Influence of processes of nonlinear transformations of waves in the coastal zone on the height of breaking waves

Using data from laboratory, field, and numerical experiments, we investigated regularities in changes in the relative limit height of breaking waves (the breaking index) from peculiarities of nonlinear wave transformations and type of wave breaking. It is shown that the value of the breaking index depends on the relative part of the wave energy in the frequency range of the second nonlinear harmonic. If this part is more than 35%, then the breaking index can be taken as a constant equal to 0.6. These waves are spilling breaking waves, asymmetric on the horizontal axis, and are almost symmetric on the vertical axis. If this part of the energy is less than 35%, then the breaking index increases with increasing energy in the frequency range of the second harmonic. These waves are plunging breaking waves, asymmetric on the vertical axis, and are almost symmetric on the horizontal axis. It is revealed that the breaking index depends on the asymmetry of waves on the vertical axis, determined by the phase shift between the first and second nonlinear harmonic (biphase). It is shown that the relation between the amplitudes of the second and first nonlinear harmonics for an Ursell number less than 1 corresponds to Stokes’ second-order wave theory. The empirical dependences of the breaking index on the parameters of nonlinear transformation of waves are proposed.     

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.     

On the possibility of biphase parametrization for wave transformation in the coastal zone

Based on experimental data, we study the possibility of parametrizing the spatial variation in the phase shift (biphase) between the first and second nonlinear harmonics of wave motion during wave transformation over an inclined bottom in the coastal zone. It is revealed that the biphase values vary in the range [–π/2, π/2]. Biphase variations rigorously follow fluctuations in amplitudes of the first and second harmonics and the periodicity of energy exchange between them. Wave breaking influences the biphase value, retaining its variations in the negative domain in the range [–π/2, 0]. The formula applied in modern practice to calculate the biphase, which depends on the Ursell number, is incorrect for calculating the biphase for wave evolution in the coastal zone, because it does not take into account periodic energy exchange between the nonlinear harmonics. We propose a linear approximation of the biphase values from the size of the ratio of the current distance to the coast to the possible spacial duration of the exchange period, which is determined by the dispersion relation. We reveal the dependence of biphase variations on the wave transformation scenario and demonstrate the possibility of constructing a separate parameterization of the biphase for each scenario. Our research and the obtained biphase parameterizations can be used to simulate the sea state in the coastal zone, as well as in problems of predicting the development of coasts under the impact of storm waves.     

Higher-harmonic focused-wave forces on a vertical cylinder

This paper considers higher-harmonic forces due to wave focusing on a vertical circular cylinder. A series of experiments has been conducted in a wave flume. The first six-harmonic components of the measured wave forces are analyzed using the scale-averaged wavelet spectrum. It is noted that due to the transient nature of focused (freak) waves, Fourier analysis would not provide equivalent information to that gleaned from the analyses used herein. The results for the experiments with very steep wave crests show significant amplitudes at the fourth and fifth harmonics. These harmonics exhibit amplitudes that are the same order as the second harmonic, but much larger than those of the third harmonic. The wavelet-based bicoherence is used to detect the quadratic nonlinearity of the measured forces. And the bicoherence spectra reveal the primary mathematical reason for the existence of the striking amplitudes of the fourth and fifth harmonics: the interaction between the lower-harmonic components couple more strongly with the fourth and fifth harmonics, thus the fourth and fifth harmonics glean more energy than those of the third-harmonic components. However, the physical explanation for this remains elusive.     

珊瑚礁破碎带附近波浪演化和波生流实验研究

为了研究珊瑚岸礁破碎带附近波浪演化和波生流特性,通过水槽实验对规则波浪作用下珊瑚岸礁上沿礁分布的水位和流速进行了详细的测量。在典型卷破波条件下,测试了礁冠存在与不存在的两种情况。实验结果表明,多重波浪反射作用引起岸礁上形成不完全驻波,而破碎带附近的浅化作用则产生高次谐波,波浪破碎所耗散的波能主要来源于主频波,礁坪上透射波成分中二次谐波与主频波的能量相当;礁冠的存在引起破碎带宽度减小、礁坪上增水变大以及礁坪上各次谐波变小;礁冠不存在时,岸礁上波生流的沿礁分布与平直海岸相似,而礁冠的存在一定程度上阻碍了礁坪上水体向外海的回流。研究成果将丰富和发展珊瑚礁水动力学理论,并为岛礁工程的建设和维护提供一定的理论参考。     

Applicability of numerical models to nonlinear dispersive waves

The applicability of three different wave-propagation models in nonlinear dispersive wave fields has been investigated. The numerical models tested here are based on three different wave theories: a fully nonlinear potential theory, a Stokes second-order theory, and a Boussinesq-type theory with an improved dispersion relation. Physical experiments and computations were conducted for wave evolutions during passage over a submerged shelf under various wave conditions. As expected, the fully nonlinear solutions agree better with the measurements than do the other solutions. Although the second-order solution has sufficient accuracy for smaller-amplitude wave cases, the truncation after the third harmonics causes significant discrepancies in wave form for larger waves. In addition, the second-order model markedly overestimates the first- and second-harmonic amplitudes in transmitted waves. The Boussinesq model provides excellent predictions of wave profile over the shelf even in larger wave cases. However, this model also overestimates the magnitudes of several higher harmonics in transmitted waves. These facts may indicate that energy transfer from bound components into free waves in these higher harmonics cannot be accurately evaluated by the Boussinesq-type equations.     

Simulation of wave-energy converter with octagonal linear generator

To extract electrical energy from sea waves in a commercially and technologically acceptable manner, a number of issues have to be solved. Electricity generation by means of direct conversion of the oscillating gravitational potential energy of a floating buoy can be anticipated, provided a proper design of a generator could be made. This paper deals with the simulation of a novel design for a linear generator aimed for the extraction of energy from ocean waves. The ocean waves are modeled by 4-m-height sinusoidal waves with a characteristic period of 7 s. A wide range of the geometrical sizes, permanent magnets, stator winding, and spring forces acting on the buoy are possible. This paper presents simulations of octagonal three-phase linear generators in the 100-kW power range. The beneficial effects of a stator of octagonal shape are briefly investigated, but not studied in depth. The main emphases in the present study have been to decrease power fluctuations and suppress voltage harmonics. In conventional rotating machines, well-known measures are to use a fractional number of slots per pole and phase, and an additional method is to make the pole edges smoother. These methods are here simulated for the first time on a linear machine aimed for ocean wave-energy conversion and a substantial reduction in power fluctuations and voltage harmonics are predicted.     

Nonlinear effects of the distribution of amplitudes of sea waves

The effect of nonlinearities, such as wave-breaking and vertical asymmetry associated with sea waves, on the distribution of wave amplitudes is explored. Semiclosed theoretical expressions are derived to describe the distributions of breaking-limited crest and trough amplitudes for Stokes-type nonlinear sea waves. These are compared with the conventional Rayleigh distribution appropriate to linear wave amplitudes. The construction of nonlinear wave envelopes with the fast Fourier transform technique is described. The technique can be utilized to enlarge the data base in empirical analyses of field records which typically contain limited information on amplitude characteristics. The theoretical distributions and the proposed data enlargement technique are demonstrated with the analysis of a nonlinear wave record.     

Rhines效应对高度计观测的海洋中尺度涡空间分布特征的影响

Rhines效应是指Rossby波和大湍流（中尺度涡）相互作用,将涡动能量以波的形式传播出去,从而使中尺度涡发生形变,最终消亡的一种动力学机制。本文通过比较海洋里涡特征速度和Rossby长波波速的方法,研究了一种广义的Rhines效应对高度计观测的海洋中尺度涡空间分布特征的影响。结果显示,广义Rhines效应比只考虑行星涡度梯度的传统形式对中尺度涡的分布具有更显著的影响。大部分中尺度涡分布在涡特征速度（U_e）大于由广义Beta值计算的Rossby长波波速（U_cg）的区域。这些涡可以由动能反向串级过程获取能量,成长为振幅和空间尺度较大的涡。热带海域以外的“涡旋沙漠”区域,中尺度涡的数量稀少,强度很弱,大都分布于U_ecg的海域。广义Rhines效应可能是这些海域中尺度涡难以成长的动力学机制。     

First and second baroclinic mode responses of the tropical Indian Ocean to interannual equatorial wind anomalies

The combined and individual responses of the first and second baroclinic mode dynamics of the tropical Indian Ocean to the well-known Indian Ocean Dipole mode (IOD) wind anomalies are investigated. The IOD forced first baroclinic Rossby waves arrive at the western boundary in three months, while the reflected component from the eastern boundary with opposite phase arrives in five to six months, both carry input energy to the west. The inclusion of the second baroclinic mode slows down the wave propagation by mode coupling and stretches the energy spectrum to a relatively longer time scale. The total energy exists in the equatorial wave guide for at least five months from the forcing, as much as 10% of that of the atmospheric input, which mainly dissipates at the western boundary. The individual responses of the ocean to IOD interannual wind anomaly show that the significant modes of oceanic anomalies are confined to a wave guide of 10° on either side of the equator.     

Internal wave spectrum in shallow water: measurement and comparisonwith the Garrett-Munk model

The covariance matrix of sound-speed variations is determined from yo-yo CTD data collected during the SWARM 95 experiment at a fixed station. The data covered approximately 2 h and were collected during a period when nonlinear solitary internal waves were absent or negligible. The method of empirical orthogonal functions (EOF) is applied to the sound-speed covariance matrix assuming that the internal wave modes are uncorrelated. The first five eigenvectors are found to agree well with the theoretically modeled eigenfunctions based on the measured buoyancy frequency and the internal wave eigenmode equation. The mode amplitudes for the first five modes are estimated from the corresponding eigenvalues. They agree with the Garrett-Munk model if j*=1 is used instead of j*=3. A second method is used to deduce the mode amplitudes and mode frequency spectra by projecting the sound-speed variation (as a function of time) onto the theoretical mode depth functions. The mode amplitudes estimated with this method are in agreement with the EOF results. A modified Garrett-Munk model is proposed to fit the frequency spectrum of linear internal waves in shallow water     

Cross-shore variations of wave groupiness by wavelet transform

This paper proposes a new definition of groupiness factor (GF) based on the local wavelet energy density of the wave time series to describe the groupiness degree of waves. The main advantage of this new GF is that the effect of the operational definition on it is smaller than that on SIWEH-based GF or envelop-based GF. Then, the new GF is used to study the groupiness variations of mechanically generated irregular waves in a wave flume propagating on a slope of 1:45. The results of present study show that the decrease of groupiness in the coast is triggered by breaking. And energy distribution along the record time for the first harmonics of waves in the surf zone, which becomes more uniform than that out the surf zone, is the main reason causing the decrease of groupiness.     

Radar scattering on gravity-capillary waves: Laboratory investigation

The mechanism of the resonance (Bragg) scattering of microwaves by gravity-capillary waves (GCWs) is checked experimentally. Resonant regularities of backscattered signal intensity are obtained as functions of frequency and GCW propagation direction. It is shown that the width of the resonance curves is determined by the width of the directivity pattern of the scatterometer’s antenna. The excitation of the GCW second harmonic and the spatial structure of the wave field at the GCW doubled frequency are investigated. The ratio of the amplitudes of the free wave and forced harmonic that originated during excitation of the primary wave is determined. The resonance curve is obtained for the second-order scattering of radio waves (on forced harmonics and free GCWs). The correction to the backscattering cross section is investigated in the second order of smallness relative to the Bragg term.     

Surf zone dynamics simulated by a Boussinesq type model. Part II: surf beat and swash oscillations for wave groups and irregular waves

This is the second of three papers on the modelling of various types of surf zone phenomena. In the first paper the general model was described and it was applied to study cross-shore motion of regular waves in the surf zone. In this paper, part II, we consider the cross-shore motion of wave groups and irregular waves with emphasis on shoaling, breaking and runup as well as the generation of surf beats. These phenomena are investigated numerically by using a time-domain Boussinesq type model, which resolves the primary wave motion as well as the long waves. As compared with the classical Boussinesq equations, the equations adopted here allow for improved linear dispersion characteristics and wave breaking is modelled by using a roller concept for spilling breakers. The swash zone is included by incorporating a moving shoreline boundary condition and radiation of short and long period waves from the offshore boundary is allowed by the use of absorbing sponge layers. Mutual interaction between short waves and long waves is inherent in the model. This allows, for example, for a general exchange of energy between triads rather than a simple one-way forcing of bound waves and for a substantial modification of bore celerities in the swash zone due to the presence of long waves. The model study is based mainly on incident bichromatic wave groups considering a range of mean frequencies, group frequencies, modulation rates, sea bed slopes and surf similarity parameters. Additionally, two cases of incident irregular waves are studied. The model results presented include transformation of surface elevations during shoaling, breaking and runup and the resulting shoreline oscillations. The low frequency motion induced by the primary-wave groups is determined at the shoreline and outside the surf zone by low-pass filtering and subsequent division into incident bound and free components and reflected free components. The model results are compared with laboratory experiments from the literature and the agreement is generally found to be very good. Finally the paper includes special details from the breaker model: time and space trajectories of surface rollers revealing the breakpoint oscillation and the speed of bores; envelopes of low-pass filtered radiation stress and surface elevation; sensitivity of surf beat to group frequency, modulation rate and bottom slope is investigated. Part III of this work (Sørensen et al., 1998) presents nearshore circulations induced by the breaking of unidirectional and multi-directional waves.     

Nonlinear interaction of internal waves in the coastal zone of the Sea of Japan

Nonlinear interactions of the internal waves of a tidal period with low-frequency synoptic-scale internal waves are studied using instrumental measurements of the current in the coastal zone of the Sea of Japan. In the course of spectral analysis of the data of instrumental measurements, it is found that a maximum in the spectrum of the kinetic energy of coastal waters in the vicinity of the semidiurnal frequency ω₀ is surrounded by satellite maxima whose frequencies obey the relation ω _s = ω₀ ± Ω, where Ω is the characteristic frequency of synoptic-scale internal waves. The spectrum of the anticyclonic current component has a similar structure in the vicinities of the frequency ω₀ and its first and second harmonics. The general theory of nonlinear interactions of weakly dispersive waves is used to solve the problem of modulation and the parametric amplification of tidal internal waves in the coastal zone using low-frequency narrow-band internal waviness. As can be judged from the literature, the effect of parametric modulation of tidal internal waves by low-frequency synoptic-scale internal waves has been recorded in the coastal zone of a tidal sea for the first time.     

Height variability from the MIROC – IPCC model for the 20th century compared to that of the TOPEX/POSEIDON altimeter

Planetary waves are key to large-scale dynamical adjustment in the global ocean as they transfer energy from the east to the west side of oceanic basins; they connect the forcing in the ocean interior with the variability at its boundaries; and they change the local heat content, thus coupling oceanic, atmospheric, and biological processes. Planetary waves, mostly of the first baroclinic mode, are observed as distinctive patterns in global time series of sea surface height anomaly (SSHA) and heat storage. The goal of this study is to compare and validate large-scale SSHA signals from coupled ocean-atmosphere general circulation Model for Interdisciplinary Research on Climate (MIROC) with TOPEX/POSEIDON satellite altimeter observations. The last decade of the models’ time series is selected for comparison with the altimeter data. The wave patterns are separated from the meso- and large-scale SSHA signals by digital filters calibrated to select the same spectral bands in both model and altimeter data. The band-wise comparison allows for an assessment of the model skill to simulate the dynamical components of the observed wave field. Comparisons regarding both the seasonal cycle and the Rossby wave field differ significantly among basins. When carried within the same basin, differences can occur between equal latitudes in opposite hemispheres. Furthermore, at some latitudes the MIROC reproduces biannual, annual and semiannual planetary waves with phase speeds and average amplitudes similar to those observed by the altimeter, but with significant differences in phase.     

多个淹没水平圆柱诱发高阶谐波特性的数值与试验研究

基于势流理论采用时域高阶边界元方法建立了模拟非线性波浪与淹没水平双、三圆柱作用的数值水槽模型,其中采用混合欧拉-拉格朗日方法更新自由水面,四阶Runga-Kutta方法进行时间步进。利用两点法分离得到潜体下游高阶自由波,进而研究潜体间距、布置形式等对高倍频自由波的影响。同时在物理水槽内开展多潜体高阶谐波物理模型试验,并将试验结果与数值结果进行对比,吻合较好。研究发现:淹没双圆柱下游高阶谐波幅值随潜体间距呈现周期性振荡变化,其振荡的重现距离约为波长一半。而淹没三圆柱下游的高阶谐波随第一个间距呈周期性振荡变化,随第二个间距的增大而减小。     

Critical Latitude in Tidal Dynamics Using the Kara Sea as an Example

It is well known that, within the linear nonviscous equations of tidal dynamics, the amplitudes of oscillations of the barotropic and baroclinic tidal velocity components unlimitedly increase when approaching the critical latitude. It is also known that the linear equations of tidal dynamics with a constant and specified vertical eddy viscosity indicate the occurrence of significant tidal velocity shears in the near-bottom layer, which are responsible for increasing the baroclinic tidal energy dissipation, the turbulent kinetic energy, and the thickness of the bottom boundary layer. The first circumstance—the growth of the amplitudes of oscillations of the barotropic and baroclinic tidal velocity components—is due to the elimination in the original equations of small terms, which are small everywhere except for the critical latitude zone. The second circumstance—the occurrence of significant tidal velocity shears—is due to the fact that internal tidal waves, which induce the dissipation of the baroclinic tidal energy and the diapycnal diffusion, are either not taken into account or described inadequately. It is suggested that diapycnal diffusion can lead to the degeneration (complete or partial) of tidal velocity shears, with all the ensuing consequences. The aforesaid is confirmed by simulation results obtained using the QUODDY-4 high-resolution three-dimensional finite-element hydrostatic model along the 66.25° E section, which passes in the Kara Sea across the critical latitude.     

空波包理论在海浪群性研究中的应用

用 Monte Carlo方法数值模拟海浪 ,研究其波包曲线跨某参考水平的波包中空波包所占的平均比例 ,并与 Ditlevsen和 L indgren关于空波包的理论相比较。结果表明 ,在二阶近似下该理论近似适用于海浪。在此基础上对 Longuet- Higgins的群性波包理论进行修正。修正后的理论与数值结果的比较表明本文所做的修正是十分有效的。修正后的群性波包理论克服了原理论的某些固有缺陷     

A note on the theoretical comparison of wave staffs and wave rider buoys in steep gravity waves

The sea surface displacements measured by a wave staff, fixed in horizontal position, and a wave rider buoy, which moves with the water particles, are compared. Second-order theory shows that, to this approximation, the second harmonic in the surface displacement is not measured by the wave rider. Here a simple approximation to maximum amplitude gravity waves is considered; then a freely floating wave rider gives an approximately sinusoidal record at a lower frequency than the wave, while a tethered wave rider gives results depending on the mooring system. The horizontal motion of a tethered buoy still affects the measurements, particularly of the second and higher harmonics.