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.     

Monin–Obukhov Similarity Functions of the Structure Parameter of Temperature and Turbulent Kinetic Energy Dissipation Rate in the Stable Boundary Layer

The Monin–Obukhov similarity theory (MOST) functions f_ε and f_T, of the dissipation rate of turbulent kinetic energy (TKE). ε, and the structure parameter of temperature, C_T², were determined for the stable atmospheric surface layer using data gathered in the context of CASES-99. These data cover a relatively wide stability range, i.e. ζ=z/L of up to 10, where z is the height and L the Obukhov length. The best fits were given by f_ε = 0.8 + 2.5ζ and f_T= 4.7[ 1+1.6(ζ)^2/3], which differ somewhat from previously published functions. ε was obtained from spectra of the longitudinal wind velocity using a time series model (ARMA) method instead of the traditional Fourier transform. The neutral limit f_ε =0.8 implies that there is an imbalance between TKE production and dissipation in the simplified TKE budget equation. Similarly, we found a production-dissipation imbalance for the temperature fluctuation budget equation. Correcting for the production-dissipation imbalance, the ‘standard’ MOST functions for dimensionless wind speed and temperature gradients (φ_m and φ_m) were determined from f_ε and f_T and compared with the φ_m and φ_h formulations of Businger and others. We found good agreement with the Beljaars and Holtslag [J. Appl. Meteorol. 30, 327–341 (1991)] relations. Lastly, the flux and gradient Richardson numbers are discussed also in terms of f_ε and f_T.     

Monin–Obukhov Similarity Functions for the Structure Parameters of Temperature and Humidity

Monin–Obukhov similarity functions for the structure parameters of temperature and humidity are needed to derive surface heat and water vapour fluxes from scintillometer measurements and it is often assumed that the two functions are identical in the atmospheric surface layer. Nevertheless, this assumption has not yet been verified experimentally. This study investigates the dissimilarity between the turbulent transport of sensible heat and water vapour, with a specific focus on the difference between the Monin–Obukhov similarity functions for the structure parameters. Using two datasets collected over homogeneous surfaces where the surface sources of sensible heat and water vapour are well correlated, we observe that under stable and very unstable conditions, the two functions are similar. This similarity however breaks down under weakly unstable conditions; in that regime, the absolute values of the correlations between temperature and humidity are also observed to be low, most likely due to large-scale eddies that transport unsteadiness, advection or entrainment effects from the outer layer. We analyze and demonstrate how this reduction in the correlation leads to dissimilarity between the turbulent transport of these two scalars and the corresponding Monin–Obukhov similarity functions for their structure parameters. A model to derive sensible and latent heat fluxes from structure parameters without measuring the friction velocity is tested and found to work very well under moderately to strongly unstable conditions (−z/L > 0.5). Finally, we discuss the modelling of the cross-structure parameter over wet surfaces, which is crucial for correcting water vapour effects on optical scintillometer measurements and also for obtaining surface sensible and latent heat fluxes from the two-wavelength scintillometry.     

On the Kinetic Energy Budget of the Unstable Atmospheric Surface Layer

We present a new account of the kinetic energy budget within an unstable atmospheric surface layer (ASL) beneath a convective outer layer. It is based on the structural model of turbulence introduced by McNaughton (Boundary-Layer Meteorology, 112: 199–221, 2004). In this model the turbulence is described as a self-organizing system with a highly organized structure that resists change by instability. This system is driven from above, with both the mean motion and the large-scale convective motions of the outer layer creating shear across the surface layer. The outer convective motions thus modulate the turbulence processes in the surface layer, causing variable downwards fluxes of momentum and kinetic energy. The variable components of the momentum flux sum to zero, but the associated energy divergence is cumulative, increasing both the average kinetic energy of the turbulence in the surface layer and the rate at which that energy is dissipated. The tendency of buoyancy to preferentially enhance the vertical motions is opposed by pressure reaction forces, so pressure production, which is the work done against these reaction forces, exactly equals buoyant production of kinetic energy. The pressure potential energy that is produced is then redistributed throughout the layer through many conversions, back and forth, between pressure potential and kinetic energy with zero sums. These exchanges generally increase the kinetic energy of the turbulence, the rate at which turbulence transfers momentum and the rate at which it dissipates energy, but does not alter its overall structure. In this model the velocity scale for turbulent transport processes in the surface layer is (kzɛ)^1/3 rather than the friction velocity, u_*. Here k is the von Kármán constant, z is observation height, ɛ is the dissipation rate. The model agrees very well with published experimental results, and provides the foundation for the new similarity model of the unstable ASL, replacing the older Monin–Obukhov similarity theory, whose assumptions are no longer tenable.     

The Budget of Turbulent Kinetic Energy in the Urban Roughness Sublayer

Full-scale observations from two urban sites in Basel, Switzerland were analysed to identify the magnitude of different processes that create, relocate, and dissipate turbulent kinetic energy (TKE) in the urban atmosphere. Two towers equipped with a profile of six ultrasonic anemometers each sampled the flow in the urban roughness sublayer, i.e. from street canyon base up to roughly 2.5 times the mean building height. This observational study suggests a conceptual division of the urban roughness sublayer into three layers: (1) the layer above the highest roofs, where local buoyancy production and local shear production of TKE are counterbalanced by local viscous dissipation rate and scaled turbulence statistics are close to to surface-layer values; (2) the layer around mean building height with a distinct inflexional mean wind profile, a strong shear and wake production of TKE, a more efficient turbulent exchange of momentum, and a notable export of TKE by transport processes; (3) the lower street canyon with imported TKE by transport processes and negligible local production. Averaged integral velocity variances vary significantly with height in the urban roughness sublayer and reflect the driving processes that create or relocate TKE at a particular height. The observed profiles of the terms of the TKE budget and the velocity variances show many similarities to observations within and above vegetation canopies.     

Turbulent kinetic energy budgets from a large-eddy simulation of airflow above and within a forest canopy

The output of a large-eddy simulation was used to study the terms ofthe turbulent kinetic energy (TKE) budget for the air layers above andwithin a forest. The computation created a three-dimensional,time-dependent simulation of the airflow, in which the lowest third ofthe domain was occupied by drag elements and heat sources to representthe forest. Shear production was a principal source of TKE in theupper canopy, diminishing gradually above tree-top height and moresharply with depth in the canopy. The transfer of energy to subgridscales (dissipation) was the main sink in the upper part of the domainbut diminished rapidly with depth in the canopy. Removal ofresolved-scale TKE due to canopy drag was extremely important,occurring primarily in the upper half of the forest where the foliagedensity was large. Turbulent transport showed a loss at the canopytop and a gain within the canopy. These general features have beenfound elsewhere but uncertainty remains concerning the effects ofpressure transport. In the present work, pressure was calculateddirectly, allowing us to compute the pressure diffusion term. Wellabove the canopy, pressure transport was smaller than, and opposite insign to, the turbulent transport term. Near the canopy top andbelow, pressure transport acted in concert with turbulent transport toexport TKE from the region immediately above and within the uppercrown, and to provide turbulent energy for the lower parts of theforest. In combination, the transport terms accounted for over half ofthe TKE loss near the canopy top, and in the lowest two-thirds of thecanopy the transport terms were the dominant source terms in thebudget. Moreover, the pressure transport was the largest source ofturbulent kinetic energy in the lowest levels of the canopy, beingparticularly strong under convective conditions. These resultsindicate that pressure transport is important in the plant canopyturbulent kinetic energy budget, especially in the lowest portion ofthe stand, where it acts as the major driving force for turbulentmotions.     

On Monin–Obukhov Similarity In The Stable Atmospheric Boundary Layer

Atmospheric measurements from several field experiments have been combined to develop a better understanding of the turbulence structure of the stable atmospheric boundary layer. Fast response wind velocity and temperature data have been recorded using 3-dimensional sonic anemometers, placed at severalheights (1 m to 4.3 m) above the ground. The measurements wereused to calculate the standard deviations of the three components of the windvelocity, temperature, turbulent kinetic energy (TKE) dissipation andtemperature variance dissipation. These data were normalized and plottedaccording to Monin–Obukhov similarity theory. The non-dimensional turbulencestatistics have been computed, in part, to investigate the generalapplicability of the concept of z-less stratification for stable conditions. From the analysis of a data set covering almost five orders ofmagnitude in the stability parameter = z/L (from near-neutral tovery stable atmospheric stability), it was found that this concept does nothold in general. It was only for the non-dimensional standard deviation oftemperature and the average dissipation rate of turbulent kinetic energythat z-less behaviour has been found. The other variables studied here(non-dimensional standard deviations of u, v, and w velocity components and dissipation of temperature variance) did not follow the concept of z-less stratification for the very stable atmospheric boundary layer. An imbalance between production and dissipation of TKE was found for the near-neutral limit approached from the stable regime, which matches with previous results for near-neutral stability approached from the unstable regime.     

Large-eddy Simulation of Turbulent Flow across a Forest Edge. Part II: Momentum and Turbulent Kinetic Energy Budgets

Momentum and turbulent kinetic energy (TKE) budgets across a forest edge have been investigated using large-eddy simulation (LES). Edge effects are observed in the rapid variation of a number of budget terms across this vegetation transition. The enhanced drag force at the forest edge is largely balanced by the pressure gradient force and by streamwise advection of upstream momentum, while vertical turbulent diffusion is relatively insignificant. For variance and TKE budgets, the most important processes at the forest edge are production due to the convergence (or divergence) of the mean flow, streamwise advection, pressure diffusion and enhanced dissipation by canopy drag. Turbulent diffusion, pressure redistribution and vertical shear production, which are characteristic processes in homogeneous canopy flow, are less important at the forest transition. We demonstrate that, in the equilibrated canopy flow, a substantial amount of TKE produced in the streamwise direction by the vertical shear of the mean flow is redistributed in the vertical direction by pressure fluctuations. This redistribution process occurs in the upper canopy layers. Part of the TKE in the vertical velocity component is transferred by turbulent and pressure diffusion to the lower canopy levels, where pressure redistribution takes place again and feeds TKE back to the streamwise direction. In this TKE cycle, the primary source terms are vertical shear production for streamwise velocity variance and pressure redistribution for vertical velocity variance. The evolution of these primary source terms downwind of the forest edge largely controls the adjustment rates of velocity variances.     

On the Velocity Gradient in Stably Stratified Sheared Flows. Part 2: Observations and Models

Observations of the dependence of the dimensionless wind speed gradient f_m{\phi_m} as a function of the Monin–Obukhov stability parameter z/L _o under strong stability diverge from results of large-eddy simulation (LES) modelling. A kinetic energy budget analysis indicates that it is likely caused by violations of the assumptions of stationarity and/or homogeneity of turbulence in the field experiments rather than in imperfections of the LES. This confirms the validity of the widely used linear approximation for f_m{\phi_m} not only at weak to moderate stability, but also under strong stability. The new interpretation of the linear approximation of f_m{\phi_m} is given in terms of turbulent scales, which gives hope for its applicability to the free atmosphere as well.     

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

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

A Similarity Model of Subfilter-Scale Energy for Large-Eddy Simulations of the Atmospheric Boundary Layer

A scale-similarity model to estimate the subfilter-scale energy using the trace of the Leonard stress tensor is proposed and evaluated for large-eddy simulations of the atmospheric boundary layer (ABL). The model is derived from a stability-dependent model of the energy spectrum in the ABL, which accounts for the effects of buoyancy and mean shear as a function of z/L, the Monin–Obukhov stability variable. An a priori test using ABL turbulence data demonstrates that the model has accurate performance for dimensionless filter widths of Δ/z = 2, 1, and 0.5 for stabilities of −1 ≤ z/L ≤ 0.5, and improves considerably upon a similar model that is derived using an infinite κ ^−5/3 spectrum. This improvement is especially significant in the first several grid points near the surface in large-eddy simulations of the ABL, where Δ/z is necessarily large. The modelling procedure is then extended to develop a similarity model for the subfilter-scale scalar variance; it is shown to have robust performance for temperature.     

On the flow structure and turbulence during sweep and ejection events in a wind-tunnel model canopy

Particle image velocimetry (PIV) data obtained in a wind-tunnel model of a canopy boundary layer is used to examine the characteristics of mean flow and turbulence. The vector spacing varies between 1.7 and 2.5 times the Kolmogorov scales. Conditional sampling based on quadrants, i.e. based on the signs of velocity fluctuations, reveals fundamental differences in flow structure, especially between sweep and ejection events, which dominate the flow. During sweeps, the downward flow generates a narrow, highly turbulent, shear layer containing multiple small-scale vortices just below canopy height. During ejections, the upward flow expands this shear layer and the associated small-scale flow structures to a broad region located above the canopy. Consequently, during sweeps the turbulent kinetic energy (TKE), Reynolds stresses, as well as production and dissipation rates, have distinct narrow peaks just below canopy height, whereas during ejections these variables have broad maxima well above the canopy. Three methods to estimate the dissipation rate are compared, including spectral fits, measured subgrid-scale (SGS) energy fluxes at different scales, and direct measurements of slightly underresolved instantaneous velocity gradients. The SGS energy flux is 40–60% of the gradient-based (direct) estimates for filter sizes inside the inertial range, while decreasing with scale, as expected, within the dissipation range. The spectral fits are within 5–30% of the direct estimates. The spectral fits exceed the direct estimates near canopy height, but are lower well above and below canopy height. The dissipation rate below canopy height increases with velocity magnitude, i.e. it has the highest values during sweep and quadrant 1 events, and is significantly lower during ejection and quadrant 3 events. Well above the canopy, ejections are the most dissipative. Turbulent transport during sweep events acts as a source below the narrow shear layer within the canopy and as a sink above it. Transport during ejection events is a source only well above the canopy. The residual term in the TKE transport equation, representing mostly the effect of pressure–velocity correlations, is substantial only within the canopy, and is dominated by sweeps.     

Large-Eddy Simulation of the Daytime Boundary Layer in an Idealized Valley Using the Weather Research and Forecasting Numerical Model

A three-dimensional numerical meteorological model is used to perform large-eddy simulations of the upslope flow circulation over a periodic ridge-valley terrain. The subgrid-scale quantities are modelled using a prognostic turbulence kinetic energy (TKE) scheme, with a grid that has a constant horizontal resolution of 50 m and is stretched along the vertical direction. To account for the grid anisotropy, a modified subgrid length scale is used. To allow for the response of the surface fluxes to the valley-flow circulation, the soil surface temperature is imposed and the surface heat and momentum fluxes are computed based on Monin–Obukhov similarity theory. The model is designed with a symmetrical geometry using periodic boundary conditions in both the x and y directions. Two cases are simulated to study the influence of along-valley geostrophic wind forcing with different intensities. The presence of the orography introduces numerous complexities both in the mean properties of the flow and in the turbulent features, even for the idealized symmetric geometry. Classical definitions for the height of the planetary boundary layer (PBL) are revisited and redefined to capture the complex structure of the boundary layer. Analysis of first- and second-moment statistics, along with TKE budget, highlights the different structure of the PBL at different regions of the domain.     

A numerical study of the convective boundary layer

Computations of the buoyantly unstable Ekman layer are performed at low Reynolds number. The turbulent fields are obtained directly by solving the three-dimensional time-dependent Navier-Stokes equations (using the Boussinesq approximation to account for buoyancy effects), and no turbulence model is needed. Two levels of heating are considered, one quite vigorous, the other more moderate. Statistics for the vigorously heated case are found to agree reasonably well with laboratory, field, and large-eddy simulation results, when Deardorff's mixed-layer scaling is used. No indication of large-scale longitudinal roll cells is found in this convection-dominated flow, for which the inversion height to Obukhov length scale ratio –z _i /L _*=26. However, when heating is more moderate (so that –z _i /L _*=2), evidence of coherent rolls is present. About 10% of the total turbulent kinetic energy and turbulent heat flux, and 20% of the Reynolds shear stress, are estimated to be a direct consequence of the observed cells.     

Outlier Problem in Evaluating Similarity Functions in the Stable Atmospheric Boundary Layer

The gradient-based similarity approach removes turbulent fluxes as governing parameters and replaces them with vertical gradients of mean wind speed and potential temperature. As a result, the gradient Richardson number, Ri, appears as a stability parameter instead of the Monin–Obukhov stability parameter z/L (L is the Obukhov length). The gradient-based scaling is more appropriate for moderate and very stable conditions when the gradients are large and their errors are relatively small whereas z/L becomes ambiguous in these conditions because turbulent fluxes are small. However, the gradient-based formulation is faced with a problem related to the influence of Ri outliers: outliers with high values of Ri can exist in conditions that are really near-neutral. These outliers are mapped into the very stable range in plots in which Ri is the independent variable and may lead to spurious dependencies for bin-averaged data (spurious bin-averaging). This effect is quite large for functions that are steep for the gradient-based scaling. The present study uses the Surface Heat Budget of the Arctic Ocean (SHEBA) data to examine the problem and proposes two methods, conditional analysis and independent binning, to limit the influence of outliers on bin-averaging. A disadvantage of the conditional analysis is associated with eliminating outliers based on criteria that could be considered as subjective. The independent bin-averaging method does not have this disadvantage, but the scatter of the bin-averaged points is higher than for the conditional analysis, rendering data analysis and interpretation difficult.     

Consistent Two-Equation Closure Modelling for Atmospheric Research: Buoyancy and Vegetation Implementations

A self-consistent two-equation closure treating buoyancy and plant drag effects has been developed, through consideration of the behaviour of the supplementary equation for the length-scale-determining variable in homogeneous turbulent flow. Being consistent with the canonical flow regimes of grid turbulence and wall-bounded flow, the closure is also valid for homogeneous shear flows commonly observed inside tall vegetative canopies and in non-neutral atmospheric conditions. Here we examine the most often used two-equation models, namely and E − ω (where is the dissipation rate of turbulent kinetic energy, E, and is the specific dissipation), comparing the suggested buoyancy-modified closure against Monin–Obukhov similarity theory. Assessment of the closure implementing both buoyancy and plant drag together has been done, comparing the results of the two models against each other. It has been found that the E − ω model gives a better reproduction of complex atmospheric boundary-layer flows, including less sensitivity to numerical artefacts, than does the model. Re-derivation of the equation from the ω equation, however, leads to the model implementation that produces results identical to the E − ω model. Overall, numerical results show that the closure performs well, opening new possibilities for application of such models to tasks related to the atmospheric boundary layer—where it is important to adequately account for the influences of both vegetation and atmospheric stability.     

Turbulence Characteristics of the Shear-Free Convective Boundary Layer Driven by Heterogeneous Surface Heating

Large-eddy simulations (LESs) are employed to investigate the turbulence characteristics in the shear-free convective boundary layer (CBL) driven by heterogeneous surface heating. The patterns of surface heating are arranged as a chessboard with two different surface heat fluxes in the neighbouring patches, and the heterogeneity scale Λ in four different cases is taken as 1.2, 2.5, 5.0 and 10.0 km, respectively. The results are compared with those for the homogeneous case. The impact of the heterogeneity scale on the domain-averaged CBL characteristics, such as the profiles of the potential temperature and the heat flux, is not significant. However, different turbulence characteristics are induced by different heterogeneous surface heating. The greatest turbulent kinetic energy (TKE) is produced in the case with the largest heterogeneity scale, whilst the TKE in the other heterogeneous cases is close to that for the homogeneous case. This result indicates that the TKE is not enhanced unless the scale of the heterogeneous surface heating is large enough. The potential temperature variance is enhanced more significantly by a larger surface heterogeneity scale. But this effect diminishes with increasing CBL height, which implies that the turbulent eddy structures are changed during the CBL development. Analyses show that there are two types of organized turbulent eddies: one relates to the thermal circulations induced by the heterogeneous surface heating, whilst the other identifies with the inherent turbulent eddies (large eddies) induced by the free convection. At the early stage of the CBL development, the dominant scale of the organized turbulent eddies is controlled by the scale of the surface heterogeneity. With time increasing, the original pattern breaks up, and the vertical velocity eventually displays horizontal structures similar to those for the homogeneous heating case. It is found that after this transition, the values of λ/z _i (λ is the dominant horizontal scale of the turbulent eddies, z _i is the boundary-layer height) ≈1.6, which is just the aspect ratio of large eddies in the CBL.     

Estimates of velocity-pressure and velocity-pressure gradient interactions in the surface layer over plant canopies

The budget equations of turbulent kinetic energy and shear stress contain interaction terms of velocity-pressure and velocity-pressure gradient. These terms were estimated in the surface layer using the air pressure observed at the surface and wind velocity components over plant canopies. The magnitude of the pressure interaction terms was significantly large; it was not negligible compared with the production terms in each budget equation. The present results obtained over a rough surface also confirmed previous results that pressure terms play an important role in the turbulent kinetic energy budgets and the shear stress budget. The height dependency of nondimensional pressure terms versus (z - d)/z ₀ was not clear.     

The Alignment of the Mean Wind and Stress Vectors in the Unstable Surface Layer

A significant non-alignment between the mean horizontal wind vector and the stress vector was observed for turbulence measurements both above the water surface of a large lake, and over a land surface (soybean crop). Possible causes for this discrepancy such as flow distortion, averaging times and the procedure used for extracting the turbulent fluctuations (low-pass filtering and filter widths etc.), were dismissed after a detailed analysis. Minimum averaging times always less than 30 min were established by calculating ogives, and error bounds for the turbulent stresses were derived with three different approaches, based on integral time scales (first-crossing and lag-window estimates) and on a bootstrap technique. It was found that the mean absolute value of the angle between the mean wind and stress vectors is highly related to atmospheric stability, with the non-alignment increasing distinctively with increasing instability. Given a coordinate rotation that aligns the mean wind with the x direction, this behaviour can be explained by the growth of the relative error of the u–w component with instability. As a result, under more unstable conditions the u–w and the v–w components become of the same order of magnitude, and the local stress vector gives the impression of being non-aligned with the mean wind vector. The relative error of the v–w component is large enough to make it undistinguishable from zero throughout the range of stabilities. Therefore, the standard assumptions of Monin–Obukhov similarity theory hold: it is fair to assume that the v–w stress component is actually zero, and that the non-alignment is a purely statistical effect. An analysis of the dimensionless budgets of the u–w and the v–w components confirms this interpretation, with both shear and buoyant production of u–w decreasing with increasing instability. In the v–w budget, shear production is zero by definition, while buoyancy displays very low-intensity fluctuations around zero. As local free convection is approached, the turbulence becomes effectively axisymetrical, and a practical limit seems to exist beyond which it is not possible to measure the u-w component accurately.