distribution of the significant wave height. The long-term variations of the annual average significant wave height and period as well as the extreme value distribution of the annual maxium wave height are estimated from 10 years of hindcasting data. Gumbel fitting can be used for the estimation of the extreme wave height in tropical cyclone prone region. The relationship between extreme wave height and period is consistent with IEC61400-3. However, when all the event is included, the effect of the swell cannot be neglected and a model proposed by Goda should be used. A model to describe the joint probability distribution of wind speed, wave height and wave period is proposed. Predicted joint probability distribution based on this model shows good agreement with measurement.

Generating potential risk maps for typhoon-induced waves along the coast of Taiwan

Typhoon-induced waves threaten the coastal areas of Taiwan each year. The roaring waves caused by typhoon Meranti (2016) even destroyed a lighthouse in a fishing port on the southwestern coast of Taiwan. Therefore, there has been increased interest in creating potential risk maps for typhoon-induced waves along the coast of Taiwan. In this study, the highest intensity and lowest intensity typhoons (HITs and LITs, respectively) for each category from 1977 to 2016 were selected. A fully coupled tide-surge-wave model was utilized to map the distributions of maximum significant wave heights for 18 typhoon events. Each map was classified into one of 5 levels and employed to generate maximum and minimum potential risk maps for typhoon-induced waves. Our results demonstrate that the northern and the eastern coasts of Taiwan are threatened by violent waves (significant wave heights (SWHs) range from 7 to 11.5 m) over a coastline length of 236.4 km and roaring waves (SWHs exceeding 11.5 m) over a coastline length of 298.1 km under HIT conditions. The lengths of coastline threatened by big waves (SWHs ranging from 3 to 6 m) and small waves (SWHs less than 3 m) are 637.7 km and 553.4 km, respectively, under LIT conditions.

Numerical Simulation of Wave Breaking Over a Submerged Step with SPH Method

Wave breaking over a submerged step with a steep front slope and a wide horizontal platform is studied by smoothed particle hydrodynamic (SPH) method. By adding a momentum source term and a velocity attenuation term into the governing equation, a nonreflective wave maker system is introduced in the numerical model. A suitable circuit channel is specifically designed for the present SPH model to avoid the nonphysical rise of the mean water level on the horizontal platform of the submerged step. The predicted free surface elevations and the spatial distributions of wave height and wave setup over the submerged step are validated using the corresponding experimental data. In addition, the vertical distributions of wave-induced current over the submerged step are also investigated at both low and high tides.

Wave energy distribution along the Portuguese continental coast based on a thirty three years hindcast

Abstract An assessment of wave power resource for Portugal continental coast is presented, using thirty three years (1979–2012) of wave hindcast. The hindcast system is based in two spectral wave models, WWIII (WAVEWATCHIII) for the wave generation and SWAN (Simulating WAves Nearshore) for wave transformation in coastal areas. To improve the SWAN model performance, six high resolution areas (Agucadoura, Figueira da Foz, Sao Pedro de Moel, Peniche, Cascais and Sines) were nested into one large area (Iberian coast). The assessment of wave power potential was done by studying the spatial and temporal behaviour using average for different time scales (monthly, seasonal, inter-annual and total time). Moreover, the wave climate variability was determined through the calculation of the coefficient of variation, seasonal variability index and monthly variability index. The results have shown that in Portugal the wave energy is higher in the northwest, that the wave climate variability has more impact in the south and that the month with the highest energy is January. A bivariate distribution of significant wave height and energy period was also determined, showing that the sea states where more wave power can be extracted are different than the ones that occur more frequently.

Wave energy assessement for Northern Spain from a 33-year hindcast

Abstract The wave energy resource along the Northern Spanish coast is determined, for a period of 33 years (1979–2012) based on hindcast results of WAVEWATCH III for the Atlantic ocean area, coupled with the SWAN model for the coastal areas, and using surface winds from ECMWF's ERA- Interim data base. Results are validated with buoy data, in order to evaluate the model's accuracy. Statistical analysis of the wave parameters and wave power results are presented. The analysis was carried out for a coarse grid, on the North of Spain, and for three local areas with a finer mesh: Northwest Galicia, the area between Cape San Adrian and Cape Ortegal and Santander. The case studies include a more realistic perspective of the amount of energy that can be extracted with a wave energy converter, by adopting examples of a large and a small device. A seasonal variability evaluation is done for the nested areas through the analyzes of the seasonal average wave power resource for the different locations, throughout the 33 year hindcast, and the joint distributions of significant wave height and peak period for the number of occurrences and wave power.

Wave statistics in the Adriatic Sea based on 24 years of satellite measurements

Based on a database of sea state parameters in the Adriatic Sea, wave statistics has been developed at three different geographic locations along the busy Adriatic shipping route. The underlying database consists of a systematic numerical hindcast calibrated by 24-year satellite measurements and represents the most comprehensive data source of wave parameters available for the region. Joint probability distributions of significant wave heights and peak spectral periods are derived, thus enabling extreme sea states to be determined. Furthermore, waves have been evaluated with respect to the well-known dominant wind patterns in the Adriatic and compared to the Tabain's spectrum, a single parametric spectral description developed for local conditions and widely used in engineering practice. Statistics for three geographical locations are compared with one another as well as with data from other available references. These long-term statistics and regression equations developed can be used not only for fatigue and extreme wave loads of ships, offshore installations and renewable energy devices intended for the Adriatic Sea, but also for planning and carrying out various marine operations in that specific environment. At the end of the study, some consideration is given to the effects of climate changes on the variability of sea states in the Adriatic.

Statistical Analyses of Wave Height and Wind Velocity Distributions for the Sea of Marmara

The Sea of Marmara, a relatively small basin located between the Black Sea and the Aegean Sea with narrow seaways of the Bosphorus and the Dardanelles, has vital importance in coastal and ocean engineering activities due to substantial industrial development surrounding it and related sea traffic. Despite this importance, measurements concerning coastal engineering applications, in particular the wave climate, are quite scarce. Lack of essential data impairs the reliability of engineering works such as breakwaters and coastal protection structure designs. For a period of approximately one year between February 2013 and January 2014 Turkish Petroleum International Company (TPIC) carried out wave and wind measurements in the Sea of Marmara. The data were collected in 30-minute intervals throughout the one year span at the location of 41o04’ N and 28o19’ E with 50 m water depth. The present work analyses this rather unique data statistically for wind, wave height and energy conditions in the Sea of Marmara.

Numerical simulation of wave transformation over platform reef using SPH method

Different from the fringing reef along the coastal beach, the reef face of the deep-sea platform reef is very steep and the water depth varies rapidly from a few thousand meters at the reef toe to a few meters at the reef edge. The special geomorphology of the deep-sea platform reef makes its hydrodynamics much different from that of the fringing reef with a gentle reef-face. Besides, the wave-generated flow induced by wave breaking has a great influence on sediment movement, nutrient transport and habitats for marine organisms on the reef flat. Therefore, the knowledge of the hydrodynamics of the deep-sea platform reef is of great importance to guide the subsequent development and protection of the deep-sea platform reefs. An improved smoothed particle hydrodynamic (SPH) model using the periodic boundary conditions and a water circulation technology as well as a non-reflected momentum source wave maker is developed to study wave transformation over the deep-sea platform reef. The Favre average N-S equations along with the sub-particle scale (SPS) turbulence closure model are employed as the governing equations to describe the wave motion. The idealized model of the platform reef is composed of a reef face with a slope of 1:1 and a horizontal reef flat. To verify the SPH model, the corresponding physical model tests are also carried out in the nonlinear wave flume of the state key laboratory of coastal and offshore engineering, Dalian university of technology. A good agreement is obtained between the numerical and experimental results, which indicates the present SPH model can accurately simulate wave transformation over the deep-sea platform reef and that it also can reasonably capture the spatial distributions of wave height and wave set-up on the reef flat. In addition, the unphysical rise and fall of the water level in the lagoon and the ocean due to the overtopping on the reef-flat, which usually occur for the long-time simulation using the vertical 2D numerical model, can be overcome by using the periodic boundary condition and a water flow circuit instead of the solid boundary condition to simulate the left and right boundaries of the flume. The wave height decay tempestuously due to wave energy dissipation by wave breaking. As observed both in the physical and numerical tests, wave breaking occurs on the reef-face or the reef-rim at the low water levels and emerges over the reef-rim or the reef-flat at the high water levels. For the coral reef with a gentle reef-face, the previous studies pointed that the basic type of wave breaking at the reef-rim is collapsing for a small relative wave height ( H / h r) and is plunging for a large relative wave height. While for the coral reef with a very steep reef-face focused in the present study, collapsing wave is the main type of wave breaking at the reef-rim as observed both in the physical and numerical tests.The wave height after breaking decreases dramatically and the broken wave propagates along the reef-flat as a bore-like wave, and the amplitude of wave setup is almost constant along the reef-flat.

Joint description of waves and currents applied in a simplified load case

In order to perform a more accurate analysis of marine structures, joint probability distributions of different metocean parameters have received an increasing interest during the last decade, facilitated by improved availability of reliable joint metocean data. There seems to be no general consensus with regard to the approach of estimating joint probability distributions of metocean parameters and a general overview of recent studies exploring different joint models for metocean parameters is presented. The main objective of this article is twofold: first to establish a joint distribution of significant wave height and current speed and then to assess the possible conservatism in the Norwegian design standard by applying this joint distribution in a simplified load case. Based on NORA10 wave data and simulated current data, a joint model for significant wave height and current speed at one location in the northern North Sea is presented. Since episodes of wind-generated inertial oscillations are governing the current conditions at this location, a joint conditional model with current speed conditional on significant wave height is suggested. A peak-over-threshold approach is selected. The significant wave height is found to be very well modelled by a 2-parameter Weibull distribution for significant wave height exceeding 8 m, while a log-normal distribution describes the current speed well. This model is used to Monte-Carlo simulate joint significant wave heights and current speeds for periods corresponding to the ultimate and accidental limit states (ULS and ALS), i.e. 100 and 10 000 years. The possible conservatism in the Norwegian design standard is assessed by a simplified case study. The results give a clear indication that the Norwegian design standard in not necessarily conservative, neither at ULS nor ALS level.

An entropy theory for the spatiotemporal patterns of the environmental matrix in the nearshore parameters

Understanding the temporal and spatial variability of sediment transport at regional scales is of great value for various purposes; however, relevant studies are still limited and have yielded inconsistent findings about the primary controls on regional temporal and spatial variability of bed load sediment transport rate (BLSTR). In this paper, sediment transport was investigated by a combination of spatiotemporal patterns and entropy theory modelling. Effort has been made in this paper to develop an entropy theory by assuming the time-averaged wave height as a random variable and applying the disorder power index to derive wave height distribution and rate of bed load sediment transport in coastal zones for ocean management planning. To analyze the spatio-temporal patterns, this study uses: A method that needs a disorder power index (entropy-power), uses the long time (1970–2015 monthly) data were retrieved of six large-scale monitoring networks located in the Makran coastal area (located in southeastern Iran), and shows the index suitability for hydrological/marine ecosystem components analyses using a coastal area case study, and studies flow velocity speed, wave and spatio-temporal patterns of BLSTR each of which is dependent on time and has an important role in terrestrial coastal/ocean studies. The combined performance of the above mentioned variables is difficult to describe (although their spatio-temporal variability is mostly meaningful) and, regarding this context, entropy-power index shows signals useful to assess coastal engineering issues and makes the understanding of wave height and sediment transport parameters interactions possible at nearshore zone scales. The advantage of using the entropy theory is that it permits quantification of uncertainty associated with concentration and determination of parameters in terms of specified information, such as mean concentration.

Some probabilistic properties of deep water wave steepness

Summary This paper provides some probabilistic properties of the deep water wave steepness and the spectral wave steepness by using distributions based on data from the Norwegian continental shelf. Here the average statistical properties represented by the mean value and the standard deviation of the two steepness parameters are considered. Examples of results for the wave steepness are given for a Phillips spectrum and a family of JONSWAP spectra for wind sea, and for sea states described by a joint frequency table of significant wave height and mean zero-crossing wave period for combined wind sea and swell. The results for the spectral wave steepness are obtained by using a joint distribution of significant wave height and spectral wave steepness, and the average statistical features are given for joint frequency tables of significant wave height and mean zero-crossing wave period from three locations on the Norwegian continental shelf.

A numerical investigation on nonlinear transformation of obliquely incident random waves on plane sloping bottoms

The statistical properties of obliquely incident irregular waves over a planar sloping bottom were investigated numerically by the well-known numerical wave model FUNWAVE 2.0. Irregular waves based on the averaged JONSWAP spectra with various wave heights and peak periods were simulated to propagate over planar bottoms. A wide range of incident angles from 0° though 75° were considered to study the influence of incident angles. It was found that incident angles have a significant influence on the wave nonlinearity. The wavelet-based bicoherence revealed that the degree of triad wave interactions of primary waves and the higher harmonics was apparently weakened by increasing wave incident angles. Importantly, using the simulated data, the Klopman wave height distribution was improved by incorporating the influence of obliquely incident angles. It was found that the improved Klopman distribution shows better performance for describing wave height distributions in shallow water depth. Moreover, two empirical formulae are recommended to reflect the relationship between the skewness and the asymmetry of waves and the Ursell number for obliquely incident waves on plane sloping bottoms.

An Approximation to the Statistical Characteristics of Wind Waves in Front and from the Toe of the Structure to the Toe of the Crown of Nonovertopped Breakwaters

According to the main recommendations and technical criteria along the world, the design project of a breakwater must address the verification of the different failure modes that can affect the breakwater stability. This research focuses on the estimation of the statistical characteristics of the wind waves interacting with different breakwater types and their evolution from the toe of the structure to the toe of the crown. This knowledge is essential to calculate the reliability of the structure in its useful life. Partial standing wave patterns are likely to occur in front and along the section of the breakwater depending on their typology. Based on Rice's theory of envelope amplitude, we present an approximate solution of the total wave height distribution in front of the breakwater. The experimental results in 2D confirmed that the incident, the reflected and the total wave height in front of the breakwater followed a Rayleigh distribution in which the parameter is the root-mean-square total wave height. Its value depends on the modulus and phase of the reflection coefficient. The probability density function of the total wave height evolves from a Rayleigh to a Weibull distribution, whose scale and shape parameters vary from the toe to the crown of the structure, and depend on the breakwater type, the relative grain diameter, and the relative water depth. The largest deviation from the Rayleigh distribution occurs at the toe of the crown. These findings are weakly dependent on the incident wave steepness, and are valid for narrow-banded incident wave trains, impinging perpendicularly on nonovertopped breakwaters with a steep frontface, which ensures the rapid evolution of the wave train. Then, each mode of failure that might occur in the breakwater section can be formally checked against the same incident wave height, but verified (calculated) with the actual wave height, locally transformed by the specific typology.