Internal flow structure of short wind waves

Characteristic features of the internal flow field of short wind waves are described mainly on the basis of streamline patterns measured for four different cases of individual wave. In some waves a distinct high vorticity region, with flow in excess of the phase speed in the surface thin layer, is formed near the crest as shown in Part I of this study, but the streamlines are found to remain quite regular even very near the water surface. The characteristics of flow in the high vorticity region are investigated, and it is argued that the high vorticity region is not supported steadily in individual waves but that growth and attenuation in individual waves repeats systematically, without no severe wave breaking. Below the surface vorticity layer a quite regular wave motion dominates. However, this wave motion is strongly affected by the presence of the high vorticity region. By comparing the measured streamline profiles with those predicted from wave profiles by the use of a water-wave theory, it is found that the flow of the wind waves studied cannot be predicted, even approximately, from the surface displacements, in contrast to the case of pure irrotational water waves.     

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.     

Modeling drift currents in a shallow basin with allowance for variations in tangential stresses induced by wind waves

A coupling model for calculating wind-driven currents and waves in a shallow basin with allowance for current-wave interactions is introduced. The model is constructed on the basis of the three-dimensional σ-coordinate model of currents [3] and the SWAN (Simulating Waves Nearshore) spectral wave model [4]. The effect of waves on currents is taken into account in the coefficients of surface and bottom friction through roughness parameters. Results of combined modeling of stationary fields of currents and waves generated by spatially homogeneous wind are correlated with the corresponding results of separate modeling for a cylindrical basin of constant depth and the water area of Lake Donuzlav (the northwestern coast of the Crimea). The allowance for the effect of waves during calculation of tangential wind stresses in the model of currents is shown to be among major factors intensifying water circulation and forming spatial inhomogeneities of the vortex type. In addition, some cases of local decreases in tangential wind stresses are revealed; they appear when the lake is penetrated from the side of the open sea by relatively long waves, which significantly decrease the roughness of the water surface.     

Internal flow structure of short wind waves

The internal flow structure of wind waves in a wind-wave tunnel was investigated on the bases of the measured vorticity distributions, streamline patterns, internal pressure fields, and stress distributions at the water surface for some waves in the field. In part I the experimental method and the internal vorticity structure relative to the individual wave crests are described. The measured vorticity distributions of distinct waves (waves with waveheight comparable with or larger than that of significant wavesH _1/3) in the field indicate that the surface vorticity layer is extraordinarily thickened near the crest, and the vorticity near the water surface shows a particularly large value below the crest. The flow near the crest of distinct waves is found to be in excess of the phase speed in a very thin surface layer, and the tangential stress distribution has a dominant peak near the crest. It is argued that the occurrence of the region of high vorticity in distinct waves is associated with the local generation of vorticity near the crest by tangential stress which attains a peak, under the presence of excess flow.     

Interaction of higher-order water waves with uniform currents

The interaction between current-free higher-order water waves with a wave-free uniform current normal to the wave crests is considered. The combined wave-current motion resulting from the interaction is assumed stable and irrotational. The velocity potential, dispersion relation, the particle kinematics and pressure distribution up to the third order in wave amplitude are developed. The conservation of mean mass, momentum and energy, together with the dispersion relation on the free surface are used to derive a set of four nonlinear equations, through which the relationship between wave-free current, current-free wave and the combined wave-current parameters is established. Numerical results for a range of current values are also presented.     

Forced convection accompanying wind waves

Wind-wave tunnel experiments reveal, by use of techniques of the flow visualization, that wind waves are accompanied by the wind drift surface current with large velocity shear and with horizontal variation of velocity relative to the wave profile. The surface current converges from the crest to a little leeward face of the crest, making a downward flow there, even though the wave is not breaking. Namely, wind waves are accompanied by forced convections relative to the crests of the waves. Since the location of the convergence and the downward flow travels on the water surface as the crest of the wave propagates, the motion as a whole is characterized by turbulent structure as well as by the nature of water-surface waves. In this meaning, the term of real wind waves is proposed in contrast with ordinary water waves. The study of real wind waves will be essential in future development of the study of wind waves.     

On the vertical distribution of

The aim of this paper is to present an analytical expression for the vertical distribution of the correlation between the horizontal ( ) and vertical ( ) wave velocity components. This quantity, , which appears explicitly in the time-averaged momentum balance equations, has been shown to play an important role in the vertical distribution of wave-induced currents.The proposed formulation for is based on an identity that relates the effective (wave) shear stress to the effective (wave) normal stresses ( ² and ²) and to the vorticity of the oscillatory flow gw. This general expression has been applied to simplified situations and has been shown to degenerate into other existing formulations with comparable simplifying assumptions, viz. irrotational waves in shallow water over an arbitrary bottom topography and breaking waves over a horizontal bottom.The model has also been applied to the case of waves interacting with a depth-varying current over a horizontal bottom, in which preliminary results have been obtained for a simplified situation invoking linear (small-amplitude) wave theory.     

Local balance in the air-sea boundary processes

A new growth equation for wind waves of simple spectrum is presented upon three basic concepts. The period and the wave height of significant waves in dimensionless forms, which are considered to correspond to the peak frequency and the energy level, respectively, are used as representative quantities of wind waves. One of the three basic concepts is the concept of local balance, and the other two concern the acquisition of wave energy and the dissipation of wave energy, respectively. It is shown from some actual data that the equation, together with two universal constants concerning the acquisition and the dissipation of wave energy (B=6.2×10^?2 andK=2.16×10^?5, respectively), is applied universally to wide ranges of wind waves from those in a wind-wave tunnel to fully developed sea in the open ocean. “The three-second power law for wind waves of simple spectrum”, and a few relations as the lemmas, are derived, such that the mean surface transport due to the orbital motion of wind waves is always proportional to the friction velocity in wind, and that the steepness is inversely proportional to the root of the wave age. It is also derived that the portion of wind stress which directly enters the wind waves decreases exponentially with increasing wave age and is 7.5 % of the total wind stress for very young waves. Also, equations are presented as to the increase of momentum of drift current, and as to the supply of turbulent energy by wind waves into the upper ocean.     

Internal flow structure of short wind waves

For wind waves generated in a wind-wave tunnel, the surface pressure and also the pressure distribution along the internal streamlines were calculated from the measured internal velocity field. In distinct waves, with wave height comparable with or larger than the mean, the surface pressure is found to vary drastically in a narrow region around the crest, showing a dominant minimum near the crest. On the other hand, the pressure distribution along the streamline shows systematic variations that are nearly in phase with the streamline profile. It is shown that the occurrence of the pressure in phase with the streamline profile is linked with the internal vorticity distribution, especially with the presence of a high vorticity region below the crest described in Part I of this study. As a result of the occurrence of such pressure variations, the dispersion relation is modified by about 10% from that for linear irrotational waves. It is argued from the present measurements that the dispersion relation and also the energy transfer from wind into wind waves are strongly affected by the internal vortical structure so that the assumption of irrotational gravity waves cannot be applied to the wind waves being studied.     

Modulation of short wind waves by a long surface wave: a mechanism for feedback with an air flow

The paper concentrates on the evolution of a spectrum of short wind waves (SW) along the profile of a long surface wave (LW). Short wave spectral variations are considered in the relaxation approximation. The SW spectrum is modulated by the orbital velocities of long waves and by the variations of wind stress along the LW profile. The latter effect occurs due to wind flux perturbations induced by both the long wave proper and variations of the sea surface roughness induced by the SW modulations. To describe this effect, a feedback mechanism is introduces—the growth of energy of short waves results in the larger roughness of the sea surface, thereby contributing to the local wind stress, which facilitates, in turn, the growth of short waves. With moderate and strong winds being involved, this effect (aerodynamic feedback) is shown to be dominant in the short wave spectrum modulation. The mechanism becomes more efficient with intensification of the wind and decreasing of the long waves' frequency. Results of model calculations are in agreement with the known experimental data. Translated by Vladimir A. Puchkin.     

非线性流函数在实验风浪场计算中的应用

风作用于水面产生风浪, 其中由于波流紊动产生的动量和能量的交换机制是一个很复杂的过程。风应力一般用来描述这种能量交换, 可以分为3个部分: 水面的剪切力、波生应力以及紊动应力。采用一种有效的非线性波流分离方法——NSFM(Nonlinear Stream Function Method)对波流运动的动量和能量输移进行定性描述。构造能够有效表达非线性波浪的解析流函数, 摄动求解使其满足拉普拉斯方程、动力边界条件和运动边界条件, 结合实验室风浪数据, 分离出波生速度场。通过交叉谱分析, 得到波生雷诺应力在不同风速下对风应力的贡献。结果表明: NSFM对不同工况条件下的风浪的处理具有较高的精度, 模型适应性良好; 且风速越大, 波生应力沿着水深衰减得越快, 且自由面波生应力在动量输移中的比重会逐渐减弱。     

Wind-induced motion of water in shallow-water closed basins

The motion of water induced by tangential wind stresses in a circular basin of finite depth is studied by the finite-element method. The dependences of vertical displacements of the surface of the basin and the field of horizontal wave currents on the topography of the bottom and the direction of the wind are analyzed. The transformations of wind-induced disturbances observed after termination of the wind action are investigated.Translated from Morskoi Gidrofizicheskii Zhurnal, No. 5, pp. 35–44, September–October, 2004.This revised version was published online in May 2005 with corrections to cover date.     

Influence of the type of sea waves on the backscattered radar cross section at medium incidence angles

We consider the influence of the sea surface state on the backscattered radar cross section and the accuracy of the wind speed retrieval from the scatterometer data. We used a joint set of radars and buoys to determine the type of sea waves. Three types of sea waves were distinguished: developing wind waves, fully developed wind waves, and mixed sea. It is shown that the retrieval error of the near surface wind speed using a one-parameter algorithm is minimal in the case of fully developed wind waves. We compared these data with the results of radio-altimeter data analysis and showed that in both cases underestimation of the retrieval wind speed exists for developing wind waves and overestimation occurs for mixed sea. A variety of swell parameters (length of the dominating wave, swell height, swell age) significantly influence the backscattered radar cross section, leading to a growth in the mean square error of the retrieved wind speed during vertical sounding (radio-altimeter data), and only slightly influence the mean square error of the scatterometer data (medium incidence angles). It is necessary to include the information about the parameters of sea waves in the algorithms and take into account the regional wave properties to increase the accuracy of wind speed retrieval.     

Quasi-linear model of interaction of surface waves with strong and hurricane winds

A quasi-linear model for determining the aerodynamic drag coefficient of the sea surface and the growth rate of surface waves under a hurricane wind is proposed. The model explains the reduction (stabilization) in the drag coefficient during hurricane winds. This model is based on the solution of the Reynolds equations in curvilinear coordinates with the use of the approximation of the eddy viscosity, which takes into account the presence of the viscous sublayer. The profile of the mean wind velocity is found with consideration for nonlinear wave stresses (wave momentum flux), whereas wave disturbances induced in air by waves on the water surface are determined in the context of linear equations. The model is verified by comparing the calculation results with experimental data for a wide range of wind velocities. The growth rate and drag coefficient for hurricane winds are calculated both with and without consideration for the shortwave portion of the windwave spectrum. On the basis of calculations with the quasi-linear model, a simple parametrization is proposed for the drag coefficient and the growth rate of surface waves during hurricane winds. This model is convenient for use in models of forecasting winds and waves.     

Coupled modeling of currents and wind waves in the Kerch Strait

We present a numerical model of the dynamics of the Kerch Strait allowing one to perform the coordinated analysis of the fields of currents and wind waves. The model includes the spectral wave module and the hydrodynamic block of currents. The influence of waves on the currents is taken into account in the hydrodynamic block both via the surface and bottom tangential stresses and via the radiation stresses. In order to take into account the inverse influence of currents upon the waves, we use the fields of currents and sea level from the hydrodynamic block in the wave module. The specific features of the structure of currents and wind waves in the strait are studied for the typical wave situations. The results of the coupled and separate simulation are compared and the importance of taking into account the mechanisms of interaction between waves and currents in the analysis of the dynamic processes in the strait is demonstrated. __________ Translated from Morskoi Gidrofizicheskii Zhurnal, No. 5, pp. 3–20, September–October, 2007.     

Taylor-Görtler vortices expected in the air flow on sea surface waves—I

Taylor-Grörtler vortices are longitudinal vortices resulting from a centrifugal instability. They are generated in the flow having a curved streamline with an increasing velocity in the direction of decreasing curvature.It is shown that the air flow above wind waves and swells also satisfies locally the condition of the centrifugal instability. Numerical calculations indicate the possibility of generation of Taylor-Görtler vortices on the trough of sea waves. For example, when a wind of about 12.2 m/s at 10-m level is blowing over sea waves of the wave length of 15 m like the swell, the critical water wave height beyond which the vortices may be generated is about 0.5 m, and the critical wave length and the height of center of the generated vortices are about 24 m and 3.7 m, respectively. Further, about the relations between the generation of vortices and wind waves, it is shown that the condition of their generation is satisfied at the trough of waves for early stages of the wave generation.In conclusion, it is expected that the Taylor-Görtler vortices change the wind profile along the sea surface, and also, play some part in the growth of wind waves, especially in the formation of their three dimensional structure.     

Local balance in the air-sea boundary processes

In the course of the new treatment of the growth process of wind waves presented in part I of the present series of the articles, there was a point where the wave energy and wave momentum were not related correctly. This point has been revised with critical argument, and at the same time, the form of the ratior, between the wind stress that directly enter the wind waves and the total wind stress, has been derived analytically. The growth equation, under the condition that the wind stress is constant, is still the same with that derived in part I, with the exception that the ratior is given analytically.A comparison between the ratior obtained analytically and that estimated empirically in part I, raises a problem to be studied about the wave current of the actual wind waves.     

Hydroelastic interaction between obliquely incident waves and a semi-infinite elastic plate on a two-layer fluid

The hydroelastic response of a semi-infinite thin elastic plate floating on a two-layer fluid of finite depth due to obliquely incident waves is investigated. The upper and lower fluids with different densities separated by a sharp and stable interface are assumed to be inviscid and incompressible and the motion to be irrotational. Simply time-harmonic incident waves of the surface and interfacial wave modes with a given angular frequency are considered within the framework of linear potential flow theory. With the aid of the methods of matched eigenfunction expansion and the inner product of the two-layer fluid, a closed system of simultaneous linear equations is derived for the reflection and transmission coefficients of the series solutions. Based on the dispersion relations for the gravity waves and the flexural–gravity waves in a two-layer fluid and Snell’s law for refraction, we obtain a critical angle for the incident waves of the surface wave mode and three critical angles for the incident waves of the interfacial wave mode, which are related to the existence of the propagating waves. Graphical representations of the series solutions show the interaction between the water waves and the plate. The effects of several physical parameters, including the density and depth ratios of the fluid and the thickness of the plate, on the wave scattering and the hydroelastic response of the plate are studied. It is found that the variation of the thickness of the plate may change the wave numbers and the critical angles. The density ratio is the main factor to influence the wave numbers of the interfacial wave modes. Finally, the stress state is considered.     

A fast convolution approach to the transformation of surface gravity waves: Linear waves in 1DH

The transformation of irrotational surface gravity waves in an inviscid fluid can be studied by time stepping the kinematic and dynamic surface boundary conditions. This requires a closure providing the normal surface particle velocity in terms of the surface velocity potential or its tangential derivative. A convolution integral giving this closure as an explicit expression is derived for linear 1D waves over a mildly sloping bottom. The model has exact linear dispersion and shoaling properties. A discrete numerical model is developed for a spatially staggered uniform grid. The model involves a spatial derivative which is discretized by an arbitrary-order finite-difference scheme. Error control is attained by solving the discrete dispersion relation a priori and model results make a perfect match to this prediction. A procedure is developed by which the computational effort is minimized for a specific physical problem while adapting the numerical parameters under the constraint of a predefined tolerance of damping and dispersion error. Two computational examples show that accurate irregular-wave transformation on the kilometre scale can be computed in seconds. Thus, the method makes up a highly efficient basis for a forthcoming extension that includes nonlinearity at arbitrary order. The relation to Boussinesq equations, mild-slope wave equations, boundary integral equations and spectral methods is briefly discussed.