Thermal plumes and turbulence spectra in the atmospheric boundary layer

Experimental observations on the temperature and wind fields above flat grassy terrain have been obtained with an instrumented 92-m tower during intervals of strong insolation about midday. The turbulence characteristics of the air confirm that free convection prevailed at heights between 16 and 48 m, with some tendency for departure at higher levels. The spectra of temperature and vertical velocity contain gaps at wave numbers in the range 0.01–0.025 m^–1. These are attributed to natural thermal plumes that act as sources of extra energy input to the Kolmogorov-Obukhov-Corrsin scheme of turbulence in or at the low-wave number limit of the inertial subrange. Modified forms of the K-O-C spectral laws for thermally unstable air are derived which agree with the observed spectra over the whole range of wave numbers examined, and which contain the spectral gap at wave numbers corresponding to the thermal plume diameters.     

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.     

Estimation of the Lagrangian Velocity Structure Function Constant C0 by Large-Eddy Simulation

The inertial subrange Kolmogorov constant C ₀, which determines the effective turbulent diffusion in velocity space, plays an important role in the Lagrangian modelling of pollutants. A wide range of values of the constant are found in the literature, most of them determined at low Reynolds number and/or under different assumptions. Here we estimate the constant C ₀ by tracking an ensemble of Lagrangian particles in a planetary boundary layer simulated with a large-eddy simulation model and analysing the Lagrangian velocity structure function in the inertial subrange. The advantage of this technique is that it easily allows Reynolds numbers to be achieved typical of convective turbulent flows. Our estimates of C ₀ is C ₀=4.3±0.3 consistent with values found in the literature     

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.     

On the existence of a relation between the Kolmogoroff and von Kármán constants

A model is described, in which the mean vertical wind profile and turbulence spectra at different heights are calculated for a turbulent boundary layer without thermal stratification. The model makes use of Heisenberg's formula for the transfer of turbulent energy and is based on the assumption of a constant shearing stress in that boundary layer. As a result, a logarithmic wind profile follows with 0.39 as the value of von Kármán's constant, which is — in this model — strongly related to the inertial subrange of the turbulent energy spectra and therefore to the Kolmogoroff constant.This paper is based on studies done by the author during a one-year visit to CSIRO Division of Meteorological Physics, Aspendale, Australia, and was presented at the AGARD Specialists Meeting on The Aerodynamics of Atmospheric Shear Flows sponsored by the Fluid Dynamics Panel at Munich, Germany, during 15–17 Sept. 1969.     

Further results from a laboratory model of the convective planetary boundary layer

The turbulence in a laboratory convective mixed layer is probed more extensively than in the preliminary study of Willis and Deardorff (1974), and results presented. Turbulence intensities, spectra and probability distributions using mixed-layer scaling compare favorably with similarly scaled field measurements not available or plentiful in 1974. However, the velocity spectra in the convection tank exhibit only a short inertial subrange due to the close proximity of the dissipation subrange to the energy-containing range.The turbulence budget suggests that the convergence of the vertical transport of pressure fluctuations is a rather important term.Results on the entrainment rate are also presented, using both mixed-layer scaling and local interfacial scaling.     

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.     

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.     

Measurements of internal waves and turbulence in two-dimensional stratified shear flows

A turbulent stratified shear flow is generated in a towing tank by towing a grid or a circular cylinder through a tank of stratified salt water. The internal waves and turbulence generated in these flows are visualized with shadowgraphs and measured with quartz-coated hot-film probes (up to four probes for velocity fluctuations) and single-electrode conductivity probes (up to four probes for salinity fluctuations) which are towed at the same speed as the obstacle. The velocity and salinity signals are recorded on magnetic tapes. A portion of these signals is processed directly-on-line with a digital computer. From these shadowgraphs and probe measurements, we observe that

1. Far downstream of the obstacle where the turbulence has already subsided, the stratified fluid always has a layered structure. This layered structure persists for a long time, and is a result of the convection of turbulently mixed layers by the mean flow. These results indicate that in the regions of a stably stratified atmosphere and ocean where the turbulence has subsided, one could often find layered structure.
2. There are spectral peaks and valleys in the measured velocity and salinity autospectra when the stratifications are sufficiently strong. Under certain conditions, these spectral peaks tend to lift up the spectral curves to show substantialf ^?5/3 subranges, although the turbulence Reynolds numbers are too low for the flows to have recognizable inertial subranges. This anomalousf ^?5/3 subrange demonstrates the pitfalls of using spectral measurements in thef ^?5/3 subrange to predict the turbulent energy dissipation rate through the Kolmogorov hypothesis.
3. A diagnostic method is developed for distinguishing internal waves from turbulence, utilizing their phase characteristics. The phase characteristics can be conveniently examined from the cospectra and quadrature spectra measurements of: (a), two vertically separated velocity probes; (b), two vertically separated density probes; and (c), a velocity probe and a density probe. This method is demonstrated to be useful in the laboratory and can be applied directly to atmospheric and oceanic measurements to distinguish internal waves from turbulence.
4. From the coherency measurements, it is found that the entire turbulent stratified wake is actually whipping up and down at a frequency corresponding to the Brunt-Väisälä frequency. This indicates that similar stratified shear flows in the atmosphere and in the ocean, such as the jet streams in the atmosphere and the Cromwell current in the ocean, may oscillate vertically, which in turn can induce horizontal oscillation and meandering.

     

Some Thoughts On Local Isotropy And The 4/3 Lateral To Longitudinal Velocity Spectrum Ratio

Applications of Kolmogorov's universal equilibrium hypothesis and the Taylor transform to velocity spectra derived from measurements within the low frequency portion of the atmospheric surface layer (ASL) inertial subrange are examined. The measured ratios of lateral to longitudinal velocity component spectra exhibit considerable scatter, but suggest convergence towards 1.0 rather than towards the 4/3 expected from theory. Shear and buoyancy introduce anisotropy to the inertial subrange, thereby contributing to the observed scatter. The apparent discrepancy between the 4/3 velocity component spectrum ratio expected from theory and the measurements could arise as a consequence of the processing used to produce spectra. These processing effects must be considered from the perspective of the propagating eddy. Spectral averaging used with sonic anemometer data is done over time periods that are large with respect to inertial subrange eddy correlation decay times. This averaging causes energy from larger scale eddies to appear as `local convection' that dominates the Taylor transform. Spectrum ratio convergence and cospectra approaching zero are necessary, but not sufficient, conditions for onset of local isotropy. Measurements of spectrum ratios and cospectra over the entire inertial subrange are needed to determine whether or not local isotropy might occur within the ASL.     

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     

Aircraft-Based Measurements Of Turbulence Structures In The Katabatic Flow Over Greenland

Turbulence structures in the katabatic flow in the stable boundary layer (SBL) over the ice sheet are studied for two case studies with high wind speeds during the aircraft-based experiment KABEG (Katabatic wind and boundary layer front experiment around Greenland) in the area of southern Greenland. The aircraft data allow the direct determination of turbulence structures in the katabatic flow. For the first time, this allows the study of the turbulence structure in the katabatic wind system over the whole boundary layer and over a horizontal scale of 80 km.The katabatic flow is associated with a low-level jet (LLJ), with maximum wind speeds up to 25 m s^-1. Turbulent kinetic energy (TKE) and the magnitude of the turbulent fluxes show a strong decrease below the LLJ. Sensible heat fluxes at the lowest level have values down to -25 W m^-2. Latent heat fluxes are small in general, but evaporation values of up to +13 W m^-2 are also measured. Turbulence spectra show a well-defined inertial subrange and a clear spectral gap around 250-m wavelength. While turbulence intensity decreases monotonously with height above the LLJ for the upper part of the slope, high spectral intensities are also present at upper levels close to the ice edge. Normalized fluxes and variances generally follow power-law profiles in the SBL.Terms of the TKE budget are computed from the aircraft data. The TKE destruction by the negative buoyancy is found to be very small, and the dissipation rate exceeds the dynamical production.     

Turbulence spectra in the near-neutral surface layer over the Loess Plateau in China

We present the power spectra of wind velocity and the cospectra of momentum and heat fluxes observed for different wind directions over flat terrain and a large valley on the Loess Plateau. The power spectra of longitudinal (u) and lateral (v) wind speeds satisfy the −5/3 power law in the inertial subrange, but do not vary as observed in previous studies within the low frequency range. The u spectrum measured at 32 m height for flow from the valley shows a power deficit at intermediate frequencies, while the v spectrum at 32 m downwind of the valley reaches another peak in the low frequency range at the same frequency as the u spectrum. The corresponding peak wavelength is consistent with the observed length scale of the convective outer layer at the site. The v spectrum for flat terrain shows a spectral gap at mid frequencies while obeying inner layer scaling in its inertial subrange, suggesting two sources of turbulence in the surface layer. All the spectra and cospectra from the valley direction show a height dependency over the three levels.     

A group-kinetic theory with a closure by memory loss for modeling turbulence in the atmosphere and the oceans

Summary The principle of the group-kinetic method is elucidated. This method of renormalization serves as the basis for analyzing the spectral structure of turbulence. The spectral distributions include the Kolmogoroff lawk ^–5/3 for isotropic turbulence, the power lawk ^–1 for shear turbulence, the spectrum for stratified turbulence not in the power law form, the power lawk ^–3 for two-dimensional geostrophic turbulence, and the power lawsk ^–3,k ^–2 andk ^–5 for two-dimensional Rossby wave turbulence with uniform and differential rotations. We discuss a spectrum-dependent modeling in reference to the problems of the universal functions and parameters in the similarity theories for the atmospheric surface layer and the planetary boundary layer. A renormalization-based modeling of atmospheric turbulence is proposed.     

Temperature and humidity spectra in the atmospheric surface layer

Temperature and humidity spectra have been measured at 3 and 12 m above the ground, together with profiles of wind, temperature and humidity, and flux measurements. Both temperature and humidity spectra appear to follow Monin-Obukov similarity as well as Kolmogorov's prediction for the inertial subrange. The standard deviations of temperature and humidity fluctuations support Monin-Obukov similarity and the predictions of local free convection. The spectral constants for the inertial subrange have been estimated as 0.8 for temperature and 0.6 for humidity.     

A general hydrodynamic theory for mixed-phase microphysics. Part I: drag and fall speed of hydrometeors

A general parameterization for solid and liquid hydrometeors is presented. hydrometeors basically are viewed as porous spheroids with the following variable parameters: diameter, axial ratio, mass, and porosity. Based on this parameterization a functional dependence on the Reynolds number of the drag of hydrometeors is derived, which is based on boundary layer theory. The basic form of this functional dependence is consistent with viscous theory and the inertial drag at low Reynolds numbers is predicted with good accuracy by matching the results from the boundary layer theory with Oseen's theory of creeping motion. Based on this solution a general semi-empirical expression for the Reynolds number and fall speed of particles is found. The results from the present theory are in remarkable agreement with experiments: The errors generally are < 5–10% for a wide variety of hydrometeors in the range of Reynolds numbers 0<N_Re<5×10⁵, including columnar and variously branched planar ice crystals, rimed and unrimed aggregates, lump, conical, and hexagonal graupel, hail, and rain drops. The present parameterisation aims far beyond the limits of the conventional methods since it is suitable for mixed-phase models of the microphysics of precipitation with continuously varying particle mass and shape characteristics and including processes such as depositional growth of ice crystals under varying environmental conditions, collisional growth of particles, and melting.     

Reynolds-Number Dependence of Gas Dispersion Over a Wavy Wall

Direct numerical simulations of an Ekman layer are performed to study flow evolution during the response of an initially neutral boundary layer to stable stratification. The Obukhov length, L, is varied among cases by imposing a range of stable buoyancy fluxes at the surface to mimic ground cooling. The imposition of constant surface buoyancy flux , i.e. constant-flux stability, leads to a buoyancy difference between the ground and background that tends to increase with time, unlike the constant-temperature stability case where a constant surface temperature is imposed. The initial collapse of turbulence in the surface layer owing to surface cooling that occurs over a time scale proportional to \(L/u_*\), where \(u_*\) is the friction velocity, is followed by turbulence recovery. The flow accelerates, and a “low-level jet” (LLJ) with inertial oscillations forms during the turbulence collapse. Turbulence statistics and budgets are examined to understand the recovery of turbulence. Vertical turbulence exchange, primarily by pressure transport, is found to initiate fluctuations in the surface layer and there is rebirth of turbulence through enhanced turbulence production as the LLJ shear increases. The turbulence recovery is not monotonic and exhibits temporal intermittency with several collapse/rebirth episodes. The boundary layer adjusts to an increase in the surface buoyancy flux by increased super-geostrophic velocity and surface stress such that the Obukhov length becomes similar among the cases and sufficiently large to allow fluctuations with sustained momentum and heat fluxes. The eventual state of fluctuations, achieved after about two inertial periods (\(ft \approx 4\pi \)), corresponds to global intermittency with turbulent patches in an otherwise quiescent background. Our simplified configuration is sufficient to identify turbulence collapse and rebirth, global and temporal intermittency, as well as formation of low-level jets, as in observations of the stratified atmospheric boundary layer.     

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.     

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.     

Local isotropy and anisotropy in the sheared and heated atmospheric surface layer

Longitudinal velocity and temperature measurements above a uniform dry lakebed were used to investigate sources of eddy-motion anisotropy within the inertial subrange. Rather than simply test the adequacy of locally isotropic relations, we investigated directly the sources of anisotropy. These sources, in a daytime desert-like climate, include: (1) direct interaction between the large-scale and small-scale eddy motion, and (2) thermal effects on the small-scale eddy motion. In order to explore these two anisotropy sources, we developed statistical measures that are sensitive to such interactions. It was found that the large-scale/small-scale interaction was significant in the inertial subrange up to 3 decades below the production scale, thus reducing the validity of the local isotropy assumption. The anisotropy generated by thermal effects was also significant and comparable in magnitude to the former anisotropy source. However, this thermal anisotropy was opposite in sign and tended to counteract the anisotropy generated by the large-scale/smallscale interaction. The thermal anisotropy was attributed to organized ramp-like patterns in the temperature measurements. The impact of this anisotropy cancellation on the dynamics of inertial subrange eddy motion was also considered. For that purpose, the Kolmogorov-Obukhov structure function equation, as derived from the Navier-Stokes equations for locally isotropic turbulence, was employed. The Kolmogorov-Obukhov structure function equation in conjunction with Obukhov's constant skewness closure hypothesis reproduced the measured second- and third-order structure functions. Obukhov's constant skewness closure scheme, which is also based on the local isotropy assumption, was verified and was found to be in good agreement with the measurements. The accepted 0.4 constant skewness value derived from grid turbulence experiments overestimated our measurements. A suggested 0.26 constant skewness value, which we derived from Kolmogorov's constant, was found to be adequate.