Similarity parameters from first-order closure and data

A two-layer, first-order closure model for the Planetary Boundary Layer (PBL) is developed with the objective of parameterizing the surface stress with respect to the synoptic scale. The model includes stability effects by considering stratification-dependent secondary flow in the outer layer and empirical corrections to the surface layer flow. It shows the compatibility of simple eddy viscosity closure solutions with similarity theory by producing the now well-known Rossby similarity equations. It allows further insight into the Rossby similarity parameters by relating them to a single similarity parameter which is the ratio of the characteristic scales of the PBL and the surface layer.The measured and derived values of the similarity parameters A and B are compared with AIDJEX data and other published values. The variation in these values in stably stratified conditions is predicted and two alternate similarity parameters are calculated, one a constant and the other with a small variation and decreasing influence on the drag coefficient in stable stratification. The result is an empirical resistance law for a geostrophic drag coefficient variation which parameterizes an observed order-of-magnitude change in surface stress with changes in roughness or PBL stratification. This variation is related to similarity parameters characteristic of the region and to measurable changes in the geostrophic departure angle.     

Effects of local accelerations and baroclinity on the mean structure of the atmospheric boundary layer over the sea

Substitution of the geostrophic wind by the actual upper wind in the equations of motion for the boundary layer implies less sensitivity of the mean wind to inertial effects. This is confirmed by observations, although the problem of computing time or spatial derivatives from scattered data reduces the accuracy and the clarity of the results. It is found that acceleration (deceleration) increases (decreases) the cross-isobar angle whereas the geostrophic drag coefficient is a minimum (maximum) for crosswind acceleration (deceleration). On the other hand, cold air advection increases the cross-isobar angle whereas the geostrophic drag coefficient is a maximum when the thermal wind is parallel to the surface wind. The universal functions A _m and B _m based on vertically averaged winds are also rather insensitive to inertial influences.     

Dependence of geostrophic drag on intensity of convection,baroclinicity, and acceleration

This paper examines the practical importance of stability, baroclinicity, and acceleration in the bulk ABL similarity formulations, in light of the random errors inherent in field measurements. This is done by propagating the measurement uncertainties through a theoretical model for the bulk ABL similarity functionsA ₀ andB ₀, under a range of assumed (but always unstable) conditions. It is shown that random measurement errors and acceleration effects may overwhelm most effects of baroclinicity and stability, once conditions are at least slightly unstable. Because of this, it is hard to discern a clear functional dependence ofA ₀ andB ₀ on degree of instability. Thus, for a given value ofh _i/z₀, whereh _i is the inversion height andz ₀ is the surface roughness, the geostrophic drag coefficient, which depends onA ₀ andB ₀, and weakly onh _i/z₀, may also be taken to be nearly independent of degree of instability.     

Some characteristics of the PBL over the ocean during the Indian summer monsoon

Data collected during the Indo-Soviet Monsoon-77 expedition are used to determine quadratic expressions for the universal constants A and B, as functions of the stability parameter, . A quadratic expression has also been obtained for u _*, in terms of the surface wind u _s. It is shown, from the mean values of q and _E , that the entire area covered by the expedition could be divided into four regions around the point 13° N, 78° E. The mean thermal characteristic of each region differs. It is shown that the northeastern quadrant is most favourable for the sustenance of a tropical storm once it has formed.     

The limitations on stress parameterization imposed by intermittency in turbulent flow

The indirect dissipation technique is used to estimate 1-min averages of friction velocity u _*in the surface layer over the tropical ocean. These estimates are compared to estimates of u _*obtained using a drag coefficient and the relative difference between the two is examined in relation to stability and averaging time. Plumes and downdrafts are found to be responsible for an anomalous behavior of the drag coefficient estimates. Certain factors relating to plume properties, derived using conditional sampling as described in Khalsa (1980), are shown to be related to the variance between the two estimates of friction velocity. An investigation into the effects of increasing the averaging time reveals that plume spacing, which is dependent on stability, and the mean wind speed determine the minimum time for smoothing the influence of plumes and downdrafts.Department of Atmospheric Sciences contribution number 513.     

Measurements of the humidity structure function parameters,C q 2and C Tq,over the ocean

Atmospheric turbulence measurements, including temperature and humidity fluctuations, were made from the R/V Acania off the coast of California in June, 1979. The purpose of the experiment was to investigate the scaling properties of the humidity structure function parameter (C _q ²) and temperaturehumidity cospectrum structure parameter (C _Tq) in the marine surface layer. The bulk parameterization method was used to obtain Monin-Obukhov Similarity (MOS) scaling parameters u _*, T _*, q _*and L. Assuming a neutral stability humidity drag coefficient c _qn = 1.3 × 10^-3the dimensionless humidity structure function parameter C _q ²Z^2/3/q_* ²was found to be 18% lower than the corresponding temperature function obtained by Wyngaard et al. (1971). Furthermore, the measurements indicate that the temperature-humidity fluctuations are highly coherent well into the inertial subrange. The results have direct application to turbulent scattering of waves propagating in the atmosphere (particularly microwaves) and methods of estimating air-sea surface fluxes.     

Interaction between the atmospheric and oceanic boundary layers

The two-layer system of an atmosphere over water bodies is reduced to a single-layer problem. Values of the interfacial quantities, such as the friction velocity, the surface velocity, the angles, and , between the surface shear stress and the geostrophic wind velocity and the surface wind velocity, respectively, and the surface roughness, all of which depend upon external parameters, such as the geostrophic wind and stratifications, are obtained. The geostrophic drag coefficient C _d, the geostrophic wind coefficient C _f, and the angles , and , of the turbulent flow at the sea-air interface are functions of a dimensionless number, mfG/kg, with S ₁ and S ₂ as two free stratification parameters. The surface roughness is uniquely determined from the geostrophic wind rather than from the wind profile in the boundary layer.Formerly Visiting Research Associate, Applied Physics Branch, Earth Observations Division, NASA-Manned Spacecraft Center, Houston, Texas.     

The influence of the geostrophic wind advection approximation on a well-mixed layer

Numerical results indicate that advection of momentum in the boundary layer may significantly alter both the structure of the planetary boundary layer and its influence on the overlying free atmosphere. However, due to the nonlinearity of the inertial terms, it is always difficult to obtain the analytical solution of the boundary-layer model that retains the flow acceleration. In order to overcome this difficulty, the geostrophic momentum (hereafter GM) approximation has been introduced into boundary-layer models. By replacing the advected momentum with the geostrophic wind, the effect of the flow acceleration is partially considered and the original nonlinear partial differential equation set is converted to ordinary differential equations, the solutions of which can be obtained easily with standard techniques. However, the model employing GM fails to capture the features of the boundary layer when the spatio-temporal variation of the boundary-layer flow cannot be properly approximated by the geostrophic wind. In the present work, a modified boundary-layer model with the inertial acceleration in a different approximate form is proposed, in which the advecting wind instead of the advected momentum is approximated by the geostrophic wind (hereafter GAM).Comparing the horizontal velocity and boundary-layer pumping obtained from the classical Ekman theory, and the model incorporating (i) GM and (ii) GAM, it is found that the model with GAM describes most facets of the steady well-mixed layer beneath a north-westerly flow with embedded mesoscale perturbations that is considered in the present work. Inspection of the solution of the model with GAM shows that, within the limit of the validation of the model (i.e., the Rossby number R_O is not very large and the drag coefficient C_D is not too small), the horizontal convergence (divergence) is strengthened by the effect of the inertial acceleration in the region of maximum positive (negative) geostrophic vorticity. Consequently, the boundary-layer pumping there is intensified. It is found that the intensification is firstly strengthened and then weakened as R_O or C_D increases.     

Determination of similarity functions of the resistance laws for the planetary boundary layer using surface-layer similarity functions

Functional forms of the universal similarity functions A, B (for wind components parallel and normal to the surface stress), and C (for potential temperature difference) are determined based on the generalized theory of the resistance laws for the Planetary Boundary Layer (PBL). The similarity-profile functions for the surface layer are matched with the velocity and temperature-defect profiles that are assumed to have shapes modified by certain powers of nondimensional height z/h, where h is the PBL height. The powers of the outer-layer profile functions are determined, so that the functions become negligible in the surface layer. To close the temperature defect law, an assumption that the temperature gradient across the top of the PBL is continuous with the stratification of the overlying atmosphere is used. The result of this assumption is that nondimensional momentum and temperature profiles in the PBL can be described in terms of four basic ratios: (1) roughness ratio = /h (2) scale-height ratio =|f|h/u_*, (3) ambient stratification parameter =h/_*, and (4) stability parameter =h/L, where L is the Monin-Obukhov length, z₀ is the surface roughness, is the upper-air stratification, u _* is the friction velocity, and _* is the temperature scale at the surface. For stable conditions, the scale-height ratio can be related to the atmospheric stability and the upperair stratification, and the generalized similarity and Rossby number similarity theories become identical. Under appropriate boundary conditions, function A is explicitly dependent on the stability parameter , while B is a function of scale-height ratio , which in turn depends on the stability. Function C is shown to be dependent on the stability and the upper-air stratification, due to the closure assumption used for the temperature profile.The suggested functional forms are compared with other empirical approximations by several authors. The general framework used to determine the functional forms needs to be tested against good boundary-layer measurements.     

Comments on ‘the characteristics of turbulent velocity components in the surface layer under convective conditions’, by H. A. Panofsky,H. Tennekes,D. H. Lenschow,and J. C. Wyngaard

Panofsky et al. (1977) have presented an analysis which seems to show a clear dependence of the dimensionless turbulence statistics _u /u _* and _v/u _* on the planetary boundary-layer stability parameter z _i/L. However it is possible that much of the apparent relationship results from artificial correlations introduced by the use of inter-related dimensionless parameters. Apparent dependencies of similar statistical quantities on z/L in the surface boundary layer might also be contaminated.This work was supported by the U.S. Department of Energy, and is a contribution of the Multistate Atmospheric Power Production Pollution Study (MAP3S).     

The steady-state height and resistance laws of the nocturnal boundary layer: Theory compared with cabauw observations

An expression is derived for the height of the stationary boundary layer during stable lapse rate conditions. It satisfies the conventional limits for neutral conditions and for large values of stability. Comparison with acoustic sounder observations near the meteorological mast at Cabauw (the Netherlands) shows that the steady-state height is not attained for large stability values. The observations are also used to investigate how the similarity functions A and B in the resistance laws depend on the stability parameters ₀ = u _*/f L and = h/L. The function B shows a clear trend as a function of stability, which can be described in terms of . The dependence of A is masked by scatter in the data points. The general conclusion leads to the concept of a non-steady boundary layer during stable lapse rate conditions.     

A boundary-layer model for the determination of hourly surface wind characteristics in a representative tropical African region

A quantative transposition model is introduced which determines hourly wind speeds in a representative tropical region (Central Sudan). The model consists of two parts. Firstly, a local boundary-layer model, based on the energy balance equation and the Businger-Dyer equations, is used to compute the average diurnal cycle of various characteristic boundary-layer parameters. Secondly, a horizontal transposition method is introduced to calculate wind speed behaviour at an arbitrary station from that at a reference station. This method is based on assumed spatial constancy of the turbulence parameter u _* in the period November–April in a region of about (700 × 800) km² in Central Sudan. The constancy of u _* is concluded from the very stationary character of the climate. Model-computed hourly wind speeds are consistent with the potential wind speeds (at 10 m over open country) calculated from the measured data, and provide better local wind estimates than the conventional procedure which assumes constant regional hourly wind speeds.     

Geostrophic drag coefficients

Data on the relationship of the surface wind to the geostrophic wind at Porton Down, Salisbury Plain, are presented for various stability conditions and analysed in the light of the Rossbynumber similarity theory. For near-neutral conditions, the geostrophic drag coefficients for geostrophic wind speeds 5 to 15 m s^-1 are close to those found by other workers but at higher speeds the values are low. Comparisons of geostrophic and radar wind speeds for ⋍900-m height, suggest that undetectably small mean cyclonic curvatures of the trajectories of the air are responsible for this departure. A value of the geostrophic drag coefficient for the open sea at wind speeds around 8 m s^-1 (neutral conditions) is deduced from recent observations of the drag in relation to the surface wind, combined with the ratios of 900-mb radar wind to surface wind obtained from the North Atlantic weather ship data tabulations of Findlater et al. (1966).     

Fluctuations of the horizontal wind under unstable conditions

The similarity relations for _u/u_* proposed by Panofskyet al. (1977) and Højstrup (1982) have been verified using eddy-correlation data collected during the EFEDA-experiment, conducted over the extensive plain of La Mancha (Spain), where vine plants form a primary crop. Also, the standard Monin-Obukhov relation is considered. It is found that the expressions by Panofskyet al. and Højstrup both yield almost identical results, and are better than the Monin-Obukhov expression. Also, _u measured with a cup anemometer obeys the similarity expressions well, provided that the effect of the variation of wind direction on propellor wind speed is accounted for. The relationship of Panofskyet al. works rather well even when the boundary-layer height scaleh is replaced by a fixed height,h _c. Best results were obtained forh _c=1800 m. This height scale is possibly associated with the horizontal variability in the surface sensible heat flux pattern.     

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.     

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.     

Note on turbulent scaling parameters for the convective planetary boundary layer

Scaling velocities relevant for turbulent flows in the planetary boundary layer are discussed. It is suggested that the scaling parameters should be determined by integrated bulk properties of the respective turbulent production terms. According to this concept, a new velocity scale, replacing the friction velocityu*, is proposed depending on bothu* and the geostrophic windu _g . The convective velocity scalew* can be determined by the integral of the buoyancy production term and is therefore an appropriate velocity scale. Examination of Minnesota and Kansas data shows that these data do not give the possibility of verifying whether the new scaling velocity is more appropriate thanu*. This is because the range of variability of atmospheric stability during the field measurements is too small. However, theoretical considerations based on integrated properties of the turbulence, through the depth of the planetary boundary layer, are given in support of the new scaling velocity.     

Observational evidence for the Monin-Obukhov similarity under all stability conditions

Data collected in the surface layer in a northern suburban area of Nanjing from 15 November to 29 December 2007 were analyzed to examine the Monin-Obukhov similarity for describing the turbulent fluctu- ations of 3D winds under all stability conditions and to obtain the turbulence characteristics under different weather conditions. The results show that the dimensionless standard deviations of turbulent velocity com- ponents (σ u /u* , σ v /u* , σ w /u * ) and dimensionless turbulent kinetic energy (TKE) can be well described by "1/3" power law relationships under stable, neutral, and unstable conditions, with σ u /u * > σ v /u * > σ w /u* . Land use and land cover changes mainly impact dimensionless standard deviations of horizontal component fluctuations, but they have very little on those of the vertical component. The dimensionless standard devi- ations of wind components and dimensionless TKE are remarkably affected by different weather conditions; the deviations of horizontal wind component and dimensionless TKE present fog day > clear sky > overcast > cloudy; the trend of the vertical wind component is the reverse. The surface drag coefficient at a Nan- jing suburban measurement site during the observation period was obviously higher than at other reported plains and plateau areas, and was approximately one order larger in magnitude than the reported plains areas. Dimensionless standard deviation of temperature declined with increasing |z /L| with an approximate "-1/3" slope in unstable stratification and "-2/3" slope in stable stratification.     

Surface-Layer Characteristics in the Stable Boundary Layer with Strong and Weak Winds

An extensive meteorological dataset obtained from the plumevalidation experiment conducted by the Electric Power Research Institute (EPRI) atKincaid during 1980–1981 is analysed for studying the characteristic differences in thesurface-layer parameters in strong and weak wind stable conditions. The surface-layerparameters are computed using the similarity functions _m and _h proposed byBeljaars and Holtslag. The weak winds are characterized using the geostrophic wind speedas well as the wind speed at the 10-m level. The surface fluxes are found to be finitein weak wind conditions.Empirical formulations for the eddy diffusivities of momentum(K_M) and heat (K_H), and drag (C_D) and heat exchange (C_H) coefficients, as powerlaw functions of the bulk Richardson number (Ri_B), are proposed under both strong andweak wind conditions. Results are close to those based on observations taken from the IndianInstitute of Technology low wind diffusion experiment, the Land surface processes experiment,the Hanford diffusion experiment, the Cabauw field experiment and the Cooperative Atmospheric SurfaceExchange Study 1999 (CASES-99) experiment. In addition, the fluxes obtained fromthe proposed empirical relations are in good agreement with those based on similarity theory as wellas the turbulence measurements taken from the CASES-99 experiment.     

Momentum and sensible heat fluxes calculated by the dissipation technique during the Ocean Storms Project

High frequency measurements of wind velocity and temperature were made during the Ocean Storms Project in November 1987. The dissipation method was applied to the resulting time series in order to determine friction velocities,u _*, and the characteristic temperature scale,t _*, at 1-min intervals. These values were then compared to the 1-min mean wind speed and air-sea temperature differences to determine relationships for the drag coefficient (C _d ) and Stanton number (C _h ). The drag coefficient was comparable to other values reported in the literature, although the variation with wind speed was greater than reported by other investigators. An examination of the residual time series indicated a systematic low frequency periodicity of about 2-hr duration which was attributed to a fluctuating wind interacting with the surface gravity wave field. The temperature fluctuations did not produce meaningful estimates ofC _h for stable conditions. For unstable conditions, a value of 1.09±0.02×10^–3 was found.