A Lagrangian Decorrelation Time Scale in the Convective Boundary Layer

A new method for deriving the Lagrangian decorrelation time scales for inhomogeneous turbulence is described. The expression for the time scales here derived for the convective boundary layer is compared to those estimated by Hanna during the Phoenix experiment. Then the values of C₀, the Lagrangian velocity structure function constant, and of B_i, the Lagrangian velocity spectrum constant, were evaluated from the Eulerian velocity spectra and from the Lagrangian time scales derived, under unstable conditions, from Taylor's statistical diffusion theory. The numerical coefficient of the lateral and vertical Lagrangian spectra in the inertial subrange was found equal to 0.21, in good agreement with previous experimental estimates.     

Comments on the 'Universality of the Lagrangian Velocity Structure Function Constant (C0) ACROSS DIFFERENT KINDS OF TURBULENCE’

Recently Du ( Boundary-Layer Meteorology 83, 207–219, 1997) estimated the value of the Lagrangian velocity structure constant, C0, in the inertial subrange by comparing experimental diffusion data and simulation results obtained with the one-dimensional form of Thomson's model ( J. Fluid Mech. 180, 529–556, 1987). Du reported that for several different flows (grid turbulence, a wind-tunnel boundary layer and the atmospheric surface layer under neutral stratification) the value of C0 is 3.0±0.5. Here, it is shown that optimal model agreement with experimental diffusion data for the wind-tunnel boundary layer is, in fact, obtained when C0=5.0 ± 0.5. It is also shown that accounting for the skewness of velocity statistics and finite Reynolds number effects does not significantly change this estimate for the value of C0. It is suggested that one-dimensional Lagrangian stochastic models are inconsistent with the supposed universality of C0.     

UNIVERSALITY OF THE LAGRANGIAN VELOCITY STRUCTURE FUNCTION CONSTANT (C0) ACROSS DIFFERENT KINDS OF TURBULENCE

In this paper, we evaluate the Lagrangian velocity structure function constant, C0, in the inertial subrange by comparing experimental diffusion data and simulation results obtained with applicable Lagrangian stochastic models. We find in several different flows (grid turbulence, laboratory boundary-layer flow and the atmospheric surface layer under neutral stratification) the value for C0 is 3.0 ± 0.5. We also identify the reasons responsible for earlier studies having not reached the present result.     

Drag coefficients and turbulence spectra within three boreal forest canopies

Atmospheric turbulence was measured within a black spruce forest, a jack pine forest, and a trembling aspen forest, located in southeastern Manitoba, Canada. Drag coefficients (C _d ) varied little with height within the pine and aspen canopies, but showed some height dependence within the dense spruce canopy. A constant C _d of 0.15, with the measured momentum flux and velocity profiles, gave good estimates of leaf-area-index (LAI) profiles for the pine and aspen canopies, but underestimated LAI for the spruce canopy.Velocity spectra were scaled using the Eulerian integral time scales and showed a substantial inertial subrange above the canopies. In the bottom part of the canopies, the streamwise and cross-stream spectra showed rapid energy loss whereas the vertical spectra showed an apparent energy gain, in the region where the inertial subrange is expected. The temperature spectra showed an inertial subrange with the expected -2/3 slope at all heights. Cospectra of momentum and heat flux had slopes of about -1 in much of the inertial subrange. Possible mechanisms to explain some of the spectral features are discussed.     

Estimation Of The Lagrangian Structure Function Constant C0 From Surface-Layer Wind Data

Eulerian turbulence observations, madein the surface layer under unstable conditions (z/L > 0),by a sonic anemometer were used to estimatethe Lagrangian structure function constant _O. Twomethods were considered. The first one makes use of arelationship, widely used in the Lagrangian stochasticdispersion models, relating _O to the turbulent kineticenergy dissipation rate , wind velocity variance andLagrangian decorrelation time. The second one employsa novel equation, connecting _O to the constant of thesecond-order Eulerian structure function. Beforeestimating _O, the measurements were processed in orderto discard non-stationary cases at least to a firstapproximation and cases in which local isotropy couldnot be assumed. The dissipation was estimated eitherfrom the best fit of the energy spectrum in theinertial subrange or from the best fit of the third-orderlongitudinal Eulerian structure function. Thefirst method was preferred and applied to the subsequentpart of the analysis. Both methods predict thepartitioning of _O in different spatial components as aconsequence of the directional dependence of theEulerian correlation functions due to the isotropy.The _O values computed by both methods are presented anddiscussed. In conclusion, both methods providerealistic estimates of _O that compare well withprevious estimations reported in the literature, evenif a preference is to be attributed to the second method.     

Spectral scaling in a high reynolds number laboratory boundary layer

Spectra and co-spectra of the streamwise (u) and normal or vertical (w) velocity fluctuations have been measured in the inner region of a large Reynolds number laboratory boundary layer over a rough wall. There is reasonable evidence of ak ₁ ^–1 range in theu spectrum (wherek ₁ is the streamwise wavenumber). Such a range results from an overlap between a spectral region dominated by largescale, inactive motion, which scales on the boundary-layer thickness, and a region dominated by smaller-scale, active motion which scales on the distance from the wall. Spectra ofw, anduw cospectra, scale in a manner consistent with the dominance by active motion. The present spectral data do not support local isotropy over the inertial subrange. A comparison between measuredw spectra and calculations based on isotropy indicates that the inertial subrange anisotropy is only slightly affected by the magnitude of the non-dimensional mean shear.     

Correlated humidity and temperature measurements in the urban atmospheric boundary layer

Summary To investigate the effect of atmospheric turbulence on microwave communication links, temperature and water vapor pressure have been measured and radio refractivity has been computed, during different meteorological conditions, in the atmospheric boundary layer of an urban site. The cospectra between temperature (T) and water vapor pressure (e) have been found to be either negative over the whole range of frequencies, or the low-frequency end of the cospectrum is of opposite sign relative to higher frequency end. In both cases cospectra follow a–5/3 law in the inertial subrange, in agreement with the theoretical predictions. The coherence spectra clearly show that the temperature and humidity fluctuations are highly coherent within the inertial subrange under both convective and stable conditions. The relative contribution ofC _T ² ,C _eT andC _e ² to the real refractive index structure parameterC _n ² is examined and discussed.With 4 Figures     

The streamwise Kolmogoroff constant

Observations over grassland of the turbulent kinetic energy in a band of frequencies in the inertial subrange of the spectrum of the streamwise wind component are related to the stress indicated by the wind profiles. The object is to determine the effective Kolmogoroff constant, _UB , which accords with the assumption of balance between turbulent energy production and dissipation. The mean from 60 half-hour runs made under stability conditions ranging from neutral to moderate instability is _UB = 0.62.This result is compared with those from other studies; four over grassland and six over the sea or a lake. There is no significant difference between the means of the land and sea values, but the latter are more scattered, partly because of difficulties in securing suitable exposure of the instruments for the stress measurements. The mean value from all ten sets of observations is _UB = 0.59 ± 0.025. So the dissipation method should be capable of giving the drag coefficient of the sea in strong winds with an uncertainty of no more than about 10%.     

Statistics of atmospheric turbulence within a natural black spruce forest canopy

Turbulence statistics were measured in a natural black-spruce forest canopy in southeastern Manitoba, Canada. Sonic anemometers were used to measure time series of vertical wind velocity (w), and cup anemometers to measure horizontal wind speed (s), above the canopy and at seven different heights within the canopy. Vertical profiles were measured during 25 runs on eight different days when conditions above the canopy were near-neutral.Profiles of s and of the standard deviation ( _w ) of w show relatively little scatter and suggest that, for this canopy and these stability conditions, profiles can be predicted from simple measurements made above the canopy. Within the canopy, a negative skewness and a high kurtosis of the w-frequency distributions indicate asymmetry and the persistence of large, high-velocity eddies. The Eulerian time scale is only a weak function of height within the canopy.Although w-power spectra above the canopy are similar to those in the free atmosphere, we did not observe an extensive inertial subrange in the spectra within the canopy. Also, a second peak is present that is especially prominent near the ground. The lack of the inertial subrange is likely caused by the presence of sources and sinks for turbulent kinetic energy within our canopy. The secondary spectral peak is probably generated by wake turbulence caused by form drag on the wide, horizontal spruce branches.     

Instability in Lagrangian stochastic trajectory models,and a method for its cure

A number of authors have reported the problem of unrealistic velocities (“rogue trajectories”) when computing the paths of particles in a turbulent flow using modern Lagrangian stochastic (LS) models, and have resorted to ad hoc interventions. We suggest that this problem stems from two causes: (1) unstable modes that are intrinsic to the dynamical system constituted by the generalized Langevin equations, and whose actual triggering (expression) is conditional on the fields of the mean velocity and Reynolds stress tensor and is liable to occur in complex, disturbed flows (which, if computational, will also be imperfect and discontinuous); and, (2) the “stiffness” of the generalized Langevin equations, which implies that the simple stochastic generalization of the Euler scheme usually used to integrate these equations is not sufficient to keep round-off errors under control. These two causes are connected, with the first cause (dynamical instability) exacerbating the second (numerical instability); removing the first cause does not necessarily correct the second, and vice versa. To overcome this problem, we introduce a fractional-step integration scheme that splits the velocity increment into contributions that are linear (U _i ) and nonlinear (U _i U _j ) in the Lagrangian velocity fluctuation vector U, the nonlinear contribution being further split into its diagonal and off-diagonal parts. The linear contribution and the diagonal part of the nonlinear contribution to the solution are computed exactly (analytically) over a finite timestep Δt, allowing any dynamical instabilities in the system to be diagnosed and removed, and circumventing the numerical instability that can potentially result in integrating stiff equations using the commonly applied explicit Euler scheme. We contrast results using this and the primitive Euler integration scheme for computed trajectories in a drastically inhomogeneous urban canopy flow.     

A Two-Dimensional Lagrangian Stochastic Dispersion Model for Convective Boundary Layers with Wind Shear

Rotach, Gryning and Tassone constructed a two-dimensional Lagrangian stochastic model to describe the dispersion of passive tracers in turbulent boundary layers with stabilities ranging from ideally-neutral (w_* = 0) to fully-convective (u_* = 0). They found that the value of the Kolmogorov constant, C₀, as determined by optimizing model agreement with the measured spread of passive tracers, was dependent upon stability. Here, it is shown that the non-uniqueness, associated with satisfaction of the well-mixed condition, can be exploited to construct an alternative version of the model of Rotach et al. for which C₀ = 3 is universally applicable over the entire range of stabilities under consideration. This alternative model is shown to be in very close agreement with predictions, obtained in large-eddy simulations, for the dispersion of passive tracers in turbulent boundary layers with stabilities ranging from ideally-neutral to fully-convective.     

Turbulent Pressure Statistics in the Atmospheric Boundary Layer from Large-Eddy Simulation

We use large-eddy simulation (LES) to study the turbulent pressure field in atmospheric boundary layers with free convection, forced convection, and stable stratification. We use the Poisson equation for pressure to represent the pressure field as the sum of mean-shear, turbulence–turbulence, subfilter-scale, Coriolis, and buoyancy contributions. We isolate these contributions and study them separately. We find that in the energy-containing range in the free-convection case the turbulence–turbulence pressure dominates over the entire boundary layer. That part dominates also up to midlayer in the forced-convection case; above that the mean-shear pressure dominates. In the stable case the mean-shear pressure dominates over the entire boundary layer.We find evidence of an inertial subrange in the pressure spectrum in the free and forced-convection cases; it is dominated by the turbulence–turbulence pressure and has a three-dimensional spectral constant of about 4.0. This agrees well with quasi-Gaussian predictions but is a factor of 2 less than recent results from direct numerical simulations at moderate Reynolds numbers. Measurements of the inertial subrange pressure spectral constant at high Reynolds numbers, which might now be possible, would be most useful.     

Time series analyses of concentration and wind fluctuations

Analyses of concentration fluctuation (C) spectra from boundary-layer smoke plume experiments at six separate locations show that the spectra from these experiments generally exhibit an inertial subrange at high frequencies with a slope of -5/3 and indicate peak energy at a time period of about 50 to 100 s. These periods of peak energy are a factor of two to five less than those for the peak of the wind speed fluctuation (u or v) spectra. A general spectral formula fits normalized spectra from the U.S. and Australia, where the frequency, n, is made dimensionless by multiplying by the plume dispersion parameter, _y , and dividing by the wind speed, u. Peak energy occurs at a dimensionless frequency of n _y/u equal to about 0.15. The Kolmogorov constant in the inertial subrange is estimated from a set of averaged spectra. Cross-spectra indicate little relation between concentration and wind fluctuations. However, most of the correlation that exists is due to periods larger than about 10 or 20 s.     

Turbulence mechanisms at an agricultural site

An extensive set of turbulence data from the 3- and 12-m heights taken over an agricultural site (Marsta, Sweden) are analyzed and compared with data from ideal sites.In unstable air, Monin-Obukhov similarity is found to be valid for the non-dimensional gradients of wind, _m , temperature, _h , and humidity, _e , for (only a few data), for _T /|T _*|,/ _E /|E _*| and for the non-dimensionalized inertial subrange spectra of temperature and humidity. Where comparison is possible, the unstable data also agree with those found in the Kansas study, with one remarkable exception, the inertial subrange constant of the temperature spectrum, ₁, being only 0.39, compared to the value 0.80 found at the Kansas site.On the stable side, most similarity predictions break down, with most of the data differing systematically from the corresponding Kansas results, the only exception being . The inertial subrange constants for temperature, ₁, and for humidity, ₁ are found to have the same values, 0.39 and 0.30, respectively, as they do on the unstable side. Remarkable similarity is found for the shape of the stable u- and - and e-spectra. In addition, this shape is found to be identical with that found in Kansas. The peak wavelength of the stable u-, and -spectra is found to be about four times larger than it is for the corresponding Kansas spectra. This is interpreted to be a result of the increased macro-roughness at the Marsta site as compared with that at the Kansas site. A possible explanation for the low ₁-value is discussed, suggesting that ₁ is not a universal constant, but instead dependent on the turbulent structure.     

Note on a Parametric Relation for Separating Flow over a Rough Hill

A parametrization method used to account for the effects of flow separation and wall roughness on the lower boundary condition for turbulent boundary layers is investigated against direct numerical simulation and laser Doppler anemometry data. The numerical simulation represents flow over a smooth, flat surface with a prescribed external adverse pressure gradient. The water-channel experiments cover flow over smooth and rough hills for two specified Reynolds numbers. Global optimization algorithms based on four different direct search methods are used to assess the parametrization function, C, in terms of local mean velocity profiles and the parametrization parameters u _* (friction velocity), ∂ _x p (local pressure gradient), z ₀ (effective roughness) and d (zero-plane displacement). The study investigates regions of attached and reversed flows, and forty-two velocity profiles are compared with the proposed expression for the function C, including two profiles that satisfy the solution of Stratford.     

Turbulent Kinetic Energy Dissipation in the Surface Layer

We estimated the turbulent kinetic energy (TKE) dissipation rate for thirty-two 1-h intervals of unstable stratification covering the stability range 0.12 ≤ −z/L ≤ 43 (z/L is the ratio of instrument height to the Obukhov length), by fitting Kolmogorov’s inertial subrange spectrum to streamwise spectra observed over a desert flat. Estimated values are compatible with the existence of local equilibrium, in that the TKE dissipation rate approximately equalled the sum of shear and buoyant production rates. Only in the neutral limit was the turbulent transport term in the TKE budget measured to be small.     

Similarity Scaling Of Surface-Released Smoke Plumes

Concentration fluctuation data from surface-layer released smokeplumes have been investigated with the purpose of finding suitable scaling parametersfor the corresponding two-particle, relative diffusion process.Dispersion properties have been measured at downwind ranges between 0.1 and 1 km from a continuous, neutrally buoyant ground level source. A combinationof SF₆ and chemical smoke (aerosols) was used as tracer. Instantaneous crosswind concentration profiles of high temporal (up to 55 Hz) and spatialresolution (down to 0.375 m) were obtained from aerosol-backscatter Lidar detectionin combination with simultaneous gas chromatograph (SF₆) reference measurements. The database includes detailed crosswind concentration fluctuation measurements. Each experiment, typically of 1/2-hour duration, containsplume mean and variance concentration profiles, intermittency profiles andexceedence and duration statistics. The diffusion experiments were accompanied by detailed in-situ micrometeorological mean and turbulence measurements. In this paper, a new distance-neighbour function for surface-released smoke plumes is proposed, accompanied by experimental evidence in its support. The new distance-neighbour function is found to scale with the surface-layer friction velocity,and not with the inertial subrange dissipation rate, over the range of distance-neighbour separations considered.     

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.     

Spectral Maxima In A Perturbed Stable Boundary Layer

Wind velocity data have been collected on Nansen Ice Sheet, Antarctica, close to the base of a steeply sloping glacier along which frequently flow katabatic winds. The aim of this study is to investigate how turbulent energy and momentum flux are perturbed by the flow interaction with topography and by the strong mechanical mixing produced by downslope flows. Spectral and cospectral analyses, performed on the wind velocity components, provide evidence that such a perturbation, at any stability, is restricted to frequencies lower than the inertial subrange. Longitudinal spectra display an energy increment, due to turbulence generated by topography and by mechanical forcing related to the katabatic wind structure. The energy, supplied by the topographic forcing, displaces the turbulent energy maximum toward lower frequencies. In near-neutral stratification the spectral maximum occurs at a reduced frequency, which seems to be consistent with the height of the steepest part of the slope, and seems to shift toward higher frequencies as a linear ,function of the local stability parameter,L_l. The parameterisation of the orographic perturbation by means of a similarity relationship allows us to scale u spectra in the same way as over uniform terrain. The scaled, perturbed spectra collapse onto a unique curve in the mid-frequency as well in the inertial subrange, while maxima are grouped in a cluster. Lateral and vertical velocity spectra exhibit shapes independent of stability, suggesting a topographic perturbation that is predominantly over stability effects.     

Comparison of Kansas Data with High-resolution Large-eddy Simulation Fields

The surface layer of an atmospheric boundary layer (ABL) is most accessible to field measurements and hence its ensemble-mean structure has been well established. The Kansas field measurements were the first detailed study of this layer, providing numerous benchmark statistical profiles for a wide range of stability states. Large-eddy simulation (LES), in contrast, is most suitable for studying the mixed layer of the ABL where the energy-containing range of the vertical velocity field is well resolved. In the surface layer, typical large-eddy simulations barely resolve the energy-containing vertical-velocity fields and hence do not provide sufficient data for a detailed analysis.We carried out a nested-mesh simulation of a moderately convective ABL (-z_i/L = 8) in which the lower 6% of the boundary layer had an effective grid resolution of 512³. We analyze the LES fields above the 6th vertical grid level (z = 23 m) where the vertical velocity field has a well formed inertial subrange, for a detailed comparison with the Kansas results. Various terms in the budgets of turbulent kinetic energy, temperature variance, Reynolds stress, temperature flux, and some higher order moments are compared. The agreement is generally quite good; however, we do observe certain discrepancies, particularly in the terms involving pressure fluctuations.