On the Dependence of the Equilibrium Range of Sea Wave Spectra with Wave Age

Donelan and Pierson have proposed a semiphysical model of the equilibrium sea wave spectrum, based upon a parameterization of wave growth and dissipation terms. Their model is applicable for fully developed seas only. In the framework of Donelan and Pierson's approach, this paper explores the dependence of the equilibrium spectrum upon wave age. To this end, we examine how the dissipation through wave breaking is expected to vary with wave age, according to the approach proposed by Longuet-Higgins in 1969. The constraint imposed by Longuet-Higgins' theory requires an increase of the equilibrium spectrum F(k,0) in the wind direction with increasing inverse wave age U/C_p. This is in accordance with Banner's empirically deduced statement that F(k,0) is proportional to (U/C_p)^0.5 in the equilibrium range. Our inferred F(k,0) tends to increase more or less linearly with U/C_p (we find F(k,0) proportional to 1 + 0.25(U/C_p - 0.83), rather than through a power law. If a power law is fitted we obtain F(k,0) approximately proportional to (U/C_p)^0.35 for the range 0.83 < U/C_p > 5. Finally, the roughness length of the air-sea interface is inferred from our modelled spectrum through integration of the form drag over wave number under rough conditions. This shows a wave age dependence that is compatible with measurements of wind stress performed in the field at various wave ages.     

Drag coefficient modeling for the near coastal zone

Using the JONSWAP spectrum for describing the surface wave state in the near coastal zone, models for the roughness length and the drag coefficient are used to simulate the dependence of the wind stress on fetch and depth. The results of each model are then compared with a compiled set of past investigations of the neutral drag coefficient over a variety of conditions. It is found that the models of Donelan, Hsu, and Kitaigorodskii correctly predict the trends in the drag coefficient with fetch and depth. Although it did not account for all the observed variations in the neutral drag coefficient. Kitaigorodskii's model, when incorporating the JONSWAP spectrum, more accurately simulated the slopes of the various C_DN regressions against windspeed.     

One-dimensional theory of the wave boundary layer

Results obtained in a 2-D modeling of the statistical structure of the wave boundary layer (WBL) are used for elaboration of the general approach to 1-D modeling taking into account the spectral properties of wave drag for an arbitrary wave field. In the case of the wave field described by the JONSWAP spectrum, the momentum and energy spectral density exchange, vertical profiles of the wave-induced momentum flux and dependence of total roughness parameter and drag coefficient on peak frequency are given. The reasons that the total roughness parameter increases with decreasing fetch are explained. The role of wind waves as an active element of the ocean-atmosphere dynamic system is also discussed.     

Field Experiments and Numerical Modeling of Wind Speed and Surface Waves in Medium-size Inland Reservoirs

An attempt is made to apply the modern methods of surface wave simulation developed for oceanic conditions to the modeling of waves in medium-size inland reservoirs (10–100 km). The results of field measurements of wind speed and waves are described, and on their basis the parameterization C _D (U ₁₀) is proposed. WAVEWATCH III spectral wave model was adapted to the conditions of a medium-size inl and reservoir. The simulated data are compared with the field data. The use of the new parameterization C _D (U ₁₀) allowed reducing the values of the wind wave growth rate that improved consistency in data from the field experiment and numerical modeling concerning the height of significant waves. Further steps towards improving the quality of prediction of the adapted WAVEWATCH III model are discussed.     

The influence of atmospheric stratification on the growth of water waves

The atmospheric surface layer over sea has a density stratification which varies with moisture content and air/sea temperature difference. This influences the growth of water waves. To study the effect quantitatively, the Reynolds equations are solved numerically. For given wind speed and surface roughness, wave growth is found to be more rapid in unstably stratified conditions than in stable conditions. This is due to an increase in turbulence, primarily caused by an increase of mixing length.Under the assumption of a Charnock relation between surface roughness and friction velocity, it is found that for large inverse wave age (u _*/c>0.07), the effect of stratification on wave growth is weell described by Monin-Obukhov scaling of the friction velocity. For smaller values ofu _*/c, Monin-Obukhov scaling overpredicts.The effect on duration-limited wave growth is studied with the third-generation WAM surface wave model driven by 10 m winds. Effects of stratification on the significant wave height are found to be of the order of 10%. The results are comparable to those of a recent reanalysis of field measurements, although the measured stratification effect is somewhat stronger. Implementation of a stratification-dependent growth in wave models is recommended, as it can lead to small but significant improvements in wave forecasts when accurate air and sea temperatures are available.     

Drag of the sea surface

It is shown how the drag of the sea surface can be computed from the wind speed and the sea state. The approach, applicable both for fully developed and for developing seas, is based on conservation of momentum in the boundary layer above the sea, which allows one to relate the drag to the properties of the momentum exchange between the sea waves and the atmosphere.The total stress is split into two parts: a turbulent part and a wave-induced part. The former is parameterized in terms of mixing-length theory. The latter is calculated by integration of the wave-induced stress over all wave numbers. Usually, the effective roughness is given in terms of the empirical Charnock relation. Here, it is shown how this relation can be derived from the dynamical balance between turbulent and wave-induced stress. To this end, the non-slip boundary conditions is assigned to the wave surface, and the local roughness parameter is determined by the scale of the molecular sublayer.The formation of the sea drag is then described for fully developed and developing seas and for light to high winds.For the Charnock constant, a value of about 0.018–0.030 is obtained, depending on the wind input, which is well within the range of experimental data.It is shown that gravity-capillary waves with a wavelength less than 5 cm play a minor role in the momentum transfer from wind to waves. Most of the momentum is transferred to decimeter and meter waves, so that the drag of developing seas depends crucially on the form of the wave spectrum in the corresponding high wavenumber range.The dependence of the drag on wave age depends sensitively on the dependence of this high wavenumbertail on wave age. If the tail is wave-age independent, the sea drag appears to be virtually independent of wave age. If the tail depends on wave age, the drag also does. There is contradictory evidence as to the actual dependence. Therefore, additional experiments are needed.The investigation was in part supported by the Netherlands Geosciences Foundation (GOA) with financial aid from the Netherlands Organization for Scientific Research (NWO).     

An extended Miles' theory for wave generation by wind

Miles' inviscid theory of surface wave generation by wind is (a) modified by replacing the logarithmic shear velocity profile with one which applies right down to the wave surface and which exhibits an explicit dependence on the roughness of the surface, and (b) extended to include the effects of the interaction of wave with air flow turbulence by considering the wave-modified mean flow as the mean of the actual turbulent air flow over water waves and using this in a mixing-length model.The surface pressure is shown to depend significantly on the flow conditions being aerodynamically smooth or rough. Its component in phase with the surface elevation is practically unaffected by the wave-turbulence interaction. However, such interaction tends to increase the rate of energy input ß from wind to waves travelling in the same direction, e.g., the increase is 2gk ² for aerodynamically rough flow, where gk is the Von Karman constant. It also provides damping of waves in an adverse wind which can be about 10% of the growth rate in a favourable wind.     

An invariant theory of the linearized shallow water equations with rotation and its application to a sphere and a plane

The system of linearized shallow water equations is formulated in this paper on any rotating and smooth surface M in terms of differential geometry. The system decouples into two separate equations: a scalar one for the height deviation and a vector one for the velocity field. For low and high frequencies these equations yield asymptotic equations whose solutions are the generalizations of the Poincare and Rossby waves to smooth surface. The application of these equations to the β-plane yields both new and previously known equations for the height deviation and for the velocity components. The application of the equations to the rotating spherical Earth shows that the meridional amplitudes of Poincare and Rossby waves are both described by the prolate angular spheroidal wave functions. The asymptotic and the power series expansions of the eigenvalues of these functions yield new approximations for the dispersion relations of these waves on a sphere. The new dispersion relations are very accurate in the physically relevant range of the single nondimensional model parameter – the square of the nondimensional gravity waves’ phase speed. The invariant formulation can also be applied to other surfaces that are of geophysical interest such as an oblate ellipsoid of revolution.     

Aerodynamic roughness of the sea surface at high winds

The role of the surface roughness in the formation of the aerodynamic friction of the water surface at high wind speeds is investigated. The study is based on a wind-over-waves coupling theory. In this theory waves provide the surface friction velocity through the form drag, while the energy input from the wind to waves depends on the friction velocity and the wind speed. The wind-over-waves coupling model is extended to high wind speeds taking into account the effect of sheltering of the short wind waves by the air-flow separation from breaking crests of longer waves. It is suggested that the momentum and energy flux from the wind to short waves locally vanishes if they are trapped into the separation bubble of breaking longer waves. At short fetches, typical for laboratory conditions, and strong winds the steep dominant wind waves break frequently and provide the major part of the total form drag through the air-flow separation from breaking crests, and the effect of short waves on the sea drag is suppressed. In this case the dependence of the drag coefficient on the wind speed is much weaker than would be expected from the standard parameterization of the roughness parameter through the Charnock relation. At long fetches, typical for the field, waves in the spectral peak break rarely and their contribution to the air-flow separation is weak. In this case the surface form drag is determined predominantly by the air-flow separation from breaking of the equilibrium range waves. As found at high wind speeds up to 60 m s⁻¹ the modelled aerodynamic roughness is consistent with the Charnock relation, i.e. there is no saturation of the sea drag. Unlike the aerodynamic roughness, the geometrical surface roughness (height of short waves) could be saturated or even suppressed when the wind speed exceeds 30 m s⁻¹.     

Fetch Limited Drag Coefficients

Measurements made at a tower located 2 km off the coast of Denmark inshallow water during the Risø Air Sea Experiment (RASEX) are analyzedto investigate the behaviour of the drag coefficient in the coastal zone.For a given wind speed, the drag coefficient is larger during conditions ofshort fetch (2-5 km) off-shore flow with younger growing waves than it isfor longer fetch (15-25 km) on-shore flow. For the strongest on-shorewinds, wave breaking enhances the drag coefficient. Variation of the neutral drag coefficient in RASEX is dominated byvariation of wave age, frequency bandwidth of the wave spectra and windspeed. The frequency bandwidth is proportional to the broadness of the waveheight spectra and is largest during conditions of light wind speeds. Usingthe RASEX data, simple models of the drag coefficient and roughness length are developed in terms of wind speed, wave age and bandwidth. An off-shoreflow model of the drag coefficient in terms of nondimensional fetch isdeveloped for situations when the wave state is not known.     

Models of the wave boundary layer

A general approach to model the structure of the wave boundary layer, based on the nonlinear Reynolds equations in a curvilinear system of coordinates, is described. Both spectral and numerical grid models are used. The energetic interactions between wind and wave in terms of Miles' parameter are studied. For waves outrunning or running against the wind, the range of the inverse flux of energy is found. For waves running slower than the wind, quadratic growth of is established. Vertical profiles of the wave momentum flux for different fetches are given. Following P. Janssen, a one-dimensional analytical model of the wave boundary layer is suggested. The effect of waves on the drag coefficient is analyzed.     

Signatures of Air–Wave Interactions Over a Large Lake

The air–water exchange of momentum and scalars (temperature and water vapour) is investigated using the Lake-Atmosphere Turbulent EXchange (LATEX) dataset. The wind waves and swell are found to affect the coupling between the water surface and the air differently. The surface-stress vector aligns with the wind velocity in the presence of wind waves, but a wide range of stress–wind misalignment angles is observed during swell. The momentum transport efficiency decreases when significant stress–wind misalignment is present, suggesting a strong influence of surface wave properties on surface drag. Based on this improved understanding of the role of wave–wind misalignment, a new relative wind speed for surface-layer similarity formulations is proposed and tested using the data. The new expression yields a value of the von Kármán constant (\(\kappa \)) of 0.38, compared to 0.36 when using the absolute wind speed, as well as reduced data fitting errors. Finally, the ratios of aerodynamic to scalar roughness lengths are computed and various existing models in the literature are tested using least-square fitting to the observed ratios. The tests are able to discriminate between the performance of various models; however, they also indicate that more investigations are required to understand the physics of scalar exchanges over waves.     

Variation of the spectrum of wind ripple on coastal waters under the action of internal waves

The mechanism of flattening of surface wind waves in the centimetre wavelength range under the action of internal waves and in the presence of surfactants is considered. It is shown that internal waves give rise to perturbations of surfactant concentration proportional to the orbital velocities of wave particles. Accordingly, the damping coefficient of centimetre wavelength ripples is also modulated. Evolution of the wind wave spectrum is described by the energy balance equation. In the right-hand side of the equation the Phillips excitation mechanism, the Miles mechanism with a nonlinear spectrum limitation, and the dissipation caused by a surface-active film are taken into account. Assuming that the energy source and sink balance of the wind wave spectrum over the centimetre wavelength range is local, relative spectra (contrasts) for the case of gentle breeze are calculated. It is shown that the spectral components of ripples in the slicks can be order less than the uperturbed spectrum values.     

应用印痕函数研究测风塔资料代表性

影响测风塔代表性的因素很多,为了合理地确定测风塔观测结果能够代表的区域范围,需要通过适当的方法综合考虑这些因素。湍流通量代表性研究中常用的印痕函数是风速、稳定度、下垫面粗糙度等的函数,可以定量地刻画单点观测结果的代表性。利用印痕函数方法对中国风能资源观测网中来自不同下垫面的44个测风塔2010年的观测结果进行了测风塔资料的代表性分析。首先,根据两层(10、70 m)的风速和温度计算稳定度参数L_p并确定中性层结,粗糙度的计算选取了中性层结条件下满足对数风廓线的资料。然后,利用中性层结下70 m高度主导风向和风向频率较大方位的资料分析了两种印痕函数方法(Schuepp和Hsieh方法)的适用性及代表性范围与粗糙度的关系。两种方法得到的印痕函数对应距离(即代表性范围)比较接近。但是,在复杂山地Schuepp方法已不适用。当粗糙度小于10 m时,粗糙度与印痕函数对应距离的关系可用对数-线性函数描述。根据经验推荐了利用印痕函数积分90%对应的距离以及下垫面显著变化位置来确定测风塔代表范围。但是,印痕函数积分阈值仍需要通过示踪实验或数值模拟检验。     

Impact of waves on air-sea exchange of sensible heat and momentum

The impact of sea waves on sensible heat and momentum fluxes is described. The approach is based on the conservation of heat and momentum in the marine atmospheric surface layer. The experimental fact that the drag coefficient above the sea increases considerably with increasing wind speed, while the exchange coefficient for sensible heat (Stanton number) remains virtually independent of wind speed, is explained by a different balance of the turbulent and the wave-induced parts in the total fluxes of momentum and sensible heat.Organised motions induced by waves support the wave-induced stress which dominates the surface momentum flux. These organised motions do not contribute to the vertical flux of heat. The heat flux above waves is determined, in part, by the influence of waves upon the turbulence diffusivity.The turbulence diffusivity is altered by waves in an indirect way. The wave-induced stress dominates the surface flux and decays rapidly with height. Therefore the turbulent stress above waves is no longer constant with height. That changes the balance of the turbulent kinetic energy and of the dissipation rate and, hence the diffusivity.The dependence of the exchange coefficient for heat on wind speed is usually parameterized in terms of a constant Stanton number. However, an increase of the exchange coefficient with wind speed is not ruled out by field measurements and could be parametrized in terms of a constant temperature roughness length. Because of the large scatter, field data do not allow us to establish the actual dependence. The exchange coefficient for sensible heat, calculated from the model, is virtually independent of wind speed in the range of 3–10 ms^-1. For wind speeds above 10 ms^-1 an increase of 10% is obtained, which is smaller than that following from the constant roughness length parameterization.The investigation was in part supported by the Netherlands Geosciences Foundation (GOA) with financial aid from the Netherlands Organization for Scientific Research (NWO).     

The roughness of wind waves

In this paper, a new dynamic roughness equation for airflow over wind waves is proposed, based on relationships of dimensionless parameters and the 1986 HEXMAX field data. The equation can be considered as a modification of the Charnock formula. The data are also compared with the parameterizations of Toba and Koga, and of Hsu, and the consequences of the new equation for aerodynamic drag are discussed. In the new equation, the drag coefficient is expressed as a function of both wind speed and wave age c _p/u_*. This testifies to the strong wave-age dependence of the drag, found in several experimental studies. It is also in good agreement with results of a theoretical model of the airflow-seawave interaction, proposed by Janssen (1989).Also: Dept. of Applied Physics, Techn. Univ. Delft, the Netherlands.     

Effect of atmospheric stability on the growth of surface gravity waves

In this paper we study the effect of atmospheric stability on the growth of surface gravity waves. To that end we numerically solved the Taylor-Goldstein equation for wind profiles which deviate from a logarithmic form because stratification affects the turbulent momentum transport. Using Charnock's relation for the roughness height z ₀ of the wind profile, it is argued that the growth rate of the wave depends on the dimensionless phase velocity c/u _* (where u _* is the friction velocity) and a measure of the effect of atmospheric stability, namely the dimensionless Obukhov length gL/u _* ², whereas it only depends weakly on gz _t /u _* ² (where z _t is the roughness height of the temperature profile). Remarkably for a given value of u _* /c, the growth rate is larger for a stable stratification (L > 0) than for an unstable one (L < 0). We explain why this is the case. If, on the other hand, one considers the growth rate as a function of c/U ₁₀ (where U ₁₀ is the windspeed at 10 m), the situation reverses for c/U ₁₀ < 1. For practical application in wave prediction models, we propose a new parameterization of the growth rate of the waves which is an improvement of the Snyder et al. (1981) proposal because the effect of stability is taken into account.     

On the growth rate of wind‐generated waves

Abstract

A new approach to fetch‐limited wave studies is taken in this paper. Using data from five towers arranged along a line from the eastern shore of Lake St Clair, the differential growth between towers is explored as a function of local wave age. It is argued that this method avoids the usual fetch‐limited pitfall of inhomogeneity over long fetches and, in particular, the changes in wind speed downfetch of an abrupt roughness change. It is found that the growth rate decreases uniformly downfetch as the waves approach full development. This differential method leads to a smooth transition from rapidly growing short fetch waves to the asymptotic invariant state of full development. When the variation in wind speed after an abrupt (land to water) roughness change is taken into account, the idea of a universal fetch‐limited growth curve is called into question.     

Sea surface wind stress and drag coefficients: The hexos results

Turbulent fluxes have been measured in the atmospheric surface layer from a boom extending upwind from the Dutch offshore research platform Meetpost Noordwijk (MPN) during HEXMAX (Humidity Exchange over the Sea Main Experiment) in October–November, 1986. We started out to study eddy flux of water vapour, but discrepancies among simultaneous measurements made with three different anemometers led us to develop methods to correct eddy correlation measurements of wind stress for flow distortion by nearby objects. We then found excellent agreement among the corrected wind stress data sets from the three anemometers on the MPN boom and with eddy correlation measurements from a mast on a tripod. Inertial-dissipation techniques gave reliable estimates of wind stress from turbulence spectra, both at MPN and at a nearby ship. The data cover a range of wave ages and the results yield new insights into the variation of sea surface wind stress with sea state; two alternative formulas are given for the nondimensional surface roughness as a function of wave age.