Comparison of PDFs, closure schemes and turbulence parameterisations in Lagrangian stochastic models

Six one-dimensional models, based on the Ito-type stochastic equation, are presented and compared. Four of these take into account up to the fourth order moment of vertical velocity fluctuations, and two up to the third order moment. Four models make use of a bi-Gaussian probability density function (PDF) and the other two are based on a Gram-Charlier series expansion truncated to the third or fourth order. All the models were run with a parameterisation of input turbulence (i.e. w², w³, and τ profiles). Concerning the fourth order moment w⁴, two different parameterisations were considered. Comparisons are made between ground-level concentrations, plume height and plume width observed in the Willis and Deardorff water tank experiments and those predicted by the different models here considered. The goal of this study was to find the models that give greater confidence in their applicability in dispersion studies and to verify the importance of considering the fourth order moment. The main conclusions are: simulation results largely depend on the turbulence parameterisation chosen; the Gram-Charlier PDF gives the best agreement with observations; some combinations of models and turbulence parameterisations perform well in simulating the shape of the ground-level concentration (g.1.c.) trend but fail in correctly simulating the form of the plume (plume height and vertical width); in the case of the Gram-Charlier PDF, the fourth order model reproduced the vertical plume width better than the third order one, whereas the two schemes yielded similar g.1.c. distributions.     

Deriving sorption indices for the prediction of potential phosphorus loss from calcareous soils

The aim of this study was to develop techniques to evaluate soil phosphorus (P) sorption capacity (PSC) and determine critical soil P levels to predict P loss potential for calcareous soils. Seventy-five soils mostly from Northern China were analyzed for soil P using four extraction methods (water, P_w; carbonate, P_Ols; ammonium oxalate, P_ox; and Mehlich 3, P_M3) as well as PSC derived from single-point (PSC₁₅₀) and multipoint sorption (S _t) isotherms. Strong correlation was found between PSC₁₅₀ and S _t (r ²=0.89, p<0.001). The sum of αCa_M3 and βMg_M3 as an index of PSC (PSC_{(CaM3 + MgM3)}) was most closely related to the maximum amount of P sorbed (S _max) as given by the sum of S _t and soil initial P setting α=0.039 and β=0.462 (r ²=0.80, p<0.001). The degree of P saturation (DPS) was thereafter calculated from PSC_{(CaM3 + MgM3)} (DPS_{(CaM3 + MgM3)}), to which Olsen P (P_Ols) was significantly correlated (r ²=0.82, p<0.001). In a split-line regression from P_w against DPS_{(CaM3 + MgM3)} (r ²=0.87, p<0.05), a change point was identified at 28.1% DPS_{(CaM3 + MgM3)}, which was equivalent to 49.2 mg kg^?1 P_Ols and corresponded to a P_w concentration of 8.8 mg kg^?1. After the change point, a sharp increase in P_w was observed. Our results reveal a new approach to approximating DPS from Ca_M3 and Mg_M3 for calcareous soils without the need to generate a S _max. We conclude that in the absence of an environmental soil test criteria for P, the DPS_{(CaM3 + MgM3)} and P_Ols could be used to predict P loss potential from calcareous soils.     

Assessment of the aerosol water content in urban atmospheric particles by the hygroscopic growth measurements in Sapporo,Japan

Aerosol water content (AWC) of urban atmospheric particles was investigated based on the hygroscopic growth measurements for 100 and 200 nm particles using a hygroscopicity tandem differential mobility analyzer in Sapporo, Japan in July 2006. In most of the humidogram measurements, presence of less and more hygroscopic mode was evident from the different dependence on relative humidity (RH). The volume of liquid water normalized by that of dry particle (V_w(RH)/V_dry) was estimated from the HTDMA data for 100 and 200 nm particles. The RH dependence of V_w(RH)/V_dry was well represented by a fitted curve with a hygroscopicity parameter κ_eff. The κ_eff values for 200 nm particles were in general higher than those for 100 nm particles, indicating a higher hygroscopicity of 200 nm particles. Based on the κ_eff values, the volume mixing ratios of water-soluble inorganic compounds (ammonium sulfate equivalent) were estimated to be on average 31% and 45% for 100 and 200 nm particles, respectively. The diurnal variation of κ_eff, with relatively higher values in the noontime and nighttime and lower values in the morning and evening hours, was observed for both particle sizes. The V_w(RH)/V_dry values under ambient RH conditions were estimated from κ_eff to range from 0.05 to 2.32 and 0.06 to 2.43 for 100 nm and 200 nm particles, respectively. The degree of correlation between κ_eff and V_w(RH)/V_dry at ambient RH suggests a significant contribution of the variation of κ_eff to atmospheric AWC in Sapporo.     

Amendment application in a multi-contaminated mine soil: effects on soil enzymatic activities and ecotoxicological characteristics

Several amendments were tested on soils obtained from an arsenopyrite mine, further planted with Arrhenatherum elatius and Festuca curvifolia, in order to assess their ability to improve soil's ecotoxicological characteristics. The properties used to assess the effects were: soil enzymatic activities (dehydrogenase, β-glucosidase, acid phosphatase, urease, protease and cellulase), terrestrial bioassays (Eisenia fetida mortality and avoidance behaviour), and aquatic bioassays using a soil leachate (Daphnia magna immobilisation and Vibrio fischeri bioluminescence inhibition). The treatment with FeSO₄ 1 % w/w was able to reduce extractable As in soil, but increased the extractable Cu, Mn and Zn concentrations, as a consequence of the decrease in soil pH, in relation to the unamended soil, from 5.0 to 3.4, respectively. As a consequence, this treatment had a detrimental effect in some of the soil enzymatic activities (e.g. dehydrogenase, acid phosphatase, urease and cellulase), did not allow plant growth, induced E. fetida mortality in the highest concentration tested (100 % w/w), and its soil leachate was very toxic towards D. magna and V. fischeri. The combined application of FeSO₄ 1 % w/w with other treatments (e.g. CaCO₃ 1 % w/w and paper mill 1 % w/w) allowed a decrease in extractable As and metals, and a soil pH value closer to neutrality. As a consequence, dehydrogenase activity, plant growth and some of the bioassays identified those as better soil treatments to this type of multi-contaminated soil.     

Octanol/water partition coefficients for environmentally important organic compounds

Partition coefficients P_O,w ⁽ⁱ⁾ describing the distribution of a solute i onto coexisting phases of 1-octanol and water are needed in a large variety of applications. They can be measured directly by HPLC as long as log P_O,w ⁽ⁱ⁾≧ 3.5. For more hydrophobic substances, several experimental procedures have been proposed in the literature. The reliability of those methods is questionable. Therefore, in the present work, P_O,w ⁽ⁱ⁾ is determined experimentally by three HPLC methods using reversed-phase HPLC [1]. Results from different procedures are compared critically. The method of Braumann [2] proved to be superior over the OECD-guidelines [3]. It was therefore applied to determine octanol/water partition coefficients for 23 substances at 25 °C. For eight of those substances (4-methylindole; 9-(hydroxymethyl)anthracene; N-ethylcarbazol; ethylcyclohexane; trans-2-octene; l,l-dimethyl-(ethy])cyclohexane; heptylbenzene; 4-dodecyl-l,3-benzenediol) no experimentally determined number for P_O,w ⁽ⁱ⁾ has been published before.     

Applied dispersion modelling based on meteorological scaling parameters

A method for calculating the dispersion of plumes in the atmospheric boundary layer is presented. The method is easy to use on a routine basis. The inputs to the method are fundamental meteorological parameters, which act as distinct scaling parameters for the turbulence. The atmospheric boundary layer is divided into a number of regimes. For each scaling regime we suggest models for the dispersion in the vertical direction. The models directly give the crosswind-integrated concentrations at the ground, x_y, for nonbuoyant releases from a continuous point source. Generally the vertical concentration profile is proposed to be other than Gaussian. The lateral concentration profile is always assumed to be Gaussian, and models for determining the lateral spread σ_y are proposed. The method is limited to horizontally homogeneous conditions and travel distances less than 10km. The method is evaluated against independent tracer experiments over land. The overall agreement between measurements and predictions is very good and better than that found with the traditional Gaussian plume model.     

Strategic guidelines for street canyon geometry to achieve sustainable street air quality

This paper is concerned with the motion of air within the urban street canyon and is directed towards a deeper understanding of pollutant dispersion with respect to various simple canyon geometries and source positions. Taking into account the present days typical urban configurations, three principal flow regimes “isolated roughness flow”, “skimming flow” and “wake interference flow” (Boundary Layer Climates, 2nd edition, Methuen, London) and their corresponding pollutant dispersion characteristics are studied for various canopies aspect ratios, namely relative height (h₂/h₁), canyon height to width ratio (h/w) and canyon length to height ratio (l/h). A field-size canyon has been analyzed through numerical simulations using the standard k-ε turbulence closure model. It is found that the pollutant transport and diffusion is strongly dependent upon the type of flow regime inside the canyon and exchange between canyon and the above roof air. Some rules of thumbs have been established to get urban canyon geometries for efficient dispersion of pollutants.     

Strategic guidelines for street canyon geometry to achieve sustainable street air quality

This paper is concerned with the motion of air within the urban street canyon and is directed towards a deeper understanding of pollutant dispersion with respect to various simple canyon geometries and source positions. Taking into account the present days typical urban configurations, three principal flow regimes “isolated roughness flow”, “skimming flow” and “wake interference flow” (Boundary Layer Climates, 2nd edition, Methuen, London) and their corresponding pollutant dispersion characteristics are studied for various canopies aspect ratios, namely relative height (h₂/h₁), canyon height to width ratio (h/w) and canyon length to height ratio (l/h). A field-size canyon has been analyzed through numerical simulations using the standard k-ε turbulence closure model. It is found that the pollutant transport and diffusion is strongly dependent upon the type of flow regime inside the canyon and exchange between canyon and the above roof air. Some rules of thumbs have been established to get urban canyon geometries for efficient dispersion of pollutants.     

Improving Oxygen Conditions in the Deeper Parts of Bornholm Sea by Pumped Injection of Winter Water

Vertical diffusivity and oxygen consumption in the basin water, the water below the sill level at about 59 m depth, have been estimated by applying budget methods to monitoring data from hydrographical stations BY4 and BY5 for periods without water renewal. From the vertical diffusivity, the mean rate of work against the buoyancy forces below 65 m depth is estimated to about 0.10 mW m⁻². This is slightly higher than published values for East Gotland Sea. The horizontally averaged vertical diffusivity κ can be approximated by the expression κ = a₀N⁻¹ where N is the buoyancy frequency and a₀ ≈ 1.25 × 10⁻⁷ m² s⁻², which is similar to values for a₀ used for depths below the halocline in Baltic proper circulation models for long-term simulations. The contemporary mean rate of oxygen consumption in the basin water is about 75 g O₂ m⁻² year⁻¹, which corresponds to an oxidation of 28 g C m⁻² year⁻¹. The oxygen consumption in the Bornholm Basin doubled from the 1970s to the 2000s, which qualitatively explains the observed increasing frequency and vertical extent of anoxia and hypoxia in the basin water in records from the end of the 1950s to present time. A horizontally averaged vertical advection–diffusion model of the basin water is used to calculate the effects on stratification and oxygen concentration by a forced pump-driven vertical convection. It is shown that the residence time of the basin water may be reduced by pumping down and mixing the so-called winter water into the deepwater. With the present rate of oxygen consumption, a pumped flux of about 25 km³ year⁻¹ would be sufficient to keep the oxygen concentration in the deepwater above 2 mL O₂ L⁻¹.     

Laboratory study of cluster ions formation in H2SO4–H2O system: Implications for threshold concentration of gaseous H2SO4 and ion-induced nucleation kinetics

Ion-induced binary H₂SO₄–H₂O nucleation is an important mechanism of aerosol formation in the atmosphere. Ions are created in the atmosphere mainly by galactic cosmic rays. The importance of ion-induced nucleation is recognized in some of the observed nucleation events in the background atmosphere. However, the predictions of current ion–aerosol models are highly uncertain mostly due to the lack of detailed experimental information concerning the thermodynamics and kinetics of ion clustering reactions. Here we continue the report of results of our laboratory experiments on the formation and growth of positive and negative cluster ions in H₂SO₄–H₂O vapours in the flow reactor started in Wilhelm et al. [2004. Ion-induced aerosol formation: new insights from laboratory measurements of mixed cluster ions HSO₄⁻(H₂SO₄)_a(H₂O)_w and H⁺ (H₂SO₄)_a(H₂O)_w. Atmospheric Environment 38, 1735–1744] and Sorokin et al. [2006. Formation and growth of sulphuric acid–water cluster ions: experiments, modelling, and implications for ion-induced aerosol formation. Atmospheric Environment 40, 2030–2045]. The main attention is given to the definition of the concentration of gaseous sulphuric acid in experiment and also to some aspects of the kinetics of small cluster ions formation. The performed analysis has indicated a threshold concentration of gaseous sulphuric acid for binary homogeneous nucleation of at least about 10¹⁰ cm⁻³ at room temperature and low relative humidity.     

Fumigation of pollutants in and above the entrainment zone into a growing convective boundary layer: a large-eddy simulation

Fumigation of a passive plume located in or above the entrainment zone (EZ) into a growing convective boundary layer (CBL) has been simulated by large-eddy simulation (LES). Three non-dimensional parameters, α(=w_e0/w_*0), z₀/z_i0, and σ_z0/z_i0, are used to classify different cases, where w_*0 is the convective velocity scale, w_e0 the initial entrainment velocity, z_i0 the initial CBL height, z₀ the initial plume height, and σ_z0 is the initial plume half-depth. Forty cases have been run and analysed. The crosswind-integrated concentrations have been compared with existing laboratory data from a saline convection tank. The results show that LES is a promising tool to reproduce fumigation processes. With a relatively coarse grid mesh near the EZ, LES derives reliable results that are in a good agreement with the laboratory data. The first parameter, α, containing the effects due to inversion strength, plays an important role in determining C₀(T), the ground-level concentration (GLC) as a function of dimensionless time, T. For large α (say >0.03, corresponding to fast entrainment), variation of α gives significant change in C₀(T); whereas for a wide range of α between 0.01 and 0.02 (corresponding to slow entrainment), C₀(T) is almost independent of α. The starting time of fumigation does not vary significantly with the second parameter, z₀/z_i0 (relative height of plume), although C₀(T) is, in general, smaller for a higher plume. This confirms laboratory findings that the traditional notion of zero fumigation for a high plume (say above 1.10z_i) is not correct. The effect of the third parameter, σ_z0/z_i0, is on the magnitude of C₀(T); thinner initial plumes have higher GLCs.     

Transport and oxidation of SO2 in a stagnant foggy valley

The fate of SO₂ emitted in the San Joaquin Valley of California under stagnant foggy conditions was determined by the release of an inert tracer and the concurrent monitoring of SO₂ and SO₄²⁻ concentrations. At night, SO₂ was found to be trapped in a dense fog layer below a strong and persistent inversion based a few hundred meters above the valley floor. This lack of ventilation led to the accumulation of SO₂ and SO₄²⁻ over a major SO₂ source region in the valley. The rate of oxidation of SO₂ to SO₄²⁻ in fog was estimated at 3 ± 2%h⁻¹. Production of acidity from the oxidation of SO₂ fully titrated the NH₃(g) present before the fog, and led to a progressive drop of the fogwater pH over the course of the night. In the afternoon, the valley was found to be efficiently ventilated by a buoyant upslope flow through the inversion. The tracer data indicated that about 40 % of the air transported upslope in the afternoon was returned to the valley in the night-time drainage flow. The fates of SO₂ and SO₄²⁻ in the valley during extended highinversion episodes appear to depend considerably on the presence of fog or stratus, and on the extent of daytime insolation.     

Internal acid buffering in San Joaquin Valley fog drops and its influence on aerosol processing

Although several chemical pathways exist for S(IV) oxidation in fogs and clouds, many are self-limiting: as sulfuric acid is produced and the drop pH declines, the rates of these pathways also decline. Some of the acid that is produced can be buffered by uptake of gaseous ammonia. Additional internal buffering can result from protonation of weak and strong bases present in solution. Acid titrations of high pH fog samples (median pH=6.49) collected in California's San Joaquin Valley reveal the presence of considerable internal acid buffering. In samples collected at a rural location, the observed internal buffering could be nearly accounted for based on concentrations of ammonia and bicarbonate present in solution. In samples collected in the cities of Fresno and Bakersfield, however, significant additional, unexplained buffering was present over a pH range extending from approximately four to seven. The additional buffering was found to be associated with dissolved compounds in the fogwater. It could not be accounted for by measured concentrations of low molecular weight (C₁–C₃) carboxylic acids, S(IV), phosphate, or nitrophenols. The amount of unexplained buffering in individual fog samples was found to correlate strongly with the sum of sample acetate and formate concentrations, suggesting that unmeasured organic species may be important contributors. Simulation of a Bakersfield fog episode with and without the additional, unexplained buffering revealed a significant impact on the fog chemistry. When the additional buffering was included, the simulated fog pH remained 0.3–0.7 pH units higher and the amount of sulfate present after the fog evaporated was increased by 50%. Including the additional buffering in the model simulation did not affect fogwater nitrate concentrations and was found to slightly decrease ammonium concentrations. The magnitude of the buffering effect on aqueous sulfate production is sensitive to the amount of ozone present to oxidize S(IV) in these high pH fogs.     

Unsteady absorption of sulfur dioxide by an atmospheric water droplet with internal circulation

Unsteady absorption characteristics of sulfur dioxide by an atmospheric water droplet in motion are predicted numerically and analyzed theoretically to recognize the physical mass transport processes inside an aerosol droplet, which is frequently encountered in the atmosphere. Considering the absorption of sulfur dioxide by a droplet in cloud or fog with various velocities, three different Reynolds numbers, viz., Re_g=0.643, 1.287, and 12.87 are studied and compared with each other. The results indicate that for the Reynolds number of 0.643, sulfur dioxide always penetrates toward the droplet centerline throughout the entire absorption period. This is due to the mass transfer dominated by diffusion along the radial direction. In contrast, when the Reynolds number is 12.87, the strength of the vortex motion inside the droplet is strong enough. It results in that, most of the time the concentration contours parallel the streamlines and the lowest SO₂ concentration is located at the vortex center. As a consequence, the diffusion distance is reduced by a factor of three and the absorption time for the droplet reaching the saturated state is shortened in a significant way. With regard to an intermediate Reynolds number such as 1.287, a two-stage mass transfer process can be clearly identified. In the first stage, it is dominated by one-dimensional diffusion, in which over 50% sulfur dioxide is absorbed before the saturated state is reached. In the second stage, the vortex motion mainly controls the mass transfer. However, the contour core is inconsistent with the vortex center. This is because the characteristic time of mass diffusion is in a comparable state with that of droplet internal circulation. The present study elucidates that the strength of a droplet's internal motion plays a vital role in determining SO₂ absorption process.     

Minimum turbulence assumptions and u* and L estimation for dispersion models during low-wind stable conditions

The U.S. Environmental Protection Agency (EPA) short-distance dispersion model, AERMOD, has been shown to overpredict by a factor of as much as 10 when compared with observed concentrations from continuous releases at the Oak Ridge, TN (OR), and Idaho Falls, ID (IF), field experiments during stable periods when wind speeds often dropped below 1 m/sec. Some of this overprediction tendency can be reduced by revising AERMOD's meteorological preprocessor's parameterizations of the friction velocity, u _* , during low-wind stable conditions, thus increasing the calculated σ _v and σ _w and hence the lateral and vertical dispersion rates. Observations show that as the mean wind speed approaches zero at night, there is always significant σ _v and σ _w over time periods of 15 to 60 min, while standard Monin–Obukhov Similarity Theory (MOST) predicts that σ _v and σ _w will approach zero. This paper focuses on the u _* estimation methods and the minimum turbulence (σ _v and σ _w ) assumptions in AERMOD (beta option 4) and two widely used U.S. operational dispersion models, AERMOD (v12345) and SCICHEM. The U.S. EPA has provided results of its tests with the OR and IF data, with its base AERMOD version and its December 2012 modified versions, which assume adjustments to the low-wind u _* and increases in the minimum σ _v parameterization. SCICHEM has relatively small mean bias for both data sets. The revised AERMOD shows much less mean bias, agreeing more with SCICHEM.

Implications:

Suggestions are made for improvements to dispersion models such as AERMOD to correct overpredictions during light-wind stable conditions. Methods for estimating u_*, L, and the minimum turbulence parameters (σ_v and σ_w) are reviewed and compared. SCICHEM and the current operational version and an optional beta version (December 2012) of AERMOD are evaluated with tracer data from low-wind stable field experiments in Idaho Falls and Oak Ridge. It is seen that the operational version of AERMOD overpredicts by a factor of 2 to 10, while the optional beta version of AERMOD and SCICHEM have much less bias.     

Removal of xylenol orange from its aqueous solution using SDS self-microemulsifying systems: optimization by Box–Behnken statistical design

The aim of present study was to develop and evaluate sodium dodecyl sulfate (SDS) self-microemulsifying systems (SMES) for the removal of an anionic dye xylenol orange (XO) from its bulk aqueous media via liquid–liquid adsorption. The composition of SDS SMES was optimized by Box–Behnken statistical design for the maximum removal of XO from its aqueous solution. Various SDS formulations were prepared by spontaneous emulsification method and characterized for thermodynamic stability, self-microemulsification efficiency, droplet size, and viscosity. Adsorption studies were conducted at 8, 16, and 24 h by mixing small amounts of SDS formulations with relatively large amounts of bulk aqueous solution of XO. Droplet size and viscosity of SDS formulations were significantly influenced by oil phase concentration (triacetin), while surfactant concentration had little impact on droplet size and viscosity. However, the percentage of removal of XO was influenced by triacetin concentration, surfactant concentration, and adsorption time. Based on lowest droplet size (35.97 nm), lowest viscosity (29.62 cp), and highest percentage of removal efficiency (89.77 %), formulation F₁₄, containing 2 % w/w of triacetin and 40 % w/w of surfactant mixture (20 % w/w of SDS and 20 % w/w of polyethylene glycol 400), was selected as an optimized formulation for the removal of XO from its bulk aqueous media after 16 h. These results indicated that SDS SMES could be suitable alternates of solid–liquid adsorption for the removal of toxic dyes such as XO from its aqueous solution through liquid–liquid adsorption.     

Aerosol indirect effect over Indo-Gangetic plain

Moderate resolution imaging spectroradiometer (MODIS) data are analyzed over the Indo-Gangetic plain (IGP) to study the effect of aerosol optical depth (AOD) on the water (R_eff,w) and ice (R_eff,i) cloud effective radius for the period 2001–2005. The temporal variation of R_eff,w and R_eff,i shows reverse trend as that of AOD for most of the time. The intensity of positive indirect effect (i.e. increase of R_eff,w/i with decrease of AOD and vice versa) is the highest in winter (ΔR_eff,w/ΔAOD∼−9.67 μm and ΔR_eff,i/ΔAOD∼−12.15 μm), when the role of meteorology is the least. The positive indirect effect is significant in 43%, 37%, 68% and 54% of area for water clouds in winter, pre-monsoon, monsoon and post-monsoon seasons, respectively, whereas the corresponding values for ice clouds are 42%, 35%, 53% and 53% for the four seasons, respectively. On the contrast, R_eff,i in some locations shows increment with the increase in AOD (negative indirect effect). The negative indirect effect is significant at 95% confidence level in 7%, 18%, 9% and 6% grids for winter, pre-monsoon, monsoon and post-monsoon seasons, respectively. The restricted spatial distribution of negative indirect effect in IGP shows that the cloud microphysical processes are very complex. Our analyses clearly identify the contrasting indirect effect, which requires further in situ investigations for better understanding of the aerosol–cloud interaction in the region.     

Observations and predictions of jet diffusion flame behaviour

A simple model has been developed for jet diffusion flames to estimate flame height (h_f) and angle of the flame to the vertical (α_B). The model is based upon the assumption that flame behaviour is dominated by momentum effects. Buoyancy influences on behaviour are assumed to be negligible.Predictions of the model were assessed against flame parameters as observed in a wind tunnel and during field tests with an industrial flare. The wind tunnel studies involved experiments with hydrocarbon (methane, propane, ethylene, butane) diffusion flames. Field experiments involved measuring h_f and α_B of flames resulting from the combustion of acid gas-fuel gas mixtures possessing molecular weights of about 37 g mol⁻¹.The one-to-one correlation coefficient between predicted and observed behaviour resulting from combustion of all hydrocarbon fuels, except methane, was about 0.85. The model significantly underestimated flame heights for methane. This seems to have been due to the neglect of buoyancy effects which could be appreciable for this relatively light gas.Evaluation of results from the tests conducted with the industrial flare showed a one-to-one correlation coefficient between observed and predicted values of h_f to be 0.92. The corresponding magnitude for the correlation coefficient between observed and predicted values of α_B was only 0.71. A partial explanation for this relatively low correlation coefficient lies in the small range of α_B values contained in the population sample.