Evaluating Models of The Neutral, Barotropic Planetary Boundary Layer using Integral Measures: Part I. Overview

Data for the cross-isobaric angle ₀, the geostrophic drag coefficient C_g, and the functions A and B of Rossby number similarity theory, obtained from meteorological field experiments, are used to evaluate a range of models of the neutral, barotropic planetary boundary layer. The data give well-defined relationships for ₀, C_g, and the integrated dissipation rate over the boundary layer, as a function of the surface Rossby number. Lettau's first-order closure mixing-length model gives an excellent fit to the data; other simple models give reasonable agreement. However more sophisticated models, e.g., higher-order closure, large-eddy simulation, direct numerical simulation and laboratory models, give poor fits to the data. The simplemodels have (at least) one free parameter in their turbulence closure that is matched toatmospheric observations; the more sophisticated models either base their closure onmore general flows or have no free closure parameters. It is suggested that all of theatmospheric experiments that we could locate violate the strict simplifying assumptionsof steady, homogeneous, neutral, barotropic flow required by the sophisticated models.The angle ₀ is more sensitive to violations of the assumptions than is C_g.The behaviour of the data varies in three latitude regimes. In middle and high latitudes the observed values of A and B exhibit little latitudinal dependence; the best estimates are A = 1.3 and B = 4.4. In lower latitudes the neutral, barotropic Rossby number theory breaks down. The value of B increases towards the Equator; the determination of A is ambiguous – the trend can increase or decrease towards the Equator. Between approximately 5° and 30° latitude, the scatter in the data is thought to be primarily due to the inherent presence of baroclinicity. The presence of the trade-wind inversion, thermal instability and the horizontal component of the Earth's rotation _H also contribute.Marked changes in the values of A and B occur in the region between the Equator andapproximately 5° latitude, as the Coriolis parameter |f| approaches zero. Although the variation of A and B with latitude suggests some similarity to the results obtained from the direct numerical simulations, the presence of additional complexities in the real atmosphere that are not included in the numerical model, precludes a meaningful direct comparison.     

Modification Of The Standard ∈-Equation For The Stable Abl Through Enforced Consistency With Monin–Obukhov Similarity Theory

A condition is derived for consistency of the standard-equation with Monin–Obukhov (MO) similarity theory of thestably-stratified surface layer. The condition is derivedby extending the procedure used to derive the analogous condition forneutral theory to stable stratification. It is shown that consistencywith MO theory requires a function of flux Richardson number, Ri_f, to be absorbed into either of two closure parameters, c ₁ or c ₂.Inconsistency, on the other hand, results if constant values of these are maintained for all Ri_f, as is done in standardapplication of the equation, and the large overpredictions ofturbulence found in such application to the one-dimensionalstable atmospheric boundary layer (1D-SBL) are traced to thisinconsistency. Guided by this, we formulate a MO-consistent-equation by absorbing the aforementioned function intoc ₁, and combine this with a Level-2.5 second-orderclosure model for vertical eddy viscosity and diffusivities.Numerical predictions of the 1D-SBL by the modified model converge to a quasi-steady state, rectifying the predictive failure of the standard -equation for the case.Quasi-steady predictions of non-dimensional variables agree stronglywith Nieuwstadt's theory. Qualitative accuracy of predictionsis inferred from comparisons to field data, large-eddy simulationresults and Rossby-number similarity relationships.     

Chronology of ‘dry fogs’ in Italy, 1374–1891

Summary A chronology and some comments about the occurrence of dry fogs i.e. clouds of volcanic aerosols trapped in the planetary boundary layer (PBL) are reported. The dry fogs have severely affected the biosphere, and they also had local climatic effects affecting the terrestrial albedo. In the most severe cases the fog persisted for months and obscured the sun, in the lesser ones a mist persisted for a few days and the sun was seen as weak and the moon with a red halo. The volcanic aerosols were trapped in the PBL in the warm season, when the atmosphere is particularly stable over the relatively cold Mediterranean waters and the Azores Anticyclone causes the absence of winds.     

Application of the E-ε closure model to simulations of mesoscale topographic effects

A mesoscale Planetary Boundary Layer (PBL) model with a simple turbulence closure scheme based on the turbulence kinetic energy (TKE) equation and the dissipation () equation is used to simulate atmospheric flow over mesoscale topography. Comparative studies with different parameterizations suggest that with a proper closure assumption for turbulence dissipation, the E-model can simulate the circulation induced by the mesoscale topography with results similar to those obtained using the E- model. On the other hand, the first-order closure using O'Brien's cubic interpolation for eddy diffusivities (K) generally produces much larger K profiles in the stable and the unstable regions, which is believed to be due to the overprediction of the height of the PBL. All models with the TKE equation yield quite similar ensemble mean fields, which are found to be little sensitive to the closure assumption for turbulence dissipation, though their predicted magnitudes of TKE and K may differ appreciably. A discussion on the diurnal evolution of the mesoscale topography-induced circulation and the spatial variations of the turbulence fluxes in the surface layer is also given based on the E- model results.     

A planetary boundary-layer numerical model containing third-order derivative terms

In this paper, the third-order derivative of velocity with respect to height is included in the traditional motion equations of the neutral PBL. The nonlinear equations are solved numerically to obtain the vertical distribution of wind in the PBL and some PBL characteristic parameters. Reasonable simulations of the Leipzig wind profile using these parameters show the success of this kind of nonlocal closure in a real PBL simulation.     

Mean stresses and velocities in the neutral,barotropic planetary boundary layer

The purpose of the paper is to find the mean velocities and stresses in the turbulent, neutral, barotropic planetary boundary layer (PBL). Correction functions are introduced similar to those used by Millikan and Hinze in discussions of flows in a pipe and in a turbulent boundary layer. The functions for the PBL are determined semi-empirically and, with a choice of constants, the resulting velocity distributions are in reasonable agreement with the Leipzig profile. The paper also discusses the correction functions for pipe and boundary-layer flows and for plane Couette flow. The results are in excellent agreement with observations.     

Stability functions correct at the free convection limit and consistent for for both the surface and Ekman layers

For the thermal stability function _h used to calculate heat and moisture fluxes in the surface layer, we choose a formulation which has the theoretically correct free convection limit % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeikaiabgk% HiTGqaciaa-PhacaqGVaGaamitaiaabMcadaahaaWcbeqaaiabgkHi% TiaaigdacaGGVaGaaG4maaaaaaa!3DFE!\[{\rm{(}} - z{\rm{/}}L{\rm{)}}^{ - 1/3} \]. We then use the experimental result that z/L Ri to deduce a formulation with an exponent -1/6 for the momentum stability function _m. This formulation also resolves the matching problem at the interface between the surface and Ekman layers. The proposed functions are found to remain reasonably close to another formulation that is well supported by observations and has exponents -1/2 for _h and -1/4 for _m. The intent of the proposals is mainly to clarify and simplify the parameterization of the convective boundary layer in present day atmospheric models, without significantly altering the results.     

The behaviour of a mixed-layer model of the convective boundary layer coupled to a big leaf model of surface energy partitioning

Daily mean values of the Priestley-Taylor coefficient, ¯, are derived from a simple model of the daily growth of a convective boundary layer. For a particular control set of driving environmental variables, ¯ is related to the prescribed bulk surface resistance, r_S by 1/¯ = 1/₀ + mr_S for parameters ₀ and m. The dependence of the parameters ₀ and m on weather is explored and a potential use of this linear relation to provide information about regional values of r_S is indicated.     

Transport-Dissipation Analytical Solutions to the E-∈Turbulence Model and their Role in Predictions of the Neutral ABL

E- turbulence model predictions of the neutralatmospheric boundary layer (NABL) are reinvestigated to determine thecause for turbulence overpredictions found in previous applications. Analytical solutions to the coupled E and equations for the case of steady balance between transport and dissipation terms, the dominant balance just below the NABL top, are derived. It is found that analytical turbulence profiles laminarizeat a finite height only for values of closure parameter ratio c ₂ /_e equal toor slightly greater than one, with laminarization as z for greater . The point = 2 is additionally foundthat where analytical turbulent length scale (l) profilesmade a transition from ones ofdecreasing ( < 2) to increasing ( > 2)values with height. Numerically predicted profiles near the NABL topare consistent with analytical findings. The height-increasingvalues of l predicted throughout the NABL with standard values ofclosure parameters thus appear a consequence of 2.5(> 2), implied by these values (c ₂ = 1.92, _{= 1.3}, _{e = 1}). Comparison of numericalpredictions with DNS data shows that turbulence overpredictions obtained with standard-valued parameters are rectifiedby resetting and _e to 1.1 and 1.6, respectively, giving, with c ₂ = 1.92, 1.3, and laminarization of the NABL's cappingtransport-dissipation region at a finite height.     

Determination of similarity functions of the resistance laws for the planetary boundary layer using surface-layer similarity functions

Functional forms of the universal similarity functions A, B (for wind components parallel and normal to the surface stress), and C (for potential temperature difference) are determined based on the generalized theory of the resistance laws for the Planetary Boundary Layer (PBL). The similarity-profile functions for the surface layer are matched with the velocity and temperature-defect profiles that are assumed to have shapes modified by certain powers of nondimensional height z/h, where h is the PBL height. The powers of the outer-layer profile functions are determined, so that the functions become negligible in the surface layer. To close the temperature defect law, an assumption that the temperature gradient across the top of the PBL is continuous with the stratification of the overlying atmosphere is used. The result of this assumption is that nondimensional momentum and temperature profiles in the PBL can be described in terms of four basic ratios: (1) roughness ratio = /h (2) scale-height ratio =|f|h/u_*, (3) ambient stratification parameter =h/_*, and (4) stability parameter =h/L, where L is the Monin-Obukhov length, z₀ is the surface roughness, is the upper-air stratification, u _* is the friction velocity, and _* is the temperature scale at the surface. For stable conditions, the scale-height ratio can be related to the atmospheric stability and the upperair stratification, and the generalized similarity and Rossby number similarity theories become identical. Under appropriate boundary conditions, function A is explicitly dependent on the stability parameter , while B is a function of scale-height ratio , which in turn depends on the stability. Function C is shown to be dependent on the stability and the upper-air stratification, due to the closure assumption used for the temperature profile.The suggested functional forms are compared with other empirical approximations by several authors. The general framework used to determine the functional forms needs to be tested against good boundary-layer measurements.     

Modelling of turbulence over low hills and varying surface roughness

Neutral surface layer flow over low hills and varying surface roughness is considered with emphasis on closure schemes in relation to the prediction of turbulence quantities. The equations are linearised, Fourier transformed in the two horizontal directions and solved by means of a finite difference method in the vertical. Three closure schemes are. employed, namely mixing length, E- and e-- closure where E, and indicate that differential equations are used for turbulent kinetic energy, dissipation rate and shear stress. Model calculations are compared with experimental data for the step in roughness problem and for the Askervein hill. The mean flow results turn out to be relatively insensitive to the closure scheme. The shear stress and the dimensionless shear, however, are much better predicted with the E- equations than with mixing length closure. In the outer layer of the hill problem, advection of shear stress becomes important. An equation for is needed here.     

Models of eddy viscosity for numerical simulation of horizontally inhomogeneous,neutral surface-layer flow

Modification of a turbulent flow due to a change from a smooth to a rough surface has been studied by means of a stream function-vorticity model. Results of four models of eddy viscosity (or turbulent exchange coefficient) K _mhave been compared. The models are: (1) K _m = l²S, where l is the mixing length and S is the deformation of mean flow; (2) K _m E/S, which is based on the assumption that turbulent momentum flux is proportional to turbulent kinetic energy E; (3) K _m lE^1/2, the so called Prandtl-Kolmogoroff approach; and (4) K _m E²/, the E — closure, where is the dissipation of turbulent kinetic energy.It is found that net-production, i.e., the difference of production and dissipation of turbulent kinetic energy counteracts the influence of mean shear on turbulent shear stress and diminishes turbulent shear stress. The reduction of mixing-length, being predicted by Model 4 only, adds to this attenuation. As a consequence, in Models 2 and 4, loss of horizontal mean momentum is concentrated close to the ground, which results in an inflexion point in the logarithmic, vertical profile of horizontal mean velocity. By contrast, in Models 1 and 3, modification of turbulent shear stress reaches larger heights causing deeper internal boundary layers. Concerning the existence of an inflexion point in U(lnz), the depth of the internal boundary layer for mean velocity, and the modification of bottom shear stress, Model 4 comes closest to experimental data.A remarkable difference of Models 1, 2, 3 and Model 4 is that only Model 4 predicts a very slow relaxation of eddy viscosity which can be attributed to the reduction of mixing-length.     

Study On Development And Application Of A Regional Pbl Numerical Model

A regional scale (a few km to a few hundred km, minutes to days)planetary boundary layer (PBL) numerical model (RPBLM) has been successfully developed to simulateprecisely the boundary-layer three-dimensional characteristics. The RPBLM model with a high-resolutionnon-hydrostatic E - turbulence closure scheme has been applied to diagnose and prognosticate the PBL characteristicsover a complex underlying surface in the Hong Kong area of 60 km × 48 km. It is shown that the RPBLMmodel can be used to simulate the PBL characteristics including wind, temperature, water vapour andturbulence over such a complex underlying surface and that the simulated result is reasonably in agreementwith observations.     

Numerical simulation of January 28 cold air outbreak during GALE Part I: The model and sensitivity tests of turbulence closures

A mesoscale planetary boundary layer (PBL) numerical model has been developed to study airflow over complex topography. Turbulence closures using the turbulent kinetic energy (TKE) and dissipation () equations are investigated in combination with the level 2.5 scheme of Mellor and Yamada (1982) to determine eddy diffusivities for momentum and heat. This modified E- closure is simpler than the level 3 one which requires more prognostic equations for moist turbulent transport.One-dimensional (1-D) model results show that the PBL mean flows under various stability conditions are not significantly sensitive to the modified Blackadar and Kolmogorov eddy mixing-length formulations used in this E- model, although the latter yields excessively large mixing lengths in the entrainment region of the upper PBL. Eddy mixing lengths in the Kolmogorov-type formulation can be better defined by introducing background dissipation. Using the same prognostic TKE equation, the 1-D model results are not significantly affected by different diagnostic formulations in the closures. The simulated results compare well with large-eddy simulations and those obtained using higher-order closure schemes including the level 3 one. The results are found to be insensitive to eddy Prandtl number, in contrast to the 2-D model results (see Part II).     

Characteristics of the Lyman-alpha humidiometer and measurements of turbulent humidity fluctuations over the ocean

The characteristics of a Lyman-alpha humidiometer have been carefully examined in an air-conditioned test chamber. The results confirm that when carefully used, this humidiometer is suitable for measurements of turbulent humidity fluctuations. Measurements with a Lyman-alpha humidiometer were carried out in the surface boundary layer over the ocean. The relation between turbulent intensity ( _a = a ^ov2) and the friction humidity (a _*) can be expressed as _a = l.6a _*. The spectrum of turbulent humidity for wind speeds larger than 3 m s ^–1 conforms to the similarity law in the surface boundary layer. The spectrum has two characteristic normalized frequencies, namely, a higher peak and a secondary peak (or a shoulder).     

A new theory of the energy spectrum

In a recent paper, the author introduced a new viscous boundary layer, called the mesolayer, in turbulent shear flow. Its importance stems from its location between the inner and outer regions which are controlled by the law of the wall and Reynolds number similarity, respectively. This intrusion prevents the classical overlap assumption which appears to be fundamental in the derivation of the classical logarithmic behavior. The mesolayer has a thickness proportional to Taylor's microscale . This, and the analogy between the energy equation for the spectrum function of isotropic turbulence and the momentum equation for shear flow, suggest the existence of a similar region in wavenumber space with wavenumber k ~ ^-1. This mesoregion separates the inner region k ~ k _s(where k _s^-1 and is the Kolmogorov length) and the outer region k k _e(where k _e ^-1 and l is the energy-containing eddy size) and again invalidates the overlap assumption which appears to be fundamental in the derivation of the classical k ^-5/3-behavior of the inertial subrange.Incorporation of the mesoregion into the argument leads to a new theory with k ^-5/3-behavior in two regions (^-1 k k _s) and (k _e k ^-1) although with two different coefficients of proportionality (Kolmogorov constants). This leads to a wandering of the spectrum curve about the classical k ^-5/3 line similar to a wandering in turbulent shear flow about the logarithmic curve. This is clearly indicated by the data for the variation of the Kolmogorov constant.Other data support the new theory. In particular, the location of the point k _mwhere the curve of the nonlinear energy-transfer function goes through zero shows agreement with the theory, i.e., k _m^-1.     

Convective Profile Constants Revisited

This paper examines the interpolation betweenBusinger–Dyer (Kansas-type) formulae,_u = (1 -1 6 )^-1/4 and_t = (1 - 16 )^-1/2, and free convection forms. Based on matching constraints, the constants, a_u and a_t, in the convective flux-gradient relations, _u = (1 - a_u )^-1/3 and _t = (1 - a_t )^-1/3, are determined. It isshown that a_u and a_t cannot be completely independent if convective forms are blended with theKansas formulae. In other words, these relationships already carryinformation about a_u and a_t. This follows because the Kansas relations cover a wide stability range (up to = - 2), which includes a lower part of the convective sublayer (about 0.1 < - < 2). Thus, there is a subrange where both Kansas and convective formulae are valid. Matching Kansas formulae and free convection relations within thesubrange 0.1 < - < 2 and independently smoothing ofthe blending function are used to determine a_u and a_t. The values a_u = 10 for velocity and a_t = 34for scalars (temperature and humidity) give a good fit. This new approacheliminates the need for additional independent model constants and yields a`smooth' blending between Kansas and free-convection profileforms in the COARE bulk algorithm.     

Formulation of the thermal internal boundary layer in a mesoscale model

Mesoscale models using a non-local K-scheme for parameterization of boundary-layer processes require an estimate of the planetary boundary layer (PBL) height z _i at all times. In this paper, two-dimensional sea-breeze experiments are carried out to evaluate three different formulations for the advective contribution in the z _i prognostic equation of Deardorff (1974).Poor representation of the thermal internal boundary layer in the sea breeze is obtained when z _i is advected by the wind at level z _i . However, significantly better results are produced if the mean PBL wind is used for the advecting velocity, or if z _i is determined simply by checking for the first sufficiently stable layer above the ground.A Lagrangian particle model is used to demonstrate the effect of each formulation on plume dispersion by the sea breeze.     

Turbulence Closures In Neutral Boundary Layer Over Complex Terrain

In this work, three turbulence closure models, Mellor andYamada level 2.5, E - l and E - implemented in a circulation model, are compared in neutral condition over complex terrain. They are firstly applied to a one-dimensional case on flat terrain and then to a schematic two-dimensional valley. The simulation results, in terms of wind field and turbulent kinetic energy, are tested against measurements from a wind-tunnel experiment. The empirical constants defining the characteristic length scales of the closures are modified based on turbulence parameters estimated in the experiment. The formulation of the diffusion coefficients is analysed to explain the differences among the various closures in the simulation results. Regarding the mean flow, both on flat and complex terrain, all the closures yield satisfactory results. Concerning the turbulent kinetic energy, the best results are obtained by E - l and E - closures.     

The Neutral, Barotropic Planetary Boundary Layer, Capped by a Low-Level Inversion

The presence of a low-level, capping inversion layer will affect the height and structure of the planetary boundary layer (PBL). Results from models of varying levels of sophistication, including analytical, turbulent kinetic energy (TKE), second-order closure (SOC), large-eddy simulation (LES) and direct numerical simulation (DNS) models, are used to investigate this influence for the neutral, barotropic PBL. Predicted and observed profiles of stress and geostrophic departure components, and integral measures, such as the parameters of Rossby-number similarity theory, are compared for the KONTUR, Marine Stratocumulus, JASIN, Leipzig, Pre-Wangara and Upavon field experiments.Analytical models of the equilibrium value of inversion height z_i, which depend on the surface friction velocity u_*, and both the Coriolis parameter f and the free-flow Brunt-Väisälä frequency N, are found to give reasonable estimates of the PBL height. They also indicate that only the KONTUR and Marine Stratocumulus experiments were strongly influenced by N. More quantitative comparisons would require larger, more comprehensive datasets. The effects of the presence of a capping inversion on the profile structure were found to be insignificant for h_* = |f|z_i/u_* > 0.15.The simple analytical model performed quite well over all values of h_*; it predicted the profiles of the longitudinal stress component (in the direction of the surface stress) better than the lateral component. The more advanced models performed well for small values of h_* (for flow over the sea), but systematically underestimated the cross-isobaric angle for flow over land. These models predicted the profiles of the lateral stress component better than the longitudinal component. The profiles of the analytical model agreed with those of the advanced models when the constant eddy viscosity of the outer layer was increased.Agreement with DNS was achieved by increasing the eddyviscosity of the analytical model by a factor of 5.Zilitinkevich and Esau(2002, Boundary-Layer Meteorology, 104, 371–379)suggest that the neutral, barotropic values of A and B of Rossby-numbersimilarity theory are not universal constants, but depend on the ratio N/|f|. The dependence for A and B is calculated using the analytical model and TKE models. Over the sea (h_* 0.1; N/|f| 100, where we have used the Zilitinkevich-Esau relation to convert between h_* and N/|f|) there is agreement between the model predictions and observations; however over land where the equilibrium boundary-layer height is greater (h_* 0.35; N/|f| 10) the inconsistency between the advanced model predictions (TKE, SOC, LES, and DNS) and observations, as noted previously by Hess and Garratt, still exists. We attribute this disagreement to violations of the strict assumptions of steady, horizontally homogeneous, neutral, barotropic conditions implicit in the observations. At small values of z_i and a strongly stable background stratification (h_* 0.04; N/|f| 1000) both the TKE and analytical models predict that A and B depend significantly on h_*, however observations are unavailable to confirm these predictions. Zilitinkevich and Esau call this case the `long-lived near-neutral PBL', and state that it is found in cold weather at high latitudes.