The temperature profile and heat transfer law in a neutrally and stably stratified planetary boundary layer

A logarithmic + polynomial approximation is proposed for the vertical temperature profile in a neutrally or stably stratified planetary boundary layer (PBL) in conditions of quasi-stationarity. Using this approximation with the asymptotic logarithmic + linear law of the Monin-Obukhov similarity theory for the near-surface layer and with the Zilitinkevich formula for the PBL thickness allows one to derive an analytical expression for the function C in the heat transfer law, which permits simple parameterization of the thermal interaction between the atmosphere and the underlying medium in terms of external parameters, such as the geostrophic wind velocity and the temperature difference across the PBL.     

Internal gravity waves in a stably stratified boundary layer

Often, a combination of waves and turbulence is present in the stably stratified atmospheric boundary layer. The presence of waves manifest itself in the vertical profiles of variances of fluctuations and in low-frequency contributions to the power spectra. In this paper we study internal waves by means of a linear stability analysis of the mean profiles in a stably stratified boundary layer and compare the results with observed vertical variance profiles of fluctuating wind and temperature along a 200 m mast. The linear stability analysis shows that the observed mean flow is unstable for disturbances in a certain frequency and wavenumber domain. These disturbances are expected to the detectable in the measurements. It is shown that indeed the calculated unstable frequencies are present in the observed spectra. Furthermore, the shape of the measured vertical variance profiles, which increase with height, is explained well by the calculated vertical structure of the amplitude of unstable Kelvin-Helmholtz waves, confirming the contribution of waves to the variances. Because turbulence and waves have quite distinct transport properties, estimates of diffusion from measurements of variances would strongly overestimate this diffusion. Therefore it is important to distinguish between them.     

稳定行星边界层相似性函数A,B,C的实验研究

本文使用位于粗糙和不均匀地形上北京320米气象塔夜间边界层风速、风向和温度资料,用行星边界层u分量第一个极值高度作为行星边界层高度尺度,用边界层内的平均风速和平均位温作为速度和位温特征尺度,分析得到在稳定条件下相似性函数A_m、B_m和C_m以及行星边界层高度的参数化表达式。 分析表明,可以将广义相似性理论推广到粗糙而不均匀地形上的大气边界层中。     

The stably stratified internal boundary layer for steady and diurnally varying offshore flow

A two-dimensional numerical mesoscale model is used to investigate the internal structure and growth of the stably stratified internal boundary layer (IBL) beneath warm, continental air flowing over a cooler sea. Two situations are studied — steady-state and diurnally varying offshore flow. In the steady-state case, vertical profiles of mean quantities and eddy diffusion coefficients (K) within the IBL show small, but significant, changes with increasing distance from the coast. The top of the IBL is well defined, with large vertical gradients within the layer and a maximum in the coast-normal wind component near the top. Well away from the coast, turbulence, identified by non-zero K, decreases to insignificant levels near the top of the IBL; the IBL itself is characterised by a critical value of the layer-flux Richardson number equal to 0.18. The overall behaviour of the mean profiles is similar to that found in the horizontally homogeneous stable boundary layer over land.A simple physical model is used to relate the depth of the layer h to several relevant physical parameters viz., x, the distance from the coast and U, the large-scale wind (both normal to the coastline) and g/, being the temperature difference between continental mixed-layer air and sea surface, is the mean potential temperature and g is the acceleration due to gravity. Excellent agreement with the numerical results is found, with h = 0.014x ^1/2 U (g/)^–1/2.In the diurnally varying case, the mean profiles within the IBL show only small differences from the steady-state case, although diurnal variations, particularly in the wind maximum, are evident within a few hundred kilometres of the coast. A mesoscale circulation normal to the coast, and superimposed upon the mean offshore flow, develops seawards of the coastline with maximum vertical velocities about sunset, of depth about 2 km and horizontal scale 500 km. The circulation is related to the advection, and subsequent decay, of daytime convective turbulence over the sea.     

The structure of the stably stratified internal boundary layer in offshore flow over the sea

Observations obtained mainly from a research aircraft are presented of the mean and turbulent structure of the stably stratified internal boundary layer (IBL) over the sea formed by warm air advection from land to sea. The potential temperature and humidity fields reveal the vertical extent of the IBL, for fetches out to several hundred of kilometres, geostrophic winds of 20–25 m s^–1, and potential temperature differences between undisturbed continental air and the sea surface of 7 to 17 K. The dependence of IBL depth on these external parameters is discussed in the context of the numerical results of Garratt (1987), and some discrepancies are noted.Wind observations show the development of a low-level wind maximum (wind component normal to the coast) and rotation of the wind to smaller cross-isobar flow angles. Potential temperature () profiles within the IBL reveal quite a different structure to that found in the nocturnal boundary layer (NBL) over land. Over the sea, profiles have large positive curvature with vertical gradients increasing monotonically with height; this reflects the dominance of turbulent cooling within the layer. The behaviour is consistent with known behaviour in the NBL over land where curvature becomes negative (vertical gradients of decreasing with height) as radiative cooling becomes dominant.Turbulent properties are discussed in terms of non-dimensional quantities, normalised by the surface friction velocity, as functions of normalised height using the IBL depth. Vertical profiles of these and the normalised wavelength of the spectral maximum agree well with known results for the stable boundary layer over land (Caughey et al., 1979).     

Similarity of structure-function parameters in the stably stratified boundary layer

The structure-function parametersC _T ² andC _v ² of temperature and velocity, respectively, from the 1973 Minnesota experiments and from large-eddy and direct numerical simulations show a smooth transition from M–O similarity to the local scaling hypothesized by Nieuwstadt for the outer regions of the stable boundary layer. Under that hypothesis, turbulence statistics aloft depend on the local vertical fluxes of momentum and temperature, so these results suggest that remote-sensing measurements ofC _T ² andC _v ² could be used to infer vertical profiles of those fluxes. We argue that the sensitivity of the fluxes to unsteadiness, baroclinity, terrain slope, and breaking gravity waves precludes the universality of the vertical profiles of structure-function parameters in the stable PBL. We find that theC _T ² profile is particularly sensitive to these effects, which is consistent with observations that it varies considerably from case to case.     

Study of near-surface models for large-eddy simulations of a neutrally stratified atmospheric boundary layer

In large-eddy simulations (LES) of the atmospheric boundary layer (ABL), near-surface models are often used to supplement subgrid-scale (SGS) turbulent stresses when a major fraction of the energetic scales within the surface layer cannot be resolved with the temporal and spatial resolution at hand. In this study, we investigate the performance of both dynamic and non-dynamic eddy viscosity models coupled with near-surface models in simulations of a neutrally stratified ABL. Two near-surface models that are commonly used in LES of the atmospheric boundary layer are considered. Additionally, a hybrid Reynolds- averaged/LES eddy viscosity model is presented, which uses Prandtl’s mixing length model in the vicinity of the surface, and blends in with the dynamic Smagorinsky model away from the surface. Present simulations show that significant portions of the modelled turbulent stresses are generated by the near-surface models, and they play a dominant role in capturing the expected logarithmic wind profile. Visualizations of the instantaneous vorticity field reveal that flow structures in the vicinity of the surface depend on the choice of the near-surface model. Among the three near-surface models studied, the hybrid eddy viscosity model gives the closest agreement with the logarithmic wind profile in the surface layer. It is also observed that high levels of resolved turbulence stresses can be maintained with the so-called canopy stress model while producing good agreement with the logarithmic wind profile.     

A multi-limit formulation for the equilibrium depth of a stably stratified boundary layer

Currently no expression for the equilibrium depth of the turbulent stably-stratified boundary layer is available that accounts for the combined effects of rotation, surface buoyancy flux and static stability in the free flow. Various expressions proposed to date are reviewed in the light of what is meant by the stable boundary layer. Two major definitions are thoroughly discussed. The first emphasises turbulence and specifies the boundary layer as a continuously and vigorously turbulent layer adjacent to the surface. The second specifies the boundary layer in terms of the mean velocity profile, e.g. by the proximity of the actual velocity to the geostrophic velocity. It is shown that the expressions based on the second definition are relevant to the Ekman layer and portray the depth of the turbulence in the intermediate regimes, when the effects of static stability and rotation essentially interfere. Limiting asymptotic regimes dominated by either stratification or rotation are examined using the energy considerations. As a result, a simple equation for the depth of the equilibrium stable boundary layer is developed. It is valid throughout the range of stability conditions and remains in force in the limits of a perfectly neutral layer subjected to rotation and a rotation-free boundary layer dominated by surface buoyancy flux or stable density stratification at its outer edge. Dimensionless coefficients are estimated using data from observations and large-eddy simulations. Well-known and widely used formulae proposed earlier by Zilitinkevich and by Pollard, Rhines and Thompson are shown to be characteristic of the above interference regimes, when the effects of rotation and static stability (due to either surface buoyancy flux, or stratification at the outer edge of the boundary layer) are roughly equally important.     

An examination of local similarity theory in the stably stratified boundary layer

A stable boundary layer is investigated in terms of local similarity theory. A study is based on a set of seven runs from the BAO tower (Colorado, U.S.A.). It is shown that a theoretical prediction of constant-with-height similarity functions applies only to ensemble-averaged quantities. Scatter of observational data is analysed.On leave from: Institute of Environmental Engineering, Warsaw Technical University, 00653, Warsaw, Poland.     

Some aspects of the structure of the stably stratified atmospheric boundary layer

Results of an analysis based on the Leipzig wind data are presented. The computed stress profiles and the proportionality constant in the inverse relationship between the non-dimensionalized stable boundary-layer height and the square root of the stability parameter seem to agree well with the model results of Businger and Arya (1974).     

The temperature structure in a stably stratified internal boundary layer over a cold sea

Data from the Öresund experiment are used to investigate the structure of the stably stratified internal boundary layer (SIBL) which develops when warm air is advected from a heated land surface over a cooler sea. The present study is based on a theory developed by Stull (1983a, b, c). He proposed that the turbulence and the mean structure of the nocturnal boundary layer is controlled by the time-integrated value of surface heat flux and that the instantaneous heat flux is of less importance.Dimensional arguments are used to define simple, physically consistent, temperature, velocity and length scales. The dimensionless surface heat flux has a high value immediately downwind of the shoreline and it decreases rapidly in magnitude with increasing distance from the coast. Farther away, it is essentially constant. The dimensionless potential temperature change exhibits an exponential profile. It is estimated that turbulence accounts for 71% of boundary-layer cooling while clear-air radiational cooling is responsible for the remaining 29%.Finally it is found that theoretical predictions for the height of the SIBL are in a good agreement with observations.     

A similarity model for maximum ground-level concentration in a height-invariant,stably stratified atmospheric boundary layer

A model is presented for determining the location and magnitude of the maximum ground-level concentration arising from an elevated buoyant source in a very stable atmospheric boundary layer. The development combines the turbulent structure of such a boundary layer, Lagrangian similarity of the diffusion process, and similarity solutions of the conservation equations of the buoyant plume with mass conservation to produce a simple, experimentally verifiable formulation. Functional analogy with previous results for the constant flux layer and a deep convectively unstable layer suggest a heuristic model by which to visualize the process.     

A numerical study of second-order turbulent moments in the stably stratified nocturnal boundary layer

The structures and the vertical profiles of turbulent variance and covariance of the stably stratified boundary layer (SBL) are simulated with a second-order closure turbulence model. The results confirm that the vertical profiles of the dimensionless turbulence variance and covariance can be well represented by the form F = A(1 - Z / h)x. Here h is the height of SBL. and both exponent a and coefficient A are the functions of terrain, baroclinicity, radiation cooling and the state of temporal development of SBL. Comparing with Minnesota and Cabauw experiment data, we have analysed the value of a and expounded the main reasons that great difference in a exists among different literatures.     

Similarity theory and resistance laws for the planetary boundary layer

Methods are developed for the determination of parameters of the atmospheric planetary boundary layer, within the framework of similarity theory based on the external parameters — wind velocity at the upper boundary of the layer, its thickness, air temperature difference between the upper and the lower boundaries, roughness of the underlying surface, and buoyancy forces. The form of the resistance laws is discussed. Determination of the thickness of the stationary and horizontally homogeneous (Ekman) boundary layer is analyzed and generalizations of the latter are suggested for non-stationary and inhomogeneous boundary layers.     

Three-dimensional numerical study of the height and mean structure of a heated planetary boundary layer

The heated boundary layer for DAY 33 of the Wangara data of southeast Australia (Clarke et al., 1971) is studied numerically with a three-dimensional model using 64000 grid points within a volume 5 km on a side and 2 km deep. Subgrid-scale transport equations were utilized in place of eddy-coefficient formulations. The rate of growth of the mixed layer is examined and parameterized, and the vertical profiles of heat flux, moisture flux and momentum fluxes are examined. The momentum boundary layer is found to coincide essentially with the mixed layer, and to grow with the latter during the hours of solar heating of the surface.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Concentration profiles of particles settling in the neutral and stratified atmospheric boundary layer

An expression for the vertical equilibrium concentration profile of heavy particles, including the effects of canopy on the eddy diffusivity as well as corrections for atmospheric stability, is proposed. This expression is validated against measurements of vertical concentration profiles of corn pollen above a corn field. The fitted theoretical profiles show that particle settling is correctly accounted for. The sensitivity to variations in the turbulent Schmidt number, settling velocity and stability corrections are explicitly characterized. The importance of independent measurements of the surface flux of pollen in future experiments is noted.     

Structure of mean winds and turbulence in the planetary boundary layer over rural terrain

A detailed analysis has been carried out of the temporal and spatial structure of mean winds and turbulence in the neutrally-stable planetary boundary layer over typically rural terrain. The data were obtained from a horizontal array of tower-mounted propeller anemometers (z = 11 m) during a five-hour period for which the mean wind direction was virtually perpendicular to the main span of the array. Various turbulence characteristics have been obtained for all three components of velocity and have been compared with idealized models for such a flow and with some of the other available atmospheric results.Considerable tower-to-tower and block-to-block variability has been observed in many of the measured results, particularly in those for the horizontal-component integral scales. Surface shear stress, roughness length and turbulence intensities were in good agreement with expected values for such a site. Power spectra for all components displayed significantly more energy at middle and lower frequencies than that observed by Kaimal et al. (1972) over flat, relatively featureless terrain. This is felt to be a result of the generally rougher gross features of the terrain in the present case and has led to the development of a modified version of the Kaimal-spectral model which fits the observed data better than either the original Kaimal model or the von Kármán model. It is suggested that it may in future be possible to represent power spectra over a wide range of terrain types by using such a modified spectral model.Integral scales of turbulence were calculated by three different techniques and in most cases displayed a strong dependence on the technique used. Averaged values of scale showed reasonable agreement with most of the available atmospheric data and with the values suggested by ESDU (1975). The anticipated elongation of turbulent eddies in the longitudinal direction was confirmed for all three velocity components, although it was found to be not as large as some other observations.     

Observed generation of an atmospheric gravity wave by shear instability in the mean flow of the planetary boundary layer

Observations of a single boundary-layer event — the generation of an atmospheric gravity wave by an unstable shear flow at Haswell, Colorado on November 12, 1971 — are briefly described and discussed. The observations were made using: (a) an acoustic echo sounder, (b) anemometers mounted at two fixed levels on a 150-m tower, (c) an anemometer and a thermometer mounted on a movable carriage on the tower, and (d) a microbarograph array, including one microbarograph mounted atop the tower. The wave phase velocity (–3.5–4.0 m s^–1) was found to equal the wind velocity in the middle of the shear flow, as assumed by other authors. The wave-associated vertical fluxes of momentum and energy measured just above the wave critical layer were estimated to be –5 dyn cm^–2 and –800 erg cm^–2 s^–1, respectively. These are large values. The annual average vertical flux of momentum at temperate and high latitudes is –0.25 dyn cm^–2, while the average kinetic energy dissipation rate in a unit column of atmosphere is –5 × 10³ erg cm^–2 s^–1. If the region of wave generation was itself propagating horizontally, its propagation velocity was large compared with the horizontal phase speed of the small-scale waves generated. Wave generation appeared to occur over an area large compared with the size of the microbarograph array (i.e., 2 km).     

Similarity theory and calculation of turbulent fluxes at the surface for the stably stratified atmospheric boundary layer

In this paper we revise the similarity theory for the stably stratified atmospheric boundary layer (ABL), formulate analytical approximations for the wind velocity and potential temperature profiles over the entire ABL, validate them against large-eddy simulation and observational data, and develop an improved surface flux calculation technique for use in operational models.     

The resistance law for the atmospheric boundary layer over a wavy surface

In analogy with two-dimensional turbulent layers, the surface layer (where wall similarity is fulfilled) and the region near the outer edge of the boundary layer (where the flow described by the velocity defect belongs to then-parameter family) may be postulated to exist for the atmospheric boundary layer over a wavy surface. The matching of the two regions yields a resistance law.