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    Idealized ILS dispersing in the simulated CBL at a certain time t after the release. The spatial evolution along x of the ILS is equivalent to the time evolution of a continuous point source (CPS) at a distance χ = Ut from the source. The vertical position z, the mean plume height z, and the instantaneous plume centerline position zm are shown together with their respective instantaneous fluctuations (z′ and zm). Note that zr = zr. The relative coordinate system is defined with respect to the instantaneous plume centerline position zm.

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    (a) Normalized vertically integrated mean concentration as a function of the dimensionless time t* and (b) normalized crosswind-integrated mean concentration. The normalized mean plume height z/zi (continuous line) and the position of the maximum concentration (dashed line) are also shown. (c) Normalized horizontal (σy) and vertical (σz) dispersion parameters. The following experimental and numerical data are also shown: Willis and Deardorff (1978) (Δ); Hibberd (2000) (*); Nieuwstadt (1992) (⋄). (d) Skewness of the vertical plume position Sz = /σ3z. The one-particle Lagrangian stochastic model results of Luhar et al. (2000) are also shown (*).

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    (a) Horizontal cross section of the concentration fluctuation intensity ic at ground (z/zi = 0.007). (b) Vertical cross section of ic in the horizontal plume centerline y. (c) Concentration fluctuation intensity along the horizontal plume centerline (y) at the ground. The following experimental and numerical data are also shown: Weil et al. (2002) (×); Deardorff and Willis (1984) (□ and Δ); Henn and Sykes (1992) (*).

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    (a) Example of trajectories of the plume instantaneous centerline position zm. The plume mean height position z is also shown. (b) Normalized horizontal (σym) and vertical (σzm) variances of the plume centerline position (meandering). The laboratory data by Hibberd (2000) (*) and the LES data by Nieuwstadt (1992) (⋄) are also shown. (c) Skewness of the vertical meandering Szm = /σ3zm (continuous line). The total skewness Sz is also shown for comparison (dashed line).

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    PDF of the plume centerline instantaneous position (pym and pzm) as a function of the normalized relative distance [ym/σym = (ymy)/σym and zm/σzm = (zmz)/σzm] calculated by the LES according to Eq. (24) at different distances from the source (t* = 0.5, 1.2, 3). The Gaussian parameterization (29) is also shown (dashed line).

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    (a) Normalized vertically integrated relative concentration as a function of the dimensionless time t*. (b) Normalized crosswind-integrated relative concentration. The positions of the maximum concentration (dashed line) are also shown. (c) Normalized horizontal (σyr) and vertical (σzr) relative diffusion parameters. The following experimental and numerical data are also shown: Hibberd (2000) (*), Nieuwstadt (1992) (⋄). (d) Skewness of the vertical relative plume position Szr = /σ3zr. The numerical results of Luhar et al. (2000) are also shown (*).

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    Vertical cross sections of averaged concentration in (a), (c), (e), (g) absolute and (b), (d), (f), (h) relative coordinates system at different distances from the source (t* = 0.5, 1, 2, 3, respectively). The white crosses indicate the mean plume position.

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    (a) Evolution of the third-order moments of the total (, continuous line), meandering (, dashed–dotted line), and relative (, dashed line) vertical plume position. The nonlinear cross-correlation term 3 is also shown as dotted line. Note that, for clarity, all the terms have been divided by the factor 106. (b) Normalized crosswind-integrated mean concentration at the ground as simulated by the LES.

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    (a) Time evolution of an idealized plume at a certain fixed position downwind of the source in the absolute coordinate system. The continuous line zm(t) represents the instantaneous plume centerline position. The shaded areas between t1 < t < t2 and t3 < t < t4 represent the concentration reflected by the surface. (b) Same as (a) but in the coordinate system relative to the plume position zm(t) (relative coordinate system). The continuous solid line at zr = 0 is the plume centerline [corresponding to zm(t) in absolute coordinates]. The shaded areas represent the concentration reflected by the surface. The dashed line represent the path of integration z1zm(t) (see section 4d).

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    (a) Horizontal cross section of the relative concentration fluctuation intensity icr in the plume mean position z. (b) Vertical cross section of icr in the horizontal plume centerline y. (c) Relative concentration fluctuation intensity along the plume mean position (y, z). The numerical data by Luhar et al. (2000) are also shown (*).

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    PDF of relative concentration in the mean plume position (y, z) at different distance from the source (t* = 0.1, 0.5, 1.5, 2.5). Parameterization (34) is also shown as a dashed line.

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    (a) Normalized crosswind-integrated mean concentration as calculated by the LES. The position of the maximum concentration is also shown (dashed–dotted line). (b) Normalized crosswind-integrated mean concentration as calculated by Gifford’s formula [Eq. (6)] using the relative mean concentration calculated by the LES. (c) Normalized crosswind-integrated mean concentration as calculated by Eq. (6) using the parameterization (39) for the relative mean concentration. (d) Normalized crosswind-integrated mean concentration as calculated by Gifford’s Eq. (6) using the parameterization (39) for the relative mean concentration and the parameterization (41) for the skewness of the relative vertical position. (e) Absolute mean concentration at ground in the plume horizontal mean position (y) as calculated by the LES (continuous line); by using Gifford’s formula with the relative mean concentration (dotted line); by using Gifford’s formula with parameterization (39) (dashed-dotted line); and by using Gifford’s formula with parameterizations (39) and (41) (dashed line). The water-tank data by Willis and Deardorff (1978) are also shown (*).

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Statistics of Absolute and Relative Dispersion in the Atmospheric Convective Boundary Layer: A Large-Eddy Simulation Study

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  • 1 Meteorology and Air Quality Section, Wageningen University, Wageningen, Netherlands
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Abstract

The influence of the different scales of turbulent motion on plume dispersion in the atmospheric convective boundary layer (CBL) is studied by means of a large-eddy simulation (LES). In particular, the large-scale (meandering) and small-scale (relative diffusion) contributions are separated by analyzing dispersion in two reference systems: the absolute (fixed) coordinate system and the coordinate system relative to the plume’s instantaneous center of mass. In the relative coordinate system, the (vertically) inhomogeneous meandering motion is removed, and only the small-scale, homogeneous turbulent motion contributes to the dispersion process.

First, mean plume position, dispersion parameters (variance), and skewness of the plume position are discussed. The analysis of the third-order moments shows how the structure and the symmetry of scalar distribution are affected with respect to the turbulent characteristics of the CBL (inhomogeneity of the large-scale vertical motion) and the presence of the boundary conditions (surface and top of the CBL). In fact, the reflection of the plume by the CBL boundaries generates the presence of nonlinear cross-correlation terms in the balance equation for the third-order moments of the plume position. As a result, the third-order moment of the absolute position is not balanced by the sum of the third-order moments of the meandering and relative plume position.

Second, mean concentration and concentration fluctuations are calculated and discussed in both coordinate systems. The intensity of relative concentration fluctuation icr, which quantifies the internal (in plume) mixing, is explicitly calculated. Based on these results, a parameterization for the probability distribution function (PDF) of the relative concentration is proposed, showing very good agreement with the LES results. Finally, the validity of Gifford’s formula, which relates the absolute concentration’s high-order moments to the relative concentration and the PDF of the plume centerline, is studied. It is found that due to the presence of the CBL boundaries, Gifford’s formula is not able to reproduce correctly the value of the absolute mean concentration near the ground. This result is analyzed by showing that, when the plume is reflected by the CBL boundaries, the instantaneous relative plume width z2r(t) departs from its mean value σ2r. By introducing the skewness of the relative plume position into a parameterization for the relative mean concentration, the results for the calculated mean concentration are improved.

* Current affiliation: Centre for Ecology and Hydrology—Edinburgh, Penicuik, Scotland, United Kingdom

Corresponding author address: A. Dosio, Centre for Ecology and Hydrology—Edinburgh, Bush Estate, Penicuik EH26 0QB, Scotland, United Kingdom. Email: aled@ceh.ac.uk

Abstract

The influence of the different scales of turbulent motion on plume dispersion in the atmospheric convective boundary layer (CBL) is studied by means of a large-eddy simulation (LES). In particular, the large-scale (meandering) and small-scale (relative diffusion) contributions are separated by analyzing dispersion in two reference systems: the absolute (fixed) coordinate system and the coordinate system relative to the plume’s instantaneous center of mass. In the relative coordinate system, the (vertically) inhomogeneous meandering motion is removed, and only the small-scale, homogeneous turbulent motion contributes to the dispersion process.

First, mean plume position, dispersion parameters (variance), and skewness of the plume position are discussed. The analysis of the third-order moments shows how the structure and the symmetry of scalar distribution are affected with respect to the turbulent characteristics of the CBL (inhomogeneity of the large-scale vertical motion) and the presence of the boundary conditions (surface and top of the CBL). In fact, the reflection of the plume by the CBL boundaries generates the presence of nonlinear cross-correlation terms in the balance equation for the third-order moments of the plume position. As a result, the third-order moment of the absolute position is not balanced by the sum of the third-order moments of the meandering and relative plume position.

Second, mean concentration and concentration fluctuations are calculated and discussed in both coordinate systems. The intensity of relative concentration fluctuation icr, which quantifies the internal (in plume) mixing, is explicitly calculated. Based on these results, a parameterization for the probability distribution function (PDF) of the relative concentration is proposed, showing very good agreement with the LES results. Finally, the validity of Gifford’s formula, which relates the absolute concentration’s high-order moments to the relative concentration and the PDF of the plume centerline, is studied. It is found that due to the presence of the CBL boundaries, Gifford’s formula is not able to reproduce correctly the value of the absolute mean concentration near the ground. This result is analyzed by showing that, when the plume is reflected by the CBL boundaries, the instantaneous relative plume width z2r(t) departs from its mean value σ2r. By introducing the skewness of the relative plume position into a parameterization for the relative mean concentration, the results for the calculated mean concentration are improved.

* Current affiliation: Centre for Ecology and Hydrology—Edinburgh, Penicuik, Scotland, United Kingdom

Corresponding author address: A. Dosio, Centre for Ecology and Hydrology—Edinburgh, Bush Estate, Penicuik EH26 0QB, Scotland, United Kingdom. Email: aled@ceh.ac.uk

1. Introduction

To understand and predict the dispersion of compounds in the atmospheric boundary layer, it is very important to determine with great accuracy statistics of plume position and concentration such as mean, variance, and skewness. In fact, the variance of the plume position is a direct measure of the plume spread (dispersion parameter), whereas the variance of the concentration (concentration fluctuations) quantifies the variability of the compound concentration. Concentration fluctuations are a consequence of the complex and inhomogeneous distribution of the compound within the plume, and they can be of the same order of the mean concentration, even at large distances from the emission source (Fackrell and Robins 1982; Deardorff and Willis 1984; Sykes and Henn 1992).

Third-order moments, such as the skewness of the plume position, provide information on the structure and the shape of the plume, and quantify the asymmetry of the plume distribution with respect to its mean position. In the convective boundary layer (CBL), the main factor responsible for this asymmetric distribution is the inhomogeneous and non-Gaussian large-scale turbulent motion. Moreover, a further contribution to the skewness of the plume shape is given by the reflection of the plume at the CBL boundaries, which tend to accumulate the scalar near the surface and in the entrainment zone at the top of the CBL.

Briefly, turbulence in the CBL is characterized by eddies with a large range of temporal and spatial scales, from the Kolmogorov scale (10−4 m) to the entire depth of the CBL (103 m). By relating the length scale l of the turbulent eddies to a characteristic length scale of the plume σr (the plume width in a coordinate system relative to the centerline instantaneous position), Yee and Wilson (2000) proposed three regimes that characterize the growth and structure of the plume. If σrl, typically close to the source, the larger eddies cause the plume meandering, which is the large-scale motion of the plume as a whole and the sweeping of the plume centerline. This large-scale motion is highly inhomogeneous and characterized by a vertical velocity that is positively skewed. When σrl, the scalar concentration is affected by the increasing entrainment of air into the body of the plume. This process is known as relative diffusion and it is the main factor responsible for the growth of the instantaneous plume width.

Finally, when σrl, the turbulent eddies are mainly responsible for the internal (in plume) mixing of contaminant. This internal mixing is quantified by the relative concentration fluctuation intensity icr (defined as the ratio of the standard deviation to the mean concentration in relative coordinates), which is the main contribution to the production of concentration fluctuation at large distances from the source where the meandering component becomes small.

To distinguish and quantify the different contributions of large-scale (meandering) and small-scale motion (relative diffusion) to the evolution of a dispersing plume, it is useful to calculate second- and third-order moments in two different coordinate systems (e.g., Munro et al. 2003b). By analyzing dispersion in an absolute framework, that is, in a coordinate system relative to a fixed point (e.g., the source location), the statistical properties are influenced by the full spectrum of turbulent eddies. As a result, the plume spread σ2 is the sum of the meandering and the relative diffusion. In the relative framework, the coordinate system moves with the instantaneous plume centerline position. In this framework, therefore, the (vertically) inhomogeneous meandering motion is removed, and the concentration statistics are only dependent on the small turbulent eddies (lσr), which are assumed to be homogeneous and isotropic.

Meandering and relative diffusion were assumed to be statistically independent in the pioneering study by Gifford (1959), who developed an analytical model for the calculation of higher-order absolute concentration statistics as a function of two independent terms: the probability distribution function (PDF) of the plume centerline position and the PDF of the concentration in relative coordinates. Although there is not a clear physical separation between large and small scales, Hanna (1986) showed that it is possible to represent a continuous spectrum of motions as a combination of two spectra with two main characteristic time scales: one corresponding to the motions that result in plume meander and the other related to relative dispersion. Gifford’s analysis provided the basis for later studies (Yee et al. 1994; Yee and Wilson 2000; Luhar et al. 2000; Reynolds 2000; Franzese 2003), which incorporated in-plume fluctuations (that were neglected in Gifford’s model) by specifying the PDF of the relative concentration, and took into account the inhomogeneous and skewed turbulence structure of the CBL by calculating the position of the plume centerline from one-particle Lagrangian model trajectories. In all of these studies, the intensity of relative concentration fluctuations icr is the key variable for the determination of the relative concentration PDF, but neither observational nor numerical data in the atmospheric CBL is currently available for an appropriate estimation of icr.

Measurements of dispersion statistics (PDFs) in absolute and relative frameworks have been recently discussed by Munro et al. (2003a, b who analyzed seven different datasets of lidar measurements. Unfortunately, measurements were only available in the horizontal (crosswind) direction at a fixed height. As a result, the vertical non-Gaussian inhomogeneous structure of the plume could not be analyzed. In the study by Ott and Jørgensen (2002) a lidar was used to obtain vertical cross sections of a dispersing plume at 100 m from the source. Mean and variances of absolute and relative concentrations were calculated, but only vertically and horizontally averaged results were presented.

In our study, a large-eddy simulation (LES) is used to analyze the dispersion of a passive scalar in absolute and relative coordinate systems. More explicitly, we first calculate the mean concentration, concentration fluctuations, dispersion parameters, and skewness of the vertical position in both reference coordinate systems. To the best of our knowledge, this is the first time that a complete three-dimensional field of concentration statistics of a dispersing plume is analyzed in both frameworks in the atmospheric CBL. Second, we calculate the PDF of the plume centerline and the PDF of the relative concentration. These results are further analyzed and related in terms of the skewness of the centerline position and the intensity of relative concentration fluctuation, which are critical variables in the formulation of operational models. Third, following Gifford’s analysis and by using the estimated PDFs, we derive concentration statistics that are finally compared with the ones directly calculated by the LES.

The paper is structured as follows: The theoretical background is exposed in section 2. In section 3 the numerical model and the experimental setup are described. The results are presented and discussed in section 4. Finally, conclusions and remarks are discussed in section 5.

2. Theoretical background

Let c = c(x, y, z, t) be the instantaneous concentration of a scalar at the downwind position x from the source. By definition, the nth moment of concentration in a fixed (absolute) reference frame reads:
i1520-0469-63-4-1253-e1
where pc is the concentration PDF.
At each time t, and any downwind distance x, let (ym, zm) be the position of the plume centerline. It is now possible to define the system of coordinates relative to the instantaneous plume centerline position as
i1520-0469-63-4-1253-e2
Consequently, for each downwind position x, the relative concentration is defined as
i1520-0469-63-4-1253-e3
Similarly to Eq. (1), the nth moment of the relative concentration is calculated as
i1520-0469-63-4-1253-e4
where pcr is the PDF of the relative concentration.
Assuming that the meandering and relative diffusion contributions are statistically independent, Gifford (1959) related the absolute concentration PDF to the relative concentration PDF by
i1520-0469-63-4-1253-e5
where pm is the PDF of the instantaneous plume centroid position.
By substituting Eqs. (5) and (4) into (1) we obtain:
i1520-0469-63-4-1253-e6
which relates the absolute and the relative concentration statistics.

By simulating the plume dispersion with a LES, in this paper we investigate the validity of Eq. (6). In particular, we first discuss the shape, structure, and symmetry of the plume by calculating first-, second-, and third-order moments of the plume distribution in both absolute and relative coordinate systems. Second, the PDFs of both the plume centerline (pm) and the relative concentration (pcr) are calculated and discussed. These results are first used to test the validity of current parameterizations for pm and pcr. Subsequently, these PDFs are used to derive the absolute mean concentration by calculating the rhs of Eq. (6), which is compared with the mean concentration directly calculated by the LES [lhs of Eq. (6)].

3. Numerical setup

a. Model description

The LES code used here is the parallelized version of the one described by Cuijpers and Duynkerke (1993) and Siebesma and Cuijpers (1995) in which a set of filtered prognostic equations for the dynamic variables (wind velocity, potential temperature, turbulent kinetic energy) is solved on a staggered numerical grid. The space and time integrations are computed with Kappa (Vreugdenhil and Koren 1993) and leapfrog numerical schemes, respectively.

The subgrid fluxes are closed by relating them to the gradient of the solved variable by means of an exchange coefficient, which depends on the subgrid turbulent kinetic energy, and a length scale, that is related to the grid size. A conservation equation for a passive tracer is added to the governing set of equations. It reads:
i1520-0469-63-4-1253-e7
where c is the mean (filtered) scalar concentration, ui is the mean wind, and is the subgrid flux.

The horizontal numerical domain covers an area of 5.120 × 5.120 km2. To solve the largest spectrum range of motion, a very fine numerical grid is prescribed with a resolution of 10 m in all the directions (512 grid points in each horizontal direction and 128 in the vertical one). Although the grid size may have an effect on the results, especially for the in-plume fluctuations that are driven by scales of motion smaller than the plume size, these effects will be significant only very close to the source when the plume size is comparable to the numerical grid size.

The aspect ratio, that is, the ratio between the horizontal domain dimension to the CBL height zi, is around 6.6 (with zi ∼ 780 m). Lateral periodic boundary conditions are imposed for all variables. However, as soon as the edges of the plume reach the lateral boundaries, the simulation is ended. A time step of 0.25 s is used.

b. Flow characteristics

At the top of the CBL, an inversion strength of Δθ = 5 K is imposed, which maintains the height of the CBL fairly constant with time. A geostrophic wind of 2 m s−1 aligned in the x direction and a heat flux of 0.1 K m s−1 are prescribed as constant forcing (lateral and surface boundary conditions). The simulation is run for an initialization period of 2 h [i.e., the period of CBL development needed to ensure that a (quasi) stationary state is reached]. After this period, the gradients of the mean variables are independent on time and the turbulent kinetic energy has become constant. The average values of the convective velocity scale w* is 1.38 m s−1 and the shear/buoyancy ratio (u*/w*) is equal to 0.14 (where u* is the friction velocity). The value of the stability parameter −zi/L is ∼136. According to the classification used in Holtslag and Nieuwstadt (1986) this simulated flow is mainly driven by convective turbulence.

c. Plume concentration calculation

After the initialization period, an instantaneous line source (ILS) of scalar (nonbuoyant tracer) is emitted along the x axis at zs/zi = 0.28. The line source measures one grid spacing in both the vertical and horizontal (y) direction. As the numerical grid moves with the mean wind along the x direction, the ILS results at time t can be interpreted as those for a continuously released point source (CPS) at corresponding downwind distances χ = Ut (where U is the mean wind speed in the CPS case), as explained by Willis and Deardorff (1981) and Nieuwstadt and de Valk (1987). Therefore the ILS results on a sequence of yz planes with different x at time t are equivalent to the CPS results of time evolution at χ.

To obtain statistically sound results, nine different realizations are performed in which the horizontal position of the instantaneous release is changed. The results are subsequently ensemble-averaged over the different realizations. As explained earlier, each simulation ends as soon as the plume reaches the lateral boundaries. The minimum end period is used as common simulation duration for the calculation of the ensemble-average.

d. Definition of statistical parameters

Figure 1 shows a vertical cross section of an idealized ILS c(x, z) at a certain time t after the release. As explained in the previous section, owing to the equivalence between ILS and CPS, all statistics can be interpreted as a function of the downwind distance χ = Ut.

Let z be the instantaneous (vertical) position of a particle in the plume. The instantaneous centerline position zm and the mean plume height z are defined as follows:
i1520-0469-63-4-1253-e8
i1520-0469-63-4-1253-e9
where dV = dxdydz.
From these mean quantities, the fluctuation of the absolute (z′), relative (zr), and centerline (zm) positions are calculated as follows:
i1520-0469-63-4-1253-e10
i1520-0469-63-4-1253-e11
i1520-0469-63-4-1253-e12
The absolute vertical dispersion parameters σz is defined according to Nieuwstadt (1992) as follows:
i1520-0469-63-4-1253-e13
The absolute dispersion σz is decomposed into meandering (σzm) and relative dispersion (σzr) according to
i1520-0469-63-4-1253-e14
where
i1520-0469-63-4-1253-e15
i1520-0469-63-4-1253-e16
Similarly, the third-order moments of the vertical position are defined as
i1520-0469-63-4-1253-e17
i1520-0469-63-4-1253-e18
i1520-0469-63-4-1253-e19
Similar expressions to (8)(19) hold for the horizontal position y.
The absolute and relative concentration fluctuations (variance of concentration) are derived using
i1520-0469-63-4-1253-e20
i1520-0469-63-4-1253-e21
where t = x/u, Lx is the domain size and c is
i1520-0469-63-4-1253-e22
A similar expression holds for cr.
From the instantaneous centerline position, the PDF pym and pzm are derived as
i1520-0469-63-4-1253-e23
i1520-0469-63-4-1253-e24
where the horizontal and vertical increments (dym and dzm) are defined by the numerical grid size (10 m).

4. Results and discussion

a. Dispersion in absolute coordinate system

The plume statistics (first-, second-, and third-order moments) in the absolute coordinate system are now discussed. As explained earlier, the statistical properties of the scalar concentration in the absolute coordinates system are influenced by both meandering and relative diffusion.

1) Mean concentration, variance, and skewness

Figures 2a and 2b show the vertically integrated and crosswind-integrated concentrations as a function of the normalized distance t*, defined as
i1520-0469-63-4-1253-e25
Figure 2a shows that, in the horizontal plane, the mean concentration has a Gaussian shape. The movement of the centerline position, also shown in the picture, is due to a small wind shear.

The crosswind-integrated concentration is shown in Fig. 2b, where both the plume mean height position z and the position of the maximum concentration are also shown. The value of z calculated by the LES overestimates slightly the water-tank results by Willis and Deardorff (1978) (not shown), probably because of the small difference in the initial plume position (zs/zi = 0.28 in our study compared with zs/zi = 0.25 in Willis and Deardorff 1978). The ground-level maximum occurs at t* = 0.65, in close agreement with the water-tank experiment. The elevated maximum, due to the fast rise of the plume caught by updrafts, occurs at t* = 1.7, and the correspondent surface minimum is present around t* = 1.75. At larger distances (t* ∼ 2.6), the position of the maximum concentration descends below the plume mean position z, meaning that the plume is not perfectly well mixed yet. This result is in agreement with the data by Willis and Deardorff (1978) and Deardorff and Willis (1982), who stated that the well-mixed condition is achieved only at very large distances from the source (t* = 6). The total horizontal and vertical dispersion parameters (σy and σz) are shown in Fig. 2c, where they are compared with laboratory data (Willis and Deardorff 1978; Hibberd 2000) and other LES results (Nieuwstadt 1992), showing satisfactory agreement. The vertical dispersion σz reaches an asymptotic limit around 0.29 when the plume is nearly uniformly well mixed. The horizontal dispersion parameter, on the contrary, follows closely Taylor’s relation of turbulent dispersion; that is, σyt at short times and σyt1/2 at longer times.

The skewness of the vertical position is defined as
i1520-0469-63-4-1253-e26
where is calculated according to Eq. (17). In a flow characterized by the inhomogeneity and skewness of the turbulence, the position skewness quantifies the asymmetry of the plume with respect to its mean position (first-order moment).

The evolution of Sz is shown in Fig. 2d. The skewness is positive in the range 0 < t* < 1.3. In fact, since the CBL is characterized by a positively skewed vertical velocity, the plume is more likely to be transported toward the surface. As a result, the plume is more likely to be below its mean position z, as shown by the maximum plume concentration, which is closer to the ground than the plume mean height (Fig. 2b). If the plume is, for most of the time, below its mean position, z′ = zz will have large probability to be negative. Since by definition = 0, z′ must have also small probability to have very high positive values. As a consequence, the tail of the PDF of the plume position has to be highly positive and the value of the skewness Sz is positive. For 1.3 < t* < 2.5 the skewness has a negative value. As Fig. 2b shows, when the plume reaches the ground, it is reflected by the ground. Near the ground the (downward) vertical motion is transformed into the horizontal, and the plume remains close to the surface until it is transported upward by thermals. For 1.3 < t* < 2.2 the position of the maximum plume concentration lies above its mean position and an elevated maximum is present. The skewness, therefore, must have a negative value. At t* = 2.6 the skewness becomes slightly positive, corresponding to the descent of the plume maximum concentration. As explained earlier, only at large distances is the plume well mixed and Sz approaches zero. The evolution of Sz is in good agreement with the results of Luhar et al. (2000), calculated from their Lagrangian particle model.

2) Concentration fluctuations

Figure 3a shows the horizontal cross section of the concentration fluctuation intensity ic = σc/c near the ground (z/zi = 0.007). Figure 3b shows the contour of ic in the horizontal plume centerline (y/zi = 0). The LES results agree with previous experimental and numerical studies (Deardorff and Willis 1984; Luhar et al. 2000; Weil et al. 2002). As expected, the concentration fluctuation is larger at the edges of the plume, and it has the smallest values near the plume centroid. It is noteworthy that even at relatively large distances (t* = 2) the fluctuation concentration is of the same order as the mean concentration.

The concentration fluctuation intensity along the horizontal plume centerline (y) at ground is shown in Fig. 3c. Our results agree generally with all the previous studies and are in good agreement with the water-tank experiment by Weil et al. (2002). The results show that ic decreases rapidly with distance from the source and reaches an asymptotic limit of about 0.35. However, there is still a large uncertainty on the value of ic at short distances. The numerical results by Franzese (2003) (not reported here) show a minimum at t* = 0.4 and a maximum at t* = 1.2, which are not present in our results. Franzese (2003) explained the presence of the maximum as a combination of two different effects: the decay of ic (in the plume’s centerline) with distance from the source and the growth of ic with distance from the centerline at any fixed position from the source. At any distance t* from the source, the value of the ground concentration (and consequently the value of ic) depends strongly on the plume spread σz and the position of the plume mean height z. Since in the work by Franzese (2003) the value of z overestimates the experimental results, especially between 0.5 < t* < 2, the edge of the plume is located close to the ground. Since σc reaches its maximum value at the plume’s edges, as a result the value of ic may be overestimated.

Also the water-tank experiments by Deardorff and Willis (1984) show a relative maximum of ic around t* = 1.4, but they doubted the validity of their data. If the anomalous maximum in the data by Deardorff and Willis (1984) is excluded (as suggested by the same authors) good agreement is found among all the experimental data and the LES results. The LES results by Henn and Sykes (1992) are in contrast with the other results at short distance (t* < 2) probably due to the low vertical resolution (40 m) and the relative high ratio between the source size and the eddy dimension.

b. Meandering component

In this section the plume meandering is discussed by analyzing the second- and third-order moments, and the PDF of the plume instantaneous centerline position. An example of trajectories of the plume centerline position zm is shown in Fig. 4a. The meander trajectory distribution is very similar to that determined by Luhar et al. (2000). Close to the source (t* < 0.8), the shape of the ensemble of trajectories is similar to the mean crosswind-integrated concentration (Fig. 2b), because meandering is the main contribution to the plume motion. The spread of the instantaneous plume centerline position reaches its maximum around t* = 0.5, and then it slowly diminishes. Far from the source, when the plume is nearly uniformly well mixed vertically, the instantaneous plume position becomes similar to the mean plume height z, which has reached the asymptotic value of 0.5.

The second-order moment of the horizontal and vertical centerline position (σym and σzm) calculated by the LES [Eq. (15)] agrees satisfactorily with previous laboratory experiments and numerical simulation (Fig. 4b). The vertical meandering component σzm reaches a maximum value around t* = 0.5, and then it decays quickly to a very small value when the plume vertical motion is constrained by the CBL boundaries and zmz. In the horizontal direction the plume motion is not limited, and the meandering component σym is in agreement with the theoretical analysis by Csanady, as reported by Weil et al. (2002),
i1520-0469-63-4-1253-e27
which implies that σym approaches a constant value at large distances (t* > 1).
The evolution of the skewness of the instantaneous centerline (vertical) position
i1520-0469-63-4-1253-e28
is shown in Fig. 4c. The skewness of the meandering position provides information on the distribution of the plume as transported by the large-scale turbulent motions. As pointed out by Luhar et al. (2000), there are no currently available data to validate the evolution of Szm. The LES results provide an estimation of the downwind variation of the meandering skewness and they can be useful to derive a suitable parameterization. Close to the source, as meandering is the main contribution to plume dispersion, the concentration distribution is mainly affected by the large-scale motions. As a result, at t* < 0.5 the meandering skewness is very similar to the total skewness Sz. The meandering skewness follows closely the motion of the plume carried by the large-scale eddies because, by definition, meandering represents the contribution of the large-scale motion to the plume total dispersion. In particular, Szm is positive when the plume is transported downward by the downdrafts (t* < 1 and t* > 2.4) and it becomes negative when the plume is transported upward by thermals, as shown by the position of the maximum concentration in Fig. 2b. The meandering skewness has a different behavior than the total skewness at distances t* > 0.5. For example, at t* = 1.1, Szm is negative, whereas Sz is still positive. This difference will be further discussed below when the third-order moment of the relative diffusion is discussed.

The evolution of the meandering skewness is closely related to the value of the PDF of the plume’s centerline position. From the instantaneous centerline position (8), the PDFs of the plume centerline pzm and pym are calculated at any downstream distance. The results are shown in Fig. 5 at selected distances from the source (t* = 0.5, 1.2, 3), where the PDF are shown as a function of the normalized relative position (zmz)/σzm (and equivalently for the horizontal PDF).

The horizontal PDF pym is well reproduced by the Gaussian function
i1520-0469-63-4-1253-e29
which is commonly used as parameterization for the plume horizontal mean position PDF (Yee and Wilson 2000; Luhar et al. 2000; Franzese 2003) based on the water-tank experiment by Willis and Deardorff (1978). This result is explained by the homogeneity of the horizontal motion in the atmospheric CBL.

The LES results for the pzm are shown in Figs. 5b,d,f. Three different shapes of the PDF can be distinguished: at short distances from the source, when the plume is transported by the downdraft, the PDF is positively skewed (Fig. 5b). At larger distances (t* = 1.2, Fig. 5d), the plume is transported upward by thermal motions. Therefore it is now more likely that the plume instantaneous centerline lies above its mean value. As a result, pzm becomes negatively skewed. Finally, the PDF becomes once again positively skewed (t* = 3, Fig. 2f). It must be noticed that, for a uniformly well-mixed plume, the PDF would have a Gaussian shape centered on the plume mean position. Our LES results are consistent with the water-tank data by Willis and Deardorff (1978), who showed that the well-mixed condition is reached only at very large distances from the source (t* > 6).

Although out of the scope of this work, finding a suitable parameterization for the PDF of the vertical plume position would be of great practical interest since it would allow the estimation of concentration statistics through Gifford’s (1959) Eq. (6) without the need of calculating the position of the plume centerline by means of a LES or Lagrangian stochastic models. As the LES results show, the shape of pzm is highly complex and may depend on different factors such as the turbulent characteristics of the CBL, plume characteristics (such as σzm and Szm), distance from the source, and height of the release.

c. Dispersion in relative coordinate system

1) Mean concentration, variance, and skewness

Figures 6a and 6b show the normalized vertically and crosswind-integrated mean concentration in the relative coordinate system. The vertically integrated concentration (Fig. 6a) has a Gaussian shape similar to the one in absolute coordinates (Fig. 2a), as expected. In the relative coordinates system, the concentration pattern is always aligned along the line yr = 0, which is the position of the centerline. The crosswind-integrated concentration is shown in Fig. 6b. Moreover, to allow a direct comparison, vertical cross sections of averaged absolute and relative concentrations at different distances from the sources (t* = 0.5, 1, 2, 3, respectively) are shown in Fig. 7.

Since in the relative coordinate system the meandering, which characterized the large-scale inhomogeneous and skewed motion, has been removed, one expects that the relative concentration has a more homogeneous and Gaussian distribution. This is evident especially close to the source (t* < 1), where the plume is still very narrow (σrl) and has not reached the boundaries yet. For instance, at t* = 0.5, the scalar distribution in absolute coordinates is vertically highly inhomogeneous. The distribution is skewed, and the maximum concentration is located close to the ground (Fig. 7a). In relative coordinates, on the contrary, the plume is very narrow and shows a Gaussian distribution in both the horizontal and vertical direction (Fig. 7b). At t* = 1 the plume in relative coordinates has reached the CBL boundaries and still shows a quasi-Gaussian distribution (Fig. 7d), but the maximum concentration is somehow below the plume centerline (Fig. 6b). This deviation form the Gaussian distribution is an effect of the reflection of the plume by the ground, which occurs at t* = 0.5 (Fig. 2b), and causes the relative concentration to be positively skewed (Fig. 6d). At larger distances (t* > 1.2), the maximum lies above the relative mean height. Both the absolute and relative concentration present an elevated maximum around t* = 2 caused by the reflection with the CBL top (Figs. 7e and 7f). It is noteworthy that the relative concentration shows a distribution very similar to the absolute concentration (Figs. 7g and 7h). Since at large distances from the source the meandering component becomes small, the relative diffusion is the main contribution to the plume dispersion, as shown by Fig. 6c where the vertical and horizontal relative dispersion parameters are shown. As a result, the concentration distribution in absolute coordinates is influenced mainly by the in-plume, small-scale motions, and the absolute concentration is very similar to the relative concentration for t* > 1.5.

This result is corroborated by the analysis of the skewness of the vertical relative plume position Szr = /σ3zr, shown in Fig. 6d. Generally the relative skewness has a small value because the relative concentration distribution is driven mainly by the small-scale motion, which is homogeneous and Gaussian. These small asymmetries in the relative concentration distribution are due solely to the reflections by the CBL boundaries. The reflection by the ground is the cause of the small positive value of Szr at distances 0 < t* < 1.1. At larger distances (t* > 1.5), the relative skewness becomes negative owing to the reflection by the CBL top, and Szr becomes very similar to total skewness Sz. In fact, at t* > 1.5 the meandering contribution becomes very small and consequently . The numerical data by Luhar et al. (2000) overestimate significantly the LES results, especially close to the source (t* < 1). In their work, Luhar et al. (2000) assumed the following balancing relationship between the meandering and relative third-order moment:
i1520-0469-63-4-1253-e30
From the definition of the third-order moments [Eqs. (17), (18), and (19)], it can be shown that
i1520-0469-63-4-1253-e31
If meandering is independent of relative dispersion, then the cross-correlation terms C1 and C2 are zero [as it is the case for the cross term in the expression of the second-order moments; Eq. (14)]. Our LES results confirm, indeed, that both the cross terms (in the second-order moment equation) and C1 are zero. In fact, since by definition = = 0, the average of the local relative position zr is not influenced by either the position of the plume mean height zm, or by its variance z2m, so that both the second-order cross term and the third-order cross term C1 are zero.
With respect to the cross-correlation term C2, our LES results show that its value varies with the distance from the source. The term C2 can be decomposed as
i1520-0469-63-4-1253-e32
where the value of the cross-correlation term C2 depends on the relative importance of the value of the local (instantaneous) relative width z2r, with respect to its average value σ2zr. The different terms in Eq. (31) are shown in Fig. 8a as a function of t*. In Fig. 8b the absolute mean ground concentration is shown for comparison.

Three separate regimes can be distinguished:

  1. Close to the source (t* < 0.2), the total dispersion is dominated by meandering. Since the plume is very narrow, zr ∼ 0 and . Since the plume has not reached the ground yet (the ground concentration for t* < 0.2 is zero as shown in Fig. 8b), the cross-correlation term C2 is zero.
  2. When the plume reaches the ground and the scalar is reflected (0.2 < t* < 2), the concentration distribution changes. To visualize the physical process, one can consider the time evolution of the plume at a fixed distance from the source, as illustrated schematically in Fig. 9a. The centerline position zm(t) shows the instantaneous vertical motion of the plume in the absolute coordinate system. The relative concentration (Fig. 9b) has a Gaussian distribution centered on the plume centerline only when the plume is not affected by the boundaries (e.g., between t0 < t < t1 and t2 < t < t3). When the plume is reflected by the surface (e.g., between t1 < t < t2 and t3 < t < t4), the relative concentration is modified as illustrated by the shaded areas in Fig. 9b. As a consequence, the instantaneous mean plume position zm changes and the instantaneous relative plume width z2r(t) between t1 < t < t2 departs from its mean value σ2zr. Consequently, zσ2zr and the cross-correlation term C2 is different than 0. The term C2 reaches a maximum around t* = 0.6 when the plume is closest to the ground (as shown by the maximum in the ground concentration). As soon as the plume is lifted up by the thermals, the effect of the ground on the plume concentration becomes smaller and the cross-correlation C2 becomes zero.
  3. Far from the source (t* > 2), the meandering contribution becomes small because the instantaneous plume centerline position zm approaches its mean value z (Fig. 4a). As a result, zm ≈ 0 and . The cross-correlation term C2 becomes zero because the plume is confined between the bottom and the top of the CBL, and the main contribution to the vertical spread is given by the relative diffusion (z2rσ2zr). The variation of the cross-correlation term C2 with distance from the source can be interpreted as an effect of the meandering (i.e., position of the mean plume height) on the relative dispersion (value of the local relative variance), which, therefore, are not statistically independent in case of reflection of the plume by the boundaries. This result will be further discussed when the validity of Gifford’s formula is analyzed.

2) Concentrated fluctuations

Figures 10a and 10b show the contour of the relative concentration fluctuation intensity icr = σcr/cr in the plume mean height (z) and in the horizontal plume centerline (y), respectively. To our knowledge, this is the first time that complete horizontal and vertical two-dimensional fields of relative concentration fluctuations are shown. In their study, Fackrell and Robins (1982) showed relative intensity of concentration fluctuation calculated as the maximum rms of concentration (σ̂) divided by the maximum mean concentration (ĉ) at any downstream position. This approach is valid in neutral conditions when the plume centerline coincides with the position of the maximum concentration. As shown earlier, in the CBL these two heights are different, and icr may be different than σ̂/ĉ.

The horizontal cross section of relative concentration fluctuation (Fig. 10a) has a shape similar to the one in absolute coordinates (Fig. 3a) because the horizontal motion is homogeneous and Gaussian in both the reference coordinate systems. The vertical cross section (Fig. 10b) shows a pattern similar to the relative mean concentration (Fig. 6b): at short distances icr spreads uniformly until it is influenced by the boundaries, around t* = 1.

The value of icr in the plume centerline (y, z) is shown in Fig. 10c. Very close to the source, σcr is zero because the plume is so narrow that only the very small eddies can generate in-plume fluctuations. As a result, cr(t) ∼ cr. As the plume grows, the mixing process is driven by an increasing number of eddies, and σcr grows consequently. The intensity of relative concentration fluctuations reaches a maximum of icr = 1.4 around 0.5 < t* < 1 when the plume size is of the same order of the turbulent length scale and all the turbulent eddies participate to the internal mixing. At larger distances, icr slowly decreases with the distances from the source. At very large distance, for an ideally well-mixed condition, the scalar concentration must be close to its mean value so that σcr is relatively small.

In the previous studies by Yee et al. (1994) and Yee and Wilson (2000), the value of icr was assumed to be constant in a crosswind cross section of the instantaneous plume at any distance because their model was develop to be used in homogeneous and isotropic turbulence. In the study by Luhar et al. (2000), the value of icr was parameterized in order to obtain the best fit of the total calculated concentration fluctuation ic. The values of icr calculated by Luhar et al. (2000) are also shown in Fig. 10c. Although the general shape is similar, the parameterization predicts the maximum at larger distances from the source than the LES results. It must be noticed that, whereas the LES results are obtained by direct calculation of icr, the parameterization is explicitly calculated in order to obtain the best fit of the results for the total intensity ic.

3) PDF of relative concentration

The PDF of the relative concentration is calculated from the evolution of cr as
i1520-0469-63-4-1253-e33
where the concentration increment dc is taken equal to one hundredth of the maximum concentration. Figure 11 shows the PDF calculated in the plume mean position (y, z) at different distance from the source (t* = 0.1, 0.5, 1.5, 2.5, respectively). The LES results show that pcr has mainly two characteristics shapes: a unimodal type with the mode at nonzero value of cr and an exponential type [or unimodal with the mode at cr = 0 (Fig. 11b)]. As expected, the former type is the most probable in the plume centerline, as shown by Munro et al. (2003a), because the relative concentration in the mean plume position is very often different than zero. On the contrary, far from the centerline small concentrations are more probable, and consequently the PDF shows an exponential shape. However, the LES results show that at distances for 0.5 < t* < 1 pcr has an exponential shape also in the plume centerline (Fig. 11b). This may be explained by the fact that the at distances for 0.5 < t* < 1.5, when dispersion is dominated by meandering, the plume instantaneous vertical position zm is very unlikely to coincide with its mean position z (Fig. 4a). As a result the time evolution of the relative concentration cr is highly intermittent. When the instantaneous plume position zm is closer to its mean value (t* > 1.5, Fig. 4a) the PDF of relative concentration is once again unimodal with the mode at nonzero (Figs. 11c and 11d).
The LES results are compared with the following Gamma parameterization (Yee and Wilson 2000; Luhar et al. 2000):
i1520-0469-63-4-1253-e34
where λ = 1/i2cr. As shown in Fig. 11, parameterization (34) is able to reproduce correctly the shape of pcr at any distances from the source. Since the parameterization depends heavily on the value of the intensity of relative concentration fluctuation icr, the agreement between the LES results and relationship (34) demonstrates the accuracy of the calculation of icr.

d. Validation of Gifford’s formula

As explained in section 2, the absolute and relative mean concentrations are related by Gifford’s formula [Eq. (6)]. In his study, Gifford (1959) pointed out that possible statistical variations in the shape of cnr are not taken into account since cr should approach its average value if the sampling time is long enough. However, Eq. (6) is strictly true only if the plume is not affected by the boundary conditions. As discussed previously, when the plume is reflected by the CBL boundaries, the relative concentration becomes skewed and meandering and relative diffusion are not statistically independent. As a consequence, the cross-correlation term C2 in the equation for the third-order moments (31) is not zero. This result implies that Gifford’s formula [Eq. (6)] has to be modified, as will be shown.

Referring to Figs. 9a and 9b, the absolute mean concentration at height z = z1 is related to the relative mean concentration by the equation:
i1520-0469-63-4-1253-e35
by the definition of the relative coordinate system. The rhs of Eq. (35) implies that one has to integrate the relative concentration along the path z1zm(t) illustrated by the dashed line in Fig. 9b. By considering a finite sampling time, one can write
i1520-0469-63-4-1253-e36
where N is the number of the samplings ti, Nzj is the number of times that the integration path z1zm(t) encounters the height zj (illustrated by the points A, B, C, and D in Fig. 9b), l is the relative concentration locally averaged over the points A, B, C, and D, and finally p(zj) = Nzj/N is the PDF of the plume centerline in the position zj.

It is evident that Eq. (36) is equivalent to Eq. (6) only if the local average l coincides with the mean relative concentration . This is true only if the relative concentration cr is not perturbed by the CBL boundaries. To study the effect of the boundaries on the absolute mean concentration, Gifford’s formula [Eq. (6)] is used to calculate crosswind integrated absolute mean concentration, which is then compared with the LES results.

Figure 12a shows the crosswind integrated absolute mean concentration calculated directly from the LES results (note that this figure is the same as Fig. 2b). Figure 12b shows the crosswind integrated absolute mean concentration calculated through Eq. (6) from the relative mean concentration and the PDF of the plume centerline pm calculated by the LES. It is evident that Gifford’s formula gives a satisfactory result for distances t* < 0.4, before the impact of the plume on the surface. For t* > 0.4, Gifford’s formula reproduces correctly the general plume behavior but it underestimates significantly the mean concentration, especially close to the ground (z/zi < 0.2). As explained previously, this result is due to the use in Eq. (6) of the mean relative concentration cr instead of the more correct local average l. Since the calculation of the local average requires the knowledge of the time evolution of the relative concentration, it is evident that its direct calculation for use in Eq. (6) is unsuitable for practical use. However, a more practical way to compute the absolute mean concentration can be found in which, instead of l, a parameterization of cr is used that takes into account the skewness of the relative plume position. This methodology is explained in the next section.

1) Parameterization of the relative mean concentration

In their model for isotropic dispersion Yee et al. (1994) assumed for cr a simple circular symmetric Gaussian, both in y and z. Both Luhar et al. (2000) and Franzese (2003) distinguished between horizontal and vertical motion by assuming
i1520-0469-63-4-1253-e37
where Q is the amount of compound emitted per unit time, and pyr and pzr are the PDF of mean particle positions relative to the plume centerline. In the horizontal direction a simple Gaussian relationship is assumed:
i1520-0469-63-4-1253-e38
The Gaussian parameterizations (38) fits closely the values of calculated by the LES normalized by its value in the plume centerline (z = zm) (not shown).
For the vertical PDF, both Luhar et al. (2000) and Franzese (2003) assumed a Gaussian form with multiple reflections due to the CBL boundaries. Moreover, Luhar et al. (2000) incorporated the skewness of the relative position by assuming the same closure procedure used in Luhar et al. (1996) for the vertical velocity PDF. As a result, the parameterization for the vertical relative position PDF reads:
i1520-0469-63-4-1253-e39
where N is the number of reflections (three in this work) and the other parameters are defined following Luhar et al. (2000):
i1520-0469-63-4-1253-eq1
It must be noticed that, since the parameterization (39) already includes reflections, the relative dispersion parameter σ̃zr is parameterized similarly to Franzese (2003):
i1520-0469-63-4-1253-e40
where the parameters a = 1/2(0.4w3*/zi) and b = 100 are chosen in a way that σ̃zσzr close to the source and σ̃zt at large distances.

If parameterization (39) is used in Gifford’s formula, the results are only very slightly improved, as shown in Fig. 12c. This result is that, by using the skewness of the relative vertical position Szr = /σ3zr, the dependence of the relative diffusion to a meandering component in the case of plume reflection is not taken into account.

In fact, by analyzing Fig. 8a, it is evident that the total skewness is larger than the sum of the meandering and relative skewness since the cross-correlation term C2 in the third-order moments Eq. (31) is not zero.

To take into account this effect, the relative skewness Szr is replaced by
i1520-0469-63-4-1253-e41
By doing so, the instantaneous variability of the relative concentration is taken into account by means of the cross-correlation term 3. Expression (41) is similar to the parameterization for the relative skewness used by Luhar et al. (2000), but in their formulation the value of the skewness Sz was assumed equal to the meandering one, Szm, which is valid only close to the source when dispersion is dominated by meandering, as discussed previously (Fig. 4c).

If parameterization (41) is used in Eq. (6), the shape and value of the resulting absolute mean concentration is closer to the value of c directly calculated by the LES, as shown in Fig. 12d. In particular, it is evident that the position of the maximum concentration (indicated in the figure by the dashed line) is better reproduced, especially at short distances (t* < 1) where the reflection by the ground occurs. The importance of calculating correctly the position of the maximum concentration is corroborated by analyzing the evolution of the ground concentration shown in Fig. 12e. The value of c at the ground calculated directly by the LES (continuous line) agrees satisfactorily with the water-tank data by Willis and Deardorff (1978). If the relative concentration cr is used directly in Eq. (6), the resulting absolute ground concentration (dashed–dotted line) underestimates significantly the LES results. If parameterization (41) is used instead, the resulting absolute ground concentration (dashed line) is closer to the LES results. Discrepancies still exist especially for distances 0.5 < t* < 1, which may indicate the need for a better parameterization of the relative concentration. Differences between the calculated mean concentration and the LES results are also found near the top of the CBL. This result is explained by the fact that a simple parameterization like Eq. (39) does not consider the difference between the reflection at the ground and in the inversion zone. When the plume reaches the ground it is simply reflected by the surface. The entrainment zone, on the contrary, does not act like a simple reflecting barrier because the structure of the turbulence in that area is very complex and the plume can be transported above zi. This effect is evident at distances t* > 1.5, when the position of the maximum concentration calculated by using parameterization (41) (Fig. 12d) is underestimated when compared to the one directed calculated by the LES (Fig. 12a). In this case, the direct use of the relative concentration cr gives better results (Fig. 12b).

5. Conclusions

By means of large-eddy simulation (LES), plume dispersion in the atmospheric convective boundary layer (CBL) was studied in two reference systems: the absolute coordinate system and the coordinate system relative to the plume’s instantaneous center of mass. By so doing, it was possible to separate the different contributions of small- and large-scale motions on the plume’s evolution. In the relative coordinate system, the (vertically) inhomogeneous meandering motion was removed, and only the small homogeneous turbulent eddies contributed to the dispersion process.

The evolution, shape, and symmetry of the scalar distribution were first analyzed by calculating the mean plume position, the dispersion parameters (variance), and the skewness of the plume position in both the coordinate systems. In particular, the analysis of the third-order moments showed that the structure and symmetry of the scalar distribution were affected by both the turbulent characteristics of the CBL (inhomogeneity of the vertical large-scale motion) and the presence of the boundary conditions (surface and top of the CBL). The skewness of the plume’s centerline position was mainly influenced by meandering of the plume as it was transported by the updrafts and downdrafts. On the contrary, the skewness of the relative position was mainly affected only by the reflection of the plume by the CBL boundaries because in the relative coordinate system the large-scale motion was removed. However, it is noteworthy that the third-order moment of the absolute position is not balanced by the sum of the meandering and relative diffusion contributions, but nonlinear cross terms generated by the reflection of the plume by the boundaries have to be taken into account.

Mean concentrations and concentration fluctuations were also studied in absolute and relative coordinate systems. In particular, the internal (in plume) mixing of the scalar within the plume was analyzed by calculating the intensity of relative concentration fluctuation icr. The evolution of icr was used in the parameterization of the probability distribution function (PDF) of the relative concentration, showing good agreement with the LES results.

Finally, the validity of Gifford’s formula, which relates the absolute concentration’s high-order moments to the relative concentration and the PDF of the plume centerline, was studied. It was found that, owing to the presence of the CBL boundaries, Gifford’s formula is not able to reproduce correctly the value of the absolute mean concentration, especially near the ground. This result was explained in terms of the effect of the reflection of the plume by the boundaries, which causes the relative concentration to depart from a Gaussian distribution and to become skewed. Consequently, the relative diffusion and the meandering component are not statistically independent. As a result, the need to use a local average for the calculation of the relative mean concentration in Gifford’s formula was shown. Alternatively, the results were improved by using a parameterization for the relative mean concentration that takes into account the skewness of the relative plume position.

However, the present study indicates the need for further investigation of the effect of the CBL boundaries on dispersion. In particular, it is necessary to study the evolution, the structure, and the shape of the relative concentration in order to find a suitable parameterization for the locally averaged relative concentration, to be introduced in Gifford’s formula when the plume is affected by the CBL boundaries. Finally, it must be noticed that plume dispersion is highly dependent on the release height and higher-order statistics of absolute and relative concentration as a function of the release height should be thoroughly investigated. However, the dependence of meandering to relative dispersion in case of plume reflection by the CBL boundaries is a general result and is valid for any source height.

Acknowledgments

The comments and suggestions by the anonymous referees have been very useful to improve clarity of the paper and are deeply appreciated. A. Dosio was supported by the Centre of Expertise Emissions and Assessment, a cooperation between TNO and Wageningen University. All numerical simulations have been performed on the TERAS supercomputer of the National Computing Center SARA (Project 2003/00302)

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Fig. 1.
Fig. 1.

Idealized ILS dispersing in the simulated CBL at a certain time t after the release. The spatial evolution along x of the ILS is equivalent to the time evolution of a continuous point source (CPS) at a distance χ = Ut from the source. The vertical position z, the mean plume height z, and the instantaneous plume centerline position zm are shown together with their respective instantaneous fluctuations (z′ and zm). Note that zr = zr. The relative coordinate system is defined with respect to the instantaneous plume centerline position zm.

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3689.1

Fig. 2.
Fig. 2.

(a) Normalized vertically integrated mean concentration as a function of the dimensionless time t* and (b) normalized crosswind-integrated mean concentration. The normalized mean plume height z/zi (continuous line) and the position of the maximum concentration (dashed line) are also shown. (c) Normalized horizontal (σy) and vertical (σz) dispersion parameters. The following experimental and numerical data are also shown: Willis and Deardorff (1978) (Δ); Hibberd (2000) (*); Nieuwstadt (1992) (⋄). (d) Skewness of the vertical plume position Sz = /σ3z. The one-particle Lagrangian stochastic model results of Luhar et al. (2000) are also shown (*).

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3689.1

Fig. 3.
Fig. 3.

(a) Horizontal cross section of the concentration fluctuation intensity ic at ground (z/zi = 0.007). (b) Vertical cross section of ic in the horizontal plume centerline y. (c) Concentration fluctuation intensity along the horizontal plume centerline (y) at the ground. The following experimental and numerical data are also shown: Weil et al. (2002) (×); Deardorff and Willis (1984) (□ and Δ); Henn and Sykes (1992) (*).

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3689.1

Fig. 4.
Fig. 4.

(a) Example of trajectories of the plume instantaneous centerline position zm. The plume mean height position z is also shown. (b) Normalized horizontal (σym) and vertical (σzm) variances of the plume centerline position (meandering). The laboratory data by Hibberd (2000) (*) and the LES data by Nieuwstadt (1992) (⋄) are also shown. (c) Skewness of the vertical meandering Szm = /σ3zm (continuous line). The total skewness Sz is also shown for comparison (dashed line).

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3689.1

Fig. 5.
Fig. 5.

PDF of the plume centerline instantaneous position (pym and pzm) as a function of the normalized relative distance [ym/σym = (ymy)/σym and zm/σzm = (zmz)/σzm] calculated by the LES according to Eq. (24) at different distances from the source (t* = 0.5, 1.2, 3). The Gaussian parameterization (29) is also shown (dashed line).

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3689.1

Fig. 6.
Fig. 6.

(a) Normalized vertically integrated relative concentration as a function of the dimensionless time t*. (b) Normalized crosswind-integrated relative concentration. The positions of the maximum concentration (dashed line) are also shown. (c) Normalized horizontal (σyr) and vertical (σzr) relative diffusion parameters. The following experimental and numerical data are also shown: Hibberd (2000) (*), Nieuwstadt (1992) (⋄). (d) Skewness of the vertical relative plume position Szr = /σ3zr. The numerical results of Luhar et al. (2000) are also shown (*).

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3689.1

Fig. 7.
Fig. 7.

Vertical cross sections of averaged concentration in (a), (c), (e), (g) absolute and (b), (d), (f), (h) relative coordinates system at different distances from the source (t* = 0.5, 1, 2, 3, respectively). The white crosses indicate the mean plume position.

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3689.1

Fig. 8.
Fig. 8.

(a) Evolution of the third-order moments of the total (, continuous line), meandering (, dashed–dotted line), and relative (, dashed line) vertical plume position. The nonlinear cross-correlation term 3 is also shown as dotted line. Note that, for clarity, all the terms have been divided by the factor 106. (b) Normalized crosswind-integrated mean concentration at the ground as simulated by the LES.

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3689.1

Fig. 9.
Fig. 9.

(a) Time evolution of an idealized plume at a certain fixed position downwind of the source in the absolute coordinate system. The continuous line zm(t) represents the instantaneous plume centerline position. The shaded areas between t1 < t < t2 and t3 < t < t4 represent the concentration reflected by the surface. (b) Same as (a) but in the coordinate system relative to the plume position zm(t) (relative coordinate system). The continuous solid line at zr = 0 is the plume centerline [corresponding to zm(t) in absolute coordinates]. The shaded areas represent the concentration reflected by the surface. The dashed line represent the path of integration z1zm(t) (see section 4d).

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3689.1

Fig. 10.
Fig. 10.

(a) Horizontal cross section of the relative concentration fluctuation intensity icr in the plume mean position z. (b) Vertical cross section of icr in the horizontal plume centerline y. (c) Relative concentration fluctuation intensity along the plume mean position (y, z). The numerical data by Luhar et al. (2000) are also shown (*).

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3689.1

Fig. 11.
Fig. 11.

PDF of relative concentration in the mean plume position (y, z) at different distance from the source (t* = 0.1, 0.5, 1.5, 2.5). Parameterization (34) is also shown as a dashed line.

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3689.1

Fig. 12.
Fig. 12.

(a) Normalized crosswind-integrated mean concentration as calculated by the LES. The position of the maximum concentration is also shown (dashed–dotted line). (b) Normalized crosswind-integrated mean concentration as calculated by Gifford’s formula [Eq. (6)] using the relative mean concentration calculated by the LES. (c) Normalized crosswind-integrated mean concentration as calculated by Eq. (6) using the parameterization (39) for the relative mean concentration. (d) Normalized crosswind-integrated mean concentration as calculated by Gifford’s Eq. (6) using the parameterization (39) for the relative mean concentration and the parameterization (41) for the skewness of the relative vertical position. (e) Absolute mean concentration at ground in the plume horizontal mean position (y) as calculated by the LES (continuous line); by using Gifford’s formula with the relative mean concentration (dotted line); by using Gifford’s formula with parameterization (39) (dashed-dotted line); and by using Gifford’s formula with parameterizations (39) and (41) (dashed line). The water-tank data by Willis and Deardorff (1978) are also shown (*).

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3689.1

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