Turbulent structure in water under laboratory wind waves

The structure of the turbulent boundary layer underneath laboratory wind waves was studied by using a combination of a high-sensitivity thermometer array with a two-component sonic flowmeter. The temperature fluctuations are used to detect movements of water parcels, with temperature as a passive quantity. The turbulence energy was dominant in the frequency range (0.01 0.1 Hz), which was much smaller than the wind-wave frequency (2 5 Hz), and in which the turbulence was anisotropic. There was a frequency range (0.2 2 Hz for velocity, 0.2 5 Hz for temperature fluctuation) where the turbulence was isotropic and had a –5/3 slope in the energy spectrum. These points are the same as those in previous works. However, by analyses of the time series by using a variable-interval time-averaging technique (VITA), it has been found that conspicuous events in this main turbulence energy band are the downward bursting from the vicinity of the water surface. Thus the structure of the water layer underneath the wind waves has characters which are similar to the familiar turbulent boundary layer over a rough solid wall, as already conceived. It has been found that, at the same time, the turbulence energy can be related to quantities of the wind waves (the root mean squared water level fluctuation and the wave peak frequency), for different wind and wave conditions. That is, the turbulence underneath the wind waves develops under a close coupling with the wind waves.     

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.     

The energy balance in the wind-wave spectrum

The spectral energy balance in the wind-wave spectrum is studied with taking into account the energy input from turbulent wind to waves, the energy rearrangement due to conservative nonlinear wave-wave interaction and the energy dissipation due to water turbulence. Using the Ichikawa's (1978) model on the turbulent wind field over wind-waves and assuming that the energy dissipation is times greater than that due to molecular viscosity of water, the energy input and dissipation are determined so as to satisfy the condition that the nonlinear-transfers of momentum and energy conserve the total momentum and energy of waves. The nonlinear energy-transfer is estimated from the energy balance at each frequency. It is found that the energy input and dissipation satisfying the condition on the conservative nonlinear-transfer are determined by the characteristic height of wind-wave field and the friction velocity of air, and that the spectral distribution of the nonlinear energy-transfer estimated in this paper is qualitatively similar to that estimated by the non-linear wave-wave interaction theory ofHasselmann (1962).     

Irrotational,progressive surface gravity waves near the limiting height

Numerical solutions of irrotational, progressive surface gravity waves in water of a constant depth are obtained by means of an iterative method. Our results suggest that waves with the surface slope angle greater than/6 may exist. The calculated phase velocity of deep water waves near the wave steepness 0.14 is significantly smaller than the value given by the Stokes' fourth approximation.In order to check our method, we apply it to the problem proposed byDavies (1951), which is hypothetical but similar to the present problem, and for which the exact solution is known. In this case our results show good agreement with the exact solution.     

Downward transfer of momentum by wind-driven waves

A model for the downward transfer of wind momentum is derived for growing waves. It is shown that waves, which grow due to an uneven pressure distribution on the water surface or a wave-coherent surface shear stress have horizontal velocities out of phase with the surface elevation. Further, if the waves grow in the x-direction, while the motion is perhaps time-periodic at any fixed point, the Reynolds stresses associated with the organized motion are positive. This is in agreement with several field and laboratory measurements which were previously unexplained, and the new theory successfully links measured wave growth rates and measured sub-surface Reynolds stresses. Wave coherent air pressure (and/or surface shear stress) is shown to change the speed of wave propagation as well as inducing growth or decay. From air pressure variations that are in phase with the surface elevation, the influence on the waves is simply a phase speed increase. For pressure variations out of phase with surface elevation, both growth (or decay) and phase speed changes occur. The theory is initially developed for long waves, after which the velocity potential and dispersion relation for linear waves in arbitrary depth are given. The model enables a sounder model for the transfer to storm surges or currents of momentum from breaking waves in that it does not rely entirely on ad-hoc turbulent diffusion. Future models of atmosphere-ocean exchanges should also acknowledge that momentum is transferred partly by the organized wave motion, while other species, like heat and gasses, may rely totally on turbulent diffusion. The fact that growing wind waves do in fact not generally obey the dispersion relation for free waves may need to be considered in future wind wave development models.     

Laboratory wave reflection analysis using co-located gages

Most coastal and ocean engineering laboratories employ techniques that use two or three spatially separated wave gages to estimate reflection of irregular waves in two-dimensional wave flumes. This paper presents a frequency domain method for separating incident and reflected wave spectra from co-located gages (gages located on the same vertical line). The technique is based on linear wave theory, and it can be applied to time series of sea surface elevation and horizontal water velocity collected in a vertical array, or it can be used with horizontal and vertical water velocity time series collected at the same point in the water column. Application of the method is limited to those frequencies showing good coherence between time series signals. Outside the range of good coherence, gross inaccuracies occur.The utility of the co-located gage method is illustrated using water velocity data collected in a wave flume with a laser Doppler velocimeter, and the method is validated for the case of complete reflection by a vertical wall. Side-by-side comparison to the spatially-separated wave gage method of Goda and Suzuki (1976) exhibited close agreement for a variety of irregular wave trains being weakly reflected by a mild sloping beach. The co-located gage method is useful in situations where there are spatial variations in the wavelength, such as on a mildly sloping bottom, or in the region close to highly reflective structures where errors arising from spatial variations in characteristic wave parameters would corrupt estimates made using the spatially-separated wave gage method.     

Transformation of a Baroclinic Tide in the Process of Motion over an Unevenness of the Sea Bottom

Within the framework of the linear theory of long waves, we study the process of incidence of a baroclinic tide on an unevenness of the sea bottom simulating an oceanic ridge. The liquid is assumed to be stratified and the action of the Coriolis force is taken into account. We determine the characteristics of wave disturbances in the region of the ridge induced by the incident baroclinic waves depending on the parameters of the bottom topography and tidal period. Thus, in particular, over the ridge, we reveal the existence of both the regions, where the amplitudes of internal waves are much higher than the amplitude of the incident baroclinic tide, and shaded regions, where the opposite phenomenon is observed.__________Translated from Morskoi Gidrofizicheskii Zhurnal, No. 6, pp. 3–9, November–December, 2004.     

Dynamic properties of green water event in the overtopping of extreme waves on a fixed dock

A statistical model is developed to predict wave overtopping volume and rate of extreme waves on a fixed deck. The probability density function for the volume and rate of overtopping water are formulated based on the truncated Weibull distribution with the assumption of local sinusoidal profile for small amplitude waves. Sensitivity to the wave nonlinearity parameter and deck clearance is discussed. The statistical model is compared to laboratory data of the instantaneous free surface elevation measured in front of a fixed deck, and overtopping volume and overtopping rate measured at the leading edge of the deck. The statistical theory compared well with the measured exceedance probability seaward of the deck. The model prediction of the exceedance probability of deck overtopping gave qualitatively good agreement for large overtopping values.     

Wave pattern induced by a moving atmospheric pressure disturbance

A moving low atmospheric pressure is a main feature of tropical cyclones, which can induce a system of forced water waves and is an important factor that cause water level rise during a storm. A numerical model based on the nonlinear shallow water equations is applied to study the forced waves caused by an atmospheric pressure disturbance moving with a constant velocity over water surface. The effects of the moving speed, the spatial scale and the central pressure drop of the pressure disturbance are discussed. The results show that the wave pattern caused by a moving low-pressure is highly related with its moving speed. The wave pattern undergoes a great change as the moving speed approaches the wave velocity in shallow water. When the moving speed is less than the wave velocity, the distribution of water surface elevation is nearly the same as that of the pressure disturbance, and the maximum of the water surface elevation is located at the center of pressure. When the moving speed is larger than the wave velocity, a triangle shaped wave pattern is formed with a depression occurs in front of the pressure center, and the maximum of the water surface elevation lags behind the center of pressure. As the moving speed increases, the maximum of the water surface elevation firstly increases and then decreases, which reaches a peak when the moving speed is close to the wave velocity. The maximum of water surface elevation is approximately in proportion to the central pressure drop, and slightly affected by the spatial scale of pressure disturbance. Both the central pressure drop and the spatial scale of the pressure disturbance do not significantly affect the forced wave pattern. However, a clear difference can be noticed on the ratio of the maximum water surface elevation in moving pressure situation to that in static situation, when the moving speed is close to the wave velocity. A pressure disturbance with smaller spatial scale and smaller central pressure drop will give a larger ratio when the moving speed is close to the wave velocity.     

次重力波对宽刈幅高度计海表面高度观测的影响

次重力波(Infragravity Wave,IGW)是一种频率较低(0.05～0.005 Hz),波长较长(约10 km)的表面重力波。由IGW引起的海表面高度变化会被宽刈幅干涉高度计SWOT(Surface Water and Ocean Topography,SWOT)卫星观测到,因此在使用SWOT观测的海表面高度来反演中尺度、次中尺度大洋环流时,IGW是一种重要的误差来源。根据数值模型模拟的全球IGW时空分布特征,本文以IGW最为活跃的东北太平洋和欧洲西北陆架附近大西洋为研究海域,估算了上述海域由IGW所引起的海表面高度变化,并将计算结果与SWOT Simulator模拟的轨道噪声(±5 cm)比较,首次定量地估算了IGW在SWOT观测海表面高度时的干扰程度。研究表明,IGW所引起的厘米量级的海表面高度变化在SWOT卫星观测海表面流场时是一种重要的,不可忽略的误差来源。在大西洋欧洲西北陆架海域,冬季IGW对海表面高度的贡献可达到SWOT卫星噪声要求水平的25%;然而,对于大陆架狭窄的美国西岸太平洋而言,由岸线产生的IGW将迅速传入深海海域,在广阔的范围内产生显著的"噪声"影响,在SWOT反演海表面流场时由IGW引起的误差将达到SWOT卫星噪声要求水平的15%。     

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.     

Very low frequency Ocean Bottom ambient seismic noise and coupling on the shallow continental shelf

Sources of very low frequency (0.01 to 1.0 Hz) ambient seismic noise in the shallow (<100 m) water continental margin sediments are investigated using Ocean Bottom Seismometers (OBS). The predominant seismic motions are found to be due to surface gravity (water) waves and water-sediment interface waves. Actual experimental measurements of seabed acceleration and hydrodynamic pressure are given, including side by side comparisons between buried and plate-mounted OBS units. OBS-sediment resonant effects are found to be negligible at the low frequencies under investigation. Wherever there exists relative motion between the seabed and the water, however, an exposed OBS is subject to added mass forces that cause it to move with the water rather than the sediments. Calculations based on measured seabed motions show that a neutral density, buried seismometer has superior sediment coupling charactersitics to any exposed OBS design.     

Experimental Study on Crescent Waves Diffracted by A Circular Cylinder

Crescent waves often observed on the sea surface are unusual wave pattern induced by the instability of Stokes wave. The paper presents the experimental results of the wave field around a circular cylinder generated by the diffraction of crescent wave in order to examine the difference of diffracted crescent waves from the commonly-used diffracted Stokes waves. The results show that with the existence of the cylinder, the crescent wave pattern can still get fully developed, and with the presence of this type of wave pattern, the symmetry breaking of the wave amplitude distribution occurs and there are extra wave components at the frequencies of 0.5ω₀, 1.5ω₀ and 2.5ω₀ (ω₀ is the frequency of Stokes waves) appearing in the wave amplitude spectrum.     

数值波浪水槽中完全非线性浅水波仿真特性研究

应用基于势流理论的时域高阶边界元方法,建立一个完全非线性的三维数值波浪水槽,通过实时模拟推板造波运动的方式产生波浪。通过混合欧拉-拉格朗日方法和四阶Runge-Kutta方法更新自由水面和造波板的瞬时位置。利用所建模型分别模拟了有限水深波和浅水波,与试验结果、相关文献结果和浅水理论结果吻合较好,且波浪能够稳定传播。系统地讨论造波板的运动圆频率、振幅和水深等对波浪传播和波浪特性的影响,并对波浪的非线性特性进行分析,研究发现造波板运动频率、运动振幅以及水深均将对波浪形态和波浪非线性产生显著影响。结果为真实水槽造波机的运动控制以及波浪生成试验提供了依据,便于实验室设置更合理的参数来准确模拟不同条件下的波浪。     

Focused Wave Properties Based on A High Order Spectral Method with A Non-Periodic Boundary

In this paper,a numerical model is developed based on the High Order Spectral(HOS) method with a non-periodic boundary.A wave maker boundary condition is introduced to simulate wave generation at the incident boundary in the HOS method.Based on the numerical model,the effects of wave parameters,such as the assumed focused amplitude,the central frequency,the frequency bandwidth,the wave amplitude distribution and the directional spreading on the surface elevation of the focused wave,the maximum generated wave crest,and the shifting of the focusing point,are numerically investigated.Especially,the effects of the wave directionality on the focused wave properties are emphasized.The numerical results show that the shifting of the focusing point and the maximum crest of the wave group are dependent on the amplitude of the focused wave,the central frequency,and the wave amplitude distribution type.The wave directionality has a definite effect on multidirectional focused waves.Generally,it can even out the difference between the simulated wave amplitude and the amplitude expected from theory and reduce the shifting of the focusing points,implying that the higher order interaction has an influence on wave focusing,especially for 2D wave.In 3D wave groups,a broader directional spreading weakens the higher nonlinear interactions.     

Occurrence probability of a freak wave in a nonlinear wave field

The Edgeworth’s form of a cumulative expansion of the probability density function (PDF) of surface elevation expands the maximum wave height distribution to predict the occurrence probability of freak waves. This study investigated the enhancement of the occurrence probability of freak waves due to the fourth order moment of surface elevation, kurtosis, change and found that the nonlinear effects on the occurrence probability of a freak wave linearly depends on kurtosis for a small number of waves N=250. The statistical theory was compared with field data, and freak waves sometimes appear when not expected by the Rayleigh theory, but they were predicted by the proposed theory.     

Statistical characteristics of long waves nearshore

The random long wave runup on a beach of constant slope is studied in the framework of the rigorous solutions of the nonlinear shallow water theory. These solutions are used for calculation of the statistical characteristics of the vertical displacement of the moving shoreline and its horizontal velocity. It is shown that probability characteristics of the runup heights and extreme values of the shoreline velocity coincide in the linear and nonlinear theory. If the incident wave is represented by a narrow-band Gaussian process, the runup height is described by a Rayleigh distribution. The significant runup height can also be found within the linear theory of long wave shoaling and runup. Wave nonlinearity nearshore does not affect the Gaussian probability distribution of the velocity of the moving shoreline. However the vertical displacement of the moving shoreline becomes non-Gaussian due to the wave nonlinearity. Its statistical moments are calculated analytically. It is shown that the mean water level increases (setup), the skewness is always positive and kurtosis is positive for weak amplitude waves and negative for strongly nonlinear waves. The probability of the wave breaking is also calculated and conditions of validity of the analytical theory are discussed. The spectral and statistical characteristics of the moving shoreline are studied in detail. It is shown that the probability of coastal floods grows with an increase in the nonlinearity. Randomness of the wave field nearshore leads to an increase in the wave spectrum width.     

Physical Investigation of Directional Wave Focusing and Breaking Waves in Wave Basin

An experimental scheme for the generation of directional focusing waves in a wave basin is established in this paper. The effects of the directional range, frequency width and center frequency on the wave focusing are studied. The distrihution of maximum amplitude and the evolution of time series and spectra during wave packet propagation and the variation of water surface parameters are extensively investigated. The results reveal that the characteristics of focusing waves are significantly influenced hy wave directionality and that that breaking criteria for directional waves are distinctly different from those for unidirectional waves.     

A field study of the development of wind-waves

This paper presents the results of observation on the development of wind-waves which were generated in a lake water about 420 cm deep with a fetch 12 km long. Measurements of surface elevation were carried out at the end of an observational pier where the water depth was 80 cm. The wave momentum flux, i.e., the growth rate of the wave momentum, was estimated from both significant waves and power spectral densities for the wave records. The values obtained by the two ways accorded fairly well and they were 57 % as large as the wind stress measured simultaneously. The exponential growth rate of spectral densities for a frequency component was in good accord with that observed bySnyder andCox (1966) and by others. If these growth rates are applied to all the components of the spectrum, the wave momentum flux must exceed the wind stress. This cannot explain the experimental results nor can be physically accepted. The difference of spectral densities between the two successive runs showed that the increase of spectral densities was. limited in several bands of frequency. The phenomena are discussed in relation with the overshoot-undershoot effects studied byBarnett andSutherland (1968).Observational results suggest that the spectral growth of a certain component is closely related to the spectral densities of other components. Energy exchange among componented waves has not been considered in the theories for generation and development of wind-waves established by Phillips, Miles and others.New generation mechanism suggested byLonguet-Higgins (1969) was found to be able to describe the observed growth rates of the form(f)={(1/2)(t–t_1/2)}²: the spectral density(f) was proportional to the square of durationt. However, the mechanism can not explain the overshoot-undershoot effects peculiar to the equilibrium spectrum of windwaves.Three frequencies characterizing the discrete distributions of frequency bands where spectral densities increased were examined and three waves corresponding to these frequencies were found to be satisfying the resonance conditions for the wave-wave interactions among three sinusoidal wave trains as studied byPhillips (1960),Longuet-Higgins (1962) andBenny (1962). The interactions are suggested to predict well both the spectral growth proportional to squares of duration and the ceaseless oscillations of spectral densities in an equilibrium spectrum.