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.     

Rapid vertical sampling in stably stratified turbulent shear flows

A new method for obtaining instantaneous vertical profiles of two components of velocity and temperature in thermally stratified turbulent shear flows is presented. In this report, the design and construction of the traversing system will be discussed and results to date will be presented. The method is based on rapid vertical sampling whereby probe sensors are moved vertically at a high speed such that the measurement is approximately instantaneous. The system is designed to collect many measurements for the calculation of statistics such as vertical wave number spectra, mean square vertical gradients, and Thorpe scales. Results are presented for vertical profiles of temperature and compared to vertical profiles measured by single-point Eulerian time averages. The quality of the vertical profiles is found to be good over many profiles. Some comparisons are made between vertical measurements and standard single-point Eulerian measurements for three cases of stably stratified turbulent shear flow in which the initial microscale Reynolds number, Re_λ≈30. In case 1, the mean conditions are characterized by a gradient Richardson number, Ri_g=0.015, for which the flow is “unstable”, meaning the spatially evolving turbulent kinetic energy (E_k) grows. In case 2, Ri_g=0.095, for which the evolving turbulent kinetic energy is almost constant. In case 3, the flow is highly stable, where Ri_g=0.25 and E_k decays with spatial evolution. The measurements indicate anisotropy in the small scales for all cases. In particular, it is found that the ratio grows initially to a maximum and then decays with further evolution. Maximum Thorpe displacements are measured and compared to single-point measures of the vertical scales. It is found that vertical length scales derived from single-point measurements, such as the Ozmidov scale, L_O=(ε/N³)^1/2 and the overturn scale, L_t=θ′/(dT/dz), do not represent well the wide range of overturning scales which are actually present in the turbulence.     

Cloud drop spectra at different levels and with respect to cloud thickness and rain

Measurements on drop size were made in cumulus clouds over Pune (inland region) during the summer monsoon seasons. In this paper, the measurements of the cloud drop spectra made in non-raining clouds at different levels and for different thickness have been studied. Also, those on the days with rain and with no rain (the rain being observed within the clouds) have been compared. The average spectra broadened with height. The concentration of drops >50 μm (N_L), liquid water content (LWC), mean volume diameter (MVD) and dispersion increased with height. The concentration of drops <20 μm (N_S) and total concentration (N_T) decreased with height. The spectra were broader, while N_S and N_T are smaller and the other parameters are greater for thicker clouds as compared to those for thinner. The spectra were broader, while N_S and N_T are smaller and the other parameters are greater on the days with rain with respect to those on the days with no rain. The distributions were bimodal at higher levels, for thicker clouds and on the days with rain, while they were unimodal at lower levels, for thinner clouds and on the days with no rain. The variations of the cloud drop spectra, preceding rain, at initial stage of rain and following rain are discussed.     

Century-scale variability in a randomly forced,two-dimensional thermohaline ocean circulation model

The response of a two-dimensional thermohaline ocean circulation model to a random freshwater flux superimposed on the usual mixed boundary conditions for temperature and salinity is considered. It is shown that for a wide range of vertical and horizontal diffusivities and a box geometry that approximates the Atlantic Ocean, 200–300 yr period oscillations exist in the basic-state, interhemispheric meridional overturning circulation with deep convection in the north. These fluctuations can also be described in terms of propagating salinity anomalies which travel in the direction of the thermohaline flow. For large horizontal (K _h = 15 × 103 m²/s) and small vertical (K _v = 0.5 × 10^–4 m²/s) diffusivities, the random forcing also excites deca-millennial oscillations in the basic structure of the thermohaline circulation. In this case, the meridional circulation pattern slowly oscillates between three different stages: a large positive cell, with deep convection in the North Atlantic and upwelling in the south; a symmetric two-cell circulation, with deep convection in both polar regions and upwelling near the equator; and a large negative cell, with deep convection in the South Atlantic and upwelling in the north. Each state can persist for 0 (10 kyr).     

Model of convective heat exchange due to isolated thermals in the atmospheric boundary layer

A numerical model of convective heat transfer due to isolated thermals in the atmospheric boundary layer is used to describe the temperature profile transformation in undisturbed conditions as a result of intensive dry free convection. Based on some assumptions, the heat transfer Equation (2) is transformed to the form (14) in which the coefficients and the function F are expressed by (d/dz)(ln ) and by parameters of thermals. Equation (14) has been solved numerically with the help of Equation (15) obtained from the statics equation because of Equation (8). The size distribution function f(z, r, t) of the thermals is discrete (Table I), according to Vulf'son (1961). On Figures 1 and 2 are plotted successive temperature profiles for a ground inversion, transformed due to free convection and turbulence (Figures 1a and 2a), and due to turbulence only (Figures 1b and 2b). The profiles are computed from Equation 14 (Figures 1a and 2a) and Equation 16 (Figures 1b and 2b) for k _z= 1 m² s^–1 (Figure 1) and k _z= 10 m² s^–1 (Figure 2). On Figure 3 the real temperature profiles in Sofia for June 22nd 1976 are compared with the profiles computed using the real initial profile for 4.30 h local time. Good qualitative agreement can be seen.     

On the effects of rotation on fluid motions in cylindrical containers of various shapes and topological characteristics

Previous theoretical and laboratory studies of mechanically driven fluids in general rotation relative to an inertial frame have shown that there is a special class of flows for which the (Eulerian) flow field u(r, t) relative to the rotating frame of reference is unaffected by gyroscopic (Coriolis) forces, and therefore remains the same for all values of the rotation vector Ω. (Here t denotes time and r the position of a general point R in a reference frame attached to the rotating apparatus.) Such flows occur when (a) Ω is independent of time t; (b) u(r, t) is independent of the coordinate z (say) parallel to Ω, (c) the fluid has constant density and is therefore ‘barotropic’ (i.e. no density variations on horizontal surfaces) and (d) the topology of the cross-section of the (cylindrical) container, in planes z = constant, is such that the bounding surfaces can support the concomitant field of (kinematic) pressure P₁ satisfying P₁ + 2 Ω × U = 0 Condition (d) is equivalent to the requirement that any fluid sources or siks within the system be multipole in character, but not monopole. In the present study the ‘baroclinic’ case is treated, where buoyancy forces due to the action of gravity (and centripetal forces) on horizontal density variations have to be taken into account. These include investigations of flows due entirely to buoyancy forces, such as thermal convection in fluids in rotating cylindrical containers of various shapes and topological characteristics subject to horizontal temperature gradients. The implications for the impressed temperature field of the mathematical requirements that the fields of kinematic pressure P₁ and density (where denotes the mean density) be everywhere single-valued are guiding such investigations and facilitating the interpretation of their findings. The investigations include laboratory studies, reported elsewhere, of convection in a rotating fluid annulus with a circular cross-section blocked by a radial barrier, where it is found inter alia that advective heat transfer is virtually independent of |Ω| over a wide range of conditions. They also include (as yet unpublished) studies of thermal convection in rotating systems with topologically triply connected cross-sections which can be rendered doubly or simply connected by the insertation of suitable barriers.     

The temporal evolution of a geostrophic flow in a rotating stratified basin

A laboratory study in a rotating stratified basin examines the instability and long time evolution of the geostrophic double gyre introduced by the baroclinic adjustment to an initial basin-scale step height discontinuity in the density interface of a two-layer fluid. The dimensionless parameters that are important in determining the observed response are the Burger number S=R/R₀ (where R is the baroclinic Rossby radius of deformation and R₀ is the basin radius) and the initial forcing amplitude (H₁ is the upper layer depth). Experimental observations and a numerical approach, using contour dynamics, are used to identify the mechanisms that result in the dominance of nonlinear behaviour in the long time evolution, τ>2⁻¹ (where τ is time scaled by the inertial period T_I=2π/f). When the influence of rotation is moderate (0.25≤S≤1), the instability mechanism is associated with the finite amplitude potential vorticity (PV) perturbation introduced when the double gyre is established. On the other hand, when the influence of rotation is strong (S≤0.1), baroclinic instability contributes to the nonlinear behaviour. Regardless of the mechanism, nonlinearity acts to transfer energy from the geostrophic double gyre to smaller scales associated with an eddy field. In the lower layer, Ekman damping is pronounced, resulting in the dissipation of the eddy field after only 40T_I. In the upper layer, where dissipative effects are weak, the eddy field evolves until it reaches a symmetric distribution of potential vorticity within the domain consisting of cyclonic and anticyclonic eddy pairs, after approximately 100T_I. The functional dependence of the characteristic eddy lengthscale L_E on S is consistent with previous laboratory studies on continuously forced geostrophic turbulence. The cyclonic and anticyclonic eddy pairs are maintained until viscous effects eventually dissipate all motion in the upper layer after approximately 800T_I. The outcomes of this study are considered in terms of their contribution to the understanding of the energy pathways and transport processes associated with basin-scale motions in large stratified lakes.     

Upstream stagnation points in stratified flow past obstacles

An experimental study has been made of stagnation points and flow splitting on the upstream side of obstacles in uniformly stratified flow. A range from small to large values of Nh/U (where N is the buoyancy frequency, h_m is the maximum obstacle height and U is the undisturbed fluid velocity) has been covered, for three obstacle shapes which are, respectively, axisymmetric, and elongated in the across-stream and in the downstream directions. Upstream stagnation for the first two of these models does not occur until Nh_m/U > 1.05, where it occurs at z ≈ h_m/2. On the central line below this point the flow descends and diverges, and we term this ‘flow splitting’. For the third model (elongated in the downstream direction), stagnation upstream first occurs at Nh_m/U ≈ 1.43, at z ≈ 0. Results for this obstacle are not consistent with the ‘Sheppard criterion’, and this upstream flow stagnation is not apparently related to lee wave overturning, in contrast to flow over two-dimensional obstacles.     

Buoyancy and The Sensible Heat Flux Budget Within Dense Canopies

In contrast to atmospheric surface-layer (ASL) turbulence, a linear relationship between turbulent heat fluxes (F_T) and vertical gradients of mean air temperature within canopies is frustrated by numerous factors, including local variation in heat sources and sinks and large-scale eddy motion whose signature is often linked with the ejection-sweep cycle. Furthermore, how atmospheric stability modifies such a relationship remains poorly understood, especially in stable canopy flows. To date, no explicit model exists for relating F_T to the mean air temperature gradient, buoyancy, and the statistical properties of the ejection-sweep cycle within the canopy volume. Using third-order cumulant expansion methods (CEM) and the heat flux budget equation, a “diagnostic” analytical relationship that links ejections and sweeps and the sensible heat flux for a wide range of atmospheric stability classes is derived. Closure model assumptions that relate scalar dissipation rates with sensible heat flux, and the validity of CEM in linking ejections and sweeps with the triple scalar-velocity correlations, were tested for a mixed hardwood forest in Lavarone, Italy. We showed that when the heat sources (S_T) and F_T have the same sign (i.e. the canopy is heating and sensible heat flux is positive), sweeps dominate the sensible heat flux. Conversely, if S_T and F_T are opposite in sign, standard gradient-diffusion closure model predict that ejections must dominate the sensible heat flux.     

A simple heat-salt oscillator

A theoretical study is made of a simple mixed-layer model, in the form of a well-mixed constant-depth layer, forced from above by a heat flux k_T(T_A−T) and salinity flux k_S(S_A−S), where T_A and S_A are two reference values and T and S the temperature and salinity of the layer. The layer has a turbulent exchange of heat and salt with underlying water, kept at constant temperature and salinity, which is small in a statically stable case; large in a statically unstable case. If k_T>k_S, self-sustained oscillations may occur. In one cycle, a fast temperature rise, a slower salinity increase, and a final relaxation when the layer adjusts to the conditions of the underlying water, are observed.     

Measurements of spray at an air-water interface

The spray content in the surface boundary layer above an air—water interface was determined by a series of measurements at various feteches and wind speeds in a laboratory facility. The droplet flux density N(z) can be described in terms of the scaling flux density N_* and von Karman constant K throguh the equation, N(z)/N_* = −(1/K) ln(z/z_0d) where z is height above the mean water level and z_0d is the droplet boundary layer thickness. N_* is given by a unique relationship in terms of the roughness Reynolds number u_*σ/ν where σ is the root-mean-square surface displacement. Spray inception occurred for u_* 0.3. The dominant mode of spray generation in the present and most other laboratory tests, as well as in available field data, appears to be bubble bursting.     

Analysis of atmospheric turbidity levels at Taichung Harbor near the Taiwan Strait

In this study, two universal turbidity parameters, the Angstrom turbidity coefficient and Linke turbidity factor, are applied to study the atmospheric turbidity characteristics of Taichung Harbor. Meteorological parameter values were measured during 2004 and 2005 at the Wuchi weather station of the Taiwan Central Weather Bureau, near the Taiwan Strait. Results based on the Angstrom turbidity models (β_Lou, β_Pin, and β_Vis) indicated that annual mean values of the Angstrom turbidity coefficients were 0.174, 0.21 and 0.201, respectively. Four sets of Linke turbidity factors (T_Lin, T_Lou, T_Pin and T_Vis) were calculated using the original Linke method and the Dogniaux method, incorporating the computed Angstrom turbidity coefficients (β_Lou, β_Pin and β_Vis); the resultant values were 4.30, 6.40, 7.10 and 6.95, respectively. The monthly average values, frequency of occurrence, and cumulative frequency distributions were calculated using different models to describe the clear-sky atmospheric conditions at Taichung Harbor. The frequency results show that for over 50% of the dataset, three sets of Angstrom turbidity coefficients fell between 0.15 and 0.18, and four sets of Linke turbidity factors (T_Lin, T_Lou, T_Pin and T_Vis) fell between 4.0 and 6.5. Thus, for 50% of cloudless days, the sky can be between turbid and clear over Taichung Harbor. Furthermore, the results reveal that for 30% of the dataset, three Angstrom sets of turbidity coefficients (β_Lou, β_Pin, and β_Vis) exceed 0.2 and four sets of Linke turbidity factors (T_Lin, T_Lou, T_Pin and T_Vis) exceed 5.0. This indicates that 30% of cloudless sky conditions can be considered turbid to very turbid.     

Statistics of Scalar Fields in the Atmospheric Boundary Layer Based on Large-eddy Simulations. Part 1: Free Convection

Convection in a quasi-steady, cloud-free, shear-free atmospheric boundary layer is investigated based on a large-eddy simulation model. The performed tests indicate that the characteristic (peak) values of statistical moments at the top of the mixed layer are proportional to the interfacial scales (from gradients of scalars in the interfacial layer). Based on this finding a parameterization is proposed for profiles of scalar variances. The parameterization employs two, semi-empirical similarity functions F_m(z/z_i) andF_i(z/z_i), multiplied by a combination of the mixed-layer scales and the interfacial scales.     

Derivation of the relationship between the Obukhov stability parameter and the bulk Richardson number for flux-profile studies

Using the relationship between the bulk Richardson numberR _z and the Obukhov stability parameterz/L (L is the Obukhov length), formally obtained from the flux-profile relationships, methods to estimatez/L are discussed. Generally,z/L can not be uniquely solved analytically from flux-profile relationships, and it may be defined using routine observations only by iteration. In this paper, relationships ofz/L in terms ofR _z obtained semianalytically were corrected for variable aerodynamic roughnessz ₀ and for aerodynamic-to-temperature roughness ratiosz ₀/z _T, using the flux-profile iteration procedure. Assuming the so-called log-linear profiles to be valid for the nearneutral and moderately stable region (z/L<1), a simple relationship is obtained. For the extension to strong stability, a simple series expansion, based on utilisation of specified universal functions, is derived.For the unstable region, a simple form based on utilisation of the Businger-Dyer type universal functions, is derived. The formulae yield good estimates for surfaces having an aerodynamic roughness of 10^–5 to 10^–1 m, and an aerodynamic-to-temperature roughness ratio ofz ₀/z _T=0.5 to 7.3. When applied to the universal functions, the formulae yield transfer coefficients and fluxes which are almost identical with those from the iteration procedure.     

Formation of horizontal convective rolls in urban areas

The formation of horizontal convective rolls (HCRs) in urban areas is investigated in this paper using observations and fine-scale numerical simulations. Cloud streets organized parallel to the mean boundary-layer wind (a manifestation of HCRs) are seen in the Fengyun-2C satellite imagery around local noon in Beijing. Observed vertical velocity and horizontal wind fields from an urban wind profiler suggest that the time scale for alternating updraft and downdraft in the boundary layer is about 30 min, and the length of the updraft/downdraft is about 9 km. Numerical simulations show that most HCRs occur in the urban areas with − z_i / L < 25 (z_i: the boundary-layer depth, L: the Monin–Obukhov length). Sensitivity tests reveal that HCRs are common in urban boundary layers, while rural areas are more conducive to forming cellular convection; the aspect ratio of HCRs in urban areas is smaller than the typical value over natural landscapes.     

Approximation formulas for the microphysical properties of saline droplets

Microphysical theory has proven essential for explaining sea spray's role in transferring heat and moisture across the air–sea interface. But large-scale models of air–sea interaction, among other applications, cannot afford full microphysical modules for computing spray droplet evolution and, thus, how rapidly these droplets exchange heat and moisture with their environment. Fortunately, because the temperature and radius of saline droplets evolve almost exponentially when properly scaled, it is possible to approximate a droplet's evolution with just four microphysical endpoints: its equilibrium temperature, T_eq; the e-folding time to reach that temperature, τ_T; its equilibrium radius, r_eq; and the e-folding time to reach that radius, τ_r.Starting with microphysical theory, this paper derives quick approximation formulas for these microphysical quantities. These approximations are capable of treating saline droplets with initial radii between 0.5 and 500 μm that evolve under the following ambient conditions: initial droplet temperatures and air temperatures between 0 and 40 °C, ambient relative humidities between 75% and 99.5%, and initial droplet salinities between 1 and 40 psu.Estimating T_eq, τ_T, and τ_r requires only one-step calculations; finding r_eq is done recursively using Newton's method. The approximations for T_eq and τ_T are quite good when compared to similar quantities derived from a full microphysical model; T_eq is accurate to within 0.02 °C, and τ_T is typically accurate to within 5%. The estimate for equilibrium radius r_eq is also usually within 5% of the radius simulated with the full microphysical model. Finally, the estimate of radius e-folding time τ_r is accurate to within about 10% for typical oceanic conditions.     

An experimental study of sulfur and NOx fluxes over grassland

Three independent sulfur sensors were used in a study of sulfur eddy fluxes to a field of wheat stubble and mixed grasses, conducted in Southern Ohio in September, 1979. Two of these sensors were modified commercial instruments; one operated with a prefilter to measure gaseous sulfur compounds and the other with a denuder system to provide submicron particulate sulfur data. The third sensor was a prototype system, used to measure total sulfur fluxes. The data obtained indicated that the deposition velocity for gaseous sulfur almost always exceeded that for particulate sulfur; average surface conductances were about 1.0 cm s^–1 for gaseous sulfur in the daytime and about 0.4 cm s^–1 for particulate sulfur. The data indicate that nighttime values were probably much lower. The total sulfur sensor provided support for these conclusions. The boundary-layer quantity ln(z ₀ /z _H )was found to be 2.75 ± 0.55, in close agreement with expectations and thus providing some assurance that the site was adequate for eddy flux studies. However, fluxes derived using a prototype NO_x sensor were widely scattered, partially as a consequence of sensor noise but also possibly because of the effects of nearby sources of natural nitrogen compounds.Formerly with Argonne National Laboratory.     

Rate constants for the reactions of the NO3 radical with HCOOH/HCOO− and CH3COOH/CH3COO− in aqueous solution between 278 and 328 K

The kinetics of the aqueous phase reactions of NO₃ radicals with HCOOH/HCOO^– and CH₃COOH/CH₃COO^– have been investigated using a laser photolysis/long-path laser absorption technique. NO₃ was produced via excimer laser photolysis of peroxodisulfate anions (S₂O ₈ ^2– ) at 351 nm followed by the reactions of sulfate radicals (SO ₄ ^– ) with excess nitrate. The time-resolved detection of NO₃ was achieved by long-path laser absorption at 632.8 nm. For the reactions of NO₃ with formic acid (1) and formate (2) rate coefficients ofk ₁=(3.3±1.0)×10⁵ l mol^–1 s^–1 andk ₂=(5.0±0.4)×10⁷ l mol^–1 s^–1 were found atT=298 K andI=0.19 mol/l. The following Arrhenius expressions were derived:k ₁(T)=(3.4±0.3)×10¹⁰ exp[–(3400±600)/T] l mol^–1 s^–1 andk ₂(T)=(8.2±0.8)×10¹⁰ exp[–(2200±700)/T] l mol^–1 s^–1. The rate coefficients for the reactions of NO₃ with acetic acid (3) and acetate (4) atT=298 K andI=0.19 mol/l were determined as:k ₃=(1.3±0.3)×10⁴ l mol^–1 s^–1 andk ₄=(2.3±0.4)×10⁶ l mol^–1 s^–1. The temperature dependences for these reactions are described by:k ₃(T)=(4.9±0.5)×10⁹ exp[–(3800±700)/T] l mol^–1 s^–1 andk ₄(T)=(1.0±0.2)×10¹² exp[–(3800±1200)/T] l mol^–1 s^–1. The differences in reactivity of the anions HCOO^– and CH₃COO^– compared to their corresponding acids HCOOH and CH₃COOH are explained by the higher reactivity of NO₃ in charge transfer processes compared to H atom abstraction. From a comparison of NO₃ reactions with various droplets constituents it is concluded that the reaction of NO₃ with HCOO^– may present a dominant loss reaction of NO₃ in atmospheric droplets.     

Monin–Obukhov Similarity Functions of the Structure Parameter of Temperature and Turbulent Kinetic Energy Dissipation Rate in the Stable Boundary Layer

The Monin–Obukhov similarity theory (MOST) functions f_ε and f_T, of the dissipation rate of turbulent kinetic energy (TKE). ε, and the structure parameter of temperature, C_T², were determined for the stable atmospheric surface layer using data gathered in the context of CASES-99. These data cover a relatively wide stability range, i.e. ζ=z/L of up to 10, where z is the height and L the Obukhov length. The best fits were given by f_ε = 0.8 + 2.5ζ and f_T= 4.7[ 1+1.6(ζ)^2/3], which differ somewhat from previously published functions. ε was obtained from spectra of the longitudinal wind velocity using a time series model (ARMA) method instead of the traditional Fourier transform. The neutral limit f_ε =0.8 implies that there is an imbalance between TKE production and dissipation in the simplified TKE budget equation. Similarly, we found a production-dissipation imbalance for the temperature fluctuation budget equation. Correcting for the production-dissipation imbalance, the ‘standard’ MOST functions for dimensionless wind speed and temperature gradients (φ_m and φ_m) were determined from f_ε and f_T and compared with the φ_m and φ_h formulations of Businger and others. We found good agreement with the Beljaars and Holtslag [J. Appl. Meteorol. 30, 327–341 (1991)] relations. Lastly, the flux and gradient Richardson numbers are discussed also in terms of f_ε and f_T.     

A sodar study of the temperature structure parameter in the convective boundary layer

From sodar measurements gathered during the Voves experiment (France, summer 1977), the variations of the temperature structure parameter C _T ² were studied in the morning planetary boundary layer. Dimensionless profiles of C _T ² are consistent with the mixed-layer scaling of Kaimal et al. (1976); however, for z < 0,5 z _i, the decrease of C _T ² as z ^4/3 should be weighted according to Frisch and Ochs (1975).When the final breakup of the nocturnal inversion is achieved, the variations of the maximum of the C _T ² profile are in good agreement with those predicted by Wyngaard and Le Mone (1980). Discrepancies are observed mainly when the mixed layer is shallow and mechanical turbulence is important compared with buoyancy-driven turbulence.