Joint modelling of wave spectral parameters for extreme sea states

Characterising the dependence between extremes of wave spectral parameters such as significant wave height (H_S) and spectral peak period (T_P) is important in understanding extreme ocean environments and in the design and assessment of marine structures. For example, it is known that mean values of wave periods tend to increase with increasing storm intensity. Here we seek to characterise joint dependence in a straightforward manner, accessible to the ocean engineering community, using a statistically sound approach.Many methods of multivariate extreme value analyses are based on models which assume implicitly that in some joint tail region each parameter is either independent of or asymptotically dependent on other parameters; yet in reality the dependence structure in general is neither of these. The underpinning assumption of multivariate regular variation restricts these methods to estimation of joint regions in which all parameters are extreme; but regions where only a subset of parameters are extreme can be equally important for design. The conditional approach of Heffernan and Tawn (2004), similar in spirit to that of Haver (1985) but with better theoretical foundation, overcomes these difficulties.We use the conditional approach to characterise the dependence structure of H_S and T_P. The key elements of the procedure are: (1) marginal modelling for all parameters, (2) transformation of data to a common standard Gumbel marginal form, (3) modelling dependence between data for extremes of pairs of parameters using a form of regression, (4) simulation of long return periods to estimate joint extremes. We demonstrate the approach in application to measured and hindcast data from the Northern North Sea, the Gulf of Mexico and the North West Shelf of Australia. We also illustrate the use of data re-sampling techniques such as bootstrapping to estimate the uncertainty in marginal and dependence models and accommodate this uncertainty in extreme quantile estimation.We discuss the current approach in the context of other approaches to multivariate extreme value estimation popular in the ocean engineering community.     

海洋工程设计风速与波高的联合分布

选取我国渤海某处21a的风暴过程后报资料,考虑风暴发生频次的影响,提出泊松二维逻辑分布,并且将其用于海洋石油工程设计中极值风速与波高的联合概率计算,给出了计算海域的风浪设计参数,并与传统的设计标准进行了比较.计算结果表明,新的统计模式适用于受风暴影响海区的海洋工程结构设计,特别是边际油田的开发建设.     

On the long-term joint distribution of characteristic wave height and period and its application

By analysing the scatter diagrams of characteristic the wave height H and the period T on the basis of instrumental data from various ocean wave stations, we found that the conditional expectation and standard deviation of wave period for a given wave height can be better predicted by using the equations of normal linear regression rather than by those based on the log- normal law. The latter was implied in Ochi' s bivariate log-normal model(Ochi. 1978) for the long-term joint distribution of H and T. With the expectation and standard deviation predicted by the normal linear regression equations and applying proper types of distribution, we have obtained the conditional distribution of T for given H. Then combining this conditional P(T / H) with long-term marginal distribution of the wave height P(H) we establish a new parameterized model for the long-term joint distribution P(H,T). As an example of the application of the new model we give a method for estimating wave period associated with an extreme w     

Multiple thermohaline states due to variable diffusivity in a hierarchy of simple models

The effect of variable vertical diffusivity is investigated in dynamically reduced models of the thermohaline circulation (THC) in a rectangular basin. In a simple box model, sufficiently strong variation of the diffusivity κ_v with stability G can lead to the existence of two stable equilibria. Related behaviour is found in well-resolved frictional geostrophic (FG) models. A hierarchy of under-resolved FG models is constructed, the simplest of which is an 8-cell cube, to connect the two extremes of resolution. Multiple solutions in low-order models are found to correspond to the formation of high-gradient layers which are unlikely to be resolved by current ocean models. Physical arguments show that layering and multiple solutions require κ_v to decrease more rapidly than 1/G and sensitivity experiments suggest that, in addition, κ_v must vary by a factor of 10–100. In two-hemisphere runs with salinity forcing included, the dependence of diffusivity on stratification is found to marginally favour equatorially symmetric states. Finally, such variation is shown to have a profound effect on the periodic, flush-collapse cycle under strong saline forcing; specifically, if diffusivity is taken to be a function of stratification rather than depth, regime transitions can occur much more easily. It will therefore be important for climate modelling to determine which is more realistic.     

Long-term variation of storm surge-associated waves in the Bohai Sea

When investigating the long-term variation of wave characteristics as associated with storm surges in the Bohai Sea,the Simulating Waves Nearshore(SWAN)model and Advanced CIRCulation(ADCIRC)model were coupled to simulate 32 storm surges between 1985 and 2014.This simulation was validated by reproducing three actual wave processes,showing that the simulated significant wave height(SWH)and mean wave period agreed well with the actual measurements.In addition,the long-term variations in SWH,pattems in SWH extremes along the Bohai Sea coast,the 100-year retum period SWH extreme distribution,and waves conditional probability distribution were calculated and analyzed.We find that the trend of SWH extremes in most of the coastal stations was negative,among which the largest trend was-0.03 m/a in the western part of Liaodong Bay.From the 100-year return period of the SWH distribution calculated in the Gumbel method,we find that the SWH extremes associated with storm surges decreased gradually from the center of the Bohai Sea to the coast.In addition,the joint probability of wave and surge for the entire Bohai Sea in 100-year return period was determined by the Gumbel logistic method.We therefore,assuming a minimum surge of one meter across the entire Bohai Sea,obtained the spatial SWH distribution.The conclusions of this study are significant for offshore and coastal engineering design.     

Wave model predictions in the Black Sea: Sensitivity to wind fields

This paper evaluates the impact of using different wind field products on the performance of the third generation wave model SWAN in the Black Sea and its capability for predicting both normal and extreme wave conditions during 1996. Wind data were obtained from NCEP CFSR, NASA MERRA, JRA-25, ECMWF Operational, ECMWF ERA40, and ECMWF ERA-Interim. Wave data were obtained in 1996 at three locations in the Black Sea within the NATO TU-WAVES project. The quality of wind fields was assessed by comparing them with satellite data. These wind data were used as forcing fields for the generation of wind waves. Time series of predicted significant wave height (H_mo), mean wave period (T_m02), and mean wave direction (DIR) were compared with observations at three offshore buoys in the Black Sea and its performance was quantified in terms of statistical parameters. In addition, wave model performance in terms of significant wave height was also assessed by comparing them against satellite data.The main scope of this work is the impact of the different available wind field products on the wave hindcast performance. In addition, the sensitivity of wave model forecasts due to variations in spatial and temporal resolutions of the wind field products was investigated. Finally, the impact of using various wind field products on predicting extreme wave events was analyzed by focussing on storm peaks and on an individual storm event in October 1996. The numerical results revealed that the CFSR winds are more suitable in comparison with the others for modelling both normal and extreme events in the Black Sea. The results also show that wave model output is critically sensitive to the choice of the wind field product, such that the quality of the wind fields is reflected in the quality of the wave predictions. A finer wind spatial resolution leads to an improvement of the wave model predictions, while a finer temporal resolution in the wind fields generally does not significantly improve agreement between observed and simulated wave data.     

异地海域年极值风暴增水同现规律的探讨

以塘沽和龙口海洋观测站20年极值增水值为样本，基于二维冈贝尔逻辑分布模式，探讨了不同海域风暴潮增水极值的联合分布规律。通过对二维分布的联合概率密度、条件概率密度和同现概率的计算，给出了相应的工程设计参数，供有关部门在防潮规划时参考。     

Storm evolution characterization for analysing stone armour damage progression

Storm evolution is fundamental for analysing the damage progression of the different failure modes and establishing suitable protocols for maintaining and optimally sizing structures. However, this aspect has hardly been studied and practically the whole of the studies dealing with the subject adopt the Equivalent triangle storm. As against this approach, two new ones are proposed. The first is the Equivalent Triangle Magnitude Storm model (ETMS), whose base, the triangular storm duration, D, is established such that its magnitude (area describing the storm history above the reference threshold level which sets the storm condition), H_T, equals the real storm magnitude. The other is the Equivalent Triangle Number of Waves Storm (ETNWS), where the base is referred in terms of the real storm's number of waves, N_z. Three approaches are used for estimating the mean period, T_m, associated to each of the sea states defining the storm evolution, which is necessary to determine the full energy flux withstood by the structure in the course of the extreme event. Two are based on the Jonswap spectrum representativity and the other uses the bivariate Gumbel copula (H_s, T_m), resulting from adjusting the storm peaks. The representativity of the approaches proposed and those defined in specialised literature are analysed by comparing the main armour layer's progressive loss of hydraulic stability caused by real storms and that relating to theoretical ones. An empirical maximum energy flux model is used for this purpose. The agreement between the empirical and theoretical results demonstrates that the representativity of the different approaches depends on the storm characteristics and point towards a need to investigate other geometrical shapes to characterise the storm evolution associated with sea states heavily influenced by swell wave components.     

How storm size matters for surge height

The observed trend of peak storm surge η_max increasing with storm size R_max, roughly as η_max ∞ R_max^0.22, particularly on gently sloping coasts, is discussed in relation to the simple 1D analytical solutions for forced long waves due to respectively surface pressure p_s and wind stress τ_w. At constant depth h, the τ_w-driven surge is proportional to storm size while the p_s-driven part is not. This could perhaps be seen to explain why the size-dependence is stronger on flatter slopes where the τ_w-driven surge dominates. However, this direct size dependence disappears in the sloping beach scenario if the typical depth is assumed proportional to storm size. The observed size dependence is then more likely due to a combination of two 2D effects: Firstly, the sideways radiation from a travelling surge which exceeds the “stationary height” Δ_p/ρg is relatively weaker for a wider system. Secondly, the wind stress field is a dipole, and the mutual cancellation of the two poles is weaker for larger systems.     

Reliability analysis of H-infinity control for a container ship in way-point tracking

Efficient control of ships in a designed trajectory is always a significant charge for ship maneuverings. The purpose of this paper is to design a robust H_∞ controller and a reliability analysis for a container ship in a way-point tracking. First, the H_∞ controller is designed for a container ship because of model parameters’ uncertainties and external disturbances such as waves, winds and ocean currents. Then, to evaluate the reliability of the designed controller, a well-known reliability analysis technique is employed to achieve the predefined heading angle overshoot (that is less than 20%) in way-point tracking. To do this, three random variables including wind speed, wind direction and wave direction are considered as the inputs due to their significant effect on overshoot, compared to other variables. The results demonstrate the capability of the designed H_∞ controller against modeling uncertainties and external disturbances in way point tracking control.     

A relationship to describe the cumulative impact of storm clusters on beach erosion

Estimation of erosion volumes for adequate dry beach buffer zones is commonly estimated on the basis of a single extreme event, such as the 1 in 100 year storm. However, the cumulative impact of several smaller, closely spaced storms can lead to equal, if not more, dry beach loss, but this is often not quantified. Here we use a calibrated model for dune erosion, XBeach, to hindcast the cumulative erosion impact of a series of historical storms that impacted the Gold Coast, Queensland region in 1967. Over a 6-month period, four named cyclones (Dinah, Barbara, Elaine, and Glenda) and three East Coast Lows caused a cumulative erosion volume greater than the predicted 1 in 100 year event. Results presented here show that XBeach was capable of reproducing the measured dry beach erosion volume to within 21% and shoreline retreat to within 10%. The storms were then run in 17 different sequences to determine if sequencing influenced final modeled erosion volumes. It is shown that storm sequencing did not significantly affect the total eroded volumes. However, individual storm volumes were influenced by the antecedent state of the beach (i.e. prior cumulative erosion). Power-law relationships between cumulative energy density (∑ E) and eroded volume (∆V) as well as cumulative wave power ((∑ P)) and eroded volume (∆V) both explained more than 94% of the modeled dry beach erosion for the 1967 storm sequences. When the relationship was compared with observed and modeled erosion volumes for similar beaches but different storm forcing, the inclusion of pre-storm beach swash slope (β_swash) in the parameterization was found to increase the applicability of the power-law relationship over a broader range of conditions.     

东南沿海台风风暴潮增水过程中非线性机制和地形的作用研究:以1509号台风"灿鸿"为例

本文基于FVCOM（Finite Volume Community Ocean Model）构建了一个覆盖中国渤海、黄海和东海的数值模型,采用NCEP-CFSR风场数据对1509号台风“灿鸿”产生的风暴潮进行模拟,与实测水位数据的对比表明该模型可靠、模拟结果合理。基于此模型,本文对非线性作用和地形在风暴潮增水过程中的作用进行了研究。首先,重点分析了增水过程中潮汐与风暴潮的非线性作用,结果表明：高潮时非线性作用使增水值降低;低潮时非线性作用使增水值升高。另外,开边界处分别只添加M₂、S₂和K₁分潮,分析天文潮的潮高和周期对非线性作用的影响,结果表明：潮高越高,非线性作用越明显;半日潮的非线性作用较全日潮更明显;并且,增水极值附近出现的半日周期的波动也与非线性作用有关。其次,除了非线性作用,地形对风暴潮的增水也有一定影响,本文改变地形的实验结果表明：坡度越大,增水极值越小。琉球群岛的存在使得东南沿海出现风暴潮增水的面积减小,但使得风暴潮增水的高值区域扩大。     

Determining the influence of wind-wave breaking on the dissipation of the turbulent kinetic energy in the upper ocean and on the dependence of the turbulent kinetic energy on the stage of wind-wave development

New experimental data that make it possible to explain and predict the observed variability of turbulent-energy dissipation in the upper ocean are discussed. For this purpose, the dependence of the energy dissipation rate of breaking wind waves on their propagation velocity (see [1]) is used. The turbulent-energy dissipation values obtained earlier in [2, 3] by a direct method are compared to the results of radar measurements of individual breaking events presented in [1]. On the basis of this comparison, a strong dependence of the turbulent-energy dissipation value on the stage of wind-wave development, which is characterized by the ratio U _a /c _p (U _a is the wind speed and c _p is the phase speed of the peak of the wind-wave spectrum) is confirmed. This dependence was found earlier purely empirically. Moreover, it is shown that the theoretically obtained dependence (c _p /U _a )⁴, does not contradict the available empirical data. The results of this study opens possibilities for scientifically substantiated calculations of greenhouse-gas exchange (specifically, CO₂ exchange between the ocean and the atmosphere).     

Numerical study of the storm-induced circulation on the Scotian Shelf during Hurricane Juan using a nested-grid ocean model

A nested-grid ocean circulation modelling system is used to assess the upper ocean response of the Scotian Shelf and adjacent slope to Hurricane Juan in September 2003. The nested-grid system consists of a fine-grid inner model covering the Scotian Shelf/slope and a coarse-grid outer model covering the northwest Atlantic Ocean. The model-calculated upper ocean response to Hurricane Juan is characterized by large divergent surface currents forced by the local wind forcing under the storm, and intense near-inertial currents in the wake of the storm. The sea surface temperature (SST) cooling produced by the model is biased to the right of the storm track and agrees well with a satellite-derived analysis. Over the deep water, off the Scotian Shelf, some of the near-inertial energy input by the storm is advected eastward by the Gulf Stream away from the storm track. The hurricane also generates shelf waves that propagate equatorward with the coastline on their right. In comparison with the outer model results, the inner model captures more meso-scale structures, greater SST cooling and stronger near-inertial currents in the study region.     

A methodology for deriving extreme nearshore sea conditions for structural design and flood risk analysis

Extreme sea conditions in the nearshore zone are required for coastal flood risk analysis and structural design. Many multivariate extreme value methods that have been applied in the past have been limited by assumptions relating to the dependence structure in the extremes. A conditional extremes statistical model overcomes a number of these previous limitations. To apply the method in practice, a Monte Carlo sampling procedure is required whereby large samples of synthetically generated events are simulated. The use of Monte Carlo approaches, in combination with computationally intensive physical process models, can raise significant practical challenges in terms of computation. To overcome these challenges there has been extensive research into the use of meta-models. Meta-models are approximations of computationally intensive physical process models (simulators). They are derived by fitting functions to the outputs from simulators. Due to their simplified representation they are computationally more efficient than the simulators they approximate.Here, a methodology for deriving a large Monte Carlo sample of extreme nearshore sea states is described. The methodology comprises the generation of a large sample of offshore sea conditions using the conditional extremes model. A meta-model of the wave transformation process is then constructed. A clustering algorithm is used to aid the development of the meta-model. The large sample of offshore data is then transformed through to the nearshore using the meta-model. The resulting nearshore sea states can be used for the probabilistic design of structures or flood risk analysis. The application of the methodology to a case study site on the North Coast of Spain is described.     

Observation of wave-enhanced turbulence in the near-surface layer of the ocean during TOGA COARE

Dissipation rate statistics in the near-surface layer of the ocean were obtained during the month-long COARE Enhanced Monitoring cruise with a microstructure sensor system mounted on the bow of the research vessel. The vibration contamination was cancelled with the Wiener filter. The experimental technique provides an effective separation between surface waves and turbulence, using the difference in spatial scales of the energy-containing surface waves and small-scale turbulence. The data are interpreted in the coordinate system fixed to the ocean surface. Under moderate and high wind-speed conditions, we observed the average dissipation rate of the turbulent kinetic energy in the upper few meters of the ocean to be 3–20 times larger than the logarithmic layer prediction. The Craig and Banner (J. Phys. Oceanogr. 24 (1994) 2546) model of wave-enhanced turbulence with the surface roughness length from the water side z₀ parameterized according to the Terray et al. (J. Phys. Oceanogr. 26 (1996) 792) formula z₀=cH_s provides a reasonable fit to the experimental dissipation profile, where z is the depth (defined here as the distance to the ocean surface), c≈0.6, and H_s is the significant wave height. In the wave-stirred layer, however, the average dissipation profile deviates from the model (supposedly because of extensive removing of the bubble-disturbed areas close to the ocean surface). Though the scatter of individual experimental dissipation rates (10-min averages) is significant, their statistics are consistent with the Kolmogorov's concept of intermittent turbulence and with previous studies of turbulence in the upper ocean mixed layer.     

引入拖曳系数参数化的海冰自由漂流模拟研究

Many interesting characteristics of sea ice drift depend on the atmospheric drag coefficient(C_a) and oceanic drag coefficient(C_w).Parameterizations of drag coefficients rather than constant values provide us a way to look insight into the dependence of these characteristics on sea ice conditions.In the present study,the parameterized ice drag coefficients are included into a free-drift sea ice dynamic model,and the wind factor α and the deflection angle θ between sea ice drift and wind velocity as well as the ratio of C_a to C_w are studied to investigate their dependence on the impact factors such as local drag coefficients,floe and ridge geometry.The results reveal that in an idealized steady ocean,C_a/C_w increases obviously with the increasing ice concentration for small ice floes in the marginal ice zone,while it remains at a steady level(0.2-0.25) for large floes in the central ice zone.The wind factor α increases rapidly at first and approaches a steady level of 0.018 when A is greater than 20%.And the deflection angle θ drops rapidly from an initial value of approximate 80° and decreases slowly as A is greater than20%without a steady level like α.The values of these parameters agree well with the previously reported observations in Arctic.The ridging intensity is an important parameter to determine the dominant contribution of the ratio of skin friction drag coefficient(C_s' /C_s) and the ratio of ridge form drag coefficient(C_r'/C_r) to the value of C_a/C_w,α,and θ,because of the dominance of ridge form drag for large ridging intensity and skin friction for small ridging intensity among the total drag forces.Parameterization of sea ice drag coefficients has the potential to be embedded into ice dynamic models to better account for the variability of sea ice in the transient Arctic Ocean.     

A generalized approach for long-term modelling of extreme crest-to-trough wave heights

This paper deals with the long-term modelling of high sea waves. The solution is given for the return period of sea storms during which an arbitrary chosen number of waves, with crest-to-trough heights exceeding a fixed threshold, occur. This return period is derived starting from the Equivalent Triangular Storm (ETS) model, which associates a triangle to each actual storm and thus represents a significant wave height time series at a fixed location by means of a sequence of triangular storms. The short-term statistics is then applied to investigate the occurrence of large crest-to-trough wave heights during a given storm. Finally, by combining the statistical distribution of significant wave heights, the ETS model and the short-term wave statistics, the solution is given for the return periods R_N and R_N of a sea storm in which N or at least N waves higher than a fixed threshold occur. The values of R_N are then calculated, starting from data of two buoys moored in the Pacific Ocean and in the Mediterranean Sea.