Simplification and Validation of a Spectral-Tensor Model for Turbulence Including Atmospheric Stability

A spectral-tensor model of non-neutral, atmospheric-boundary-layer turbulence is evaluated using Eulerian statistics from single-point measurements of the wind speed and temperature at heights up to 100 m, assuming constant vertical gradients of mean wind speed and temperature. The model has been previously described in terms of the dissipation rate \(\epsilon \), the length scale of energy-containing eddies \(\mathcal {L}\), a turbulence anisotropy parameter \(\varGamma \), the Richardson number Ri, and the normalized rate of destruction of temperature variance \(\eta _\theta \equiv \epsilon _\theta /\epsilon \). Here, the latter two parameters are collapsed into a single atmospheric stability parameter z / L using Monin–Obukhov similarity theory, where z is the height above the Earth’s surface, and L is the Obukhov length corresponding to \(\{Ri,\eta _\theta \}\). Model outputs of the one-dimensional velocity spectra, as well as cospectra of the streamwise and/or vertical velocity components, and/or temperature, and cross-spectra for the spatial separation of all three velocity components and temperature, are compared with measurements. As a function of the four model parameters, spectra and cospectra are reproduced quite well, but horizontal temperature fluxes are slightly underestimated in stable conditions. In moderately unstable stratification, our model reproduces spectra only up to a scale \(\sim \) 1 km. The model also overestimates coherences for vertical separations, but is less severe in unstable than in stable cases.     

Water-Channel Estimation of Eulerian and Lagrangian Time Scales of the Turbulence in Idealized Two-Dimensional Urban Canopies

Lagrangian and Eulerian statistics are obtained from a water-channel experiment of an idealized two-dimensional urban canopy flow in neutral conditions. The objective is to quantify the Eulerian \((T^{\mathrm{E}})\) and Lagrangian \((T^{\mathrm{L}})\) time scales of the turbulence above the canopy layer as well as to investigate their dependence on the aspect ratio of the canopy, AR, as the latter is the ratio of the width (W) to the height (H) of the canyon. Experiments are also conducted for the case of flat terrain, which can be thought of as equivalent to a classical one-directional shear flow. The values found for the Eulerian time scales on flat terrain are in agreement with previous numerical results found in the literature. It is found that both the streamwise and vertical components of the Lagrangian time scale, \(T_\mathrm{u}^\mathrm{L} \) and \(T_\mathrm{w}^\mathrm{L} \), follow Raupach’s linear law within the constant-flux layer. The same holds true for \(T_\mathrm{w}^\mathrm{L} \) in both the canopies analyzed \((AR= 1\) and \(AR= 2\)) and also for \(T_\mathrm{u}^\mathrm{L} \) when \(AR = 1\). In contrast, for \(AR = 2\), \(T_\mathrm{u}^\mathrm{L} \) follows Raupach’s law only above \(z=2H\). Below that level, \(T_\mathrm{u}^\mathrm{L} \) is nearly constant with height, showing at \(z=H\) a value approximately one order of magnitude greater than that found for \(AR = 1\). It is shown that the assumption usually adopted for flat terrain, that \(\beta =T^{\mathrm{L}}/T^{\mathrm{E}}\) is proportional to the inverse of the turbulence intensity, also holds true even for the canopy flow in the constant-flux layer. In particular, \(\gamma /i_\mathrm{u} \) fits well \(\beta _\mathrm{u} =T_\mathrm{u}^\mathrm{L} /T_\mathrm{u}^\mathrm{E} \) in both the configurations by choosing \(\gamma \) to be 0.35 (here, \(i_\mathrm{u} =\sigma _\mathrm{u} / \bar{u} \), where \(\bar{u} \) and \(\sigma _\mathrm{u} \) are the mean and the root-mean-square of the streamwise velocity component, respectively). On the other hand, \(\beta _\mathrm{w} =T_\mathrm{w}^\mathrm{L} /T_\mathrm{w}^\mathrm{E} \) follows approximately \(\gamma /i_\mathrm{w} =0.65/\left( {\sigma _\mathrm{w} /\bar{u} } \right) \) for \(z > 2H\), irrespective of the AR value. The second main objective is to estimate other parameters of interest in dispersion studies, such as the eddy diffusivity of momentum \((K_\mathrm{{T}})\) and the Kolmogorov constant \((C_0)\). It is found that \(C_0\) depends appreciably on the velocity component both for the flat terrain and canopy flow, even though for the latter case it is insensitive to AR values. In all the three experimental configurations analyzed here, \(K_\mathrm{{T}}\) shows an overall linear growth with height in agreement with the linear trend predicted by Prandtl’s theory.     

Momentum- and Heat-Flux Parametrization at Dome C,Antarctica: A Sensitivity Study

An extensive meteorological observational dataset at Dome C, East Antarctic Plateau, enabled estimation of the sensitivity of surface momentum and sensible heat fluxes to aerodynamic roughness length and atmospheric stability in this region. Our study reveals that (1) because of the preferential orientation of snow micro-reliefs (sastrugi), the aerodynamic roughness length \(z_{0}\) varies by more than two orders of magnitude depending on the wind direction; consequently, estimating the turbulent fluxes with a realistic but constant \(z_{0}\) of 1 mm leads to a mean friction velocity bias of \(24\,\%\) in near-neutral conditions; (2) the dependence of the ratio of the roughness length for heat \(z_{0t}\) to \(z_{0}\) on the roughness Reynolds number is shown to be in reasonable agreement with previous models; (3) the wide range of atmospheric stability at Dome C makes the flux very sensitive to the choice of the stability functions; stability function models presumed to be suitable for stable conditions were evaluated and shown to generally underestimate the dimensionless vertical temperature gradient; as these models differ increasingly with increases in the stability parameter z / L, heat flux and friction velocity relative differences reached \(100\,\%\) when \(z/L > 1\); (4) the shallowness of the stable boundary layer is responsible for significant sensitivity to the height of the observed temperature and wind data used to estimate the fluxes. Consistent flux results were obtained with atmospheric measurements at heights up to 2 m. Our sensitivity study revealed the need to include a dynamical parametrization of roughness length over Antarctica in climate models and to develop new parametrizations of the surface fluxes in very stable conditions, accounting, for instance, for the divergence in both radiative and turbulent fluxes in the first few metres of the boundary layer.     

Turbulent Structure of a Simplified Urban Fluid Flow Studied Through Stereoscopic Particle Image Velocimetry

Stereoscopic particle image velocimetry was used to provide a three-dimensional characterization of the flow around a simplified urban model defined by a 5 by 7 array of blocks, forming four parallel streets, perpendicular to the incoming wind direction corresponding to a zero angle of incidence. Channeling of the flow through the array under consideration was observed, and its effect increased as the incoming wind direction, or angle of incidence (AOI), was changed from \(0^{\circ }\) to \(15^{\circ }\), \(30^{\circ }\), and \(45^{\circ }\). The flow between blocks can be divided into two regions: a region of low turbulence kinetic energy (TKE) levels close to the leeward side of the upstream block, and a high TKE area close to the downstream block. The centre of the arch vortex is located in the low TKE area, and two regions of large streamwise velocity fluctuation bound the vortex in the spanwise direction. Moreover, a region of large spanwise velocity fluctuation on the downstream block is found between the vortex legs. Our results indicate that the reorientation of the arch vortex at increasing AOI is produced by the displacement of the different TKE regions and their interaction with the shear layers on the sides and top of the upstream and downstream blocks, respectively. There is also a close connection between the turbulent structure between the blocks and the wind gusts. The correlations among gust components were also studied, and it was found that in the near-wall region of the street the correlations between the streamwise and spanwise gusts \(R_{uv}\) were dominant for all four AOI cases. At higher wall-normal positions in the array, the \(R_{uw}\) correlation decreased with increasing AOI, whereas the \(R_{uv}\) coefficient increased as AOI increased, and at \({\textit{AOI}}=45^{\circ }\) all three correlations exhibited relatively high values of around 0.4.     

Estimation of Surface-Layer Scaling Parameters in the Unstable Boundary Layer: Implications for Orographic Flow Speed-Up

A method is proposed for estimating the surface-layer depth \((z_s)\) and the friction velocity \((u_*)\) as a function of stability (here quantified by the Obukhov length, L) over the complete range of unstable flow regimes. This method extends that developed previously for stable conditions by Argaín et al. (Boundary-Layer Meteorol 130:15–28, 2009), but uses a qualitatively different approach. The method is specifically used to calculate the fractional speed-up \((\varDelta S)\) in flow over a ridge, although it is suitable for more general boundary-layer applications. The behaviour of \(z_s \left( L\right) \) and \(u_*\left( L\right) \) as a function of L is indirectly assessed via calculation of \(\varDelta S\left( L\right) \) using the linear model of Hunt et al. (Q J R Meteorol Soc 29:16–26, 1988) and its comparison with the field measurements reported in Coppin et al. (Boundary-Layer Meteorol 69:173–199, 1994) and with numerical simulations carried out using a non-linear numerical model, FLEX. The behaviour of \(\varDelta S\) estimated from the linear model is clearly improved when \(u_*\) is calculated using the method proposed here, confirming the importance of accounting for the dependences of \(z_s\left( L \right) \) and \(u_*\left( L \right) \) on L to better represent processes in the unstable boundary layer.     

Optimizing the Determination of Roughness Parameters for Model Urban Canopies

We present an objective optimization procedure to determine the roughness parameters for very rough boundary-layer flow over model urban canopies. For neutral stratification the mean velocity profile above a model urban canopy is described by the logarithmic law together with the set of roughness parameters of displacement height d, roughness length \(z_0\), and friction velocity \(u_*\). Traditionally, values of these roughness parameters are obtained by fitting the logarithmic law through (all) the data points comprising the velocity profile. The new procedure generates unique velocity profiles from subsets or combinations of the data points of the original velocity profile, after which all possible profiles are examined. Each of the generated profiles is fitted to the logarithmic law for a sequence of values of d, with the representative value of d obtained from the minima of the summed least-squares errors for all the generated profiles. The representative values of \(z_0\) and \(u_*\) are identified by the peak in the bivariate histogram of \(z_0\) and \(u_*\). The methodology has been verified against laboratory datasets of flow above model urban canopies.     

Katabatic Flow: A Closed-Form Solution with Spatially-Varying Eddy Diffusivities

The Nieuwstadt closed-form solution for the stationary Ekman layer is generalized for katabatic flows within the conceptual framework of the Prandtl model. The proposed solution is valid for spatially-varying eddy viscosity and diffusivity (O’Brien type) and constant Prandtl number (Pr). Variations in the velocity and buoyancy profiles are discussed as a function of the dimensionless model parameters \(z_0 \equiv \hat{z}_0 \hat{N}^2 Pr \sin {(\alpha )} |\hat{b}_\mathrm{s} |^{-1}\) and \(\lambda \equiv \hat{u}_{\mathrm{ref}}\hat{N} \sqrt{Pr} |\hat{b}_\mathrm{s} |^{-1}\), where \(\hat{z}_0\) is the hydrodynamic roughness length, \(\hat{N}\) is the Brunt-Väisälä frequency, \(\alpha \) is the surface sloping angle, \(\hat{b}_\mathrm{s}\) is the imposed surface buoyancy, and \(\hat{u}_{\mathrm{ref}}\) is a reference velocity scale used to define eddy diffusivities. Velocity and buoyancy profiles show significant variations in both phase and amplitude of extrema with respect to the classic constant \(\textit{K}\) model and with respect to a recent approximate analytic solution based on the Wentzel-Kramers-Brillouin theory. Near-wall regions are characterized by relatively stronger surface momentum and buoyancy gradients, whose magnitude is proportional to \(z_0\) and to \(\lambda \). In addition, slope-parallel momentum and buoyancy fluxes are reduced, the low-level jet is further displaced toward the wall, and its peak velocity depends on both \(z_0\) and \(\lambda \).     

Scale Properties of Anisotropic and Isotropic Turbulence in the Urban Surface Layer

The scale properties of anisotropic and isotropic turbulence in the urban surface layer are investigated. A dimensionless anisotropic tensor is introduced and the turbulent tensor anisotropic coefficient, defined as C, where \(C = 3d_{3}\,+\,1 (d_{3}\) is the minimum eigenvalue of the tensor) is used to characterize the turbulence anisotropy or isotropy. Turbulence is isotropic when \(C \approx 1\), and anisotropic when \(C \ll 1\). Three-dimensional velocity data collected using a sonic anemometer are analyzed to obtain the anisotropic characteristics of atmospheric turbulence in the urban surface layer, and the tensor anisotropic coefficient of turbulent eddies at different spatial scales calculated. The analysis shows that C is strongly dependent on atmospheric stability \(\xi = (z-z_{\mathrm{d}})/L_{{\textit{MO}}}\), where z is the measurement height, \(z_{\mathrm{d}}\) is the displacement height, and \(L_{{\textit{MO}}}\) is the Obukhov length. The turbulence at a specific scale in unstable conditions (i.e., \(\xi < 0\)) is closer to isotropic than that at the same scale under stable conditions. The maximum isotropic scale of turbulence is determined based on the characteristics of the power spectrum in three directions. Turbulence does not behave isotropically when the eddy scale is greater than the maximum isotropic scale, whereas it is horizontally isotropic at relatively large scales. The maximum isotropic scale of turbulence is compared to the outer scale of temperature, which is obtained by fitting the temperature fluctuation spectrum using the von Karman turbulent model. The results show that the outer scale of temperature is greater than the maximum isotropic scale of turbulence.     

On the Nature of the Transition Between Roll and Cellular Organization in the Convective Boundary Layer

Both observational and numerical studies of the convective boundary layer (CBL) have demonstrated that when surface heat fluxes are small and mean wind shear is strong, convective updrafts tend to organize into horizontal rolls aligned within 10–20\(^\circ \) of the geostrophic wind direction. However, under large surface heat fluxes and weak to negligible shear, convection tends to organize into open cells, similar to turbulent Rayleigh-Bénard convection. Using a suite of 14 large-eddy simulations (LES) spanning a range of \(-z_i/L\) between zero (neutral) and 1041 (highly convective), where \(z_i\) is the CBL depth and L is the Obukhov length, the transition between roll- and cellular-type convection is investigated systematically for the first time using LES. Mean vertical profiles including velocity variances and turbulent transport efficiencies, as well the “roll factor,” which characterizes the rotational symmetry of the vertical velocity field, indicate the transition occurs gradually over a range of \(-z_i/L\); however, the most significant changes in vertical profiles and CBL organization occur from near-neutral conditions up to about \(-z_i/L \approx \) 15–20. Turbulent transport efficiencies and quadrant analysis are used to characterize the turbulent transport of momentum and heat with increasing \(-z_i/L\). It is found that turbulence transports heat efficiently from weakly to highly convective conditions; however, turbulent momentum transport becomes increasingly inefficient as \(-z_i/L\) increases.     

Estimating Sediment Mass Fluxes on Surfaces Sheltered by Live Vegetation

We present a simple model based on already existing and widely used equations for estimating particle mass fluxes on surfaces sheltered by live vegetation. Wind-tunnel measurements of vertical profiles of mass flux in three different dense live plant canopies, and as a function of the spatially averaged skin friction velocity \({u_{\tau }}'\), provide the baseline set of data. For the bare-sand surface, the total mass flux Q shows the typical \(b({u_\tau }' - {u_{\tau t}}')^{3 }\) increase with increasing skin friction velocity \({u_{\tau }}'\), where b is a constant and \({u_{\tau t}}'\) is the threshold at the onset of particle erosion. Similar relations, however, with different values for b and \({u_{\tau t}}'\) compared to the bare-sand surface were found for experiments with 5.25 and 24.5 plants \(\hbox {m}^{-2}\) and can be explained by the spatial variations of \(u_{\tau }\) for the canopy cases. Based on the resulting parameters b and \({u_{\tau t}}'\), which are found to be functions of the roughness density \(\lambda \), we present a final simple relation \(Q(\lambda ,\, {u_{\tau }}')\) used for estimating the total mass flux for surfaces sheltered by live vegetation.     

Seasonal Variations in Drag Coefficient over a Sastrugi-Covered Snowfield in Coastal East Antarctica

The surface of windy Antarctic snowfields is subject to drifting snow, which leads to the formation of sastrugi. In turn, sastrugi contribute to the drag exerted by the snow surface on the atmosphere and hence influence drifting snow. Although the surface drag over rough sastrugi fields has been estimated for individual locations in Antarctica, its variation over time and with respect to drifting snow has received little attention. Using year-round data from a meteorological mast, seasonal variations in the neutral drag coefficient at a height of 10 m \((C_{{ DN}10})\) in coastal Adelie Land are presented and discussed in light of the formation and behaviour of sastrugi based on observed aeolian erosion patterns. The measurements revealed high \(C_{{ DN}10} \) values \((\ge \) 2 \(\times \) 10\(^{-3})\) and limited drifting snow (35% of the time) in summer (December–February) versus lower \(C_{{ DN}10} \) values \((\approx \) 1.5 \(\times \) \(10^{-3})\) associated with more frequent drifting snow (70% of the time) in winter (March–November). Without the seasonal distinction, there was no clear dependence of \(C_{{ DN}10} \) on friction velocity or wind direction, but observations revealed a general increase in \(C_{{ DN}10} \) with rising air temperature. The main hypothesis defended here is that higher temperatures increase snow cohesion and the development of sastrugi just after snow deposition while inhibiting the sastrugi streamlining process by raising the erosion threshold. This increases the contribution of the sastrugi form drag to the total surface drag in summer when winds are lighter and more variable. The analysis also showed that, in the absence of erosion, single snowfall events can reduce \(C_{{ DN}10} \) to \(1\,\times \,10^{-3}\) due to the burying of pre-existing microrelief under newly deposited snow. The results suggest that polar atmospheric models should account for spatial and temporal variations in snow surface roughness through a dynamic representation of the sastrugi form drag.     

Parametric Study of Urban-Like Topographic Statistical Moments Relevant to a Priori Modelling of Bulk Aerodynamic Parameters

For a horizontally homogeneous, neutrally stratified atmospheric boundary layer (ABL), aerodynamic roughness length, \(z_0\), is the effective elevation at which the streamwise component of mean velocity is zero. A priori prediction of \(z_0\) based on topographic attributes remains an open line of inquiry in planetary boundary-layer research. Urban topographies – the topic of this study – exhibit spatial heterogeneities associated with variability of building height, width, and proximity with adjacent buildings; such variability renders a priori, prognostic \(z_0\) models appealing. Here, large-eddy simulation (LES) has been used in an extensive parametric study to characterize the ABL response (and \(z_0\)) to a range of synthetic, urban-like topographies wherein statistical moments of the topography have been systematically varied. Using LES results, we determined the hierarchical influence of topographic moments relevant to setting \(z_0\). We demonstrate that standard deviation and skewness are important, while kurtosis is negligible. This finding is reconciled with a model recently proposed by Flack and Schultz (J Fluids Eng 132:041203-1–041203-10, 2010), who demonstrate that \(z_0\) can be modelled with standard deviation and skewness, and two empirical coefficients (one for each moment). We find that the empirical coefficient related to skewness is not constant, but exhibits a dependence on standard deviation over certain ranges. For idealized, quasi-uniform cubic topographies and for complex, fully random urban-like topographies, we demonstrate strong performance of the generalized Flack and Schultz model against contemporary roughness correlations.     

Pollutant Dispersion in Boundary Layers Exposed to Rural-to-Urban Transitions: Varying the Spanwise Length Scale of the Roughness

Both large-eddy simulations (LES) and water-tunnel experiments, using simultaneous stereoscopic particle image velocimetry and laser-induced fluorescence, have been used to investigate pollutant dispersion mechanisms in regions where the surface changes from rural to urban roughness. The urban roughness was characterized by an array of rectangular obstacles in an in-line arrangement. The streamwise length scale of the roughness was kept constant, while the spanwise length scale was varied by varying the obstacle aspect ratio l / h between 1 and 8, where l is the spanwise dimension of the obstacles and h is the height of the obstacles. Additionally, the case of two-dimensional roughness (riblets) was considered in LES. A smooth-wall turbulent boundary layer of depth 10h was used as the approaching flow, and a line source of passive tracer was placed 2h upstream of the urban canopy. The experimental and numerical results show good agreement, while minor discrepancies are readily explained. It is found that for \(l/h=2\) the drag induced by the urban canopy is largest of all considered cases, and is caused by a large-scale secondary flow. In addition, due to the roughness transition the vertical advective pollutant flux is the main ventilation mechanism in the first three streets. Furthermore, by means of linear stochastic estimation the mean flow structure is identified that is responsible for street-canyon ventilation for the sixth street and onwards. Moreover, it is shown that the vertical length scale of this structure increases with increasing aspect ratio of the obstacles in the canopy, while the streamwise length scale does not show a similar trend.     

Scaling of Wall-Normal Turbulence Intensity and Vertical Eddy Structures in the Atmospheric Surface Layer

Adequate high-quality data on three-dimensional velocities in the atmospheric surface layer (height \(\delta \)) were acquired in the field at the Qingtu Lake Observation Array. The measurement range occupies nearly the entire logarithmic layer from approximately \(0.006\delta \)–\(0.2\delta \). The turbulence intensity and eddy structures of the velocity fluctuations in the logarithmic region were primarily analyzed, and their variations in the z (wall-normal) direction were revealed. The primary finding was that the turbulent intensity of wall-normal velocity fluctuations exhibits a sharp upswing in the logarithmic region, which differs from classic scaling law and laboratory results. The upswing of the wall-normal turbulence intensity in the logarithmic region is deemed to be linear based on an ensemble of 20 sets of data. In addition, the wall-normal extent of the correlated structures and wall-normal spectra were compared to low Reynolds number results in the laboratory.     

Turbulent Flow Over Large Roughness Elements: Effect of Frontal and Plan Solidity on Turbulence Statistics and Structure

Wind-tunnel experiments were carried out on fully-rough boundary layers with large roughness (\(\delta /h \approx 10\), where h is the height of the roughness elements and \(\delta \) is the boundary-layer thickness). Twelve different surface conditions were created by using LEGO? bricks of uniform height. Six cases are tested for a fixed plan solidity (\(\lambda _\mathrm{P}\)) with variations in frontal density (\(\lambda _\mathrm{F}\)), while the other six cases have varying \(\lambda _\mathrm{P}\) for fixed \(\lambda _\mathrm{F}\). Particle image velocimetry and floating-element drag-balance measurements were performed. The current results complement those contained in Placidi and Ganapathisubramani (J Fluid Mech 782:541–566, 2015), extending the previous analysis to the turbulence statistics and spatial structure. Results indicate that mean velocity profiles in defect form agree with Townsend’s similarity hypothesis with varying \(\lambda _\mathrm{F}\), however, the agreement is worse for cases with varying \(\lambda _\mathrm{P}\). The streamwise and wall-normal turbulent stresses, as well as the Reynolds shear stresses, show a lack of similarity across most examined cases. This suggests that the critical height of the roughness for which outer-layer similarity holds depends not only on the height of the roughness, but also on the local wall morphology. A new criterion based on shelter solidity, defined as the sheltered plan area per unit wall-parallel area, which is similar to the ‘effective shelter area’ in Raupach and Shaw (Boundary-Layer Meteorol 22:79–90, 1982), is found to capture the departure of the turbulence statistics from outer-layer similarity. Despite this lack of similarity reported in the turbulence statistics, proper orthogonal decomposition analysis, as well as two-point spatial correlations, show that some form of universal flow structure is present, as all cases exhibit virtually identical proper orthogonal decomposition mode shapes and correlation fields. Finally, reduced models based on proper orthogonal decomposition reveal that the small scales of the turbulence play a significant role in assessing outer-layer similarity.     

An Analytical Formulation of the Monin–Obukhov Stability Parameter in the Atmospheric Surface Layer Under Unstable Conditions

A non-iterative analytical scheme is developed for unstable stratification that parametrizes the Monin–Obukhov stability parameter \(\zeta \) (\({=}z{/}L\), where z is the height above the ground and L is the Obukhov length) in terms of bulk Richardson number (\(Ri_B\)) within the framework of Businger–Dyer type similarity functions. The proposed scheme is valid for a wide range of roughness lengths of heat and momentum. The absolute relative error in the transfer coefficients of heat and momentum is found to be less than 1.5% as compared to those obtained from an iterative scheme for Businger–Dyer type similarity functions. An attempt has been made to extend this scheme to incorporate the similarity functions having a theoretically consistent free convection limit. Further, the performance of the scheme is evaluated using observational data from two different sites. The proposed scheme is simple, non-iterative and relatively more accurate compared to the schemes reported in the literature and can be used as a potential alternative to iterative schemes used in numerical models of the atmosphere.     

The Hurst Phenomenon in Error Estimates Related to Atmospheric Turbulence

The Hurst phenomenon is a well-known feature of long-range persistence first observed in hydrological and geophysical time series by E. Hurst in the 1950s. It has also been found in several cases in turbulence time series measured in the wind tunnel, the atmosphere, and in rivers. Here, we conduct a systematic investigation of the value of the Hurst coefficient H in atmospheric surface-layer data, and its impact on the estimation of random errors. We show that usually \(H > 0.5\), which implies the non-existence (in the statistical sense) of the integral time scale. Since the integral time scale is present in the Lumley–Panofsky equation for the estimation of random errors, this has important practical consequences. We estimated H in two principal ways: (1) with an extension of the recently proposed filtering method to estimate the random error (\(H_p\)), and (2) with the classical rescaled range introduced by Hurst (\(H_R\)). Other estimators were tried but were found less able to capture the statistical behaviour of the large scales of turbulence. Using data from three micrometeorological campaigns we found that both first- and second-order turbulence statistics display the Hurst phenomenon. Usually, \(H_R\) is larger than \(H_p\) for the same dataset, raising the question that one, or even both, of these estimators, may be biased. For the relative error, we found that the errors estimated with the approach adopted by us, that we call the relaxed filtering method, and that takes into account the occurrence of the Hurst phenomenon, are larger than both the filtering method and the classical Lumley–Panofsky estimates. Finally, we found that there is no apparent relationship between H and the Obukhov stability parameter. The relative errors, however, do show stability dependence, particularly in the case of the error of the kinematic momentum flux in unstable conditions, and that of the kinematic sensible heat flux in stable conditions.     

Ground-Based Observations of the Thermodynamic and Kinematic Properties of Lake-Breeze Fronts in Southern Manitoba,Canada

The effects on the convective boundary layer (CBL) of shading due to shallow cumulus clouds are investigated. The main question is to see whether clouds are able to produce secondary circulations by shading of the surface (dynamic heterogeneities) and how these dynamic heterogeneities interact with static heterogeneities in terms of the production of secondary circulations. Also the effects of cloud shadows on cloud-field characteristics are analyzed. The effects are studied using large-eddy simulations of a cloud-topped CBL with an idealized surface. Over a homogeneous surface, shadows trigger secondary circulations with different strengths depending on the solar zenith angle \(\vartheta \), with large \(\vartheta \) favouring the development of secondary circulations. Over a static heterogeneous surface with a simple striped pattern, the strength of secondary circulations is effectively reduced by dynamic heterogeneities at small \(\vartheta \). At large \(\vartheta \), however, the effect on secondary circulations depends on the orientation of the striped static heterogeneities to the shadow-casting direction of the clouds. The influence of shadows is only small if they are cast perpendicular to the striped heterogeneity, but if stripes and the shadow-casting direction are parallel, secondary circulations are reduced in strength also for large \(\vartheta \). Shadow effects on the cloud-field characteristics vary with \(\vartheta \) as well. The results show that small \(\vartheta \) favours the development of small clouds with a reduced lifetime while large \(\vartheta \) promotes the development of larger clouds with an extended lifetime compared to non-shading clouds.     

A Case Study of the Performance of Different Detrending Methods in Turbulent-Flux Estimation

Observations using a three-dimensional scanning coherent Doppler lidar in an urban area revealed the characteristics of streaky structures above a rough, inhomogeneous surface for a high-Reynolds-number flow. The study focused on two points: (1) the frequency of occurrence and conditions required for the presence of streaky structures, and (2) the universal scaling of the spacing of streaky structures (\(\lambda )\). The horizontal snapshots of the radial velocity were visually classified into six groups: Streak, Mixed, Fishnet, No streak, Front, and Others. The Streak category accounted for more than 50% of all possible flows and occurred when the horizontal wind speed was large and the atmospheric stratification was near-neutral. The spacing (\(\lambda )\) was estimated from the power spectral density of the streamwise velocity fluctuations along the spanwise direction. The spacing \(\lambda \) decreased with an increase in the local velocity gradient. Furthermore, it was revealed that the local velocity gradient normalized by the friction velocity and the boundary-layer height (\(z_i )\) comprehensively predicts \(\lambda /z_i \) under various experimental and environmental conditions, in terms of the scale of motion (i.e., indoor and outdoor scales), thermal stratification (i.e., from weakly unstable to stable stratification), and surface roughness (i.e., from flat to very rough surfaces).     

Observations of the Temperature and Humidity Structure Parameter Over Heterogeneous Terrain by Airborne Measurements During the LITFASS-2003 Campaign

We used numerical simulations to investigate the general relationship between urban morphology and the intensity of wind gusts in built-up areas at the pedestrian level. The simulated urban boundary layer developed over a 19.2 km (length) \(\times \) 4.8 km (width) \(\times \) 1.0 km (height) simulation domain, with 2-m resolution in all directions, to explicitly resolve the detailed shapes of buildings and the flow at the pedestrian level. This complex computation was accomplished using the lattice Boltzmann method and by implementing a large-eddy simulation model. To generalize the results, a new parameter that expresses the intensity of gusts (the gust index, \({\tilde{U}}_{ max})\) was defined as the local maximum wind speed divided by the freestream velocity. In addition, this parameter was decomposed into the mean wind-speed ratio, \({\tilde{U}} \) and turbulent gust ratio, \({\tilde{U}}^{{\prime }}\) to evaluate the qualities of gusts. These parameters were useful for quantitatively comparing the gust intensities within urban canopies at different locations or even among different experiments. In addition, the entire horizontal domain was subdivided into homogeneous square patches, in which both the simulated gust parameters and the morphological characteristics of building geometries were averaged. This procedure masked the detailed structure of individual buildings but retained the bulk characteristics of the urban morphology. At the pedestrian level, the gust index decreased with increasing building cover. Compared to \({\tilde{U}} \), the quantity \({\tilde{U}}^{{\prime }}\) notably contributed to the index throughout the range of plan area index \((\lambda _p)\) values. The dependences of all normalized wind-speed ratios transiently changed at \(\lambda _p =~0.28\). In cases where \(\lambda _p < 0.28, {\tilde{U}} \) decreased with increasing \(\lambda _p \), although \({\tilde{U}}^{{\prime }}\) was almost constant. In cases where \(\lambda _p > 0.28, {\tilde{U}}\) was almost constant and \({\tilde{U}}^{{\prime }}\) decreased with increasing \(\lambda _p \). This was explained by the change in flow regimes within the building canyon. At a higher elevation above the canopy layer, \(\lambda _p \) becomes less relevant to normalized wind-speed ratios, and instead the aerodynamic roughness length became important.