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).     

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.     

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.     

Surface pressure distribution and pressure drag on mountains

Summary A mesoscale numerical model is used to compute the different components of the pressure drag on mountains, i.e.: form drag, wave drag, hydrostatic drag, and total pressure drag, for stable stratification. The Froude number is chosen so that non-breaking lee-waves evolve.The paper explains how the different parts of the drag are computed from the numerical results and how they form part of the horizontal momentum budget.For a single mountain the drag from the evolving stationary solution is compared to the wave drag from linear inviscid theory. Wave drag turns out to be about one third of the value expected from linear theory, and nonlinear interaction between wave and form drag is found. Wave drag is responsible for about 75% of the total drag if blocking is negligible.For two obstacles with varying distance the wave drag in the stationary solution varies between 5% and 30% of the value from linear theory due to partial cancellation between the lee-waves from the two mountains.Finally in an instationary simulation the passage of a cold air mass over a mountain and the respective drag components have been computed. 1500 m above the crestline of the obstacle wave drag is only 10% to 30% of the total drag.From the present results it seems realistic that wave drag from ALPEX experimental data was only a few percent of the value expected from the surface pressure distribution and from linear theory.With 7 Figures     

Drag Partition for Regularly-Arrayed Rough Surfaces

Vegetation and other roughness elements distributed across a surface can providesignificant protection against wind erosion by extracting momentum from the flowand thereby reducing the shear stress acting at the surface. A theoretical model haspreviously been presented to specify the partition of drag forces for rough surfacesand to predict required vegetation density to suppress wind erosion. However, themodel parameters have not yet been constrained and the predictive capacity of themodel has remained uncertain. A wind-tunnel study was conducted to measure thedrag partition for a range of roughness densities and to parameterise the model inorder to improve its range of potential applicability. The drag forces acting on bothan array of roughness elements and the intervening surface were measured independentlyand simultaneously using new drag balance instrumentation. A detailed measure of thespatial heterogeneity of surface shear stresses was also made using Irwin sensors. Thedata agreed well with previous results and confirmed the general form of the model.Analysis of the drag partition confirmed the parameter definition = C_R/C_S(where C_R and C_S are roughness element and surface drag coefficients,respectively) and a constant proportional difference between the mean and maximumsurface shear stress was found. The results of this experiment suggest that the definitionfor m, the surface shear stress inhomogeneity parameter, should be revised, although thetheoretical and physical reasons for including this parameter in the model appear to bevalid. Best-fit values for m ranged from 0.53 to 0.58.     

Determination of Roughness Lengths for Heat and Momentum Over Boreal Forests

The roughness length for momentum (z_0m), zero-plane displacementheight (d), and roughness length for heat (z_0h) are importantparameters used to estimate land-atmosphere energy exchange. Although many different approaches have been developed to parameterizemomentum and heat transfer, existing parameterizations generally utilizehighly simplified representations of vegetation structure. Further, a mismatch exists between the treatments used for momentum and heat exchange and those used for radiative energy exchanges. In this paper, parameterizations are developed to estimate z_0m, d, and z_0h for forested regimes using information related to tree crown density and structure. The parameterizations provide realistic representationfor the vertical distribution of foliage within canopies, and include explicit treatment for the effects of the canopy roughness sublayer and leaf drag on momentum exchange. The proposed parameterizationsare able to realistically account for site-to-site differences in roughness lengths that arise from canopy structural properties.Comparisons between model predictions and field measurements show good agreement, suggesting that the proposed parameterizations capture the most important factors influencing turbulent exchange of momentumand heat over forests.     

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.     

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.     

A coupled mesoscale-model Fourier-method for idealized mountain-wave simulations over Hawaii

Mesoscale model simulations of representative trade winds impinging upon the Big Island of Hawaii are diagnosed for their mountain-wave characteristics by coupling a mesoscale model to a Fourier method. Localized phase-averaged wave momentum fluxes are calculated, which facilitates the study of wave generation from fine-scale topographic features. We find that the wave momentum fluxes are dominated by forcing from subsidiary topographic peaks, with the broader island topography controlling flow splitting and lee vortex generation. Waves also arise at the far northern and southern extremities of the island by acceleration of split flow. The strength of the local momentum fluxes proves to be sensitive to a small change in the incident flow direction. Areally integrated fluxes (wave drag) align closely with the incident flow direction and are an order of magnitude smaller than linear predictions and an order of magnitude larger than corresponding dividing streamline predictions. We briefly discuss the relevance of these results to the parameterization of subgrid-scale mountain-wave drag in climate and weather models.     

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 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.     

对流云街激发的重力波和波动阻力

大气边界层中的对流活动,可以在其上部稳定层中激发出重力波,并引起垂直动量输送,影响到对流层和平流层中的动量平衡过程.从二层模式中大气波动方程的线性解出发,得出了对流云街激发的重力波波阻解析表达式,并讨论了大气条件对波阻的影响.这些分析可有助于大气环流模式(GCM)中此类重力波波阻参数化表达式的建立和改进.     

Numerical Simulation of Drag Partition over Rough Surfaces

We present a numerical simulation of drag partition over rough surfaces. A computational fluid dynamics model is applied with high resolution to simulatingturbulent flows over arrays of roughness elements positioned on asmooth surface. The skin drag on the surface and the pressure drag on the roughnesselements are computed. The simulated drag partition compares well with wind-tunnelmeasurements and theoretical estimates for similar rough surfaces. This confirms that the computational approach offers an alternative to wind-tunnel and field experiments in studying drag and drag partition. The model is then applied to studying drag partition over rough surfaces with various roughness configurations. It is shown that drag partition depends not only on the magnitude of the roughness frontal area but also on the sizes and arrangement of roughness elements, because (1) the pressure drag coefficient is sensitive to roughness-element dimensions and (2) the arragement of roughness elements lead to different interferences of turbulent wakes. The impact ofthe latter factor is not insignificant.     

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.     

Heat Flux in the Coastal Zone

Various difficulties with application of Monin–Obukhov similarity theory are surveyed including the influence of growing waves, advection and internal boundary-layer development. These complications are normally important with offshore flow. The transfer coefficient for heat is computed from eddy correlation data taken at a mast two kilometres off the Danish coast in RASEX. For these coastal zone data, the thermal roughness length shows no well-defined relation to the momentum roughness length or roughness Reynolds number, in contrast to previous theories. The variation of the momentum roughness length is dominated by wave state. In contrast, the thermal roughness length shows significant dependence on wave state only for small values of wave age where the mixing is apparently enhanced by wave breaking. The development of thin internal boundary layers with offshore flow substantially reduces the heat transfer and thermal roughness length but has no obvious influence on momentum roughness length. A new formulation of the thermal roughness length based on the internal boundary-layer depth is calibrated to the RASEX data. For the very stable case, the turbulence is mainly detached from the surface and existing formulations do not apply.As an alternative to adjusting the thermal roughness length, the transfer coefficient is related directly to the stability and the internal boundary-layer depth. This avoids specification of roughness lengths resulting from the usual integration of the non-dimensional temperature function. The resulting stability function is simpler than previous ones and satisfies free convection similarity theory without introduction of the gustiness factor. The internal boundary layer also influences the moisture transfer coefficient.     

Observations and parametrization of momentum transfer in heterogeneous terrain consisting of regularly spaced obstacles

Tethered balloon data collected over a heterogeneous, semi-rural area of southern England are presented. The terrain within which the observations were taken is dominated by regularly spaced bluff obstacles that form the boundaries to grass fields. The data are compared with previous surface data to assess how representative these fixed-point measurements are of area-averaged values. It is found that there is a vertical height above which the flow approximates to being homogeneous and observations made above it are representative of the area average. The effective roughness length for momentum is then compared with that derived from a model describing the drag partition around isolated obstacles that are either non-porous or porous to the incident flow. It is shown that the method chosen to partition the drag is of the right order to produce reasonable predictions of the effective roughness length for different roughness densities.