Analysis of Vertical Turbulent Heat Flux Limit in Stable Conditions with a Local Equilibrium, Turbulence Closure Model

Assuming that the vertical turbulent heat flux vanishes at extremely stable conditions, one should expect its maximal absolute value to occur somewhere at moderate stability, between a neutral and extremely stable equilibrium. Consequently, in some situations duality of solutions may be encountered (e.g. two different values of temperature difference associated with the same values of heat flux and wind speed). A quantitative analysis of this feature with a local equilibrium Reynolds-stress model is presented. The fixed-wind / fixed-shear maximum has been identified both in the bulk and in single-point flux–gradient relationships (that is, in the vertical temperature gradient and wind-shear parameter domain). The value of the Richardson number corresponding to this maximum is derived from the model equations. To study the possible feedback in strongly stable conditions, weak and intense cooling scenarios have been simulated with a one-dimensional numerical, high-resolution atmospheric boundary-layer model. Despite the rapid cooling, flow decoupling at the surface has not been observed; instead, a stability-limited heat flux is maintained, with a gradual increase of the Richardson number towards the top of the turbulent layer, with some signs of oscillatory behaviour at intermediate heights. Vertical changes of wind shear and the Brunt–Väisälä frequency display a remarkably non-monotonic character, with some signs of a gradually developing instability.     

Investigation of the Stable Atmospheric Boundary Layer at Halley Antarctica

Boundary-layer measurements from the Brunt Ice Shelf, Antarctica are analyzed to determine flux–profile relationships. Dimensionless quantities are derived in the standard approach from estimates of wind shear, potential temperature gradient, Richardson number, eddy diffusivities for momentum and heat, Prandtl number, mixing length and turbulent kinetic energy. Nieuwstadt local scaling theory for the stable atmospheric boundary-layer appears to work well departing only slightly from expressions found in mid-latitudes. An $E$ E – $l_{\mathrm{m}}$ l m single-column model of the stable boundary layer is implemented based on local scaling arguments. Simulations based on the first GEWEX Atmospheric Boundary-Layer Study case study are validated against ensemble-averaged profiles for various stability classes. A stability-dependent function of the dimensionless turbulent kinetic energy allows a better fit to the ensemble profiles.     

Energy- and Flux-Budget Turbulence Closure Model for Stably Stratified Flows. Part II: The Role of Internal Gravity Waves

We advance our prior energy- and flux-budget (EFB) turbulence closure model for stably stratified atmospheric flow and extend it to account for an additional vertical flux of momentum and additional productions of turbulent kinetic energy (TKE), turbulent potential energy (TPE) and turbulent flux of potential temperature due to large-scale internal gravity waves (IGW). For the stationary, homogeneous regime, the first version of the EFB model disregarding large-scale IGW yielded universal dependencies of the flux Richardson number, turbulent Prandtl number, energy ratios, and normalised vertical fluxes of momentum and heat on the gradient Richardson number, Ri. Due to the large-scale IGW, these dependencies lose their universality. The maximal value of the flux Richardson number (universal constant ≈0.2–0.25 in the no-IGW regime) becomes strongly variable. In the vertically homogeneous stratification, it increases with increasing wave energy and can even exceed 1. For heterogeneous stratification, when internal gravity waves propagate towards stronger stratification, the maximal flux Richardson number decreases with increasing wave energy, reaches zero and then becomes negative. In other words, the vertical flux of potential temperature becomes counter-gradient. Internal gravity waves also reduce the anisotropy of turbulence: in contrast to the mean wind shear, which generates only horizontal TKE, internal gravity waves generate both horizontal and vertical TKE. Internal gravity waves also increase the share of TPE in the turbulent total energy (TTE = TKE + TPE). A well-known effect of internal gravity waves is their direct contribution to the vertical transport of momentum. Depending on the direction (downward or upward), internal gravity waves either strengthen or weaken the total vertical flux of momentum. Predictions from the proposed model are consistent with available data from atmospheric and laboratory experiments, direct numerical simulations and large-eddy simulations.     

Snowdrift Suspension And Atmospheric Turbulence. Part II: Results Of Model Simulations

An atmospheric surface-layer model is used to investigate the interactionbetween suspended snow particles and the near-surface flow. Themodel incorporates the effects of upward diffusion, gravitational settling and sublimation of snow particles in 48 size classes, the effects of snowdrift sublimation on the heat and moisture budget of the surface layer, and the buoyancy destruction of turbulent kinetic energy (TKE) caused by the presence of suspended particles. A new term in the E- closure model representing the buoyancy destruction due to suspended particles is included in the prognostic equation for TKE. Generally, model results indicate that the presence of suspended particles causes significant decreases in TKE, the dissipation rate, turbulent length scales and eddy exchange coefficients (up to 40%). It is found that the reduction in the eddy exchangecoefficients is due mainly to reductions in turbulent length scales. Theassociated particle Richardson number peaks near the saltation-suspensioninterface, but at higher levels in the surface layer the particle-induced buoyancy can also significantly affect the flow. A detailed analysis of the various snowdrift quantities, the TKE budget and the particle buoyancy effects on the flow is presented.     

Sensitivity of Turbine-Height Wind Speeds to Parameters in Planetary Boundary-Layer and Surface-Layer Schemes in the Weather Research and Forecasting Model

We evaluate the sensitivity of simulated turbine-height wind speeds to 26 parameters within the Mellor–Yamada–Nakanishi–Niino (MYNN) planetary boundary-layer scheme and MM5 surface-layer scheme of the Weather Research and Forecasting model over an area of complex terrain. An efficient sampling algorithm and generalized linear model are used to explore the multiple-dimensional parameter space and quantify the parametric sensitivity of simulated turbine-height wind speeds. The results indicate that most of the variability in the ensemble simulations is due to parameters related to the dissipation of turbulent kinetic energy (TKE), Prandtl number, turbulent length scales, surface roughness, and the von Kármán constant. The parameter associated with the TKE dissipation rate is found to be most important, and a larger dissipation rate produces larger hub-height wind speeds. A larger Prandtl number results in smaller nighttime wind speeds. Increasing surface roughness reduces the frequencies of both extremely weak and strong airflows, implying a reduction in the variability of wind speed. All of the above parameters significantly affect the vertical profiles of wind speed and the magnitude of wind shear. The relative contributions of individual parameters are found to be dependent on both the terrain slope and atmospheric stability.     

Energy- and flux-budget (EFB) turbulence closure model for stably stratified flows. Part I: steady-state,homogeneous regimes

We propose a new turbulence closure model based on the budget equations for the key second moments: turbulent kinetic and potential energies: TKE and TPE (comprising the turbulent total energy: TTE = TKE + TPE) and vertical turbulent fluxes of momentum and buoyancy (proportional to potential temperature). Besides the concept of TTE, we take into account the non-gradient correction to the traditional buoyancy flux formulation. The proposed model permits the existence of turbulence at any gradient Richardson number, Ri. Instead of the critical value of Richardson number separating—as is usually assumed—the turbulent and the laminar regimes, the suggested model reveals a transitional interval, , which separates two regimes of essentially different nature but both turbulent: strong turbulence at ; and weak turbulence, capable of transporting momentum but much less efficient in transporting heat, at . Predictions from this model are consistent with available data from atmospheric and laboratory experiments, direct numerical simulation and large-eddy simulation.     

The Turbulence Structure of the Stable Atmospheric Boundary Layer Around a Coastal Headland: Aircraft Observations and Modelling Results

The turbulence structure of a stable marine atmospheric boundary layer in the vicinity of a coastal headland is examined using aircraft observations and numerical simulations. Measurements are drawn from a flight by the NCAR C-130 around Cape Mendocino on the coast of northern California on June 7 1996 during the Coastal Waves 96 field program. Local similarity scaling of the velocity variances is found to apply successfully within the continuously turbulent layer; the empirical scaling function is similar to that found by several previous studies. Excellent agreement is found between the modelled and observed scaling results. No significant change in scaling behaviour is observed for the region within the expansion fan that forms downstream of the Cape, suggesting that the scaling can be applied to horizontally heterogeneous conditions; however, the precise form of the function relating scaled velocities and stability is observed to change close to the surface. This result, differences between the scaling functions found here and in other studies, and the departure of these functions from the constant value predicted by the original theory, leads us to question the nature of the similarity functions observed. We hypothesize that the form of the functions is controlled by non-local contributions to the velocity variance budgets, and that differences in the non-local terms between studies explain the differences in the observed scaling functions.     

Some aspects of the turbulent stable boundary layer

We consider the structure of the stable boundary layer using the concept of local scaling. In this scaling approach turbulence variables, non-dimensionalized with measurements taken at the same height, can be expressed as a function of a single parameter z/, where z is the height and a local Obukhov length. One of the consequences is that locally scaled variables become constant above the surface layer. This behavior is illustrated with observations of the Richardson number. With local scaling as a closure hypothesis we then formulate a model of the stable boundary layer. Its solution for steady-state conditions is given. One result we obtain is the well-known Zilitinkevich equation for the boundary-layer height. A comparison of this equation with observations results in a reasonable agreement. Also we discuss some alternative expressions for the stable boundary-layer height and compare them with observations. Another result of our model is an explicit profile for the K-coefficient as a quadratic function of height. We discuss the consequences of this expression for the dispersion of a point source emission. We find that the time scale of diffusion in this case is about 5 hours.     

Stable Boundary-Layer Scaling Regimes: The Sheba Data

Turbulent and mean meteorological data collected at five levels on a 20-m tower over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are analyzed to examine different regimes of the stable boundary layer (SBL). Eleven months of measurements during SHEBA cover a wide range of stability conditions, from the weakly unstable regime to very stable stratification. Scaling arguments and our analysis show that the SBL can be classified into four major regimes: (i) surface-layer scaling regime (weakly stable case), (ii) transition regime, (iii) turbulent Ekman layer, and (iv) intermittently turbulent Ekman layer (supercritical stable regime). These four regimes may be considered as the basic states of the traditional SBL. Sometimes these regimes, especially the last two, can be markedly perturbed by gravity waves, detached elevated turbulence (‘upside down SBL’), and inertial oscillations. Traditional Monin–Obukhov similarity theory works well in the weakly stable regime. In the transition regime, Businger–Dyer formulations work if scaling variables are re-defined in terms of local fluxes, although stability function estimates expressed in these terms include more scatter compared to the surface-layer scaling. As stability increases, the near-surface turbulence is affected by the turning effects of the Coriolis force (the turbulent Ekman layer). In this regime, the surface layer, where the turbulence is continuous, may be very shallow (< 5 m). Turbulent transfer near the critical Richardson number is characterized by small but still significant heat flux and negligible stress. The supercritical stable regime, where the Richardson number exceeds a critical value, is associated with collapsed turbulence and the strong influence of the earth’s rotation even near the surface. In the limit of very strong stability, the stress is no longer a primary scaling parameter.     

Triple-Hot-Film Anemometer Performance in Cases-99 and A Comparison With Sonic Anemometer Measurements

Two levels of triple-hot-film and sonic anemometers were deployed on a 5.5-m towerduring the Cooperative Atmospheric Surface Exchange Study (CASES-99) in October1999. Each triple-hot-film probe was collocated 50 mm from the sonic sensing path ona common boom. Various problems with using triple-hot-films in the atmosphere toresolve wind components are addressed including the derivation of a yaw angle correction using the collocated sensors. It was found that output voltage drift due to changes in environmental temperature could be monitored and corrected using an automated system. Non-unique solutions to heat transfer equations can be resolved using a collocated sonic anemometer. Multi-resolution decomposition of the hot-film data was used to estimate appropriate day and night averaging periods for turbulent flux measurements in and near the roughness sub-layer. Finally, triple-hot-film measurements of mean wind magnitude (M), turbulent kinetic energy (TKE), sensible heat flux (H), and local friction velocity (u_*) are compared to those of the collocated CSAT3 sonic anemometers. Overall, the mean wind magnitudes measured by the triple-hot-film and the collocated sonic sensorswere close, consistent and independent of stability or proximity to the ground. The turbulent statistics, TKE, u_*, and H, measured by the two sensor systems were reasonably close together at z = 5 m. However, the ratio of sonic measurement/hot-film measurement decreased toward the ground surface, especially during stable conditions.     

Statistics of Turbulence in the Stable Boundary Layer Affected by Gravity Waves

In order to investigate effects of interactions between turbulence and gravity waves in the stable boundary layer on similarity theory relationships, we re-examined a dataset, collected during three April nights in 1978 and in 1980 on the 300-m tower of the Boulder Atmospheric Observatory (BAO). The BAO site, located in Erie, Colorado, USA, 30 km east of the foothills of the Rocky Mountains, has been known for the frequent detection of wave activities. The considered profiles of turbulent fluxes and variances were normalized by two local, gradient-based scaling systems, and subsequently compared with similarity functions of the Richardson number, obtained based on data with no influence of gravity currents and topographical factors. The first scaling system was based on local values of the vertical velocity variance $\sigma _\mathrm{w}$ and the Brunt–Väisäla frequency $ N,$ while the second one was based on the temperature variance $\sigma _{\theta }$ and $N.$ Analysis showed some departures from the similarity functions (obtained for data with virtually no influence of mesoscale motions); nonetheless the overall dependency of dimensionless moments on the Richardson number was maintained.     

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.     

The Coupling State of an Idealized Stable Boundary Layer

The coupling state between the surface and the top of the stable boundary layer (SBL) is investigated using four different schemes to represent the turbulent exchange. An idealized SBL is assumed, with fixed wind speed and temperature at its top. At the surface, two cases are considered, first a constant temperature, 20 K lower than the SBL top, and later a constant 2 K h⁻¹ cooling rate is assumed for 10 h after a neutral initial condition. The idealized conditions have been chosen to isolate the influence of the turbulence formulations on the coupling state, and the intense stratification has the purpose of enhancing such a response. The formulations compared are those that solve a prognostic equation for turbulent kinetic energy (TKE) and those that directly prescribe turbulence intensity as a function of atmospheric stability. Two TKE formulations are considered, with and without a dependence of the exchange coefficients on stability, while short and long tail stability functions (SFs) are also compared. In each case, the dependence on the wind speed at the SBL top is considered and it is shown that, for all formulations, the SBL experiences a transition from a decoupled state to a coupled state at an intermediate value of mechanical forcing. The vertical profiles of potential temperature, wind speed and turbulence intensity are shown as a function of the wind speed at the SBL top, both for the decoupled and coupled states. The formulation influence on the coupling state is analyzed and it is concluded that, in general, the simple TKE formulation has a better response, although it also tends to overestimate turbulent mixing. The consequences are discussed.     

A Modulated-Gradient Parametrization for the Large-Eddy Simulation of the Atmospheric Boundary Layer Using the Weather Research and Forecasting Model

The performance of the modulated-gradient subgrid-scale (SGS) model is investigated using large-eddy simulation (LES) of the neutral atmospheric boundary layer within the weather research and forecasting model. Since the model includes a finite-difference scheme for spatial derivatives, the discretization errors may affect the simulation results. We focus here on understanding the effects of finite-difference schemes on the momentum balance and the mean velocity distribution, and the requirement (or not) of the ad hoc canopy model. We find that, unlike the Smagorinsky and turbulent kinetic energy (TKE) models, the calculated mean velocity and vertical shear using the modulated-gradient model, are in good agreement with Monin–Obukhov similarity theory, without the need for an extra near-wall canopy model. The structure of the near-wall turbulent eddies is better resolved using the modulated-gradient model in comparison with the classical Smagorinsky and TKE models, which are too dissipative and yield unrealistic smoothing of the smallest resolved scales. Moreover, the SGS fluxes obtained from the modulated-gradient model are much smaller near the wall in comparison with those obtained from the regular Smagorinsky and TKE models. The apparent inability of the LES model in reproducing the mean streamwise component of the momentum balance using the total (resolved plus SGS) stress near the surface is probably due to the effect of the discretization errors, which can be calculated a posteriori using the Taylor-series expansion of the resolved velocity field. Overall, we demonstrate that the modulated-gradient model is less dissipative and yields more accurate results in comparison with the classical Smagorinsky model, with similar computational costs.     

A Boundary-Layer Scaling for Turbulent Katabatic Flow

Scaling relationships are proposed for the turbulent katabatic flow of a stably stratified fluid down a homogeneously cooled planar slope—the turbulent analogue of a Prandtl-type slope flow. The \(\Pi \) Theorem predicts that such flows are controlled by three non-dimensional parameters: the slope angle, the Prandtl number, and a Reynolds number defined in terms of the surface thermal forcing (surface buoyancy or surface buoyancy flux), Brunt-Väisälä frequency, slope angle, and molecular viscosity and diffusivity coefficients. However, by exploiting the structure of the governing differential equations in a boundary-layer form, scaled equations are deduced that involve only two non-dimensional parameters: the Prandtl number and a modified Reynolds number. In the proposed scaling framework, the slope angle does not appear as an independent governing parameter, but merely acts as a stretching factor in the scales for the dependent and independent variables, and appears in the Reynolds number. Based on the boundary-layer analysis, we hypothesize that the full katabatic-flow problem is largely controlled by two rather than three parameters. Preliminary tests of the scaling hypothesis using data from direct numerical simulations provide encouraging results.     

Local Imbalance of Turbulent Kinetic Energy in the Surface Layer

We utilize experimental data collected in 2002 over an open field in Hanford, Washington, USA, to investigate the turbulent kinetic energy (TKE) budget in the atmospheric surface layer. The von Kármán constant was determined from the near-neutral wind profiles to be 0.36 ± 0.02 rather than the classical value of 0.4. The TKE budget was normalized and all terms were parameterized as functions of a stability parameter z/L, where z is the distance from the ground and L is the Obukhov length. The shear production followed the Businger–Dyer relation for −2 < z/L < 1. Contrary to the traditional Monin–Obukhov similarity theory (MOST), the shear, buoyancy and dissipation terms were found to be imbalanced due to a non-zero vertical transport over all stabilities. Motivated by this local imbalance, modified parameterizations of the dissipation and the turbulent transport were attempted and generated good agreement with the experimental data. Assuming stationarity and horizontal homogeneity, the pressure transport was estimated from the residual of the TKE budget.     

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.     

A numerical model study of the structure and similarity scaling of the nocturnal boundary layer (NBL)

A one-dimensional numerical model based on the equations of mean motion and turbulent kinetic energy (TKE), with Delage's (1974) mixing-length parameterization has been used to simulate the mean and turbulent structure of the evolving stably stratified nocturnal boundary layer (NBL). The model also includes a predictive equation for the surface temperature and longwave radiational cooling effects.In the absence of advective and gravity wave effects, it is found that the model-simulated structure, after a few hours of evolution, could be ordered fairly well by a similarity scaling (u _*0, _*0, L ₀, and h) based on surface fluxes and the NBL height. Simple expressions are suggested to describe the normalized profiles of momentum and heat fluxes, TKE, eddy-viscosity and energy dissipation. A good ordering of the same variables is also achieved by a local scaling (u _*0, _* and L) based on the height-dependent local fluxes. The normalized TKE, eddy viscosity and energy dissipation are unique functions of z/L and approach constant values as z/L , where L is the local Monin-Obukhov length. These constants are close to the values predicted for the surface layer as z/L , thus suggesting that the Monin-Obukhov similarity theory can be extended to the whole NBL, by using the local (height-dependent) scales in place of surface-layer scales. The observed NBL structure has been shown to follow local similarity (Nieuwstadt, 1984).     

Morning Boundary-Layer Turbulent Kinetic Energy by Theoretical Models

A major factor that influences the diurnal variation of turbulent kinetic energy (TKE) is the sensible heat flux at the surface. Here, the TKE variations are analysed during the morning transition phase because subsequent to the neutral or stable stratification during the night, peaks of concentration of scalars develop. The characteristics of the TKE during the growth phase of convection are analysed with the help of two analytical models. For this purpose, a three-dimensional spectral model of the growth of convection, starting from a neutral layer, and other formulations of micrometeorological parameters such as the convective and neutral spectra, velocity variance and dissipation rates are utilised. The peak values in the TKE spectra in the lower, middle and upper levels of the convective boundary layer showed a migration to higher wavelengths as the convection increased with time. The TKE evolutions generated by the analytical models agree fairly well with the results of large-eddy simulation for three vertical levels.     

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.