Parameterization of Sheared Entrainment in a Well-developed CBL.Part Ⅱ:A Simple Model for Predicting the Growth Rate of the CBL

Following the parameterization of sheared entrainment obtained in the companion paper, Liu et al. (2016), the present study aims to further investigate the characteristics of entrainment, and develop a simple model for predicting the growth rate of a well-developed and sheared CBL. The relative stratification, defined as the ratio of the stratification in the free atmosphere to that in the entrainment zone, is found to be a function of entrainment flux ratio (A _e). This leads to a simple expression of the entrainment rate, in which A _e needs to be parameterized. According to the results in Liu et al. (2016), A _e can be simply expressed as the ratio of the convective velocity scale in the sheared CBL to that in the shear-free CBL. The parameterization of the convective velocity scale in the sheared CBL is obtained by analytically solving the bulk model with several assumptions and approximations. Results indicate that the entrainment process is influenced by the dynamic effect, the interaction between mean shear and environmental stratification, and one other term that includes the Coriolis effect. These three parameterizations constitute a simple model for predicting the growth rate of a well-developed and sheared CBL. This model is validated by outputs of LESs, and the results show that it performs satisfactorily. Compared with bulk models, this model does not need to solve a set of equations for the CBL. It is more convenient to apply in numerical models.     

Parameterization of sheared entrainment in a well-developed CBL. Part I: Evaluation of the scheme through large-eddy simulations

The entrainment flux ratio A _e and the inversion layer (IL) thickness are two key parameters in a mixed layer model. A _e is defined as the ratio of the entrainment heat flux at the mixed layer top to the surface heat flux. The IL is the layer between the mixed layer and the free atmosphere. In this study, a parameterization of A _e is derived from the TKE budget in the firstorder model for a well-developed CBL under the condition of linearly sheared geostrophic velocity with a zero value at the surface. It is also appropriate for a CBL under the condition of geostrophic velocity remaining constant with height. LESs are conducted under the above two conditions to determine the coefficients in the parameterization scheme. Results suggest that about 43% of the shear-produced TKE in the IL is available for entrainment, while the shear-produced TKE in the mixed layer and surface layer have little effect on entrainment. Based on this scheme, a new scale of convective turbulence velocity is proposed and applied to parameterize the IL thickness. The LES outputs for the CBLs under the condition of linearly sheared geostrophic velocity with a non-zero surface value are used to verify the performance of the parameterization scheme. It is found that the parameterized A _e and IL thickness agree well with the LES outputs.     

Effects of Temporal Discretization on Turbulence Statistics and Spectra in Numerically Simulated Convective Boundary Layers

Six state-of-the-art large-eddy simulation codes were compared in Fedorovich et al. (Preprints, 16th American Meteorological Society Symposium on Boundary Layers and Turbulence, 2004b) for three airflow configurations in order to better understand the effect of wind shear on entrainment dynamics in the convective boundary layer CBL). One such code was the University of Oklahoma large-eddy simulation (LES) code, which at the time employed a second-order leapfrog time-advancement scheme with the Asselin filter. In subsequent years, the code has been updated to use a third-order Runge–Kutta (RK3) time-advancement scheme. This study investigates what effect the upgrade from the leapfrog scheme to RK3 scheme has on turbulence statistics in the CBL differently affected by mean wind shear, also in relation to predictions by other LES codes that participated in the considered comparison exercise. In addition, the effect of changing the Courant number within the RK3 scheme is investigated by invoking the turbulence spectral analysis. Results indicate that low-order flow statistics obtained with the RK3 scheme generally match their counterparts from simulations with the leapfrog scheme rather closely. CBL growth rates due to entrainment in the shear-free case were also similar using both timestepping schemes. It was found, however, that care should be given to the choice of the Courant number value when running LES with the RK3 scheme in the sheared CBL setting. The advantages of the largest possible (based on the stability criterion) Courant number were negated by degrading the energy distribution across the turbulence spectrum. While mean profiles and low-order turbulence statistics were largely unaffected, the entrainment rate was over-predicted compared to that reported in the original code-comparison study.     

Statistics of Scalar Fields in the Atmospheric Boundary Layer Based on Large-Eddy Simulations. Part II: Forced Convection

Forced convection in a quasi-steady atmospheric boundary layer is investigated based on a large-eddy simulation (LES) model. The performed simulations show that in the upper portion of the mixed layer the dimensionless (in terms of mixed layer scales) vertical gradients of temperature, humidity, and wind velocity depend on the dimensionless height z/z _i and the Reech number Rn. The peak values of variances and covariances at the top of the mixed layer, scaled in terms of the interfacial scales, are functions of the interfacial Richardson number Ri. As a result expressions for the entrainment rates, in the case when the interfacial layer has a finite depth, and a condition for the presence of moistening or drying regimes in the mixed layer, are derived. Profiles of dimensionless scalar moments in the mixed layer are proposed to be expressed in terms of two empirical similarity functions F _m and F _i , dependent on dimensionless height z/z _i , and the interfacial Richardson number Ri. The obtained similarity expressions adequately approximate the LES profiles of scalar statistics, and properly represent the impact of stability, shear, and entrainment. They are also consistent with the parameterization proposed for free convection in the first part of this paper.     

Application of Dynamic Subgrid-scale Models for Large-eddy Simulation of the Daytime Convective Boundary Layer over Heterogeneous Surfaces

The sensitivity of large-eddy simulation (LES) to the representation of subgrid-scale (SGS) processes is explored for the case of the convective boundary layer (CBL) developing over surfaces with varying degrees of spatial heterogeneity. Three representations of SGS processes are explored: the traditional constant Smagorinsky–Lilly model and two other dynamic models with Lagrangian averaging approaches to calculate the Smagorinsky coefficient (C _S ) and SGS Prandtl number (Pr). With initial data based roughly on the observed meteorology, simulations of daytime CBL growth are performed over surfaces with characteristics (i.e. fluxes and roughness) ranging from homogeneous, to striped heterogeneity, to a realistic representation of heterogeneity as derived from a recent field study. In both idealized tests and the realistic case, SGS sensitivities are mostly manifest near the surface and entrainment zone. However, unlike simulations over complex domains or under neutral or stable conditions, these differences for the CBL simulation, where large eddies dominate, are not significant enough to distinguish the performance of the different SGS models, irrespective of surface heterogeneity.     

Length Scales of Scalar Diffusion in the Convective Boundary Layer: Laboratory Observations

A laboratory study of scalar diffusion in the convective boundary layer has found results that are consistent with a 1999 large-eddy simulation (LES) study by Jonker, Duynkerke and Cuijpers. For bottom-up and top-down scalars (introduced as ‘infinite’ area sources of passive tracer at the surface and inversion, respectively) the dominant length scale was found to be much larger than the length scale for density fluctuations, the latter being equal to the boundary-layer depth h. The variance of the normalized passive scalar grew continuously with time and its magnitude was about 3–5 times larger for the top-down case than for the bottom-up case. The vertical profiles of the normalized passive scalar variance were found to be approximately constant through the convective boundary layer (CBL) with a value of about 3–8c_*² for bottom-up and 10–50c_*² for top-down diffusion. Finally, there was some evidence of a minimum in the variance and dominant length scale for scalar flux ratios (top-down to bottom-up flux) close to −0.5. All these convection tank results confirm the LES results and support the hypothesis that there is a distinct difference in behaviour between the dynamic and passive variables in the CBL.     

Dispersion of passive tracer in the atmospheric convective boundary layer with wind shears: a review of laboratory and numerical model studies

Summary Paper reviews recent laboratory and numerical model studies of passive gaseous tracer dispersion in the atmospheric convective boundary layer (CBL) with surface and elevated wind shears. Atmospheric measurement data used for validation of these two model techniques are briefly discussed as well. A historical overview is given of laboratory studies of dispersion in the atmospheric CBL. Model studies of tracer dispersion in two CBL types, the (i) non-steady, horizontally homogeneous CBL and (ii) quasi-stationary, horizontally heterogeneous CBL, are reviewed. The discussion is focused on the dispersion of non-buoyant plume emitted from a point source located at different elevations within the CBL. Approaches towards CBL modeling employed in different laboratory facilities (water tanks and wind tunnels) are described. The reviewed numerical techniques include Large Eddy Simulation (LES) and Lagrangian modeling. Numerical data on dispersion in the sheared CBL is analyzed in conjunction with experimental results from wind-tunnel CBLs.     

Parameterization of Entrainment in a Sheared Convective Boundary Layer Using a First-order Jump Model

Basic entrainment equations applicable to the sheared convective boundary layer (CBL) are derived by assuming an inversion layer with a finite depth, i.e., the first-order jump model. Large-eddy simulation data are used to determine the constants involved in the parameterizations of the entrainment equations. Based on the integrated turbulent kinetic energy budget from surface to the top of the CBL, the resulting entrainment heat flux normalized by surface heat flux is a function of the inversion layer depth, the velocity jumps across the inversion layer, the friction velocity, and the convection velocity. The developed first-order jump model is tested against large-eddy simulation data of two independent cases with different inversion strengths. In both cases, the model reproduces quite reasonably the evolution of the CBL height, virtual potential temperature, and velocity components in the mixed layer and in the inversion layer.The part of this work was done when the first author visited at NCAR.     

Effect of the Entrainment Flux Ratio on the Relationship between Entrainment Rate and Convective Richardson Number

The parameterization of the dimensionless entrainment rate (w _e /w _*) versus the convective Richardson number (Ri _δθ ) is discussed in the framework of a first-order jump model (FOM). A theoretical estimation for the proportionality coefficient in this parameterization, namely, the total entrainment flux ratio, is derived. This states that the total entrainment flux ratio in FOM can be estimated as the ratio of the entrainment zone thickness to the mixed-layer depth, a relationship that is supported by earlier tank experiments, and suggesting that the total entrainment flux ratio should be treated as a variable. Analyses show that the variability of the total entrainment flux ratio is actually the effect of stratification in the free atmosphere on the entrainment process, which should be taken into account in the parameterization. Further examination of data from tank experiments and large-eddy simulations demonstrate that the different power laws for w _e /w _* versus Ri _δθ can be interpreted as the variability of the total entrainment flux ratio. These results indicate that the dimensionless entrainment rate depends not only on the convective Richardson number but also upon the total entrainment flux ratio.     

对流边界层大涡模式的改进及对夹卷速度的研究

对已建的对流边界层(CBL)大涡模式进行了改进，将次网格闭合方案改为次网格能量闭合，并考虑了水汽的源汇项和水汽相变潜热的作用。通过对均匀下垫面上由热扰动发展的对流边界层的模拟及与实验结果的比较表明，模式较好地模拟了对流边界层的主要物理结构，较好地反映了各物理量之间的对应关系。本文在一定的对流理查森数(Ri*)范围内给出了一些算例，对无量纲夹卷速度(We/W*)进行了研究。结果表明，无量纲夹卷速度随地表热通量(Qs)的增大而增大，随对流边界层上部温度递减率（γ)的增大而减小。当9.06≤Ri*≤45.29时，无量纲夹卷速度We/W*可以拟合成A(Ri*)-1的形式，其中A=0.226。并且与我们的对流槽实验结果，Sullivan等人的大涡模拟结果以及Deardorff等人的对流槽实验结果作了比较，四者吻合较好。     

Entrainment Processes in the Convective Boundary Layer with Varying Wind Shear

Large-eddy simulations (LES) are performed to investigate the entrainment andthe structure of the inversion layer of the convective boundary layer (CBL) withvarying wind shears. Three CBLs are generated with the constant surface kinematicheat flux of 0.05 K m s^-1 and varying geostrophic wind speeds from 5 to 15m s^-1. Heat flux profiles show that the maximum entrainment heat flux as afraction of the surface heat flux increases from 0.13 to 0.30 in magnitude withincreasing wind shear. The thickness of the entrainment layer, relative to the depthof the well-mixed layer, increases substantially from 0.36 to 0.73 with increasingwind shear. The identification of vortices and extensive flow visualizations nearthe entrainment layer show that concentrated vortices perpendicular to the meanboundary-layer wind direction are identified in the capping inversion layer for thecase of strong wind shear. These vortices are found to develop along the mean winddirections over strong updrafts, which are generated by convective rolls and to appearas large-scale wavy motions similar to billows generated by the Kelvin–Helmholtzinstability. Quadrant analysis of the heat flux shows that in the case of strong windshear, large fluctuations of temperature and vertical velocity generated by largeamplitude wavy motions result in greater heat flux at each quadrant than that inthe weak wind shear case.     

Scaling Properties of Temperature Spectra and Heat-Flux Cospectra in the Surface Friction Layer Beneath an Unstable Outer Layer

Temperature variance and temperature power spectra in the unstable surface layer have always presented a problem to the standard Monin-Obukhov similarity model. Recently that problem has intensified with the demonstration by Smedman et al. (2007, Q J Roy Meteorol Soc 133: 37–51) that temperature spectra and heat-flux cospectra can have two distinct peaks in slightly unstable conditions, and by McNaughton et al. (2007, Nonlinear Process Geophys 14: 257–271) who showed that the wavenumber of the peak of temperature spectra in a convective boundary layer (CBL), closely above the surface friction layer (SFL), can be sensitive to the CBL depth, z _i. Neither the two-peak form at slight instability nor the dependence of peak position on z _i at large instability is compatible with the Monin-Obukhov model. Here we examine the properties of temperature spectra and heat-flux cospectra from between these extremes, i.e. from within the unstable SFL, in two experiments. The analysis is based on McNaughton’s model of the turbulence structure in the SFL. According to this model, heat is transported through most of the SFL by sheet plumes, created by the action of impinging outer eddies. The smallest and most effective of these outer eddies have sizes that scale on SFL depth, z _s. The z _s-scale eddies and plumes are organised within the overall convection pattern in the CBL, and in turn they organise the motion of smaller eddies within the SFL, whose sizes scale on height, z. The main experimental results are: (1) the peak amplitudes of the temperature spectra in the SFL are collapsed with a scaling factor (z_sz)^1/3e_o^2/3{(z_{\rm s}z)^{1/3}\varepsilon_{\rm o}^{2/3}} divided by the square of the surface temperature flux, where e_o{\varepsilon_{\rm o}} is the dissipation rate of turbulent energy in the outer CBL (above the SFL); (2) the peak wavenumbers of the temperature spectra are collapsed with the mixed length scale (z _i z _s)^1/2; (3) the peak wavenumbers of the heat-flux cospectra are collapsed with the doubly-mixed length scale (z _i z _s)^1/4 z ^1/2; (4) for z/z _s < 0.03, the peak in the cospectrum is replaced by another peak at a wavenumber about a magnitude larger. This peak’s position scales on z; (5) all these findings are consistent with the observations of Smedman et al.     

A Modelling Study of Flux Imbalance and the Influence of Entrainment in the Convective Boundary Layer

It is frequently observed in field experiments that the eddy covariance heat fluxes are systematically underestimated as compared to the available energy. The flux imbalance problem is investigated using the NCAR’s large-eddy simulation (LES) model imbedded with an online scheme to calculate Reynolds-averaged fluxes. A top–down and a bottom–up tracer are implemented into the LES model to quantify the influence of entrainment and bottom–up diffusion processes on flux imbalance. The results show that the flux imbalance follows a set of universal functions that capture the exponential decreasing dependence on u _*/w _*, where u _* and w _* are friction velocity and the convective velocity scale, respectively, and an elliptic relationship to z/z _i , where z _i is the mixing-layer height. The source location in the boundary layer is an important factor controlling the imbalance magnitude and its horizontal and vertical distributions. The flux imbalance of heat and the bottom–up tracer is tightly related to turbulent coherent structures, whereas for the top–down diffusion, such relations are weak to nonexistent. Our results are broadly consistent with previous studies on the flux imbalance problem, suggesting that the published results are robust and are not artefacts of numerical schemes.     

Use of obliquely rotated principal component analysis to identify coherent structures

Principal component analysis (PCA) with oblique rotation is applied to Large Eddy Simulation (LES) results to discern and quantify coherent structures within the convective boundary layer (CBL). Sensitivity tests are first conducted on a moderately convective LES run. Once the ability of PCA to generate robust results is verified, the method is applied to LES runs spanning a range of stability regimes. Interregime similarities and differences in the coherent structures are discussed. For the moderately convective LES run, three-dimensional convective cells are arrayed in two-dimensional bands aligned with the geostrophic wind. The resulting gravity waves in the free atmosphere and convective inflow and outflow in the boundary layer are also captured by the PCA. Convective modes are more sensitive to the ratio of w _* to u _* than are the dynamic modes.PCA has demonstrated advantages over previous analysis methods. PCA score maps provide information on the spatial distribution of phenomena that has not been available from traditional conditional sampling studies. Principal components provide information on the vertical structures of phenomena that would be obscured by life-cycle effects or erratic tilts from the vertical in the conventional approaches to either conditional sampling or composite analysis. Future work includes application of this technique to multi-level observational time series from a surface-layer tower for the Risø Air/Sea Experiment (RASEX).     

The Role of Shear in the Morning Transition Boundary Layer

We use large-eddy simulation (LES) to better define the early stages of the morning transition boundary layer. Previous LES studies relating to the morning transition boundary layer focus on the role of the entraining convective boundary layer (CBL). By using a combination of different domain sizes and grid lengths, the full evolution from the stable boundary layer (SBL) to the CBL is modelled here. In the early stages of the morning transition the boundary layer is shown to be a combination of a shallow mixed layer capped by a significant shear driven stable boundary layer (the so-called mixed CBL–SBL state). The mixed CBL–SBL state is the key to understanding the sensitivity to shear. Turbulent kinetic energy budgets also indicate that it is shear driven. The negative flux from the mixed CBL–SBL state extends much further above the minimum than is typically found for the CBL later in the day, and the depth of penetration scales as w _m /N _i , where w _m is the combined friction and convective velocity scale and N _i the static stability at the inversion top.     

A Comparison of Higher-Order Vertical Velocity Moments in the Convective Boundary Layer from Lidar with In Situ Measurements and Large-Eddy Simulation

We have analyzed measurements of vertical velocity w statistics with the NOAA high resolution Doppler lidar (HRDL) from about 390 m above the surface to the top of the convective boundary layer (CBL) over a relatively flat and uniform agricultural surface during the Lidars-in-Flat-Terrain (LIFT) experiment in 1996. The temporal resolution of the zenith-pointing lidar was about 1 s, and the range-gate resolution about 30 m. Vertical cross-sections of w were used to calculate second- to fourth-moment statistics of w as a function of height throughout most of the CBL. We compare the results with large-eddy simulations (LES) of the CBL and with in situ aircraft measurements. A major cause of the observed case-to-case variability in the vertical profiles of the higher moments is differences in stability. For example, for the most convective cases, the skewness from both LES and observations changes more with height than for cases with more shear, with the observations changing more with stability than the LES. We also found a decrease in skewness, particularly in the upper part of the CBL, with an increase in LES grid resolution.     

Equilibrium evaporation beneath a growing convective boundary layer

Expressions for the equilibrium surface Bowen ratio ( _s ) and equilibrium evaporation are derived for a growing convective boundary layer (CBL) in terms of the Bowen ratio at the top of the mixed layer _i and the entrainment parameter A _R . If A_R is put equal to zero, the solution for _s becomes-that previously obtained for the zero entrainment or closed box model. The Priestley-Taylor parameter is also calculated and plotted in terms ofA _R and _i . Realistic combinations of the atmospheric parameters give values of in the range 1.1 to 1.4.     

Impacts of Aerosol Shortwave Radiation Absorption on the Dynamics of an Idealized Convective Atmospheric Boundary Layer

We investigated the impact of aerosol heat absorption on convective atmospheric boundary-layer (CBL) dynamics. Numerical experiments using a large-eddy simulation model enabled us to study the changes in the structure of a dry and shearless CBL in depth-equilibrium for different vertical profiles of aerosol heating rates. Our results indicated that aerosol heat absorption decreased the depth of the CBL due to a combination of factors: (i) surface shadowing, reducing the sensible heat flux at the surface and, (ii) the development of a deeper inversion layer, stabilizing the upper CBL depending on the vertical aerosol distribution. Steady-state analytical solutions for CBL depth and potential temperature jump, derived using zero-order mixed-layer theory, agreed well with the large-eddy simulations. An analysis of the entrainment zone heat budget showed that, although the entrainment flux was controlled by the reduction in surface flux, the entrainment zone became deeper and less stably stratified. Therefore, the vertical profile of the aerosol heating rate promoted changes in both the structure and evolution of the CBL. More specifically, when absorbing aerosols were present only at the top of the CBL, we found that stratification at lower levels was the mechanism responsible for a reduction in the vertical velocity and a steeper decay of the turbulent kinetic energy throughout the CBL. The increase in the depth of the inversion layer also modified the potential temperature variance. When aerosols were present we observed that the potential temperature variance became significant already around $0.7z_i$ (where $z_i$ is the CBL height) but less intense at the entrainment zone due to the smoother potential temperature vertical gradient.     

基于风廓线雷达的对流边界层高度的确定

大气边界层高度对于天气、气候和大气污染研究是一个至关重要的参量。对流边界层(Convective Boundary Layer,CBL)顶部的夹卷过程造成温度和湿度垂直梯度增强,导致这一层的折射率结构常数C■变高。C■的这种垂直分布特征经常被用来定位出CBL高度Z_i。本文利用2010年7—8月天津大港的风廓线雷达数据推断出CBL高度Z_i,对于多重C■峰值或不明确的C■峰值,本文改进了对Z_i的测定,分别讨论了C■最大后向散射法与C■和垂直速度方差(σ■)相结合的新方法的适用性。研究显示:(1)C■廓线具有单峰时,最大后向散射强度法能正确估计CBL高度,这种情况往往对应的是晴天。CBL上存在的残留层或云层引起的温湿起伏变化导致C■廓线具有双峰甚至多峰时,最大后向散射强度法可能会错误估计CBL高度;(2)C■和σ■结合的方法不仅与晴天时C■最大后向散射法有较好的一致性,而且可以将CBL造成的C■峰值从云层造成的C■峰值中区分出来,从而正确估计CBL高度;(3)一般而言,对流边界层中存在有明显的、破碎或者分散不明显的云时,C■和σ■结合的方法都能较好地识别出CBL对应的C■峰值。但由于边界层中的情况极为复杂,C■和σ■结合法也会因不同的原因而错误估计CBL高度。     

A generalized scaling for convective shear flows

During the last two decades, different scalings for convective boundary layer (CBL) turbulence have been proposed. For the shear-free regime, Deardorff (1970) introduced convective velocity and temperature scales based on the surface potential temperature flux,Q _s , the buoyancy parameter, , and the time-dependent boundary-layer depth,h. Wyngaard (1983) has proposed decomposition of turbulence into two components, bottom-up (b) and top-down (t), the former characterized byQ _s , the latter, by the potential temperature flux due to entrainment,Q _h . Sorbjan (1988) has devised height-dependent velocity and temperature scales for both b- and t-components of turbulence.Incorporating velocity shear, the well known similarity theory of Monin and Obukhov (1954) has been developed for the atmospheric surface layer. Zilitinkevich (1971, 1973) and Betchov and Yaglom (1971) have elaborated this theory with the aid of directional dimensional analysis for a particular case when different statistical moments of turbulence can be alternatively attributed as being of either convective or mechanical origin.In the present paper, we attempt to create a bridge between the two approaches pointed out above. A new scaling is proposed on the basis of, first, decomposition of statistical moments of turbulence into convective (c), mechanical (m) and covariance (c&m) contributions using directional dimensional analysis and, second, decomposition of these contributions into bottom-up and top-down components using height-dependent velocity and temperature scales. In addition to the statistical problem, the scaling suggests a new approach of determination of mean temperature and velocity profiles with the aid of the budget equations for the mean square fluctuations.Notation ATL alternative turbulence layer - CBL convective boundary layer - CML convective and mechanical layer - FCL free convection layer - MTL mechanical turbulence layer