On nondimensional correlation between roughness length and wind-friction velocity

Physical bases for nondimensional parameters,z ₀/(u ² _*/g) andz ₀/(u _*/), characterizing wind-wave interaction are discussed; data selected to support the latter are critically reviewed. Both parameters are herewith unified, with the former describing the primary growth of roughness length with wind and the latter the secondary effects due to waves.     

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.     

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 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.     

Dependence of sea surface drag coefficient on wind-wave parameters

The relationships between sea surface roughness z 0 and wind-wave parameters are analyzed,and spurious self-correlations are found in all of the parameterization schemes.Sea surface drag coefficient C D is fitted by four wind-wave parameters that are wave age,wave steepness,windsea Reynolds number R B and R H ,and the analyzed data are divided into laboratory,field and combined data sets respectively.Comparison and analysis of dependence of C D on wind-wave parameters show that R B can fit the C D most appropriately.Wave age and wave steepness are not suitable to fit C D with a narrow range data set.When the value of wave age has a board range,R H is not suitable to fit C D either.Three relationships between C D and R B are integrated into the bulk algorithm COARE to calculate the observational friction velocity,and the results show that the relationship between C D and R B which is fitted with field data set can describe the momentum transfer in the open ocean,under low-moderate wind speed condition,most appropriately.     

On the local equilibrium of winds and wind-waves in relation to surface drag

Local equilibrium of winds and wind-waves is discussed as a basis for research of the drag coefficient of the water surface as well as for the spectral growth of wind-waves. This hypothesis assumes, in a narrow sense, that statistical properties are determined from four physical quantities, which represent winds and wind-waves: the friction velocityu _*, the gravitational accelerationg, the powerE of the surface displacement, and the peak frequency _p of a wind-wave spectrum. Then one has only one nondimensionalcontrol parameter, which may be either the wave age or wave nonlinearity (slope) of dominant waves. In a wide-sense, one can take into account viscosity and surface tension in terms of one more additional parameter by virtue of the virtual invariance of those material constants; that parameter describes the scale ratio between dominant waves and the short waves for which viscosity or surface tension is important. A unified expression for the roughness height according to this hypothesis turns out to include Charnock's and Toba's formulas as special cases. On the basis of a preliminary analysis of the experimental data, a new empirical formula is proposed.     

强风条件下海面拖曳系数和粗糙长度研究

进一步研究强风条件下海-气湍流动量交换以及海浪特征,有助于提高数值天气模式对台风强度演变、移动路径以及恶劣海况的预报能力。依照前人的方法将台风分为风向与浪向(1)相同,(2)相反,和(3)交叉3个扇形区,并结合台风路径数据,得到了浮标数据相对于台风的方位。分别对3种类型的浮标数据进行分析,进而发现了波浪高度和相速度随风速增加而变化的规律。并利用GWW参数化方案计算出摩擦速度(u_*)、拖曳系数(C_DN)和粗糙长度(z₀)。将这些结果与前人代表性的研究论文中所用观测数据和所得研究结论进行比较,结果表明二者有较强的一致性。该研究证明GWW参数化方案在强风条件下依然有很好的适用性。     

A note on a critical wind speed for air–sea boundary processes

Wind and wind-generated waves were measured in a wind-wave tank. A clear transition was found in the relation between the wind speed U ₁₀ and the wind friction velocity u _* near u _* = 0.2 m/s, where U ₁₀ is the wind speed at 10 m height extrapolated from the measured wind profile in a logarithmic layer, and u _* = 0.2 m/s corresponds roughly to U ₁₀ = 8 m/s in the present measurement. Quite a similar transition was found in the relation between the spectral density of high frequency wind waves and u _*. These results suggest the existence of the critical wind speed for air–sea boundary processes, which was proposed by Munk (J Marine Res 6:203–218, 1947) more than half a century ago. His original idea of the critical wind speed was based on the discontinuities in such phenomena as white caps, wind stress, and evaporation, which commonly appear at a wind speed near 7 m/s. On the basis of the results of our present study and those of earlier studies, we discuss the phenomena which are relevant to the critical wind speed for the air–sea boundary processes. The conclusion is that the critical wind speed exists and it is attributed to the start of wave breaking rather than the Kelvin–Helmholtz instability, but the air–sea boundary processes are not discontinuous at a particular wind speed; because of the stochastic nature of breaking waves, the changes occur over a range of wind speeds. Detailed discussions are presented on the dynamical processes associated with the critical wind speed such as wind-induced change of sea surface roughness and high frequency wave spectrum. Future studies are required, however, to clarify the dynamical processes quantitatively. In particular, there is a need to further examine the gradual change of breaking patterns of wind waves with the increase of wind speed, and the associated change of the structure of the wind over wind waves, such as separation of the airflow at the crest of wind waves, the turbulent stress, and wave-induced stress. Studies on the dynamical structure of the high frequency wave spectrum are also needed.     

A method for evaluating statistical errors associated with logarithmic velocity profiles

A logarithmic velocity profile is often fitted to velocity data in order to calculate the friction velocity (u _*) and typify the surface texture by a roughness length (z _o ). A method is given for estimating the errors in these parameters as calculated by this method. An example is given in which the size of the error is compared with the fluctuations that typically occur in the time seriesu _*(t) andz _o (t).     

Field data support of three-seconds power law andgu *σ−4-spectral form for growing wind waves

Observational data on air-sea boundary processes at the Shirahama Oceanographic Tower Station, Kyoto University, obtained in November, 1969, was analyzed and presented as an example representing the structure of growing wind-wave field. The condition was an ideal onshore wind, and the data contained continuous records of the wind speed at four heights, the wind direction, the air and water temperatures, the tides, and the growing wind waves, for more than six hours. The main results are as follows. Firstly, in both of the wind speed and the sea surface wind stress, rather conspicuous variations of about six-minute period were appreciable. Secondly, the three-seconds power law and its lemma expressed byH ^*=BT ^*3/2 and=2BT ^*–1/2, respectively, are very well supported by the data, whereH ^*(gH/u _* ²) andT ^*(gT/u _*) are the dimensionless significant wave height and period, respectively, the wave steepness,u _* the friction velocity of air,g the acceleration of gravity, andB=0.062 is a universal constant. Thirdly, the spectral form for the high-frequency side of the spectral maximum is well expressed by the form of()= _sgu_*^–4, where is the angular frequency and() the spectral density. The value of _s is determined as 0.062±0.010 from the observational data. There is a conspicuous discrepancy between the spectral shape of wind waves obtained in wind-wave tunnels and those in the sea, the former containing well-defined higher harmonics of the spectral peak, and consequently there is an apparent difference in the values of _s also. However, it is shown that the discrepancy of _s may be eliminated by evaluating properly the energy level of the spectral form containing higher harmonics.     

Local balance in the air-sea boundary processes

A combination of the three-second power law, presented in part I for wind waves of simple spectrum, and the similarity of the spectral form of wind waves, leads to a new concept on the energy spectrum of wind waves. It is well substantiated by data from a wind-wave tunnel experiment.In the gravity wave range, the gross form of the high frequency side of the spectrum is proportional tog u _* ^–4, whereg represents the acceleration of gravity,u _* the friction velocity, the angular frequency, and the factor of proportionality is 2.0×l0^–2. The wind waves grow in such a way that the spectrum slides up, keeping its similar form, along the line of the gross form, on the logarithmic diagram of the spectral density,, versus. Also, the terminal value of, at the peak frequency of the fully developed sea, is along a line of the gradient ofg ^{2 –5}.The fine structure of the spectrum from the wind-wave tunnel experiment shows a characteristic form oscillating around the ^–4-line. The excess of the energy density concentrates around the peak frequency and the second- and the third-order harmonics, and the deficit occurs in the middle of these frequencies. This form of the fine structure is always similar in the gravity wave range, in purely controlled conditions such as in a wind-wave tunnel. Moving averages of these spectra tend very close to the form proportional to ^–5.As the wave number becomes large, the effect of surface tension is incorporated, and the ^–4-line in the gravity wave range gradually continues to a ^–8/3-line in the capillary wave range, in accordance with the wind-wave tunnel data. Likewise, the ^–5-line gradually continues to a ^–7/3-line.Also, through a discussion on these results, is suggested the existence of a kind of general similarity in the structure of wind wave field.     

Tidal boundary layer measurements in the presence of waves

Measurements of tidal current and wave velocity made at 0.69 and 1.85 m above a rough seafloor exhibit large current gradients (boundary layer) in the water column. The logarithmic boundary layer flow model was fitted to the measurements, and thus roughness (z₀) and friction velocity (u^*) parameters were derived. The roughness parameter values were generally consistent with the observed upstream physical roughness. The values of both parameters for conditions in the rough turbulence flow regime are generally larger (much larger for ebb) than earlier published values for similar measurements of currents in the absence of significant waves but are comparable to values from recent measurements of currents in the presence of storm waves. The high parameter values here appear to relate more to the magnitude of the current and to the upstream physical bottom roughness than to the magnitude of the seastate. Large boundary layers in the flow at the seabed have a profound effect on the design of offshore structures such as offshore pipelines.     

Improvement and application of a saturation based wave dissipation function in SWAN model

Wave dissipation characteristics in SWAN (Simulating Waves Nearshore) model are investigated through numerical experiments. It is found that neither the fully developed integral parameters of wind waves (significant wave height and peak frequency) nor the high frequency spectral tail can be well reproduced by the default wave dissipation source terms. A new spectral dissipation source term is proposed, which comprises saturation based dissipation above two times of peak frequency and improved whitecapping dissipation at lower frequency spectrum. The reciprocal wave age (u /c p ) is involved into the whitecapping model to adjust dissipation rate at different wind speed. The Phillips higher frequency saturation parameter in the saturation-based dissipation is no longer taken as a constant, but varies with wave age. Numerical validations demonstrate that both the wind wave generation process and higher frequency spectrum of wind waves can be well simulated by the new wave dissipation term.     

A method for evaluating statistical errors associated with logarithmic velocity profiles

A logarithmic velocity profile is often fitted to velocity data in order to calculate the friction velocity (u _*) and typify the surface texture by a roughness length (z _o ). A method is given for estimating the errors in these parameters as calculated by this method. An example is given in which the size of the error is compared with the fluctuations that typically occur in the time seriesu _*(t) andz _o (t).     

Inter-comparison of model-predicted wave heights with satellite altimeter measurements in the north Indian Ocean

A numerical model based on a wind-wave energy transport formulation of Toba is developed to generate hindcast wave height data for the equatorial and the north Indian Ocean, which is otherwise a data-sparse region. The intercomparison between model-predicted wave heights for three years (1987–1989) obtained utilising analysed surface wind fields' data, and model grid averaged GEOSAT Altimeter significant wave height data showed moderate match, particularly for H_S greater than 1 m.     

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.     

Numerical Modeling of Wave-Enhanced Turbulence in the Oceanic Upper Layer

A coupled model of air-wave-sea interaction is modified based on a new roughness formulation and the latest data. The model parameters for aerodynamic roughness from below (ARB) and wave-dependent roughness from above (ARA, z _0a ) are assumed equal. The combined roughness is assumed to be a function of friction velocity, gravity, air and seawater densities, and wave age (c _w ). The model is used in a study of wave-enhanced turbulence under breaking waves to predict turbulent dissipation (), ARA, and drag coefficient (C _d ). Both waves and shear production are considered as sources of ocean turbulent energy. The atmospheric part of the model is used only to specify a correct condition at the interface. Numerical experiments are performed to study the -distribution, z _0a and C _d , and to compare with data. The major achievement is model verification using all available data. The first full application of this model is in conjunction with an ocean circulation model in a coupled circulation-wave system. Simulations show that the -distribution is strongly dependent on local wind-forced wave heights. For each wind and wave state there is a particular wave-dependent depth that is verified by data. The comparison shows that the model predicted agrees well with the observed of the z ^–4 law distribution of Gargett (1989). Simulations also show that waves have an important role in causing to differ from the classical wall-layer theory and z _0a , with a value of 0.30 for the empirical constant a _a . The model-predicted , z _0a , C _d and C _gd agree well with data.     

波流作用下海底边界层沉积物再悬浮与影响因素研究

海底沉积物再悬浮及其分布取决于海洋水动力、沉积物类型与床面形态之间复杂的相互作用,准确地理解和确定沉积物再悬浮过程对于沉积物输运的研究具有重要的意义。本文在祥云湾海洋牧场典型海域开展现场原位观测,获取研究区波浪、海流及悬浮沉积物浓度数据;分析了波、流作用下海底边界层悬浮沉积物垂向分布特征,并探究了海洋水动力和床面形态对悬浮沉积物垂向分布的影响。结果表明,研究区波流之间的相互作用不显著,沉积物再悬浮受控于风暴浪作用,风暴浪作用下底床切应力可以达到沉积物临界切应力的10～15倍,沉积物的再悬浮滞后于风暴浪作用2～3 h。在波浪荷载微小的情况下,悬浮沉积物垂向分布呈现"I"型,波浪荷载下,悬浮沉积物垂向分布呈现幂指函数分布,表现为"L"型;床面形态随波、流作用而演化,影响沉积物的再悬浮过程,u_(?w)/u_(?c)=1.00可作为波浪和海流起主导控制作用的床面形态的判别依据,纯波浪荷载作用下的u_(?w)/u_(?c)显著高于波浪主控作用下,但二者之间的界线随着波浪荷载的增加而升高。