The roughness height and drag law over the water surface based on the hypothesis of local equilibrium

A series of measurements of winds and wind-waves were carried out in wind-wave flumes. A data analysis based on the hypothesis of local equilibrium yielded a new empirical formula on the controversial quantity of roughness heightz ₀ over the water surface: , where the nondimensional roughness height is defined bygz ₀/u _* ² and the wave-wind parameterũ byω _p u _*/g, g being the gravitational acceleration,u _* the friction velocity of air,ω _p the peak frequency of wind-wave spectra. The obtained formula is compared with Charnock's (1955) and Toba's (1979) proposals; is constant in the former and inversely proportional toũ in the latter. As in Toba's, this formula immediately leads to a practically important conclusion that the drag coefficientC _d depends not merely on the usual variableU ₁₀ (wind velocity at 10m height over the water surface), but also on the surface state represented by wind-waves. An explicit expression is provided for the drag coefficient incorporating the wave-wind parameter; it covers the range ofC _d calculated from most of the previous drag formulas, by varying the wave-wind parameter.     

The mechanism of the development of wind-wave spectra

The mechanism of the development of wind-waves will be proposed on the basis of the observed wave spectra in the wind tunnels and at Lake Biwa (Imasato, 1976). It consists of two aspects: One is that the air flow over the wind-waves transfers momentum concentratively to the steepest component waves and the other is that the upper limit of the growth of a wave spectral density is given by the ultimate value in the slope spectral density. The first aspect means that the wave field has the momentum transfer filter on receiving the momentum from the air flow. Wind-waves in the stage of sea-waves receive the necessary amount of momentum by the form drag,e.g. according to the Miles' (1960) inviscid mechanism, through a very narrow frequency region around a dominant spectral peak. On the other hand, wind-waves in the stage of initial-wavelets receive it according to the Miles' (1962a) viscous model through a fairly broad frequency region around the peak. The upper limit ofS _max developing according to viscous mechanism is given byS _max =6.40×10^–4 k _max ^–2cm²s andS _max =2.03C(f _max )^–2cm²s(S _max is the power density of the wave spectral peak with the frequencyf _max ,k _max is the wave number corresponding to the frequencyf _max andC is the phase velocity).From the second aspect, the upper limit of the growth of wave spectral density is given by 33.3f ^–4cm²s in the frequency region of late stage of sea-waves. Therefore, the spectral peak, which has the largest value in the slope spectral density in the component waves of the wave spectrum, rises high over the line 4.15f ^–5cm²s. The energy is transported from the spectral peak to the high frequency part and to the forward face of a wave spectrum by nonlinear wave-wave interaction. This nonlinearity is confirmed by the bispectra calculated from the observed wind-wave data. In the stage of sea-waves, nonlinear rearrangement of the wave energy comes from a narrow momentum transfer filter, and, in the stage of initial-wavelets, it comes mainly from small corrugations and small steepness of the wave field.     

Some characteristics of the development process of the wind-wave spectrum

The development process of wind-waves of which spectral peak distributes from 0.6 cps to 9.3 cps will be discussed on the basis of the wind tunnel experiments and of the field observations performed at Lake Biwa. The characteristics of power and slope spectra are here presented. The development process of these wind-waves is characterized by three stages;i.e. initial-wavelets, transition stage and sea-waves. In the wind tunnel experiments, the transition from the stage of the initial-wavelets to the transition stage occurs when the wave spectral peak arrives at the line 6.40×10^–4 k ^–2cm²·sec (wherek is wave number) or when the slope spectral density at the frequencyf _max becomes larger than 6.40×10^–4 sec. In the stage of sea-waves, the component wave of a wave-spectral peak is steepest in the component waves. And the wave spectral peak develops along the line 1.02×10² f ^–6 cm²·sec (wheref is the frequency corresponding to the wave numberk) untill it reaches the line 33.3f ^–4cm²·sec, and thereafter develops along the latter line, which indicates the constant density of slope spectrum. It is suggested that the nonlinearity of wind-waves must become stronger as wind-waves develop. The effective momentum flux _ws from the air flow to wind-waves in this stage is evaluated to be about 49% of the total stress ₀.     

On the wave dependence of sea-surface wind stress

Analysis is made of wind and wave data, which were obtained during the passage of Typhoon 8013 at an Ocean Data Buoy Station south of Honshu operated by the Japan Meteorological Agency, in order to investigate the wave dependence of sea-surface roughness parameter in the situation where wind waves are dominant with less significant swells. The data fit better the wave-dependent expression of the wind stress,z _{0 p}/u*=, than to Charnock's formula,gz ₀/u*²=, wherez ₀ is the roughness length, _p the angular frequency of the spectral peak of wind waves,u* the friction velocity of air,g the acceleration of gravity, and are non-dimensional constants. The results are very similar to those of our previous study using data from an oil producing platform in the Bass Strait, Australia, although the type of observation system and the synoptic situation of the winds and wind waves were totally different.     

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.     

A New Empirical Formula for the Aerodynamic Roughness of Water Surface Waves

A new empirical formula for the aerodynamic roughness of water surface waves has been derived from laboratory experimental results using dimensional analysis. The formula has different forms according to wind speed: at moderate wind speeds the formula is a function of the friction velocity of wind, the surface tension, the water density, the kinematic viscosity of water and the acceleration of gravity; at strong winds the formula is expressed by the Charnock relation. The aerodynamic roughness does not depend on such wave state parameters as the spectral peak frequency or the steepness of waves, unlike almost all parameterizations that have been proposed to date. The drag coefficient at moderate winds depends on the surface tension of water and the water temperature through the temperature dependence of the kinematic viscosity of water.     

Internal flow structure of short wind waves

The minimum value of wind stress under which the flow velocity in short wind waves exceeds the phase speed is estimated by calculating the laminar boundary layer flow induced by the surface tangential stress with a dominant peak at the wave crest as observed in previous experiments. The minimum value of the wind stress is found to depend strongly on, the ratio of the flow velocity just below the boundary layer and the phase speed, but weakly onL, the wavelength. For wind waves previously studied (=0.5,L=10 cm), the excess flow appears when the air friction velocityu _* is larger than about 30 cm sec^–1. The present results confirm that the excess flow found in my previous experiments is associated with the local growth of a laminar boundary layer flow near the wave crest.     

A new expression for the production rate of sea water droplets on the sea surface

A new set of empirical formulas for the production rate and the number concentration of sea-water droplets on the sea surface are proposed, synthesizing past observation data of sea-salt particles in the sea and water droplets in wind-wave tanks. A new levelz _c is introduced as the effective wind-sea surface where seawater droplets are produced. The new formulas are expressed in linear functions in logarithmic scales ofu*²/v _p , a parameter to describe overall conditions of airsea boundary processes, whereu _* is the friction velocity of air,v the kinematic viscosity of air and _p the peak angular frequency of wind-wave part of wave spectra. A model of coexistence of spray droplets and suspended particles near the sea surface is proposed. As for the independent parameter, a comparison between the uses ofu*²/v _p and ofu _* ³ which was the traditional way of parameterization excluding wave measure, shows that the advantage of usingu*²/v _p is statistically significant with a confidence limit 89% in F-test.     

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).     

On the relation between the growth rate of water waves and wind speed

Various wind velocitiesu _*,U _/2,U andU ₁₀ are correlated to the measured growth rate of water waves , whereu _* is the friction velocity of the wind, andU _/2,U andU ₁₀ are the wind speeds respectively at the heights /2, and 10m above sea surface (: wave length). It is shown that within a range of the dimensionless wind speed, 0.1<u _* /C<0.6, there are no appreciable differences in the correlations, whereC is the phase velocity of water waves. The present relation between andU shows qualitatively similar properties as the one obtained by Al'Zanaidi and Hui (1984); the growth rate for waves with rough surface is larger than that with smooth surface. However, our present relations give, for the both waves with different surface roughness, larger values by factors 1.71.8 than those given by Al'Zanaidi and Hui's relation.     

A representation for the frequency spectrum of wind-generated waves

This paper proposes the following generalized representation for a wind-wave frequency spectrum:

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where E = ∫S(f)df is the variance of the surface displacement; f_m is the frequency of the spectral peak; and C_i's, i = 1,2,3, are dimensionless parameters that can be determined from the internal spectral parameters of a given spectrum.When applied to 234 sets of wave spectra recorded in the Great Lakes, this representation has been realistic, accurate, and capable of representing widely varied wave processes. The C_i's are clearly related to wave growth processes; they are large during early growth, decrease as waves grow, and reach approximate equilibrium when waves are fully developed.     

Simulation of the extreme waves generated by typhoon Bolaven (1215) in the East China Sea and Yellow Sea

Record-breaking high waves occurred during the passage of the typhoon Bolaven (1215) (TYB) in the East China Sea (ECS) and Yellow Sea (YS) although its intensity did not reach the level of a super typhoon. Winds and directional wave measurements were made using a range of in-situ instruments mounted on an ocean tower and buoys. In order to understand how such high waves with long duration occurred, analyses have been made through measurement and numerical simulations. TYB winds were generated using the TC96 typhoon wind model with the best track data calibrated with the measurements. And then the wind fields were blended with the reanalyzed synoptic-scale wind fields for a wave model. Wave fields were simulated using WAM4.5 with adjustment of C_d for gust of winds and bottom friction for the study area. Thus the accuracy of simulations is considerably enhanced, and the computed results are also in better agreement with measured data than before. It is found that the extremely high waves evolved as a result of the superposition of distant large swells and high wind seas generated by strong winds from the front/right quadrant of the typhoon track. As the typhoon moved at a speed a little slower than the dominant wave group velocity in a consistent direction for two days, the wave growth was significantly enhanced by strong wind input in an extended fetch and non-linear interaction.     

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.     

Some properties of surf-beats

Long ocean waves with periods of several minutes (surf-beats) were observed at a marine observation tower. We have analysed time series data of an envelope of incident swell, long period current velocity and surface elevation fluctuations. Current velocity was measued by an electromagnetic flow meter. Surf-beats amplitudeH ^(l) is shown to be proportional to 3/2 power of incident swell amplitudeH ^(s), and decreases with increase of depthh in proportional toh ^–1/2 such thatH ^(l) H ^(s) (H ^(s)/h)^1/2. Frequency energy density functionP _LL (f) of surface elevation had two dominant peaks whose frequencies were highly stable through the entire observational period. Cross-spectral analysis suggested that those peaks correspond to traveling edge waves caused by the excess momentum and mass flux in the surf zone. The forced long ocean waves predicted byLonguet-Higgins andStewart (1964) was ditected. Phase-shift and wave height of the wave with respect to those of incident swell envelope are shown to be in remarkable agreement with the predictions. However the forced long wave is only a minor component in the total energy of surf-beats. Current fields are shown to be largely composed of non-surface modes.     

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.     

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.     

Statistical study on the local equilibrium between wind and wind waves by using data from ocean data buoy stations

The local equilibrium between the wind and wind waves, which is defined by a range of the coefficient of the 3/2-power law between the non-dimensional significant wave height and period, is statistically investigated by using wind and wave data obtained at four ocean data buoy stations in the seas near Japan. The friction velocity is calculated from the wind speed measured at one height together with the significant wave period by using formulas of the wave dependent drag coefficient proposed by Tobaet al. (1990). The data for small waves or for weak winds indicate that the waves do not satisfy the criterion for the local equilibrium, because they may be affected by changing winds or remotely generated swells. In the seas near Japan, the data which satisfy the local equilibrium are about 6% through a year. Otherwise swells are dominant in most situations. Changing winds also cause deviations from the local equilibrium. The degree of satisfaction of the local equilibrium can be classified by ranges of the significant wave height. As the significant wave height exceeds 4 m, the local equilibrium is more frequently satisfied.     

Infra-gravity wave generation by the shoaling wave groups over beaches

A physical parameter, μb, which was used to meet the forcing of primary short waves to be off-resonant before wave breaking, has been considered as an applicable parameter in the infra-gravity wave generation. Since a series of modulating wave groups for different wave conditions are performed to proceed with the resonant mechanism of infra-gravity waves prior to wave breaking, the amplitude growth of incident bound long wave is assumed to be simply controlled by the normalized bed slope, βb. The results appear a large dependence of the growth rate, α, of incident bound long wave, separated by the three-array method, on the normalized bed slope, βb. High spatial resolution of wave records enables identification of the cross-correlation between squared short-wave envelopes and infra-gravity waves. The cross-shore structure of infra-gravity waves over beaches presents the mechanics of incident bound- and outgoing free long waves with the formation of free standing long waves in the nearshore region. The wave run-up and amplification of infra-gravity waves in the swash zone appear that the additional long waves generated by the breaking process would modify the cross-shore structure of free standing long waves. Finally, this paper would further discuss the contribution of long wave breaking and bottom friction to the energy dissipation of infra-gravity waves based on different slope conditions.     

Bispectra of wind-waves and wave-wave interaction

Bispectra of wind-waves in wind tunnels were calculated in order to understand the characteristics of the nonlinear wave-wave interaction in actual wind-wave field. It is shown that the nonlinearity in wind-waves increases in magnitude with the development of wind-waves and that the characteristics of nonlinearity in wind-waves in the early stage of development differ from those in the late stage. It is shown that the bispectra are classified into five types (IV), and that the bispectral type changes from the type I to the type V as the wind-waves develop from the stage of the initial-wavelets to that of the sea-waves. The relations between frequencies of the component waves interacting each other are discussed in each bispectral type.