A barotropic model of the Ekman planetary boundary layer based on the geostrophic momentum approximation

A model of a stationary planetary boundary layer is proposed based on the equation of motion with the advective term retained. The latter is modeled by means of the so-called geostrophic momentum approximation in two versions — original and modified. New expressions for the vertical velocity W at the top of the boundary layer are derived and analyzed. They underestimate W compared to the classical expression.     

Turbulence in an unstable surface layer over suburban terrain

Turbulence spectra and integral statistics measured in unstable conditions over a suburban surface are presented. The stability dependence of the integral statistics is shown to be consistent with surface layer scaling, with adiabatic limits near those over much smoother surfaces. The spectra are computed over a wider range of non-dimensional frequency than previously reported for this type of surface, and show clearly the low-frequency roll-off. The horizontal components show three distinct spectral regions as elucidated by Kaimal (1978). Due to large uncertainties in the spectral amplitudes, very little systematic dependence on the Monin-Obukhov stability parameter could be detected over a wide range of unstable conditions.     

Analytical solutions for the Ekman layer

The PBL equation that governs the transition from the constant-stress surface layer to the geostrophic wind in a neutrally stratified atmosphere for which the eddy viscosityK(z) is assumed to vary smoothly from the surface-layer value U _*z (0.4,U _*=friction velocity,z=elevation) to the geostrophic asymptoteK _GU _*d forzd is solved through an expansion in fd/U _*1 (f=Coriolis parameter). The resulting solution is separated into Ekman's constant-K solution an inner component that reduces to the classical logarithmic form forzd and isO() relative to the Ekman component forzd. The approximationKU _*d is supported by the solution of Nee and Kovasznay's phenomenological transport equation forK(z), which yieldsK–U _*d exp(–z/d), where is an empirical constant for which observation implies, 1. The parametersA andB in Kazanskii and Monin's similarity relation forG/U _* (G=geostrophic velocity) are determined as functions of . The predicted values ofG/U _* and the turning angle are in agreement with the observed values for the Leipzig wind profile. The predicted value ofB based on the assumption of asymptotically constantK is 4.5, while that based on the Nee-Kovasznay model is 5.1; these compare with the observed value of 4.7 for the Leipzig profile. A thermal wind correction, an asymptotic solution for arbitraryK(z) and 1, and an exact (unrestricted ) solution forK(z)=U _*d[1–exp(–z/d)] are developed in appendices.     

The Ekman boundary layer over orography: An analysis of vertical motion

A model of the planetary boundary layer is used to determine the field of vertical motion over large-scale orography. This model represents Ekman boundary-layer dynamics modified by the inclusion of accelerations of the geostrophic wind under the geostrophic momentum approximation. The orography is represented by a circular mountain. The inviscid solution is provided by the sum of a constant translation and a steady, uniform potential vorticity, anticyclonic vortex. The boundary-layer solution vanishes on the mountain, but is matched to the inviscid solution as the top of the boundary layer is approached. The vertical velocity field at the top of the boundary layer is determined by integration of the continuity equation. The field of motion is largely determined by descent from above into the anticyclonic circulation, as in the classical Ekman model. Contributions that arise from the inclusion of accelerations are associated with boundary-layer advection and ageostrophic divergence that produce vorticity tendencies. Finally, the boundary-layer vertical motion is shown to be comparable in magnitude to the vertical motion forced by inviscid flow over the orography, although the distributions of each are significantly different. Effects of mountain asymmetry and a changing pressure field, that can be treated more fully by numerical model simulations, are not considered in the present study.On leave at the University of Colorado, 1990.     

Wave drag in the planetary boundary layer over complex terrain

The concepts of mountain-induced wave drag are applied to the smaller scale problem of the boundary layer over complex terrain. It is found that the Reynolds stress and surface drag caused by surface-generated waves can be at least as large as those conventionally associated with turbulence. Conditions in which wave effects are important are identified.ATDD Contribution No. 88/5.     

Investigation of the planetary boundary layer height variations over complex terrain

The effects of sea-breeze interactions with synoptic forcing on the PBL height over complex terrain are investigated through the use of a 3-D mesoscale numerical model. Two of the results are as follows. First, steep PBL height gradients—order of 1500 m over a grid interval of 10 km — are associated with the sea-breeze front and are enhanced by the topography. Second, a significant horizontal shift in the maximum PBL height relative to the mountains, is induced by a corresponding displacement of the thermal ridge due to the mountains, in the presence of large scale flow.     

Virtual heat entrainment in the mixed layer over very rough terrain

Inversion fluxes of virtual heat were computed for seven clear days over the Pre-Alpine region in Switzerland with profile data from a sequence of radio soundings. Several entrainment models based on the turbulent kinetic energy equation were tested with the data. It was found that the relatively simple equation first proposed by Tennekes (1973) which contains both a convective and a mechanical term for the entrainment does as well as the more complicated parameterizations. In addition, the effect of water vapor on the magnitude of the buoyancy fluxes at the surface and at the inversion was observed to be important since the Bowen ratio normally ranged between 0.1 and 0.2.Now at the Hydrology Laboratory of the Agricultural Research Service, U.S. Department of Agriculture, Beltsville, Maryland 20705, U.S.A.     

The perturbed structure of the neutral atmospheric boundary layer over irregular terrain

The differential equations for first-order (linear) response of the planetary boundary layer are formulated for flow over periodic terrain, for variations in both surface roughness and terrain elevation. A simple second-order closure model of the turbulence is used, and Coriolis forces are neglected. Flow over a periodic terrain produces corresponding periodic structure in all meteorological fields above the surface. The periodic structure consists of two components. The first is very nearly evanescent with height. It corresponds to the motion that would be observed were the atmosphere inviscid. The second component, introduced by turbulent viscosity, exhibits a phase variation with height in addition to a decay in amplitude. W.K.B. [Wentzel-Kramers-Brillouin] approximations for the two components are developed, and the coupling between the components is discussed. The formulation for calculating solutions by numerical integration is developed, including specification of appropriate boundary conditions. Calculations are presented in a companion paper.Wave Propagation Laboratory.Environmental Science Group.     

A case study of the nocturnal boundary layer over a complex terrain

A case study of the structure of the nocturnal boundary layer (NBL) over complex terrain is presented. Observations were made during the third night of Project STABLE (Weber and Kurzeja, 1991), whose main goal was to study turbulence and diffusion over the complex terrain of the Savannah River Site (SRS) near Augusta, Georgia.The passage of a mesoscale phenomenon, defined as a turbulent meso-flow (TMF) with an explanation of the nomenclature used, and a composite structure of the lowest few hundred meters over complex terrain are presented. The spatial extent of the TMF was at least 30–50 km, but the forcing is not well understood. The TMF occurred without the presence of a synoptic-scale cold front, under clear conditions, and with no discernible discontinuity in a microbarograph pressure trace. The structure of the NBL over the complex terrain at SRS differed from the expected homogeneous terrain NBL. The vertical structure exhibited dual low level wind maxima, dual inversions, and a persistent elevated turbulent layer.The persistent elevated turbulent layer, with a spatial extent of at least 30 km, was observed for the entire night. The persistent adiabatic layer may have resulted from turbulence induced by shear instability.     

The perturbed structure of the neutral atmospheric boundary layer over irregular terrain

The first-order (linear) response of the planetary boundary layer is calculated for flow over periodic terrain, for variations in both surface roughness and terrain elevation. Calculations are made for horizontal wavenumbers varying from 10^–4m^–1 to 3 × 10^–3m^–1. A simple second-order closure model of the turbulence is used, and Coriolis and buoyancy forces are neglected. As expected, flow over a periodic terrain produces corresponding periodic structure in all meteorological fields above the surface. The periodic structure consists of two components. The first is very nearly evanescent with height, showing little vertical structure. It corresponds to the motion that would be observed were the atmosphere inviscid. The second component, introduced by turbulent viscosity, exhibits considerable vertical structure, with vertical wavelengths the order of 100 m, and thus could be responsible for the layering often seen on acoustic sounder observations of the atmospheric boundary layer.Wave Propagation Laboratory.Environmental Science Group.     

Dynamics of nonlinear baroclinic Ekman boundary layer

By the geostrophic momentum approximation, the wind structure and vertical motion within the non-linear baroclinic Ekman layer matching with the surface layer are determined. A comparison of the Ekman solution with the classical one is made. It is demonstrated that the contributions of baroclinity, stratification and nonlinear effects to the wind profile within the layer are all of definite importance.     

The depth of the internal boundary layer over an urban area under sea-breeze conditions

Field data are analyzed in order to study the development of the Thermal Internal Boundary Layer (TIBL) under sea breeze conditions. The measurements were carried out by the National Observatory of Athens (NOA) during ATHens Internal Boundary Layer Experiment (ATHIBLEX) in summer 1989 and 1990.Several formulations found in the literature are tested against the measurements in order to investigate whether they are capable of predicting the depth of the TIBL. It is found that a slab model including mechanical production of turbulence gives overall good agreement with the measurements.Finally, the concept of local equilibrium is used to explain the discrepancies found between small-and meso-scale observations and models; a formula is proposed which is intended for use over a wide range of downwind fetches.     

Simulation of three-dimensional wind flow over complex terrain in the atmospheric boundary layer

A three-dimensional model for wind prediction over rough terrain has been developed for practical use. It is a compromise between hydrodynamic and objective wind models. The proposed model includes: (1) a statistical model to predict the wind velocity and potential temperature at anemometer height at observing stations, (2) the drainage wind model expressed by Prandtl's analytic solution for the slope wind, (3) the Businger-Dyer surface-layer formulation which considers the surface energy budget and (4) the model for three-dimensional boundary-layer solutions to the stationary flow. In this model, mass consistency is guaranteed by using flow fields that satisfy the continuity equation. Model predictions show good agreement with the observations.     

On the extension of the wind profile over homogeneous terrain beyond the surface boundary layer

Analysis of profiles of meteorological measurements from a 160 m high mast at the National Test Site for wind turbines at Høvsøre (Denmark) and at a 250 m high TV tower at Hamburg (Germany) shows that the wind profile based on surface-layer theory and Monin-Obukhov scaling is valid up to a height of 50–80 m. At higher levels deviations from the measurements progressively occur. For applied use an extension to the wind profile in the surface layer is formulated for the entire boundary layer, with emphasis on the lowest 200–300 m and considering only wind speeds above 3 m s^?1 at 10 m height. The friction velocity is taken to decrease linearly through the boundary layer. The wind profile length scale is composed of three component length scales. In the surface layer the first length scale is taken to increase linearly with height with a stability correction following Monin-Obukhov similarity. Above the surface layer the second length scale (L _MBL ) becomes independent of height but not of stability, and at the top of the boundary layer the third length scale is assumed to be negligible. A simple model for the combined length scale that controls the wind profile and its stability dependence is formulated by inverse summation. Based on these assumptions the wind profile for the entire boundary layer is derived. A parameterization of L _MBL is formulated using the geostrophic drag law, which relates friction velocity and geostrophic wind. The empirical parameterization of the resistance law functions A and B in the geostrophic drag law is uncertain, making it impractical. Therefore an expression for the length scale, L _MBL , for applied use is suggested, based on measurements from the two sites.     

Structure of mean winds and turbulence in the planetary boundary layer over rural terrain

A detailed analysis has been carried out of the temporal and spatial structure of mean winds and turbulence in the neutrally-stable planetary boundary layer over typically rural terrain. The data were obtained from a horizontal array of tower-mounted propeller anemometers (z = 11 m) during a five-hour period for which the mean wind direction was virtually perpendicular to the main span of the array. Various turbulence characteristics have been obtained for all three components of velocity and have been compared with idealized models for such a flow and with some of the other available atmospheric results.Considerable tower-to-tower and block-to-block variability has been observed in many of the measured results, particularly in those for the horizontal-component integral scales. Surface shear stress, roughness length and turbulence intensities were in good agreement with expected values for such a site. Power spectra for all components displayed significantly more energy at middle and lower frequencies than that observed by Kaimal et al. (1972) over flat, relatively featureless terrain. This is felt to be a result of the generally rougher gross features of the terrain in the present case and has led to the development of a modified version of the Kaimal-spectral model which fits the observed data better than either the original Kaimal model or the von Kármán model. It is suggested that it may in future be possible to represent power spectra over a wide range of terrain types by using such a modified spectral model.Integral scales of turbulence were calculated by three different techniques and in most cases displayed a strong dependence on the technique used. Averaged values of scale showed reasonable agreement with most of the available atmospheric data and with the values suggested by ESDU (1975). The anticipated elongation of turbulent eddies in the longitudinal direction was confirmed for all three velocity components, although it was found to be not as large as some other observations.     

Maintenance of oscillations in a turbulent Ekman layer

The turbulent Ekman boundary layer is modelled by prescribing a particular behavior for the eddy viscosity and thereby making it possible to calculate velocity profiles. This flow is then examined with respect to stability. Comparisons for this system are made with the more substantial documented features of a conventional flat-plate turbulent boundary layer without rotation and the accompanying stability investigations that have been made. Unlike the results known for the nonrotating layer, it appears that viscous oscillations are possible when rotation is present. General arguments are made in terms of the physics to substantiate this finding and one conjecture is that the modelled system does not represent a fully developed or equilibrium turbulent flow.New address: Naval Research Laboratory, Washington, D.C., U.S.A.     

On the determination of the height of the Ekman boundary layer

The heighth of the Ekman turbulent boundary layer determined by the momentum flux profile is estimated with the aid of considerations of similarity and an analysis of the dynamic equations. Asymptotic formulae have been obtained showing that, with increasing instability,h increases as ¦¦^1/2 (where is the non-dimensional stratification parameter); with increasing stability, on the other hand,h decreases as ^–1/2. For comparison, a simple estimate of the boundary-layer heighth _u determined by the velocity profile is given. As is shown, in unstable stratification,h _u behaves asymptotically as ¦¦^–1, i.e., in a manner entirely different from that ofh .     

Ekman动量近似下中间边界层模式中的风场结构

发展了一个准三维的、中等复杂的边界层动力学模式，该模式包含了EKman动量近似下的惯性加速度和Blackadar的非线性湍流粘性系数，它进一步改进了Tan和Wu(1993）提出的边界层理论模型。该模型在数值计算复杂性上与经典Ekman模式相类似，但由于包含了Ekman动量近似下的惯性项，使得该模式比传统Ekman模式更近于实际过程。中详细地比较了该模式与其他简化边界层模式在动力学上的差异，结果表明：在经典的Ekman模式中，由于忽略了流动的惯性项作用，导致在气旋性切变气流（反气旋性切变气流）中风速和边界层顶部的垂直速度的高估（低估），而在半地转边界层模式中，由于高估了流动惯性项的作用，结果与经典Ekman模式相反。同样，该模式可以应用于斜压边界层，对于Ekman动量下的斜压边界层风场同时具有经典斜压边界层和Ekman动量近似边界层的特征。     

Wind profile constants in a neutral atmospheric boundary layer over complex terrain

The roughness height z ₀ and the zero-plane displacement height d ₀ were determined for a region of complex terrain in the Pre-Alps of Switzerland. This region is characterized by hills of the order of 100 m above the valley elevations, and by distances between ridges of the order of 1 km; it lies about 20 to 30 km north from the Alps. The experimental data were obtained from radiosonde observations under near neutral conditions. The analysis was based on the assumption of a logarithmic profile for the mean horizontal wind existing over one half of the boundary layer. The resulting (z ₀/h) and (d ₀/h) (where h is the mean height of the obstacles) were found to be in reasonable agreement with available relationships in terms of placement density and shape factor of the obstacles, which were obtained in previous experiments with h-scales 2 to 4 orders of magnitude smaller than the present ones.