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.     

Measurement of Prandtl Number as a Function of Richardson Number Avoiding Self-Correlation

The empirical dependence of turbulence Prandtl number (Pr) on gradient Richardson number (Ri) is presented, derived so as to avoid the effects of self-correlation from common variables. Linear power relationships between the underlying variables that constitute both Pr and Ri are derived empirically from flux and profile observations. Pr and Ri are then reconstructed from these power laws, to indicate their interdependence whilst avoiding self-correlation. Data are selected according to the stability range prior to regression, and the process is iterated from neutral to higher stability until error analysis indicates the method is no longer valid. A Butterworth function is fitted to the resulting Pr ⁻¹(Ri) regression to give an empirical summary of the analysis. The form suggests that asymptotically Pr ⁻¹ decreases as Ri ^3/2. Scatter in the data increases above Ri ~ 1, however, indicating additional constraints to Pr are not captured by Ri alone in this high stability regime. The Butterworth function is analytic for all Ri > 0, and may be included in suitable boundary-layer parameterisation schemes where the turbulent diffusivity for heat is derived from the turbulent diffusivity for momentum.     

A Hierarchy of Energy- and Flux-Budget (EFB) Turbulence Closure Models for Stably-Stratified Geophysical Flows

Here we advance the physical background of the energy- and flux-budget turbulence closures based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy, and a new relaxation equation for the turbulent dissipation time scale. The closure is designed for stratified geophysical flows from neutral to very stable and accounts for the Earth’s rotation. In accordance with modern experimental evidence, the closure implies the maintaining of turbulence by the velocity shear at any gradient Richardson number Ri, and distinguishes between the two principally different regimes: “strong turbulence” at ${Ri \ll 1}$ typical of boundary-layer flows and characterized by the practically constant turbulent Prandtl number Pr _T; and “weak turbulence” at Ri > 1 typical of the free atmosphere or deep ocean, where Pr _T asymptotically linearly increases with increasing Ri (which implies very strong suppression of the heat transfer compared to the momentum transfer). For use in different applications, the closure is formulated at different levels of complexity, from the local algebraic model relevant to the steady-state regime of turbulence to a hierarchy of non-local closures including simpler down-gradient models, presented in terms of the eddy viscosity and eddy conductivity, and a general non-gradient model based on prognostic equations for all the basic parameters of turbulence including turbulent fluxes.     

Evaluation of turbulent Prandtl (Schmidt) number parameterizations for stably stratified environmental flows

In this study, we evaluate four different parameterizations of the turbulent Prandtl (Schmidt) number Pr_t = ν_t/Γ_t where ν_t is the eddy viscosity and Γ_t is the scalar eddy diffusivity, for stably stratified flows. All four formulations of Pr_t are strictly functions of the gradient Richardson number Ri, which provides a measure of the strength of the stratification. A zero-equation (i.e. no extra transport equations are required) turbulence model for ν_t in a one-dimensional, turbulent channel flow is considered to evaluate the behavior of the different formulations of Pr_t. Both uni-directional and oscillatory flows are considered to simulate conditions representative of practical flow problems such as atmospheric boundary layer flows and tidally driven estuarine flows, to quantify the behavior of each of the four formulations of Pr_t. We perform model-to-model comparisons to highlight which of the models of Pr_t allow for a higher rate of turbulent mixing and which models significantly inhibit turbulent mixing in the presence of buoyancy forces resulting from linear (continuous) stratification as well as two-layer stratification. The basis underlying the formulation of each model in conjunction with the simulation results are used to emphasize the considerable variability in the different formulations and the importance of choosing an appropriate parameterization of Pr_t given a model for ν_t in stably stratified flows.     

Intermittent Bursting of Turbulence in a Stable Boundary Layer with Low-level Jet

The atmospheric stable boundary layer (SBL) with a low-level jet is simulated experimentally using a thermally stratified wind tunnel. The turbulence structure and flow characteristics are investigated by simultaneous measurements of velocity and temperature fluctuations and by flow visualization. Attention is focused on the effect of strong wind shear due to a low-level jet on stratified boundary layers with strong stability. Occasional bursting of turbulence in the lower portion of the boundary layer can be found in the SBL with strong stability. This bursting originates aloft away from the surface and transports fluid with relatively low velocity and temperature upward and fluid with relatively high velocity and temperature downward. Furthermore, the relationship between the occurrence of turbulence bursting and the local gradient Richardson number (Ri) is investigated. The Ri becomes larger than the critical Ri, Ri_cr = 0.25, in quiescent periods. On the other hand, the Ri number becomes smaller than Ri_cr during bursting events.     

SHEBA flux–profile relationships in the stable atmospheric boundary layer

Measurements of atmospheric turbulence made during the Surface Heat Budget of the Arctic Ocean Experiment (SHEBA) are used to examine the profile stability functions of momentum, φ _m , and sensible heat, φ _h , in the stably stratified boundary layer over the Arctic pack ice. Turbulent fluxes and mean meteorological data that cover different surface conditions and a wide range of stability conditions were continuously measured and reported hourly at five levels on a 20-m main tower for 11 months. The comprehensive dataset collected during SHEBA allows studying φ _m and φ _h in detail and includes ample data for the very stable case. New parameterizations for φ _m (ζ) and φ _h (ζ) in stable conditions are proposed to describe the SHEBA data; these cover the entire range of the stability parameter ζ = z/L from neutral to very stable conditions, where L is the Obukhov length and z is the measurement height. In the limit of very strong stability, φ _m follows a ζ ^1/3 dependence, whereas φ _h initially increases with increasing ζ, reaches a maximum at ζ ≈ 10, and then tends to level off with increasing ζ. The effects of self-correlation, which occur in plots of φ _m and φ _h versus ζ, are reduced by using an independent bin-averaging method instead of conventional averaging.     

Local Scales of Turbulence in the Stable Boundary Layer

Local, gradient-based scales, which contain the vertical velocity and temperature variances, as well as the potential temperature gradient, but do not include fluxes, are tested using data collected during the CASES-99 experiment. The observations show that the scaling based on the temperature variance produces relatively smaller scatter of empirical points. The resulting dimensionless statistical moments approach constant values for sufficiently large values of the Richardson number Ri. This allows one to derive predictions for the Monin–Obukhov similarity functions φ _m and φ _h , the Prandtl number Pr and the flux Richardson number Rf in weak turbulence regime.     

The influence of nonstationarity on the turbulent flux–gradient relationship for stable stratification

Extensive eddy-correlation datasets are analyzed to examine the influence of nonstationarity of the mean flow on the flux–gradient relationship near the surface. This nonstationarity is due to wavelike motions, meandering of the wind vector, and numerous unidentified small-scale mesoscale motions. While the data do not reveal an obvious critical gradient Richardson number, the maximum downward heat flux increases approximately linearly with increasing friction velocity for significant stability. The largest of our datasets is chosen to more closely examine the influence of stability, nonstationarity, distortion of the mean wind profile and self-correlation on the flux-gradient relationship. Stability is expressed in terms of z/L, the gradient Richardson number or the bulk Richardson number over the tower layer. The efficiency of the momentum transport systematically increases with increasing nonstationarity and attendant distortion of the mean wind profile. Enhancement of the turbulent momentum flux associated with nonstationarity is examined in terms of the nondimensional shear, Prandtl number and the eddy diffusivity.     

Intermittent turbulence in measurements of the temperature structure parameter under very stable conditions

C _T ²measurements taken over a desert in stable conditions indicate that the atmosphere remains intermittently turbulent for Ri numbers as high as 10. This is in contrast to previous results which suggest that the atmosphere is essentially nonturbulent for Ri > 2. These measurements also indicate that time-averaged C _T ²measurements do not scale with the time-averaged mean Ri number in very stable conditions. However, the standard deviation of log₁₀ C _T ²does appear to scale with Ri.     

The Critical Richardson Number and Limits of Applicability of Local Similarity Theory in the Stable Boundary Layer

Measurements of atmospheric turbulence made over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are used to determine the limits of applicability of Monin–Obukhov similarity theory (in the local scaling formulation) in the stable atmospheric boundary layer. Based on the spectral analysis of wind velocity and air temperature fluctuations, it is shown that, when both the gradient Richardson number, Ri, and the flux Richardson number, Rf, exceed a ‘critical value’ of about 0.20–0.25, the inertial subrange associated with the Richardson–Kolmogorov cascade dies out and vertical turbulent fluxes become small. Some small-scale turbulence survives even in this supercritical regime, but this is non-Kolmogorov turbulence, and it decays rapidly with further increasing stability. Similarity theory is based on the turbulent fluxes in the high-frequency part of the spectra that are associated with energy-containing/flux-carrying eddies. Spectral densities in this high-frequency band diminish as the Richardson–Kolmogorov energy cascade weakens; therefore, the applicability of local Monin–Obukhov similarity theory in stable conditions is limited by the inequalities Ri < Ri _cr and Rf < Rf _cr. However, it is found that Rf _cr  =  0.20–0.25 is a primary threshold for applicability. Applying this prerequisite shows that the data follow classical Monin–Obukhov local z-less predictions after the irrelevant cases (turbulence without the Richardson–Kolmogorov cascade) have been filtered out.     

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.     

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.     

Characteristics of intermittent turbulence in the upper stable boundary layer over Greenland

The experiment IGLOS (Investigation of the Greenland Boundary Layer Over Summit) was conducted in June and July 2002 in the central plateau of the Greenland inland ice. The German research aircraft Polar2, equipped with the turbulence measurement system Meteopod, was used to investigate turbulence and radiation flux profiles near research station “Summit Camp”. Aircraft measurements are combined with measurements of radiation fluxes and turbulent quantities made from a 50 m tower at Summit Camp operated by Eidgenössische Technische Hochschule (ETH) Zürich. During all six flight missions, well-developed stable boundary layers were found. Even in high-wind conditions, the surface inversion thickness did not exceed roughly 100 m. The turbulent height of the stable boundary layer (SBL) was found to be much smaller than the surface inversion thickness. Above the surface layer, significant turbulent fluxes occurred only intermittently in intervals on the order of a few kilometres. Turbulent event fraction in the upper SBL shows the same dependence on gradient Richardson number as reported for near-surface measurements. Clear-air longwave radiation divergence was always found to contribute significantly to the SBL heat budget. In low-wind cases, radiative cooling even turned out to be dominant.     

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.     

Intermittency of Turbulence in the Atmospheric Boundary Layer: Scaling Exponents and Stratification Influence

We show the relationship between the intermittency of turbulence and the type of stratification for different atmospheric situations during the SABLES98 field campaign. With this objective, we first demonstrate the scaling behaviour of the velocity structure functions corresponding to these situations; next, we analyze the curvature of the scaling exponents of the velocity structure functions versus the order of these functions (ζ _p vs. p), where ζ _p are the exponents of the power relation for the velocity structure function with respect to the scale. It can be proved that this curve must be concave, under the assumption that the incompressible approximation does not break down at high Reynolds numbers. The physical significance of this kind of curvature is that the energy dissipation rate increases as the scale of the turbulent eddies diminishes (intermittency in the usual sense). However, the constraints imposed by stability, preventing full development of the turbulence, allow the function ζ _p versus p to show any type of curvature. In this case, waves of high frequency trapped by the stability, or bursts of turbulence caused by the breaking up of internal waves, may produce a redistribution of energy throughout the scaling range. Due to this redistribution, the variation with the scale of the energy dissipation rate may be smaller (decreasing the intermittency) and, even in more stable situations, this rate may diminish (instead of increasing) as the scale diminishes (convex form of the curve ζ _p vs. p).     

Modelling of the Evolving Stable Boundary Layer

A single-column model of the evolving stable boundary layer (SBL) is tested for self-similar properties of the flow and effects of ambient forcing. The turbulence closure of the model is diagnostic, based on the K-theory approach, with a semi-empirical form of the mixing length, and empirical stability functions of the Richardson number. The model results, expressed in terms of local similarity scales, are universal functions, satisfied in the entire SBL. Based on similarity expression, a realizability condition is derived for the minimum allowable turbulent heat flux in the SBL. Numerical experiments show that the development of “horse-shoe” shaped, fixed-elevation hodographs in the interior of the SBL around sunrise is controlled by effects imposed by surface thermal forcing.     

The influence of stratification on heat and momentum turbulent transfer in antarctica

Data from the Antarctic winter at Halley Base have been used in order to evaluate qualitatively and quantitatively how the stratification in the low atmosphere (evaluated with the gradient Richardson number, Ri) influences the eddy transfers of heat and momentum. Vertical profiles of wind and temperature up to 32 m, and turbulent fluxes ( , and ) measured from three ultrasonic thermo-anemometers installed at 5, 17 and 32 m are employed to calculate Ri, the friction velocity (u _*) and the eddy diffusivities for heat (K _h ) and momentum (K _m ). The results show a big dependence of stability onK _m ,K _h andu _*, with a sharp decrease of these turbulent parameters with increasing stability. The ratio of eddy diffusivities (K _h /K _m ) is also analyzed and presents a decreasing tendency as Ri increases, reaching values even less than 1, i.e., there were situations where the turbulent transfer of momentum was greater than that of heat. Possible mechanisms of turbulent mixing are discussed.     

Resolution Sensitivity and Scaling of Large-Eddy Simulations of the Stable Boundary Layer

Large-eddy simulations (LES) of the continuously turbulent quasi-equilibrium stable boundary layer (SBL) are conducted with grid lengths in the range of 12.5 m to 2 m, in order to explore resolution sensitivity, and determine at what point grid convergence occurs. The structure of the mean potential temperature, winds, and turbulent fluxes varies significantly over this resolution range. The highest resolution simulations show a significant degree of convergence. The dimensionless momentum diffusivity asymptotes to a value of 0.06, corresponding to a limiting flux Richardson number of 0.15.Using the converged simulations, some scaling hypotheses underpinning first-order and second-order closure models are revisited. The effective Richardson number stability functions of the LES are compared with the forms often used in numerical weather prediction (NWP). The mixing implied by the LES is less than that used in NWP. The commonly used similarity profiles for heat and momentum fluxes, and the scalings for dissipation and pressure covariances are compared with the LES. This information could provide guidance for the next generation of SBL parametrization schemes.     

Sublayer-Stanton numbers of heat and matter for aerodynamically smooth surfaces: basic considerations and evaluation

Summary Sublayer-Stanton numbers, B_i, of heat and matter for the interfacial sublayer over aerodynamically smooth surfaces determined for forced convective conditions by elementary and numerical integration are reviewed and evaluated. The results are based on Roth’s modified Heisenberg model for the spectral energy transfer in the equilibrium range under locally isotropic conditions and the approaches of Reichardt, Elser, Deissler, van Driest, Rannie, Sheppard, and Spalding for the normalized eddy diffusivity K_m/v. The results substantiate that with the exception of Sheppard’s K_m/v-approach all formulations are appropriate to provide sublayer-Stanton numbers with a sufficient degree of accuracy. From a theoretical point of view the K_m/v-relationships of Roth, Reichardt, van Driest, and Spalding are to be preferred, when a turbulent Prandtl number Pr_t = 1 is presumed. Since within the framework of mesoscale meteorological modelling numerical integration techniques would consume too much CPU-time because of the large number of near-wall grid points, a parameterization formula for the sublayer-Stanton number is proposed and evaluated. Compared to the B_i ⁻¹-results obtained by numerical integration, this kind of parameterization leads to a relative error of less than 5 percent for roughness Reynolds numbers, η_r, ranging from 30 to 600. Received January 2001 Revised November 5, 2001