A Richardson Number criterion for the air-sea interface

It is shown that the observationally determined roughness relation z ₀ = u _* ²/g in which g is the acceleration of gravity, u _*, is the friction velocity in air, and = 0.0185 (Wu, 1982) for the wind profile over the sea surface relative to the surface current, is consistent with the existence of a Richardson Number criterion at the air-sea interface in which the critical Richardson Number, Ri_c = 1, such that all the shear energy is converted into potential energy.     

Momentum and scalar transfer coefficients over aerodynamically smooth antarctic surfaces

Vertical profiles of wind speed, temperature and humidity were used to estimate the roughness lengths for momentum (z ₀), heat (z _H ) and moisture (z _Q) over smooth ice and snow surfaces. The profile-measurements were performed in the vicinity of a blue ice field in Queen Maud Land, East Antarctica. The values ofz ₀ over ice (3·10^–6 m) seem to be the smallest ever obtained over permanent, natural surfaces. The settling of snow on the ice and the loss of momentum at saltating snow particles serve as momentum dissipating processes during snow-drift events, expressed as a strong dependence ofz ₀ on u_#.The scalar roughness lengths and surface temperature can be evaluated from the temperature and humidity profile measurements if the ratioz _H /z _Q is specified. This new method circumvents the difficult measurement of surface temperature. The scalar roughness lengths seem to be approximately equal toz0 for a large range of low roughness Reynolds numbers, despite the frequent occurrence of drifting snow. Possible reasons for this agreement with theory of non-saltating flow are discussed.     

Standard deviation of vertical two-point longitudinal velocity differences in the atmospheric boundary layer

The standard deviation of vertical two-point longitudinal velocity fluctuation differences is analyzed experimentally with eleven sets of turbulence measurements obtained at the NASA 150-m ground-winds tower site at Cape Kennedy, Florida. It is concluded that /u _*0 is proportional to (fz/u _*0)^0.22, where the coefficient of proportionality is a function of fz/u _*0 and u _*0/fL ₀. The quantities f and L₀ denote the Coriolis parameter and the surface Monin-Obukhov stability length, respectively; u _*0 is the surface friction velocity; z is the vertical distance between the two points over which the velocity difference is calculated; and zz is the mean height of the mid-point of the interval z above natural grade. The results of the analysis are valid for 20<-u _*0/fL ₀<2000.     

Three-dimensional simulations and wind-tunnel experiments on airflow over isolated forest stands

Prediction of windthrow risk to individual or groups of retained trees in harvested stands requires an improved understanding of canopy airflow dynamics. Large-eddy simulations were used to simulate wind-tunnel experiments in two and three dimensions to compare with observations for model validation and to address parameter space considerations for the design of subsequent retention pattern experiments. The three-dimensional simulations were similar to the observed wind-tunnel data for the statistical profiles for but there were greater differences in skewness and kurtosis. These results were obtained using a common leaf-area drag formulation without either skin friction or speed dependent drag that enables scaling with U ₀ (ambient wind speed) and h (height of the canopy). This scaling results in a single non-dimensional parameter h/h _c where h _c (x, y, z) is the momentum range resulting from the canopy drag. The validity of the model scaling was tested using two-dimensional simulations. The irrotational component of the flow (potential flow) was found to be important when defining vertical domain limitations and has significant implications for time dependent flow (i.e. turbulent conditions) when considering retention pattern design. The sudden onset of drag associated with the isolated stand presents some unexpected challenges. The horizontal scales of the shearing instabilities were simulated in two dimensions and found to range between 2h for early times to 7h for later times. The early-time horizontal scales are in the range of logical retention pattern scales and as such need to be taken into account as part of the parameter space, i.e. a range of retention pattern lengths need consideration.     

On the impact of thermal stability on some rough flow effects over mobile surfaces

Calculations are made of the effects of thermal stability under a range of conditions, over the sea and land, on the physical factors (including the critical wind speed) affecting dust-storm generation, snow drift, and rough sea conditions. The computational procedure involves the surface friction velocity, u _*, and its relation with the aerodynamic roughness over aerodynamically rough, mobile surfaces. The results indicated that even at relatively high wind speeds, thermal effects under extreme advection situations may be significant, particularly for those properties of the agitated surface dependent on u _* ³ and u _* ⁴.     

Momentum Absorption in Rough-Wall Boundary Layers with Sparse Roughness Elements in Random and Clustered Distributions

A wind-tunnel experiment has been used to investigate momentum absorption by rough surfaces with sparse random and clustered distributions of roughness elements. An unusual (though longstanding) method was used to measure the boundary-layer depth δ and friction velocity u _* and thence to infer the functional relationship z ₀/h = f(λ) between the normalised roughness length z ₀/ h and the roughness density λ (where z ₀ is the roughness length and h the mean height of the roughness elements). The method for finding u _* is based on fitting the velocity defect in the outer layer to a functional form for the dimensionless velocity-defect profile in a canonical zero-pressure-gradient boundary layer. For the conditions investigated here, involving boundary layers over sparse roughness with strong local heterogeneity, this velocity-defect-law method is found to be more robust than several alternative methods for finding u _* (uw covariance, momentum integral and slope of the logarithmic velocity profile).The experimental results show that, (1) there is general agreement in the relationship z ₀/h = f(λ) between the present experiment with random arrays and other wind-tunnel experiments with regular arrays; (2) the main effect of clustering is to increase the scatter in the z ₀/h = f(λ) relationship, through increased local horizontal heterogeneity; (3) this scatter obscures any trend in the z ₀/h = f(λ) relationship in response to clustering; and (4) the agreement between the body of wind-tunnel data (taken as a whole) and field data is good, though with scatter for which it is likely that a major contribution stems from local horizontal heterogeneity in the field.     

Shear-Stress Partitioning in Live Plant Canopies and Modifications to Raupach’s Model

The spatial peak surface shear stress t_S^￠￠{\tau _S^{\prime\prime}} on the ground beneath vegetation canopies is responsible for the onset of particle entrainment and its precise and accurate prediction is essential when modelling soil, snow or sand erosion. This study investigates shear-stress partitioning, i.e. the fraction of the total fluid stress on the entire canopy that acts directly on the surface, for live vegetation canopies (plant species: Lolium perenne) using measurements in a controlled wind-tunnel environment. Rigid, non-porous wooden blocks instead of the plants were additionally tested for the purpose of comparison since previous wind-tunnel studies used exclusively artificial plant imitations for their experiments on shear-stress partitioning. The drag partitioning model presented by Raupach (Boundary-Layer Meteorol 60:375–395, 1992) and Raupach et al. (J Geophys Res 98:3023–3029, 1993), which allows the prediction of the total shear stress τ on the entire canopy as well as the peak (t_S ^￠￠/t)^1/2{(\tau _S ^{\prime\prime}/\tau )^{1/2}} and the average (t_S^￠/t)^1/2{(\tau _S^{\prime}/\tau )^{1/2}} shear-stress ratios, is tested against measurements to determine the model parameters and the model’s ability to account for shape differences of various roughness elements. It was found that the constant c, needed to determine the total stress τ and which was unspecified to date, can be assumed a value of about c = 0.27. Values for the model parameter m, which accounts for the difference between the spatial surface average t_S^￠{\tau _S^{\prime}} and the peak t_S ^￠￠{\tau _S ^{\prime\prime}} shear stress, are difficult to determine because m is a function of the roughness density, the wind velocity and the roughness element shape. A new definition for a parameter a is suggested as a substitute for m. This a parameter is found to be more closely universal and solely a function of the roughness element shape. It is able to predict the peak surface shear stress accurately. Finally, a method is presented to determine the new a parameter for different kinds of roughness elements.     

Effect of atmospheric stability on the growth of surface gravity waves

In this paper we study the effect of atmospheric stability on the growth of surface gravity waves. To that end we numerically solved the Taylor-Goldstein equation for wind profiles which deviate from a logarithmic form because stratification affects the turbulent momentum transport. Using Charnock's relation for the roughness height z ₀ of the wind profile, it is argued that the growth rate of the wave depends on the dimensionless phase velocity c/u _* (where u _* is the friction velocity) and a measure of the effect of atmospheric stability, namely the dimensionless Obukhov length gL/u _* ², whereas it only depends weakly on gz _t /u _* ² (where z _t is the roughness height of the temperature profile). Remarkably for a given value of u _* /c, the growth rate is larger for a stable stratification (L > 0) than for an unstable one (L < 0). We explain why this is the case. If, on the other hand, one considers the growth rate as a function of c/U ₁₀ (where U ₁₀ is the windspeed at 10 m), the situation reverses for c/U ₁₀ < 1. For practical application in wave prediction models, we propose a new parameterization of the growth rate of the waves which is an improvement of the Snyder et al. (1981) proposal because the effect of stability is taken into account.     

Temporal and Spatial Variations of the Aerodynamic Roughness Length in the Ablation Zone of the Greenland Ice Sheet

To understand the response of the Greenland ice sheet to climate change the so-called ablation zone is of particular importance, since it accommodates the yearly net surface ice loss. In numerical models and for data analysis, the bulk aerodynamic method is often used to calculate the turbulent surface fluxes, for which the aerodynamic roughness length (z ₀) is a key parameter. We present, for the first time, spatial and temporal variations of z ₀ in the ablation area of the Greenland ice sheet using year-round data from three automatic weather stations and one eddy-correlation mast. The temporal variation of z ₀ is found to be very high in the lower ablation area (factor 500) with, at the end of the summer melt, a maximum in spatial variation for the whole ablation area of a factor 1000. The variation in time matches the onset of the accumulation and ablation season as recovered by sonic height rangers. During winter, snow accumulation and redistribution by snow drift lead to a uniform value of z ₀≈ 10⁻⁴ m throughout the ablation area. At the beginning of summer, snow melt uncovers ice hummocks and z ₀ quickly increases well above 10⁻² m in the lower ablation area. At the end of summer melt, hummocky ice dominates the surface with z ₀ > 5  ×  10⁻³ m up to 60 km from the ice edge. At the same time, the area close to the equilibrium line (about 90 km from the ice edge) remains very smooth with z ₀ = 10⁻⁵ m. At the beginning of winter, we observed that single snow events have the potential to lower z ₀ for a very rough ice surface by a factor of 20 to 50. The total surface drag of the abundant small-scale ice hummocks apparently dominates over the less frequent large domes and deep gullies. The latter results are verified by studying the individual drag contributions of hummocks and domes with a drag partition model.     

Variability of surface roughness and turbulence intensities at a coastal site in India

Surface-layer features with different prevailing wind directions for two distinct seasons (Southwest Monsoon and Northeast Monsoon) on the west coast of India are studied using data obtained from tower-based sensors at a site located about 500 m from the coast. Only daytime runs have been used for the present analysis. The surface boundary-layer fluxes have been estimated using the eddy correlation method. The surface roughnessz ₀ obtained using the stability-corrected wind profiles (Paulson, 1970) has been found to be low for the Southwest monsson season. For the other season,z ₀ is relatively high. The drag coefficientC _D varies with height in the NE monsoon season but not in the season with lowz ₀. This aspect is reflected in the wind profiles for the two seasons and is discussed in detail. The scaling behaviour of friction velocityu _* and the turbulence intensity of longitudinal, lateral and vertical winds _u, _v and _w, respectively) are further examined to study their dependence on fetch. Our study shows that for the non-dimensional case, _u/u_* and _v/u_* do not show any surface roughness dependence in either season. On the other hand, for _w/u_* for the season with lowz ₀, the values are seen to agree well with that of Panofskyet al. (1977) for homogeneous terrain whereas for the other season with highz ₀, the results seem to conform more to the values observed by Smedman and Högström (1983) for coastal terrain. The results are discussed in the light of observations by other investigators.     

The Apsley and Castro Limited-Length-Scale $${{k-\varepsilon}}$$ Model Revisited for Improved Performance in the Atmospheric Surface Layer

The limited-length-scale k-e{k-\varepsilon} model proposed by Apsley and Castro for the atmospheric boundary layer (Boundary-Layer Meteorol 83(1):75–98, 1997) is revisited with special attention given to its predictions in the constant-stress surface layer. The original model proposes a modification to the length-scale-governing e{\varepsilon} equation that ensures consistency with surface-layer scaling in the limit of small ℓ _m/ℓ _max (where ℓ _m is the mixing length and ℓ _max its maximum) and yet imposes a limit on ℓ _m as ℓ _m/ℓ _max approaches one. However, within the equilibrium surface layer and for moderate values of z/ℓ _max, the predicted profiles of velocity, mixing length, and dissipation rate using the Apsley and Castro model do not coincide with analytical solutions. In view of this, a general e{\varepsilon} transport equation is derived herein in terms of an arbitrary desired mixing-length expression that ensures exact agreement with corresponding analytical solutions for both neutral and stable stability. From this result, a new expression for C_e3{C_{\varepsilon3}} can be inferred that shows this coefficient tends to a constant only for limiting values of z/L; and, furthermore, that the values of C_e3{C_{\varepsilon3}} for z/L → 0 and z/L →∞ differ by a factor of exactly two.     

Critical Evaluation of Scalar Roughness Length Parametrizations Over a Melting Valley Glacier

We present a field investigation over a melting valley glacier on the Tibetan Plateau. In the ablation zone, aerodynamic roughness lengths (z _0M ) vary on the order of 10⁻⁴–10⁻² m, whose evolution corresponds to three melt phases with distinct surface cover and moisture exchange: snow (sublimation/evaporation), bare ice (deposition/condensation), and ice hummocks (sublimation/evaporation). Bowen-ratio similarity is validated in the stably stratified katabatic winds, which suggests a useful means for data quality check. A roughness sublayer is regarded as irrelevant to the present ablation season, because selected characteristics of scalar turbulence over smooth snow are quite similar to those over hummocky ice. We evaluate three parametrizations of the scalar roughness lengths (z _0T for temperature and z _0q for humidity), viz. key factors for the accurate estimation of sensible heat and latent heat fluxes using the bulk aerodynamic method. The first approach is based on surface-renewal models and has been widely applied in glaciated areas; the second has never received application over an ice/snow surface, despite its validity in (semi-)arid regions; the third, a derivative of the first, is proposed for use specifically over rough ice defined as z _0M > 10⁻³ m or so. This empirical z _0M threshold value is deemed of general relevance to glaciated areas (e.g. ice sheet/cap and valley/outlet glaciers), above which the first approach gives notably underestimated z _0T,q . The first and the third approaches tend to underestimate and overestimate turbulent heat/moisture exchange, respectively, frequently leading to relative errors higher than 30%. Comparatively, the second approach produces fairly low errors in energy flux estimates both in individual melt phases and over the whole ablation season; it thus emerges as a practically useful choice to parametrize z _0T,q in glaciated areas. Moreover, we find all three candidate parametrizations unable to predict diurnal variations in the excess resistances to humidity transfer, thus encouraging more efforts for improvement.     

The Parameterisation of Scalar Transfer over Rough Ice

We test a surface renewal model that is widely used over snow and ice surfaces to calculate the scalar roughness length (z _s ), one of the key parameters in the bulk aerodynamic method. For the first time, the model is tested against observations that cover a wide range of aerodynamic roughness lengths (z ₀). During the experiments, performed in the ablation areas of the Greenland ice sheet and the Vatnajökull ice cap in Iceland, the surface varied from smooth snow to very rough hummocky ice. Over relatively smooth snow and ice with z ₀ below a threshold value of approximately 10^?3 m, the model performs well and in accord with earlier studies. However, with growing hummock size, z ₀ increases well above the threshold and the bulk aerodynamic flux becomes significantly smaller than the eddy-correlation flux (e.g. for z ₀ = 0.01 m, the bulk aerodynamic flux is about 50% smaller). Apparently, the model severely underpredicts z _s over hummocky ice. We argue that the surface renewal model does not account for the deep inhomogeneous roughness sublayer (RSL) that is generated by the hummocks. As a consequence, the homogeneous substrate ice grain cover becomes more efficiently ‘ventilated’. Calculations with an alternative model that includes the RSL and was adapted for use over hummocky ice, qualitatively confirms our observations. We suggest that, whenever exceedance of the threshold occurs (z ₀  >  10^?3 m, i.e., an ice surface covered with at least 0.3-m high hummocks), the following relation should be used to calculate scalar roughness lengths, ln (z _s /z ₀)  =  1.5  ? 0.2 ln (Re _*)  ? 0.11(ln (Re _*))².     

Displaced-Beam Small Aperture Scintillometer Test. Part I: The Wintex Data-Set

The friction velocity (u_*) and the sensible heat flux density (H) determined with a displaced-beam small aperture scintillometer (DBSAS) and a hot-film eddy correlation system are compared. Random errors in the DBSAS are relatively small, compared to scatter found with two eddy-correlation systems. Assuming that the hot-film system yields the true fluxes, theDBSAS appears to overestimate u_* when u_* is less than 0.2 m s^-1 and to underestimate u_* at high wind speeds. This implies that the DBSAS measurements of theinner scale length of turbulence, l₀, a direct measure for the dissipation rate of kinetic turbulent energy, are biased. Possible causes for these results are discussedin detail. A correction procedure is presented to account for effects of random noise and of so-called inactive turbulence or sensor vibrations. The errors in u_* cause errors in the DBSAS measurements of the structure parameter of temperature C_T ². The derived H appears to be less sensitive to errors in l₀ and C_T ², because errors in these quantities tend to cancel out.     

Measurements of spray at an air-water interface

The spray content in the surface boundary layer above an air—water interface was determined by a series of measurements at various feteches and wind speeds in a laboratory facility. The droplet flux density N(z) can be described in terms of the scaling flux density N_* and von Karman constant K throguh the equation, N(z)/N_* = −(1/K) ln(z/z_0d) where z is height above the mean water level and z_0d is the droplet boundary layer thickness. N_* is given by a unique relationship in terms of the roughness Reynolds number u_*σ/ν where σ is the root-mean-square surface displacement. Spray inception occurred for u_* 0.3. The dominant mode of spray generation in the present and most other laboratory tests, as well as in available field data, appears to be bubble bursting.     

A model of airflow above changes in surface heat flux,temperature and roughness for neutral and unstable conditions

A numerical model of airflow in the lowest 50–100 m of the atmosphere above changes in surface roughness and temperature or heat flux has been developed based on boundary layer approximations, the Businger-Dyer hypotheses for the non-dimensional wind shear and heat flux and a mixing length hypothesis.Results have been obtained for several situations, in particular, airflow with neutral upstream conditions encountering a step change in surface temperature or heat flux with no roughness change. In these cases large increases in shear stress at the outer edge of the internal boundary layer are predicted. The case of unstable upstream flow encountering a step change to zero heat flux is also considered.Two situations that may be encountered near the shores of the Great Lakes are considered.Notation B Businger-Dyer constant (= 16.0) in form for _M, _H - c _p Specific heat at constant pressure - g Acceleration due to gravity - H Upward vertical heat flux - H ₀ , H ₁ Surface heat fluxes for x < 0, x 0 - k von Kármán's constant ( = 0.4) - l Mixing length - L Monin-Obukhov length - L ₀ Upstream value of L - m Ratio of roughness lengths (= z ₁/z ₀) - RL ^* Non-dimensional parameter, see Equations (20, 22 and 24) - RL ₁ ^* Same as RL ^* but with z ₁ scaling (= mRL ^*) - T Scaled temperature - T ₀ (z) Upstream temperature profile - u ₀, u ₁(x) Surface friction velocities for x < 0, x 0 - U, W Horizontal and vertical mean velocities - U ₀ (z) Upstream velocity profile - x, z Horizontal and vertical coordinates - z _i Local roughness length     

An Evaluation of the Flux–Gradient Relationship in the Stable Boundary Layer

Data collected during the SHEBA and CASES-99 field programs are employed to examine the flux–gradient relationship for wind speed and temperature in the stably stratified boundary layer. The gradient-based and flux-based similarity functions are assessed in terms of the Richardson number Ri and the stability parameter z/Λ_*, z being height and Λ_* the local Obukhov length. The resulting functions are expressed in an analytical form, which is essentially unaffected by self-correlation, when thermal stratification is strong. Turbulence within the stably stratified boundary layer is classified into four regimes: “nearly-neutral” (0 < z/Λ_* < 0.02), “weakly-stable” (0.02 < z/Λ_* < 0.6), “very-stable” (0.6 < z/Λ_* < 50), and “extremely-stable” (z/Λ_* > 50). The flux-based similarity functions for gradients are constant in “nearly-neutral” conditions. In the “very-stable” regime, the dimensionless gradients are exponential, and proportional to (z/Λ_*)^3/5. The existence of scaling laws in “extremely-stable” conditions is doubtful. The Prandtl number Pr decreases from 0.9 in nearly-neutral conditions and to about 0.7 in the very-stable regime. The necessary condition for the presence of steady-state turbulence is Ri < 0.7.     

Flux-gradient profiles for momentum and heat over an urban surface

Summary In this paper, we evaluate the applicability of flux-gradient relationships for momentum and heat for urban boundary layers within the Monin-Obukhov similarity (MOS) theory framework. Although the theory is widely used for smooth wall boundary layers, it is not known how well the theory works for urban layers. To address this problem, we measured the vertical profiles of wind velocity, air temperature, and fluxes of heat and momentum over a residential area and compared the results to theory. The measurements were done above an urban canopy whose mean height z_h is 7.3 m. 3-D sonic anemometers and fine wire thermocouples were installed at 4 heights in the region 1.5z_h < z < 4z_h. We found the following: (1) The non-dimensional horizontal wind speed has good agreement with the stratified logarithmic profile predicted using the semi-empirical Monin-Obukov similarity (MOS) function, when it was scaled by the surface friction velocity that is derived from the shear stress extrapolated to the roof-top level. (2) The scaled gradient of horizontal wind speed followed a conventional semi-empirical function for a flat surface at a level (z/z_h = 2.9), whereas, in the vicinity of the canopy height was larger than the commonly-used empirical relationship. (3) The potential temperature profile above the canopy shows dependency on the atmospheric stability and the scaled gradient of temperature is in good agreement with a conventional shear function for heat. In the case of heat, the dependency on height was not found. (4) The flux-gradient relationship for momentum and heat in the region 1.5z_h < z < 4z_h was rather similar to that for flat surfaces than that for vegetated canopies.     

Surface influence upon vertical profiles in the nocturnal boundary layer

Near-surface wind profiles in the nocturnal boundary layer, depth h, above relatively flat, tree-covered terrain are described in the context of the analysis of Garratt (1980) for the unstable atmospheric boundary layer. The observations at two sites imply a surface-based transition layer, of depth z _*, within which the observed non-dimensional profiles Φ _M ⁰ are a modified form of the inertial sub-layer relation \(\Phi _M \left( {{z \mathord{\left/ {\vphantom {z L}} \right. \kern-0em} L}} \right) = \left( {{{1 + 5_Z } \mathord{\left/ {\vphantom {{1 + 5_Z } L}} \right. \kern-0em} L}} \right)\) according to $$\Phi _M^{\text{0}} \simeq \left( {{{1 + 5z} \mathord{\left/ {\vphantom {{1 + 5z} L}} \right. \kern-\nulldelimiterspace} L}} \right)\exp \left[ { - 0.7\left( {{{1 - z} \mathord{\left/ {\vphantom {{1 - z} z}} \right. \kern-\nulldelimiterspace} z}_ * } \right)} \right]$$ , where z is height above the zero-plane displacement and L is the Monin-Obukhov length. At both sites the depth z _* is significantly smaller than the appropriate neutral value (z _*N ) found from the previous analysis, as might be expected in the presence of a buoyant sink for turbulent kinetic energy.     

A Study in Search of Interconnection between Surface Parameters and Surrounding Synoptic And Subsynoptic Features

The paper reveals that the variations in parameters like u*, the scaling velocity and θ*. The scaling tempera-ture during the various phases of monsoon might be linked with subsynoptic features. The rise in u* is mainly connected with the presence of lower tropospheric cyclonic vorticity over a subsynoptic scale of the site. However the variations in θ* is mainly linked with the various phases of monsoon and θ* shows a sharp rise in presence of low level convective cloud.Besides the correlation studies of u and u*, θv and θv* , θv-θv0 and θv* are undertaken. The correlation be?tween θv and θv* is poor. In other two cases correlations are good. Besides u/u* , has shown good coefficient of variation values within the ζ range.