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.     

Water waves induced by a fluctuating tangential stress

In the framework of linear wave theory both the effects of tangential and normal stresses on the water waves are discussed without the assumption of irrotational water motion. A formal solution initially at rest with level surface and developed under the actions of the surface stresses depending arbitrarily on time and sinusoidally on space is obtained.A progressive wave type tangential stress whose wave number and frequency are satisfying the dispersion relation of the water waves is shown to be equivalent to the normal stress of the same type on the growth of the waves except the phase relations between the stresses and the water motion. The growth rate of the waves induced by the tangential stress is also shown to be quite insensitive to the actual value of the viscosity.The rotational part of the water motion can dominate only in the early stage of wave generation and becomes negligible with the growth of the waves relative to the irrotational part of the motion even in the case where the motion is induced by the tangential stress alone. Therefore it is not reasonable to neglect effect of the tangential stress on wind waves even if the developed wind waves seem to be irrotational.     

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.     

A model for the response of wave directions in slowly turning wind fields

On the basis of the wave energy balance equation, the response model of mean directions of locally wind-generated waves in slowly turning wind fields has been derived. The results show that in a homogeneous field, the time scale of the response is not only related to the rate of wave growth, but also to the directional energy distribution and the angle between the wind direction and the mean wave direction. Furthermore, the law of change in the mean wave direction has been derived. The numerical computations show that the response of wave directions to slowly turning wind directions can be treated as the superposition of the responses of wave directions to a series of sudden small-angle changes of wind directions and the turning rate of the mean wave direction depends on the turning rate and the total turning angles of the wind direction. The response of wave directions is in agreement with the response for a sudden change of wind directions if the change in wind directions is very fast. Based on the no     

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

The instability of Taylor-Görtler vortices which are expected in the air flow on water waves was studied in part I, under the assumption that the curvature around the crest or the trough of water waves, where the instability was expected to take place first, was constant, namely that the characteristics of the vortices were affected little by the local change of the curvature along the direction of the progress of water waves (the direction ofx-axis) However, the curvature actually varies from positive to negative, or vice versa. In order to study this effect, the instability of Taylor-Görtler vortices is examined with respect to the range of the part of a constant curvature, in the model in which the curvature is positive constant near the trough and negative constant near the crest, and zero in the intermediate regions, respectively. It is shown that as the region of the constant curvature becomes narrower, the instability tends to weaken. For the same example with part I, namely, when the wind of 12.2 m s^–1 is blowing over swells of 15 m in wavelength, if the range of constant curvature near the trough is taken as a quarter of one wave length, the critical wave height becomes 0.96 m instead of 0.50 m, and conversely, the wave length and the height of center of the vortex become 11.9 m and 2.1 m instead of 24 m and 3.7 m, respectively.Further, using the energy equations, quantitative estimates are performed of the intensity of the vortices which develop when the wave height of the swell is 1.05 m in the above described example, and also of the influence of the vortices upon the wind profile when the equilibrium state is reached. When the vortices are generated and grow to attain to an equilibrium state interacting with the mean flow, the maximumx-component of velocity in the vortices is about 1.04 m s^–1. Consequently, the wind profile undergoes a considerable distortion from the logarithmic one near the level of 2 m height. This distorted wind profile has a form similar to those sometimes observed above the sea surface.     

Analytically Derived Wind Wave Growth Relations

By the use of the 3/2 power law presented by Toba combined with the significant wave energy balance equation for wind wave, wind wave growth at a limited fetch is analytically investigated. The new wind wave growth relations (WWGRs) are analytically derived with shehering coefficient and wind drag coefficient as parameters. The geometrical average of observational values of shehering coefficient and the arithmetic average of observational values of wind drag coefficient are applied to detennine the new WWGRs. Comparisons with existing empirical WWGRs are made.     

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 general theory of three-dimensional wave groups Part II: Interaction with a breakwater

The equation obtained in Part I predicts how an exceptionally high wave occurs at any fixed point within a wind wave field. The equation may be applied with a theoretical spectrum or directly with the random time series obtained by an array of wave gauges in the field. From both approaches, it emerges that a very high wave at a breakwater occurs because a well-defined three-dimensional wave group at the apex of its development hits against the breakwater, and that a very high wave at some distance before the breakwater occurs because of the collision of two wave groups: the first one going back after having been reflected, and the second one approaching the breakwater. In order to test the theory, a special breakwater was assembled off the beach at Reggio-Calabria where the significant height of the wind waves typically ranges from 0.20 to 0.40 m. When an exceptionally high wave (H = 9.6 σ) occurred at a point before this breakwater, the records made by a gauge array confirmed all the essential features of the prediction.     

Air-Sea Interactions under the Existence of Opposing Swell

Data of a comprehensive laboratory study on the coexistent system of wind waves and opposing swell (Mitsuyasu and Yoshida, 1989) have been reanalyzed to clarify the air-sea interaction phenomena under the coexistence of wind waves and swell. It is shown that the magnitude of the decay rate of swell due to an opposing wind is almost the same as that of the growth rate of swell caused by a following wind, as measured by Mitsuyasu and Honda (1982). The decay rate is much smaller than that obtained recently by Peirson et al. (2003), but the reason for the disagreement is not clear at present. The effect of an opposing swell on wind waves is very different from that of a following swell; wind waves are intensified by an opposing swell while they are attenuated by a following one. The phenomenon contradicts the model of Phillips and Banner (1974), but the reason for this is not clear at this time. The high-frequency spectrum of wind waves shows a small increase of the spectral density. Wind shear stress increases a little due to the effect of opposing swell. The intensification of wind waves by opposing swell and the small increase of the spectral density in a high-frequency region can be attributed to the increase of wind shear stress. Such organized phenomena lead to the conclusion that the hypothesis of local equilibrium for pure wind waves (Toba, 1972) can also be satisfied for wind waves that coexist with opposing swell. The recent finding of Hanson and Phillips (1999) can be explained by this mechanism.     

Generation mechanism of higher mode nondispersive shelf waves by wind forcing

The higher mode predominance in the current velocity fields associated with wind-induced shelf waves in the nondispersive regime is studied with a special attention to the effect of the geographical boundary, e.g. wide strait or wide bank areas. The effect of such large topographic change is represented by wind forcing with a finite dimension near the geographical boundary. The time development processes of the wind-induced shelf waves is examined in the context of an initial-value problem, where a spatially finite wind stress is applied att=0. Various modes of shelf waves excited at the boundary start propagating simultaneously and develop monotonically within the forcing region. After the passage of such wave, the energy of wind is used to maintain the attained equilibrium condition, i.e. the steady shelf circulation. The current evolution of the lower mode is restricted to the earlier stage because of the large propagation speed. In contrast, the higher mode waves can travel slowly within the forcing region so that the kinetic energy is supplied from wind stress for a long time before the equilibrium condition is established. Consequently, the observation at the fixed point near the geographical boundary would show that the higher mode waves gradually dominate as time goes on, i.e. for the long-term forcing.     

Statistical characteristics of individual waves in laboratory wind waves

On the basis of data on the statistical characteristics of individual waves in laboratory wind waves reported in part I of this series, a self-consistent similarity regime is found to exist among properties of the individual waves, such as the nondimensional frequency, the wave number, the phase speed, and the steepness. Also, it is shown that forms of past empirical formulas for the development of the peak wave can be derived starting from the 3/2-power law, as an extension of the persent laboratory experimental data. In the derivation, only values of the coefficient of the 3/2-power law, and the fraction of momentum transferred from the wind retained by the wind waves, remain on an empirical basis.     

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.     

Wind wave breaking under the conditions of inhomogeneous currents and of the atmospheric boundary layer

As known fromin situ observations, inhomogeneities of flows and of the atmospheric boundary layer produce variations of the intensity of wind wave breaking. A relevant phenomenological model is suggested here, describingin situ data on the breaking of waves in the presence of internal waves. The response of the wave breaking to the flow's inhomogeneity enhances with the growth of its spatial or temporal scale. For the mesoscale (10–100 km) inhomogeneities, the model is essentially simplified—wave breakings depict the local energy inputs to wind waves. The model allows us to compute currents of various type in the wave breaking intensity field. The results may have practical implications, in terms of remote sensing of the ocean. Translated by Vladimir A. Puchkin.     

稳定风浪场谱特征量间的关系

风浪宏观特征量是描述风浪场特征的重要物理量。作者基于风浪有停留在混乱运动状态的趋势的性质对风浪场特征量间的关系进行了研究。主频波频率附近的波动自风摄取能量,风浪吸收的能量通过非线性相互作用在谱中重新分配。谱中能量的重新分配产生多尺度波动,这导致风浪波面的混乱运动(风浪处于混乱运动状态)。在稳定状态,风浪运动最为混乱。当风浪状态偏离最混乱运动状态,谱中非线性相互作用引起的能量重新分配将使风浪回到该状态。基于线性海浪理论导出风浪场特征量间的关系。导出的关系与观测结果进行了对比,发现理论结果与观测结果很好地符合。风浪场宏观特征量间存在固有关系。尽管目前风浪场特征量关系的观测结果存在差异,但本文中证明,所导出的理论关系与实验结果很好地符合。     

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.     

Growth of Wind Waves with Fetch in the Sea of Japan under Winter Monsoon Investigated Using Data from Satellite Altimeters and Scatterometer

By using wind vector fields observed by the NASA Scatterometer (NSCAT) and significant wave heights observed by the TOPEX/POSEIDON and European Remote Sensing Satellite-2 (ERS-2) altimeters, one-dimensional fetch growth of wind waves has been investigated under conditions of strong wind and high waves caused by the East Asian winter monsoon in the Sea of Japan. The evolution of fetch-limited wind waves can be observed by the altimeters along their ground tracks. The fetch is estimated by using vector wind fields observed by NSCAT. The derived growth characteristics of wind waves are compared with empirical relationships between the non-dimensional fetch and significant wave height proposed by previous studies. Good agreement is discernible with Toba's fetch graph formula normalized by the friction velocity, while Wilson's well-known formula normalized by the wind speed at a height of 10 m tends to underestimate the wave height under such severe conditions of high wind and very long fetch. This discrepancy is explained by the wind-speed dependence of the drag coefficient. A simple correction to Wilson's formula for the high wind conditions is proposed and compared with the observed data.     

Instability of Two Interacting Quasi-Monochromatic Waves in Shallow Water

The nonlinear interactions of waves with a double-peaked power spectrum have been studied in shallow water.The starting point is the prototypical equation for nonlinear unidirectional waves in shallow water,i.e.the Korteweg de Vries equation.By means of a multiple-scale technique two defocusing coupled Nonlinear Schrdinger equations are derived.It is found analytically that plane wave solutions of such a system are unstable for small perturbations,showing that the existence of a new energy exchange mechanism which can influence the behavior of ocean waves in shallow water.     

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.     

A derivation of the scaling law, the power law and the scaling relation for fetch-limited wind waves using the renormalization group theory

A state of wind waves at a fetch is assumed to be transformed into another state of wind waves at a different fetch by the renormalization group transformation. The scaling laws for the covariance of water surface displacement and for the one-dimensional and two-dimensional spectrum and the power law for the growth relation are derived from the fact that the renormalization group transformation constitutes a semigroup. The scaling relation or the relation among the exponents of the power law is also derived, using the two assumptions that the renormalization group transformation is applicable to fetch-limited wind waves and that the saturated range exists, which implies that the directional distribution function of energy in the wave number region much larger than the peak wave number does not depend on wave number.     

Duality of turbulence and wave in wind waves

The three-seconds power law for wind waves of simple spectra, already derived byToba (1972 and 1973), may also be derived by introducing surface-wave properties into the form of the rate of energy dissipation in the theory of turbulence. The universal constantB, which was formerly determined empirically as 0.062 is here obtained asB=(2)^–3/2=0.0635. Thus wind waves have the duality of turbulence and wave.