Surface gravity waves from direct numerical simulations of the Euler equations: A comparison with second-order theory

When the wave spectrum is sufficiently narrow-banded and the wave steepness is sufficiently high, the modulational instability can take place and waves can be higher than expected from second-order wave theory. In order to investigate these effects on the statistical distribution of long-crested, deep water waves, direct numerical simulations of the Euler equations have been performed. Results show that, for a typical design spectral shape, both the upper and lower tails of the probability density function for the surface elevation significantly deviate from the commonly used second-order wave theory. In this respect, the crest elevation is observed to increase up to 18% at low probability levels. It would furthermore be expected that wave troughs become shallower due to nonlinear effects. Nonetheless, the numerical simulations show that the trough depressions tend to be deeper than in second-order theory.     

随机波浪下泰勒离散系数的时域解

利用Wolk提出的粒子追踪方程,通过等分频率法划分不规则波谱,利用MATLAB做粒子运动模拟计算,得到无因次化泰勒离散系数K/D随时间t变化的曲线;通过与Huang等得到的P-M谱的泰勒离散系数K/D计算结果比较证明了本计算方法的可靠性。采用该方法研究了不规则波条件下,波序列(同一谱型不同波面序列)和谱型(谱峰周期、有效波高、谱峰升高因子)对波浪离散系数的影响;计算结果表明:同一谱型不同波序列对泰勒纵向离散系数稳定值和稳定时间无影响;不规则波谱峰周期越大,纵向离散系数K/D越小,稳定时间越短;有效波高越大,纵向离散系数K/D越大,稳定时间越长;谱峰升高因子越大,泰勒离散系数K/D越大,稳定时间越长;与规则波相比,不规则波的泰勒离散系数K/D的值略小10%~30%。     

Statistical characteristics of individual waves in laboratory wind waves I. Individual wave spectra and similarity structure

Statistical characteristics of individual waves in laboratory wind waves have been studied by use of a wind-wave tunnel. The individual waves are defined by actual undulations of the water surface at any instant, and are characterized by concentrated shearing stress and strong vorticity at their crests. A conspicuous self-similarity structure is found in the individual wave field. The similarity manifests itself as a simple spectral form, and as the statistical 3/2-power law between nondimensional wave height and wave period, and further as the -1/2-power relationship between nondimensional phase speed and frequency, for waves of the high frequency side. The normalized energy spectrum, specially defined for individual waves, has a form practically equivalent to the traditional spectrum for component waves in the main frequency range from 0.7 to 1.5 in the frequency normalized by the peak frequency, but does not have secondary peaks at harmonics. The phase speed of individual waves also coincides with that of component waves in the main frequency range.     

The energy cycle of wind waves on the sea surface

The problems of wind-induced waves on the sea surface are considered. To this end, the empirical fetch laws that determine variations in the basic periods and heights of waves in relation to their fetch are used. The relation between the fetch and the physical time is found, as are the dependences of the basic characteristics of waves on the time of wind forcing. It is found that about 5% of wind energy dissipated in the near-water air layer contributes to the growth of wave heights, i.e. wave energy, although this quantity depends on the age of waves and the exponent in the fetch laws. With consideration for estimates of the probability distribution functions for the wind over the world ocean [11], it is found that the rate of wind-energy dissipation in the near-water air layer is on the order of 1 W/m². The calculations of wind waves [19] for the world ocean for 2007 have made it possible to assess the mean characteristics of the cycle of wave development and their seasonal variations. An analysis of these calculations [19] shows that about 20% of wind energy is transferred to the water surface. The remaining amount (80%) of wind energy is spent on the generation of turbulence in the near-water air layer. About 2%, i.e., one tenth of the energy transferred to water, is spent on turbulence generation due to the instability of the vertical velocity profile of the Stokes drift current and on energy dissipation in the surf zones. Of the remaining 18%, 5% is spent directly on wave growth and 13% is spent on the generation of turbulence during wave breaking and on a small-scale spectral region. These annually and globally mean estimates have a seasonal cycle with an amplitude on the order of 20% in absolute values but with a smaller amplitude in relative values. According to [19] and to the results of this study, the annually mean height of waves is estimated as 2.7 m and their age is estimated as 1.17.     

Rainfall effect on wind waves and the turbulence beneath air-sea interface

Rainfall effects on wind waves and turbulence are investigated through the laboratory experiments in a large wind-wave tank. It is found that the wind waves are damped as a whole at low wind speeds, but are enhanced at high wind speeds. This dual effect of rain on the wind waves increases with the increase of rain rate, while the influence of rainfall-area length is not observable. At the low wind speed, the corresponding turbulence in terms of the turbulent kinetic energy （TKE） dissipation rate is significantly enhanced by rain- fall as the waves are damped severely. At the high wind speed, the augment of the TKE dissipation rate is suppressed while the wind waves are enhanced simultaneously. In the field, however, rainfall usually hin- ders the development of waves. In order to explain this contradiction of rainfall effect on waves, a possibility about energy transfer from turbulence to waves in case of the spectral peak of waves overlapping the inertial subrange of turbulence is assumed. It can be applied to interpret the damping phenomenon of gas trans- fer velocity in the laboratory experiments, and the variation of the TKE dissipation rates near sea surface compared with the law of wall.     

二维海浪波包跨水平特征量及其应用

将Adler关于二维随机波场跨水平特征量的理论应用于二维线性随机海浪,严格导出了二维海浪波包的跨水平特征量表达式,并利用该式讨论了给定时刻二维海浪的一种几何结构——大波簇集情况。对于海面上波高均超过较高参考水平的一簇大波,给出了一个计算其中波峰平均个数的公式,它与Glazman所给出的公式不同另外还给出计算上述大波簇平均面积的公式。最后结合现有的海浪万向谱给出一些具体计算结果,并对这些结果进行了讨论。     

Numerical simulations of Rossby–Haurwitz waves

Simulations of Rossby–Haurwitz waves have been carried out using four different high‐resolution numerical shallow water models: a spectral model, two semi‐Langrangian models predicting wind components and potential vorticity respectively, and a finite‐volume model on a hexagonal–icosahedral grid. The simulations show that (i) unlike the nondivergent case, the shallow water Rossby–Haurwitz wave locally generates small‐scale features and so has a potential enstrophy cascade, and (ii) contrary to common belief, the zonal wavenumber 4 Rossby–Haurwitz wave is dynamically unstable and will eventually break down if initially perturbed. Implications of these results for the use of the Rossby–Haurwitz wave as a numerical model test case are discussed. The four models tested give very similar results, giving confidence in the accuracy and robustness of the results. The most noticeable difference between the models is that truncation errors in the hexagonal–icosahedral grid model excite the Rossby–Haurwitz wave instability, causing the wave to break down quickly, whereas for the other models in the configurations tested the instability is excited only by roundoff error at worst, and the Rossby–Haurwitz wave breaks down much more slowly or not at all.     

Laboratory study of deep-water breaking waves

In an attempt to elucidate the mechanics of deep-water wave breaking, a variety of breaking waves, including spilling and plunging waves, of different length scales and geometries was studied. The waves were generated through wave-wave interactions using wave packets with constant-steepness components, constant-amplitude components, and also components following the Pierson-Moskowitz distribution. Wave steepening prior to breaking were found to cause an increase in the high frequency spectral slope of the wave spectrum. The slopes were correlated to the type of breaking and the intensity of the breaking. The energy loss through breaking varied with the spectral characteristics of the wave packet. On the other hand, it was also noted that, irrespective of the wave packet, the losses were from the higher frequency end of the first harmonics.     

Disintegration of linear edge waves

It is demonstrated that offshore wavenumbers of edge waves change from imaginary wavenumbers in deep water to real wavenumbers in shallow water. This finding indicates that edge waves in the offshore direction exist as evanescent waves in deep water and as propagating waves in shallow water. Since evanescent waves can stably exist in a limited region while propagating waves cannot, energy should be released from nearshore regions. In the present study, the instability region is predicted based on both the full water wave solution and the shallow-water wave approximation.     

Wave group analysis of natural wind waves based on modulational instability theory

This paper discusses wave grouping of wind waves observed at Lake Biwa from the physical viewpoint of wave modulational instability. Amplitude modulation periods obtained from the smoothed instantaneous wave energy history (SIWEH) of the observed data proposed by Funke and Mansard are compared with the values predicted by the modulational instability theory using the Zakharov equation for a finite constant water depth derived by Stiassnie and Shemer. The modulation period normalized by the typical wave period corresponds to the length of total run. It is shown that the amplitude modulation periods of the observed data agree satisfactorily with the predicted values. Thus, it is concluded that the modulational instability is a hydrodynamical cause of grouping of high waves.     

Wind-generated waves in Hurricane Juan

We present numerical simulations of the ocean surface waves generated by hurricane Juan in 2003 as it reached its mature stage (travelling from deep waters off Bermuda to Nova Scotia and making landfall near Halifax) using SWAN (v.40.31) nested within WAVEWATCH-III (v.2.22; denoted WW3) wave models, implemented on multiple-nested domains. As for all storm-wave simulations, spectral wave development is highly dependent on accurate simulations of storm winds during its life cycle. Due to Juan’s rapid translation speed (accelerating from 2.28 m s⁻¹ on 27 September, 1200 UTC to 20 m s⁻¹ on 29 September, 1200 UTC), an interpolation method is developed to blend observed hurricane winds with numerical weather prediction (NWP) model winds accurately. Wave model results are compared to in situ surface buoys and ADCP wave data along Juan’s track. At landfall, Juan’s maximum waves are mainly swell-dominated and peak waves lag the occurrence of the maximum winds. We explore the influence of surface waves on the wind and show that the accuracy of the wave simulation is enhanced by introducing swell and Stokes drift feedback mechanisms to modify the winds, and by limiting the peak drag coefficient under high wind conditions, in accordance with recent theoretical and experimental results.     

Stokes波的谱特性分析

应用理论推导及数值计算方法，对Stokes随机波的谱特性进行了分析。首先将波面方程，海水质点水平速度用一阶波面分量的非线性组合表示，应用平稳随机高阶短的降阶计算法则，得到了波面方程及海水质点水平速度与一阶波面分量的自相关函数之间的关系，从而确定了Stokes随机波浪的波浪谱密度及海水质点水平速度和加速度谱密度，进而求得有关波浪要素的均方根值。文章还应有数值计算方法，分析了波浪基本参数对均方根值的影响。     

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.     

Dynamic pressure distribution on large circular cylinders caused by wind generated random waves

A technique has been developed to evaluate the dynamic pressures on cylindrical structures due to irregular waves based on McCamy and Fuchs linear diffraction theory and principle of superposition. Experiments have been conducted in Hydraulic Engineering Laboratory, Indian Institute of Technology, Madras to examine the dynamic pressures excited by irregular waves on circular cylinders of large diameter. The pressures around the cylinder at representative locations were measured and compared with calculated pressures. The spectral density functions computed from pressure time series measured in the laboratory agree well with the calculated pressure spectral densities. The agreement is good and encouraging.     

Spectral analysis of Stokes waves

The spectral properties of Stokes waves are shown in this paper by theoretical and numerical methods. This is done by expressing wave profiles and velocities of water particles as nonlinear combinations of the first order component of wave profiles. Under the assumption of the first order wave profiles being zero mean Gaussian processes, the relationship between autocorrelation functions of wave profiles and velocities of water particles and the first order component of wave profiles is established using the nonlinear spectral analysis. The spectral densities of nonlinear random waves, the velocities and accelerations of water particles are then obtained. Numerical computations are carried out to analyze the effect of fundamental parameters of waves. The results indicate that wave height is the most sensible parameter to the root mean squares related and wave depth is the least sensible one of all.     

Efficient computation method for two-dimensional nonlinear waves

I~IOWNUmerics wave tank is a ~ tOOl by which variouS nonlinear wave - ac interactionpwhleln can be treated in time domain. As the face stage, we f~ our efforts on the devious of an effita saution tee~ of fully nofor waves. aam numtried of water - wavesconnected PwhlemS, es~ in a wave tank, is alwayS comas Of PartS: wave generation, P~ and a~. The effi~ Of a nUm~ wave tankdepends not Only on the qwhty Of the wave abotion tecboaue, which allowS the lOng times~ and keePS finite tank leng…     

The initialization of nonlinear water waves in a numerical wave flume

This study investigates the initialization of nonlinear free-surface simulations in a numerical wave flume.Due to the mismatch between the linear input wavemaker motion and the kinematics of fully nonlinear waves,direct numerical simulations of progressive waves,generated by a sinusoidally moving wavemaker,are prone to suffering from high-frequency wave instability unless the flow is given sufficient time to adjust.A time ramp is superimposed on the wavemaker motion at the start that allows nonlinear free-surface simulations to be initialized with linear input.The duration of the ramp is adjusted to test its efficiency for short waves and long waves.Numerical results show that the time ramp scheme is effiective to stabilize the wave instability at the start of the simulation in a wave flume.     

Simulating offshore sand waves

Sand waves form a prominent regular pattern in the offshore seabeds of sandy shallow seas and pose a threat to a range of offshore activities. A two-dimensional vertical (2DV) flow and morphological simulation model describing the behaviour of these sand waves has been developed. The simulation model contains the 2DV shallow water equations, with a free water surface and a general bed load formula. The water movement is coupled to the sediment transport equation with a seabed evolution equation. The domain is non-periodic in both directions. The spatial discretisation is performed by a spectral method based on Chebyshev polynomials. A fully implicit method is chosen for the discretisation in time. Firstly, we validate the simulation model mathematically by reproducing the results obtained using a linear stability analysis for infinitely small sand waves. Hereby, we investigate a steady current situation induced by a wind stress applied at the sea surface. The bed forms we find have wavelengths in the order of hundreds of metres when the resistance at the seabed is relatively large. The results show that it is possible to model the initial evolution of sand waves with a numerical simulation model. This paper forms the necessary first step to investigate the intermediate term behaviour of sand waves.     

南海北部台风海浪谱的双峰性和成长性

本论文通过对南海北部三次台风过境期间基于浮标观测的海浪谱进行分析,发现虽然大部分成熟的台风海浪谱为单峰结构,但实际上在台风海浪的成长和衰减阶段,双峰谱占据了很大的比例。双峰谱的形成主要是由于风浪和涌浪的叠加以及不同波分量之间的非线性相互作用,我们可以通过能量密度的成长率对谱型变化进行高效的预报。此外,台风海浪的主要波向依赖于台风中心相对观测点的位置,而波向的分散情况在相距台风中心较远的区域无明显规律。本文提出了一个新的六参数波浪谱型拟合双峰谱,其拟合效果相较于前人的谱型更好。通过验证,形状参数和谱宽度之间的理论关系依然适用于单个谱峰。通过分析谱参量的变化特征,证明了谱参量不仅与台风强度和台风路径相关,还存在很强的交互相关。最后通过拟合海浪谱数据,本文得到了台风影响下海浪有效波高和有效周期之间的成长关系,这对海洋工程实际应用具有重要意义。     

Non-linear effects in the packets of internal waves in the ocean

The mechanisms for the generation of mean density and current velocity fields in a medium with a vertical shear, conditioned by the non-linearity of packets of internal waves are analysed. Modulation instability of internal waves is studied allowing for the earth's rotation. A non-linear Schrödinger's equation for the evolution of the envelope has been derived. Also corrections to the mean density and current velocity in the approximation, quadratic by the wave's amplitude, non-oscillating on the wave's time scale have been obtained. The longitudinal modulation conditions have been analysed.Translated by V. Puchkin. UDK 551.466.8.