Steepness and asymmetry of extreme waves and the highest waves in deep water

Traditional wave steepness s=H/L does not define steep asymmetric waves in a random sea uniquey. Three additional parameters characterising single zero-downcross waves in a time series are crest front steepness, vertical asymmetry factor and horizontal asymmetry factor. Results for steepness and asymmetry from zero-downcross analysis of wave data obtained from full scale measurements in deep water on the Norwegian continental shelf in 58 time series are presented. The analysis demonstrates clearly the asymmetry of both “extreme waves” and the highest waves. The period and height of the highest waves are also given together with their correlation to spectral parameters. The measured maximum wave heights are also compared with predicted values of maximum wave heights showing good agreement.     

An approach for eliminating re-reflected waves

Theoretical and experimental studies were conducted to eliminate the re-reflected waves in a wave channel by installing a wavefilter in front of the wavemaker. A thin porous mesh is installed in front of the wavemaker.to serve as a wavefilter. A porous-effect parameter [Chwang, A. T. and Li, W. (1983), A piston-type porous wavemaker theory. J. Engng Math. 17, 301–313], G₀ = bω/μL₀, is employed to characterize the transmissability of the wavefilter. Theoretical relationships are established between the amplitude of progressive wave and G₀, and the distances between the wavefilter, the wavemaker and the test structure. The proper location for the wavefilter to eliminate re-reflected waves can be determined. Several experimental tests were conducted to verify the theory.Both theoretical and experimental studies show that re-reflected waves can be effectively eliminated by placing a wavefilter at a proper position between the wavemaker and the test structure, provided G₀ ≤ 1. For G₀ > 1 however, the wavefilter would fail.     

A simple analytical model for surface water waves on a depth-varying current

A perturbation analysis is presented in which a series of small amplitude regular waves co-exist with an arbitrarily sheared current, U(z). Assuming that the current velocity is weak, i.e. U(z)/c=O(ε), the solution is extended to O(ε²), where c is the phase velocity and ε=ak the wave steepness. This provides a first approximation to the non-linear wave-current interaction, and allows simple explicit solutions for both the modified dispersion relation and the water-particle kinematics to be derived. These solutions differ from the existing irrotational models commonly used in design and, in particular, highlight the importance of the near-surface vorticity distribution. These results are shown to be in good agreement with laboratory data provided by Swan et. al. [J. Fluid Mech (2001, in press)]. Perhaps more surprisingly, good agreement is also achieved in a number of strongly non-linear wave-current combinations, where the results of the present analytical solution are compared with a fully non-linear numerical wave-current model.     

Prediction of wave pressures on smooth impermeable seawalls

Simple prediction methods are proposed to estimate the wave induced pressures on smooth impermeable seawalls. Based on the physics of the wave structure interaction, the sloped seawall is divided into a total of five zones (zones 1, 2 and 3 during run-up (corresponding pressures are called as positive pressures) and zones 4 and 5 during run-down (corresponding pressures are called negative pressures)) (Fig. 1). Zone 1 (0<z<d−H_i/2), where the wave pressure is governed by the partial reflection and phase shift; Zone 2 (d−H_i/2<z<d), where the effect of wave breaking and turbulence is significant; Zone 3 (d<z<Run-up height), where the pressure is induced by the run-up water; Zone 4 (Run-down<z<d), where the wave pressure is caused by the run-down effect and Zone 5 (0<z<d-Run down), where the negative wave pressures are due to partial reflection and phase shift effects. Here d is the water depth at the toe of the seawall, H_i is the incident wave height and z is the vertical elevation with toe of the seawall as origin and z is positive upward. For wave pressure prediction in zones 1 and 5, the empirical formula proposed by Ahrens et al. (1993) to estimate the wave reflection and Sutherland and Donoghue's recommendations (1998) for the estimation of phase shift of the waves caused by the sloped structures are used. Multiple regression analysis is carried out on the measured pressure data and empirical formulas are proposed for zones 2, 3 and 4. The recommendations of Van der Meer and Breteler (1990) and Schüttrumpf et al. (1994) for the prediction of wave run-down are used for pressure prediction at zone 4. Comparison of the proposed prediction formulas with the experimental results reveal that the prediction methods are good enough for practical purposes. The present study also shows a strong relation between wave reflection, wave run-up, wave run-down and phase shift of waves on wave pressures on the seawalls.     

深水极端波浪非线性几何特征试验分析

为了解极端波浪非线性特征,明确波群在演化过程中的水动力学特性,针对一系列高斯波群进行了深水物理试验分析。试验结果显示,增加波陡或波群宽度,均可使波面偏度Sk发生明显变化,尤其当波浪发生破碎后,在破碎区域内,波面偏度变化范围剧烈增大,说明该偏度极大值可能作为判断破碎的一个指标。波陡和波群宽度对波面不对称度影响程度不同:当波陡或波群宽度增加后,波峰不对称度所受影响最大,波峰前端波谷不对称度次之,波峰后端波谷不对称度所受影响最小,但仍不可忽略。在波浪演化过程中,幅值谱出现不同程度频带下移,波浪破碎后,会出现永久频带下移;当调制不稳定发生时,随着调制不稳定指数增加,频带下移量呈现快速增长趋势。     

Long waves in directional seas

In recent years the group-induced long waves have received an enhanced degree of attention. Especially in nearshore regions, the long waves can be of considerable height, and consequently the influence on harbour resonance, on the operation of ship terminals, on moorings of large vessels, etc. is obviously very important. It is the grouping of natural wave fields that generates the long waves, and they are proportional to the square of the short-wave height. Therefore, the expressions for the long-wave elevations can be found to include the short-wave components of the wave field and a second-order transfer function. This function is presented in a diagram with dimensionless parameters. For practical purposes a formula for rough estimate of the long-wave height is proposed.The second-order equations show that the long waves are determined by the difference of the wave-number vectors of the short waves. This is shown to imply that the spread of the long waves is larger than that of the short waves, and that the wave lengths of the long waves are dependent on the short-wave spread. Hereby it is possible to change the long-wave lengths, which seems to be a quality of great practical importance.The long waves are also expressed in spectral terms. That is, a formula for the directional long-wave spectrum is shown to comprise the transfer function squared and the short-wave amplitudes and phases.     

Characteristics of turbulent boundary layers over a rough bed under saw-tooth waves and its application to sediment transport

A large number of studies have been done dealing with sinusoidal wave boundary layers in the past. However, ocean waves often have a strong asymmetric shape especially in shallow water, and net of sediment movement occurs. It is envisaged that bottom shear stress and sediment transport behaviors influenced by the effect of asymmetry are different from those in sinusoidal waves. Characteristics of the turbulent boundary layer under breaking waves (saw-tooth) are investigated and described through both laboratory and numerical experiments. A new calculation method for bottom shear stress based on velocity and acceleration terms, theoretical phase difference, φ and the acceleration coefficient, a_c expressing the wave skew-ness effect for saw-tooth waves is proposed. The acceleration coefficient was determined empirically from both experimental and baseline k–ω model results. The new calculation has shown better agreement with the experimental data along a wave cycle for all saw-tooth wave cases compared by other existing methods. It was further applied into sediment transport rate calculation induced by skew waves. Sediment transport rate was formulated by using the existing sheet flow sediment transport rate data under skew waves by Watanabe and Sato [Watanabe, A. and Sato, S., 2004. A sheet-flow transport rate formula for asymmetric, forward-leaning waves and currents. Proc. of 29th ICCE, ASCE, pp. 1703–1714.]. Moreover, the characteristics of the net sediment transport were also examined and a good agreement between the proposed method and experimental data has been found.     

The stability of the antifer units used on breakwaters in case of irregular placement

The primary aim of the study is to experimentally investigate the stability performance of antifer units on the trunk section of breakwaters under the effect of regular and irregular waves in case of irregular placement. The stability performance tests were conducted for different slopes, i.e. cot α=1.25, 1.5, 2.0, 2.5, under irregular waves and for cot α=2.5 under regular waves. Hudson’s formula was employed in order to characterize the stability performance of antifer units for the irregular placement technique. Different representative wave height parameters, i.e. H_s, H_1/10 and H_max, were examined to determine the one best characterizing breakwater stability. Furthermore, the effects of wave period and wave steepness on the stability of the breakwater were explored.     

Trapped bottom topographic waves over the inclined bottom

In the Boussinesq approximation, we study baroclinic topographic waves trapped by the flat meridional slope. The existence of these waves is explained by stratification, inclined bottom, and Earth's rotation. We deduce the evolutionary equation for the square of the envelope of a narrow-band wave packet of trapped waves. In the second order of smallness relative to the wave amplitude, we find the mean fields of velocity and density induced by the packet. It is shown that, in the limiting case of weakly nonlinear plane waves, the induced current is zonal. In the Northern hemisphere, depending on the slope of the bottom γ₁, the sign of the phase velocity σ/k (k is the zonal wave number) is either always positive (for γ₁>γ_1cr) or always negative (for γ₁<γ_1cr). If we neglect the vertical component of the Coriolis acceleration, then γ_1cr=0. Translated by Peter V. Malyshev and Dmitry V. Malyshev     

Nonlinear properties of laboratory wind waves at energy containing frequencies

Nonlinear properties of wind waves in a wind-wave tunnel are investigated by measuring the probability density distribution of surface elevation. The surface elevation distribution of raw records are found to have a positive skewness (K ₃=0.21 to 0.43) and a negative kurtosis (K ₄=–0.74 to –0.41) with magnitude depending of fetch and wind speed. The values of skewness are in qualitative agreement with a prediction of the weak interaction theory for a random wave field incorporating the effects of second harmonics (Tayfun, 1980), but the values of kurtosis are different in sign from the prediction.To examine the nonlinear properties of energy containing components, higher harmonic components are excluded from the wave records by using a kind of a band-pass filter. The surface elevation distributions of the filtered waves show a sharp decrease in skewness , but the distributions remain highly non-Gaussian with a large negative kurtosis almost independent of the fetch and wind speed . It is concluded that the negative kurtosis is due to the non-random character of the phase and amplitude among the energy containing components, and that nonlinear interactions occur amongst the energy containing frequencies.     

A study of Coriolis effects on long waves propagating in an unbounded ocean, channel and basin

The effects of Coriolis force on long waves have been discussed based on gravity waves propagating in an unbounded ocean, channel and basin. In case of ocean, results show that the Coriolis effect will be significant and negligible, when the wave period is comparable to 2π/f and much shorter, respectively. Results also show in a channel, the wave amplitude and water particle velocity decrease exponentially in the positive y direction in the northern hemisphere (where f is positive). Moreover, in a basin, the Cotidal lines have been found as curves and rotate counterclockwise around the origin.     

Parameters of JONSWAP spectral model for surface gravity waves—II. Predictability from real data

Monte Carlo simulation of wave spectra was carried out to provide an assessment of JONSWAP spectral model and parameters. The simulation method is found to be satisfactory because (a) it excludes the spectral variability due to geophysical factors from the sampling errors in the spectral estimates and the statistical uncertainty in determining the model parameters; and (b) the simulated spectra can represent ideal spectral estimates where the sampling errors have been minimized by increasing the degrees of freedom of the spectra. The latter (b) allows both the magnitude of sampling errors to be evaluated and errors due to statistical uncertainty to be isolated. Thus, the stimulation study provides a useful error analysis to assess the JONSWAP spectral model and parameters. For instance, it is found from the results that the sampling errors could be as high as 20% while errors due to uncertainty in determining the model parameter could be as high as 17%. However, the overall errors may be reduced to the minimum of approx. 15% if the simulated spectra have 80 degrees of freedom and constant values of σ_a and σ_b i.e. σ_a = 0.07 and σ_b = 0.09. This implies that the maximum accuracy of 85% may be achieved in JONSWAP spectral model even though the α parameter has been underestimated by about 1.5%. The overestimated values of γ might come from the underestimated α and the biased φ_m estimator caused by the statistical uncertainty in the presence of a sharp spectral peak. Although the scale parameters (α and φ_m) exhibit smaller errors and variability than the shape parameters (ψ, σ_a and σ_b), they are more sensitive to the degrees of freedom of the spectra and their estimators are not better than the estimators of shape parameters. The simulation experiments have also shown that simulated spectra at 20–40 degrees of freedom contain a substantial amount of sampling errors. Therefore, the measured wave spectra at the same degrees of freedom (20–40) are not suitable and should not be used for evaluating the accuracy of any wave spectral model.     

Nonlinear effects in the process of propagation of internal waves with regard for the turbulent viscosity and diffusion

By the method of asymptotic multiscale expansions in the Boussinesq approximation, we study nonlinear effects observed in the process of propagation of internal waves with regard for the turbulent viscosity and diffusion. We determine the decrement of attenuation of waves and the boundary-layer solutions at the bottom and on the free surface. The wave-induced mean current is found in the second order of smallness in the wave steepness. The coefficients of the nonlinear Schr?dinger equation are obtained for the envelope of the wave packet. It is shown that a weakly nonlinear plane wave is stable under longitudinal modulation in the long-wave limit. If the wavelength is smaller than a certain critical value, then the wave is unstable under modulation.     

Instability of the Kuroshio in Luzon Strait: Effects of ridge topography and stratification

The spectral analyses of moored current velocities in the central Luzon Strait reveal northward (i.e., downstream of the Kuroshio) propagation of a frontal wave with a five-day period, with wave amplitude increasing northward. Estimated from both curve fitting and frequency domain Empirical Orthogonal Function methods, the characteristics of five-day variations have wave speeds ranging from 32 to 40 cm s⁻¹, wavelengths ranging from 130 to 150 km, and e-folding time scales for growth ranging from 0.8 to 3 days. An analytical two-layer model used to explore linear stability characteristics indicates that bottom topography (two meridional ridges) is important for the Kuroshio stability characteristics in the Luzon Strait. In the two-layer model with the two ridges, the flow is stabilized for the long-wave mode but destabilized for the short-wave mode (due to increasing vertical shear in the horizontal velocity). The analytical model produces wavelengths and phase speeds for the most unstable mode which is similar to the observation, but the growth rate is underestimated. However, a spectral numerical model applied with a more realistic stratification and velocity structure does obtain faster growth rates comparable to the observations. Parameter sensitivity tests were conducted using the analytical model. The characteristics of the most unstable mode are most sensitive to the surface front location relative to the bottom topography but not sensitive to varying the density difference and thickness of the upper layer.     

多向不规则波浪的确定性模拟

波浪波动时间过程及波列的模拟,对于开展实际波浪对于工程建筑物的作用具有重要的意义。本文采用线性叠加的单叠加模型,建立了多向不规则波浪的确定性模拟方法。基于理论模拟的规则波、单向不规则波和多向不规则波,验证了波浪确定性模拟方法的有效性。定性地对比分析了模拟波列和已知波列的一致性;定量地研究了模拟波浪在空间范围rr/L_s的误差分布情况(rr表示指定位置与给定位置的空间距离,L_s为有效波长)。并且建议,采用本文方法进行波浪确定性模拟时,最佳的浪高仪间距应小于0.12L_s。     

Wave interaction with ‘’-type breakwaters

The wave transmission, reflection and energy dissipation characteristics of ‘’-type breakwaters were studied using physical models. Regular and random waves in a wide range of wave heights and periods and a constant water depth were used. Five different depths of immersion (two emerged, one surface flushing and two submerged conditions) of this breakwater were selected. The coefficient of transmission, K_t, and coefficient of reflection, K_r, were obtained from the measurements, and the coefficient of energy loss, K_l was calculated using the law of balance of energy. It was found that the wave transmission is significantly reduced with increased relative water depth, d/L, whether the vertical barrier of the breakwater is surface piercing or submerged, where ‘d’ is the water depth and ‘L’ is the wave length. The wave reflection decreases and energy loss increases with increased wave steepness, especially when the top tip of the vertical barrier of this breakwater is kept at still water level (SWL). For any incident wave climate (moderate or storm waves), the wave transmission consistently decreases and the reflection increases with increased relative depth of immersion, Δ/d from −0.142 to 0.142. K_t values less than 0.3 can be easily obtained for the case of Δ/d=+0.071 and 0.142, where Δ is the height of exposure (+ve) or depth of immersion (−ve) of the top tip of the vertical barrier. This breakwater is capable of dissipating wave energy to an extent of 50–80%. The overall performance of this breakwater was found to be better in the random wave fields than in the regular waves. A comparison of the hydrodynamic performance of ‘’-type and ‘T’-type shows that ‘T’-type breakwater is better than ‘’-type by about 20–30% under identical conditions.     

Design weight of armour stone considering the effect of extreme waves

In this study, waves with the heights higher than H_1/3 in an irregular wave train are called as extreme waves and defined with the help of extreme wave parameter, α_extreme. In order to see the effect of extreme waves on the design weight of armour stone, stability analysis is carried out based on the hydraulic model test results. The test results of high α_extreme cases (HE) and low α_extreme cases (LE) are compared with currently used van der Meer's formulae with permeability factor P=0.4 and 0.45 and Hudson formula by using H_1/3 and H_1/10 in terms of the design weight of armour stone. As a result of the comparison, it is found that Hudson formula by using H_1/3 underestimates the necessary armour weight. Usage of H_1/10 instead of H_1/3 in Hudson formula doubles the weight which seems overestimated when Irribaren number is away from the transition zone in which both wave run-up and run-down forces become effective. However, it seems underestimated near the transition zone where experiment case HE gives higher armour weights. When the design weight of armour stone is calculated by van der Meer's formulae with P=0.4, it may be necessary to increase the weight up to 30% in the case of high extreme waves. On the other hand, van der Meer's formulae may overestimate the weight 14% when the extreme waves are low.     

基于Sentinel-3载荷OLCI和SRAL数据的内波同步探测研究

The ocean and land color instrument(OLCI) and synthetic aperture radar altimeter(SRAL) installed aboard the Sentinel-3 satellite have been in orbit for operational uses. In this study, data collected from Sentinel-3 are used to investigate internal waves in the South China Sea. An internal wave is detected using an OLCI image with a resolution of 300 m, and an analysis was performed with a quasi-synchronous moderate-resolution imaging spectroradiometer(MODIS) image. The opposite characteristics of OLCI and MODIS images of the same internal wave are explained by the critical angle in brightness reversals. The unique observational geometry of the OLCI image and its influence on observations of internal waves are discussed. The distribution of σ0 and sea surface height anomalies(SSHAs) induced by internal waves are studied using SRAL records. The σ0 records of SRAL occasionally show less sensitivity to the modulation of internal waves, which may be attributed to the observational geometry, while SSHAs show obvious variations. The synchronous pairing of OLCI images and SRAL records are analyzed to extract the three-dimensional sea surface signatures induced by internal waves. The analysis demonstrates that the profile of SSHAs in the surface shows an opposite phase to the profiles of internal waves in the ocean. The opposite phase relationship, observed in the remote sensing view, is also confirmed with a laboratory experiment.     

Spectral characteristics of high shallow water waves

The spectral characteristics of shallow water waves with significant wave height more than 2 m based on the data collected along the Indian coast is examined. It was found that the value of Joint North Sea Wave Project (JONSWAP) parameters (α and γ) increases with significant wave height and mean wave period and decreases with spectral peak period. The estimated average value (0.0027 and 1.63) of the JONSWAP parameters, α and γ were less than the generally recommended values of 0.0081 and 3.3, respectively. By carrying out a multi-regression analysis, an empirical equation is arrived relating the JONSWAP parameters with significant wave height, peak wave period and mean wave period. It was found that the Scott spectra underestimate the maximum spectral energy of high waves. The study shows that the measured wave spectra can be represented by JONSWAP spectra with the JONSWAP parameters estimated based on the equation proposed in this paper.