The Impact Of Air-Flow Separation On The Drag Of The Sea Surface

An approach that allows assessment ofthe impact of air-flow separation (AFS) fromwave breaking fronts on the sea-surface drag is presented. Wave breaking fronts are modelled by the discontinuities of the sea-surface slope. It is assumedthat the dynamics of the AFS from wave breaking crests is similar to thatfrom the backward facing step. The form drag supported by an individualbreaker is described by the action of the pressure drop distributed alongthe forward face of the breaking front. The total stress due to the AFS isobtained as a sum of contributions from breaking fronts of different scales.Outside the breaking fronts the drag of the sea surface is supported by theviscous surface stress and the wave-induced stress. To calculate the stressdue to the AFS and the wave-induced stress a physical model of the wind-wavespectrum is used. Together with the model of the air flow described in termsof surface stresses it forms a self-consistent dynamical system for the seasurface-atmosphere where the air flow and wind waves are strongly coupled.Model calculations of the drag coefficient agree with measurements. It is shownthat the dimensionless Charnock parameter (roughness length normalized onthe square of the friction velocity and the acceleration of gravity)increases with the increase of the wind speed in agreement with fieldmeasurements. The stress due to the AFS normalized on the square of thefriction velocity is proportional to the cube of wind speed. At low windsthe viscous surface stress dominates the drag. The role of the form drag,which is the sum of the stress due to the AFS and the wave-induced stress, isnegligible. At moderate and high winds the form drag dominates. At windspeeds higher than 10 m s^-1 the stress supported by the AFS becomescomparable to the wave-induced stress and supports up to 50% of the totalstress.     

Nonlinear Formulation of the Bulk Surface Stress over Breaking Waves: Feedback Mechanisms from Air-flow Separation

Historically, our understanding of the air-sea surface stress has been derived from engineering studies of turbulent flows over flat solid surfaces, and more recently, over rigid complex geometries. Over the ocean however, the presence of a free, deformable, moving surface gives rise to a more complicated drag formulation. In fact, within the constant-stress turbulent atmospheric boundary layer over the ocean, the total air-sea stress not only includes the traditional turbulent and viscous components but also incorporates surface-wave effects such as wave growth or decay, air-flow separation, and surface separation in the form of sea-spray droplets. Because each individual stress component depends on and alters the sea state, a simple linear addition of all stress components is too simplistic. In this paper we present a model of the air-sea surface stress that incorporates air-flow separation and its effects on the other stress components, such as a reduction of the surface viscous stress in the separated region as suggested by recent measurements. Naturally, the inclusion of these effects leads to a non-linear stress formulation. This model, which uses a variable normalized dissipation rate of breaking waves and normalized length of the separation bubble, reproduces the observed features of the drag coefficient from low to high wind speeds despite extrapolating empirical wave spectra and breaking wave statistics beyond known limits. The model shows the saturation of the drag coefficient at high wind speeds for both field and laboratory fetches, suggesting that air-flow separation over ocean waves and its accompanying effects may play a significant role in the physics of the air-sea stress, at least at high wind speeds.     

Air Flow Structure Over Short-gravity Breaking Water Waves

Despite its importance for momentum and mass transfer across the air–sea interface, the dynamics of airflow over breaking waves is largely unknown. To fill this gap, velocity and vorticity distributions above short-gravity breaking waves have been measured in a wind-wave tank. A Digital Particle Image velocimetry technique (DPIV) was developed to accomplish these measurements above single breaking waves, propagating in mechanically-generated wave groups and forced by the wind. By varying the wind speed and initial characteristics of the groups, the airflow structure was captured over waves at different stages of the breaking process, and breaking with various intensities. The instantaneous airflow that separates from a sharp breaking crest is very similar to that occurring over a backward facing step. The separation bubble is however strongly unsteady: the steeper the wave crest and the larger the Reynolds number based on the crest-height, the higher the separated layer and the farther downwind the reattachment point. Instantaneous flow topology displays specific features of three-dimensional separation patterns. The tangential stress above the wave profile does not exhibit spikes at reattachment but grows progressively downwind from zero at reattachment to a value at the next crest approximately that found at the upwind breaking crest. Static pressure measurements revealed that large pressure falls are generated by vortices in the separated layer, as found in separated flows over solids. This study may provide useful data for theoretical and numerical modelling of the flow and associated phenomena.     

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.     

Numerical simulation of air flow over breaking waves

The air flow above breaking monochromatic Stokes waves is studied using a numerical nonlinear model of the turbulent air flow above waves of finite amplitude. The breaking event (spilling breaker) is parameterized by increasing the local roughness at the downwind slope of the wave, just beyond the crest. Both moderate slope waves and steep waves are considered. Above steep breaking waves, a large increase (typically 100%) in the total wind stress — averaged over the wave profile — is found compared to nonbreaking moderate slope waves. This is due to the drastic increase of the form drag, which arises from the asymmetrical surface pressure pattern above breaking waves. Both increase of wave slope (sharpening of the crest) and increase of local roughness in the spilling breaker area cause this asymmetrical surface pressure pattern. A comparison of the numerical results with the recent experimental measurements of Banner (1990) is carried out and a good agreement is found for the structure of the pressure pattern above breaking waves and for the magnitude of enhanced momentum transfer. Also: Dept. of Applied Physics, Techn. Univ. Delft, Netherlands.     

Effects of Sea-Surface Waves and Ocean Spray on Air-Sea Momentum Fluxes

The effects of sea-surface waves and ocean spray on the marine atmospheric boundary layer(MABL) at different wind speeds and wave ages were investigated. An MABL model was developed that introduces a wave-induced component and spray force to the total surface stress. The theoretical model solution was determined assuming the eddy viscosity coefficient varied linearly with height above the sea surface. The wave-induced component was evaluated using a directional wave spectrum and growth rate. Spray force was described using interactions between ocean-spray droplets and wind-velocity shear. Wind profiles and sea-surface drag coefficients were calculated for low to high wind speeds for wind-generated sea at different wave ages to examine surface-wave and ocean-spray effects on MABL momentum distribution. The theoretical solutions were compared with model solutions neglecting wave-induced stress and/or spray stress. Surface waves strongly affected near-surface wind profiles and sea-surface drag coefficients at low to moderate wind speeds. Drag coefficients and near-surface wind speeds were lower for young than for old waves. At high wind speeds, ocean-spray droplets produced by wind-tearing breaking-wave crests affected the MABL strongly in comparison with surface waves, implying that wave age affects the MABL only negligibly. Low drag coefficients at high wind caused by ocean-spray production increased turbulent stress in the sea-spray generation layer, accelerating near-sea-surface wind. Comparing the analytical drag coefficient values with laboratory measurements and field observations indicated that surface waves and ocean spray significantly affect the MABL at different wind speeds and wave ages.     

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).     

New evidence for a relation between wind stress and wave age from measurements during ASGAMAGE

Data from the 1996 ASGAMAGE experiment, performed in the southern North Sea at research platform Meetpost Noordwijk (MPN), are analysed for the parameters affecting the momentum flux. The stress turns out to be quadratically related to the 10-m wind speed and linearly to the wind speed at a wavelength related level. The Charnock parameter (dimensionless roughness length) shows a pronounced correlation with wave age. This implies, due to a coupling between wave age and the steepness of the waves, a connection between the stress and the steepness. We find that our North Sea results are consistent withopen ocean observations. For a given wind speed the mean stress at MPN turns out to be higher because the wave age there is in general lower. We define and give an expression for a drag coefficient at a wavelength related level that can be calculated straightforwardly from the wave age and then reduced to a standard level.     

1979—2013年ERA-Interim资料的青藏高原低涡活动特征分析

为了更好地了解青藏高原多尺度地形的动力作用,并为改良数值模式中地形的表示方法奠定基础,通过采用2010年青藏高原西南部6个地面台站的观测资料以及4种不同分辨率的分析(再分析)资料,分别估算了冈底斯山及整个青藏高原主体范围内的地表气压拖曳,得出了青藏高原可能存在的拖曳类型,并且分析了青藏高原气压拖曳的一些特征。得出如下主要结论:由罗斯贝波产生的波动拖曳作为行星尺度的拖曳对青藏高原地区总拖曳的贡献最大;同时,青藏高原范围内存在着大量与天气过程密切相关的天气尺度的拖曳;对于冈底斯山对气流的中尺度动力作用的进一步分析可知,夏季基本全为气流分离,冬季500 hPa以下为气流分离,500—200 hPa为气流分离和波动破碎的混合区,而200 hPa以上的平流层则为重力波的产生及其破碎区域;冈底斯山地区的地表气压拖曳主要集中在3000—5000 m高度,并且,冈底斯山总拖曳的方向近乎与山脊垂直;地表气压和地形高度资料的分辨率越高,所能分辨出的更小波长的气压拖曳也越多,估算出的高原主体范围内的拖曳值也越大;变压梯度和地形梯度是影响气压拖曳的基本因子,但地形梯度对拖曳的影响最终是通过气压梯度来实现的。     

High-order cumulants of sea surface elevations

The variability of cumulants of the fifth and sixth orders of the sea surface elevations is studied. The investigations are carried out on the basis of the data of direct wave measurements carried out in the field conditions. It is demonstrated that the cumulants of the fifth order are weakly correlated with the variations of the mean sea surface slope generated by the dominant waves. The variations of the cumulant of the sixth order are not correlated with the mean slope variations. The errors of the sea surface elevation probability density approximated by the models based on the use of Gram-Charlier series are analyzed. The analysis is carried out for the models including various numbers of terms of the Gram-Charlier series.     

Coastal-trapped waves with finite bottom friction

Coastal-trapped waves with finite-amplitude bottom friction are explored. “Finite-amplitude” in this context means that the bottom stresses are large enough to change the wave modal structure. The importance of bottom friction is measured by the nondimensional number r/(ωh), where r is a bottom resistance coefficient, ω the wave frequency and h the water depth. Increasing bottom drag causes free wave modes to adjust by having their amplitude maxima for alongshore current translate offshore to the point that, with relatively large bottom stress, the alongshore current variance is trapped entirely on the slope, even though pressure variations remain substantial right up to the coast. In conjunction with these adjustments, wave frequency, hence propagation speed, varies and the wave damping is usually less than would be expected based on a weak-friction perturbation calculation. Stronger density stratification increases wave damping, all else being the same. A mean alongshore flow can strongly affect modal structure and wave damping, although general trends are difficult to discern. Results suggest that bottom friction may cause an observed tendency for lower frequency alongshore current fluctuations to become relatively more important with distance offshore.     

Impact of waves on the sea drag: measurements in the Baltic sea and a model interpretation

Measurements from the Baltic Sea and a wind-over-wave coupled model are used to study the wave impact on the sea drag. The study has been carried out for different wave conditions, namely a pure wind-sea, following-swell/ mixed sea and cross-swell/ mixed sea. Measurements reveal the fact that the sea drag is dependent on the sea-state. In stationary conditions and in the absence of severe cross-swell, swell reduces drag compared to wind-sea at the same wind speed. The cross-swell enhances the drag as compared to the following-swell case and the magnitude of the drag coefficient is increased with increasing the angle of swell propagation to the wind. It is shown that the agreement between the model results and measurements is good for pure wind-sea and stationary mixed-sea cases. Discrepancies occur at light winds, where most of the data represent pure swell conditions. During these pure swell conditions the data are characterized by a large variation of the drag coefficient. The variation is caused by mesoscale variability in the stress co-spectra, wind-cross-swell effects and nonstationarity in the wave and wind fields not represented in the model.     

A Note on a Parameterization of the Sea Drag

A new parameterization of the sea drag is based on a wind-over-wavescoupling theory. The parameterization accounts for the wind speed, wave ageand finite depth dependencies of the sea drag. The latter two are introducedthrough the integral parameters of the wind-wave field: the dominant waveheight and the wavenumber at the spectral peak, and the water depth. Theparameterization is checked against the wind-over-waves model results andtwo field datasets obtained in a wide range of the wind speed and wave age.The comparison is encouraging. The parameterization is aimed for use inoperational ocean-state and atmosphere models.     

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.     

DOES WIND STRESS DEPEND ON SEA-STATE OR NOT? – A STATISTICAL ERROR ANALYSIS OF HEXMAX DATA

There have been several claims, either explicit or by implication, either based on experimental evidence or on theoretical reasoning, that the wind stress is modified by the stage of development of the wind sea. However, the overall evidence is weak, because theories are still incomplete and because it is questionable whether the sea-state effect, which is of the order of 10%, can be separated from experimental noise, which is of the order of 20%. In this paper a rigorous statistical analysis of HEXMAX data is pursued in order to establish the significance of sea-state effects. It appears that the enhanced drag, especially at high winds, which has already been established by previous analyses, cannot be attributed to the effect of young waves. The analysis provides no clues for the actual mechanism, which could be related to breaking or shoaling waves. As the effect of sea-state on wind stress is much smaller than the experimental noise level, it is hard to detect. Nevertheless, HEXMAX seems to contain a wave effect that is at the edge of statistical significance. It is, however, not the wave age itself that influences the drag, but a parameter involving wave height.Because the HEXMAX evidence is only indicative, we conclude that the issue set out in this paper cannot be answered on the basis of the HEXMAX data alone. It is recommended that error analyses are also carried out for other relevant observational data sets and that new measurements with suppressed noise will be taken up.     

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.     

INFLUENCES OF TOPOGRAPHY ON BAROCLINIC SOLITARY ROSSBY WAVES IN A MULTILEVEL MODEL

The KdV equation with topography included in an N-level model is derived. It is shown that if the topography ex-ists. the KdV equation may describe the solitary Rossby waves in the case of basic current without vertical shear, and itis no necessary to introduce the MKdV equation. The results of calculations show that the change of horizontal shearpattern of basic flow may cause an important change of the streamline pattern of the solitary waves with the oddmeridional wavenumber m, and has no effect for the even meridional wavenumber m. The vertical shear increases thesteepness of the barotropic solitary modes, and it has a complicated effect on the baroclinic modes. The influences oftopographic slope on the solitary waves are very great. The southern and northern slopes of topography may cause dif-ferent solitary wave patterns, with the effect of northern slope greater. The effect of Froude number on the solitarywaves is generally to steepen the solitary waves, however, the effect also depends on the meridional wavenumber m andthe modes of solitary wave.     

Wave generation on the air-sea interface

The sea state and the air flow above the sea during active wave generation is discussed. From energy balance considerations, a relationship between the wind duration and the phase speed of the waves at the peak of the energy spectrum is derived and compared with previous experimental results. It is shown that fluid viscosity plays a negligible role in the transfer of momentum from the air to the sea. Consequently the drag coefficient for the air-sea interface is related only to the apparent roughness of the sea surface.     

A note on ‘separation’ over short wind waves

Whether or not separation occurs in the airflow over wind-generated water waves is partly a question of semantics but also has an important bearing on the wave generation process. In the present paper we use a rather formal and perhaps narrow definition of separation and show that it does not occur where the shear stress is zero but only in conjunction with wave breaking. This is unlikely to happen except in the presence of quite strong surface drift velocities in the water. A similar connection between separation, surface drift and wave breaking has recently been established by Banner and Melville (1976).The effects of increasing wave amplitude or steepness are investigated with a numerical model of the airflow over water waves. Variations in the depth of the closed streamline region are predicted. The model is also used to investigate the possible importance of surface drift velocities.On leave 1976, Atmospheric Environment Service, Downsview, Ont., Canada.     

The influence of sea waves on the wind profile

From wind profile and wave measurements performed during the JONSWAP II experiment, relations between the dimensionless profile slope and the significant wave height are derived. It is shown that the wind profile is distorted by the waves especially in the vicinity of the water surface. The wave influence on the profile seems to be restricted to heights below about three wave heights. Above this level, the dimensionless profile slope is an approximately constant value corresponding to a drag coefficient of about 1.15 × 10^–3.