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Turbulence and Dispersion Studies Using a Three-Dimensional Second-Order Closure Eulerian Model

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  • 1 Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, China
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Abstract

A three-dimensional second-order closure meteorological and pollutant dispersion model is developed, and the computed results are evaluated. A finite-element method is used to solve the governing equations because of its versatility in handling variable-resolution meshes and complex geometries. The one-dimensional version of this model is used to simulate a 24-h diurnal cycle for a horizontally homogeneous atmospheric boundary layer in neutral, stable, and unstable stratifications. The simulated turbulence fields under a convective boundary layer act as the background turbulence for simulating cases of three-dimensional pollutant dispersion from elevated point sources. The simulated turbulence and pollutant distribution compared well with experimental observations and with other numerical models, ensuring the validity of the adopted mathematical formulation as well as the developed model. The computed results provide an overview of turbulence structures in different atmospheric stabilities and are helpful to enhance understanding of the characteristics of air pollutant dispersion, such as plume rise and descent in a convective boundary layer. The current study suggests the need for an insightful and practical numerical model to perform air-quality analysis, one that is capable of overcoming the weaknesses of traditional Gaussian plume and k-theory dispersion models.

Corresponding author address: Dr. D. Y. C. Leung, Dept. of Mechanical Engineering, 7/F, Haking Wong Bldg., The University of Hong Kong, Pokfulam Road, Hong Kong, China.

ycleung@hkucc.hku.hk

Abstract

A three-dimensional second-order closure meteorological and pollutant dispersion model is developed, and the computed results are evaluated. A finite-element method is used to solve the governing equations because of its versatility in handling variable-resolution meshes and complex geometries. The one-dimensional version of this model is used to simulate a 24-h diurnal cycle for a horizontally homogeneous atmospheric boundary layer in neutral, stable, and unstable stratifications. The simulated turbulence fields under a convective boundary layer act as the background turbulence for simulating cases of three-dimensional pollutant dispersion from elevated point sources. The simulated turbulence and pollutant distribution compared well with experimental observations and with other numerical models, ensuring the validity of the adopted mathematical formulation as well as the developed model. The computed results provide an overview of turbulence structures in different atmospheric stabilities and are helpful to enhance understanding of the characteristics of air pollutant dispersion, such as plume rise and descent in a convective boundary layer. The current study suggests the need for an insightful and practical numerical model to perform air-quality analysis, one that is capable of overcoming the weaknesses of traditional Gaussian plume and k-theory dispersion models.

Corresponding author address: Dr. D. Y. C. Leung, Dept. of Mechanical Engineering, 7/F, Haking Wong Bldg., The University of Hong Kong, Pokfulam Road, Hong Kong, China.

ycleung@hkucc.hku.hk

Introduction

Transport and diffusion of an air pollutant involve complex interaction among the pollutant, approaching wind, and atmospheric turbulence. Many factors, such as source configurations, meteorological conditions, and geographical locations, influence these processes. The atmospheric pollutant transport is so complicated that its characteristics are hardly deduced on the basis of sparse and routine field measurements. Moreover, high-resolution field measurements are generally difficult and expensive to conduct. Hence, atmospheric boundary layer and air pollutant dispersion models are essential tools for analyzing and understanding the phenomenon of atmospheric pollutant dispersion.

Because of their versatility, Gaussian plume models are the most commonly used air pollutant dispersion models in the world for elevated point sources. However, experiments (Willis and Deardorff 1976, 1981) showed that some plume behaviors, such as the descent and rise of the plume under an unstably stratified atmosphere, could not be simulated accurately by either Gaussian plume or k-theory dispersion models. Enger (1986) showed that these non-Gaussian plume behaviors could be predicted by using higher-order closure dispersion models. Furthermore, higher-order closure dispersion models simulate explicitly the effects of atmospheric turbulence to pollutant dispersion instead of using empirically determined dispersion coefficients that lack universality and may be site specific. The higher-order closure dispersion models that solve mean concentration and second-order moments prognostically can overcome the above weaknesses and open up new avenues in this research area.

The modeled pollutant transport depends not only on its mathematical formulation but also is affected by the background atmospheric flow and turbulence structure. Some commonly available numerical models for studying the dynamics of atmospheric boundary layer include the first-order closure or k models (Pielke 1974; Cotton and Tripoli 1978), second-order closure models (Mellor 1973; Enger 1983a), third-order closure models (Andre et al. 1978; Briere 1987), and large-eddy simulation models (Deardorff 1972; Moeng 1984). A comparison of the relative merits and shortcomings of these models can be found in Liu (1998). The second-order closure model has been adopted in the current study because of its good compromise among its general applicability, simplicity, and computational complexity.

Lewellen (1977) derived a second-order closure model from the exact Reynolds stress equations and showed that this approach was viable for turbulent flow computation. Sykes et al. (1984) calculated the concentration fluctuations and fluxes by using another second-order closure model under a neutral turbulent flow field. Their calculated results compared well with experimental measurements. Sun and Chang (1986), Pai and Tsang (1991), and Liu and Leung (1997) computed the crosswind integrated pollutant concentration in an unstably stratified atmosphere by using different two-dimensional second-order closure Eulerian dispersion models. In general, the results of these models matched with experimental observations. A three-dimensional second-order closure model has been developed for the current study by further extending a previously developed two-dimensional model.

This study models the turbulence field and the pollutant dispersion from an elevated point source situated at different altitudes. A horizontally homogeneous atmospheric flow is simulated by varying the ground-surface potential temperature to represent a hypothetical diurnal cycle that consists of stable, neutral, and unstable stratifications. The simulated mean wind and turbulence act as background atmospheric parameters to model pollutant dispersion in unstable stratification, which is usually created on a clear day with bright sunshine. Because of the fact that the pollutant concentration gradient near the point source is very strong, variable-resolution mesh is used to obtain local refinement of the resolution near the point source. Liu and Leung (1998) demonstrated that the finite-element method was very flexible in handling variable-resolution mesh and is therefore used to discretize the mathematical model of pollutant dispersion over flat and horizontally homogeneous terrain. Apart from the ease of handling variable-resolution mesh, the finite-element method is also suitable for handling any arbitrary and complex geometries, such as terrain elevation, by employing an isoparametric mapping. A summary of the advantages of using the finite-element method over the finite-difference method in meteorological and pollutant dispersion studies is given in Liu and Leung (1998).

Although the second-order closure models require proper calibration for their numerical parameters, these parameters can be determined easily as demonstrated in this paper. The model developed provides a more realistic basis for considering the effects of turbulence and buoyancy force on pollutant dispersion that makes it superior to the conventional Gaussian and k-theory dispersion models. The validity of the model is evaluated by comparing the simulated turbulence field and pollutant concentration with experimental observations and other numerical results. The simulated results are helpful to advance our knowledge of the relationship between atmospheric turbulence and pollutant transport. In particular, the non-Gaussian plume behaviors caused by buoyancy force in an unstable stratification are well simulated. Recent development in high-speed computers makes some large-scale computations viable within reasonable time and expense. Hence, more-sophisticated models such as the second-order closure models might be used to replace those empirical models for research, practical pollutant dispersion analysis, or routine air-quality prediction purposes.

Mathematical model

The dispersion of pollutant from a point source in the atmosphere is a function of mean wind, temperature, and turbulence field. Therefore, the governing mathematical model consists of two modules: an atmospheric boundary layer module and a pollutant dispersion module.

Atmospheric boundary layer module

This module is a second-order closure wind-field model (Liu 1998), which is used to simulate the wind, turbulence, and temperature fields. Because the current study concentrates on simulating the vertical profiles of turbulence over flat and horizontally homogeneous terrain, a one-dimensional version of the wind model under dry atmosphere is used. The definitions of all variables and parameters are given in the appendix. The governing equations of mean velocity and potential temperature are
i1520-0450-40-1-92-e1
respectively.
The turbulence closure scheme used is based on the model suggested by Andrén (1990), which is basically a “Yamada–Mellor 2.5-level” model. All transport equations for the second-order turbulent quantities are reduced to diagnostic equations except the turbulent kinetic energy q2/2, which is modeled by the following prognostic equation (Yang 1991):
i1520-0450-40-1-92-e4
The components of Reynolds stress, heat flux, and potential temperature covariance are calculated diagnostically with the use of u, υ, θ, and q2.

Pollutant dispersion module

A second-order closure dispersion model is developed to solve prognostically the mean pollutant concentration and the second-order moments (Liu 1998). The dispersion model is an Eulerian dispersion model that starts from the following mass-continuity equation in tensor notation:
i1520-0450-40-1-92-e5
Equation (5) involves the calculation of second-order turbulent fluxes, thus the corresponding transport equation
i1520-0450-40-1-92-e6
is required to close (5) mathematically.
We need to parameterize the second (turbulent diffusion) and the third (pressure covariance) terms on the right-hand side of (6) to close the system of equations. The pressure covariance term is approximated by (Enger 1986 and 1990)
i1520-0450-40-1-92-e7
The turbulent diffusion term ∂(uiujc)/∂xj is parameterized by using a gradient diffusion approximation. According to Lewellen (1977),
i1520-0450-40-1-92-e8
It should be mentioned that the parameterization of turbulent diffusion term by a constant gradient diffusion approximation [(8)] instead of a time-dependent diffusion rate (to ensure a Gaussian profile) is not the best approximation (Deardorff 1978). One of the objectives of the current study is to develop a practical three-dimensional air-quality model with the simplest possible second-order closure model, so a simpler constant gradient diffusion is used in the parameterization. Moreover, the time-dependent diffusion rates are difficult to determine for those practical dispersion problems with multiple pollutant sources, so (8) is adopted in this study. Nevertheless, the reasonably good agreements with experimental measurements and previous numerical calculations indicate that the errors induced by this approximation are not significant.
With the use of the parameterization equations (7) and (8), the differential equation (6) then takes the following form:
i1520-0450-40-1-92-e9
Similarly, if the molecular destruction and turbulent diffusion terms are parameterized to (α3q/λ)cθ and a4(∂/∂xi)[(∂(cθ)/∂xi)], respectively, the transport equation of cθ will be
i1520-0450-40-1-92-e10
Equations (5), (9), and (10) form a closed set of equations for calculating the mean pollutant concentration and its second-order moments.

Turbulent length scales

In this study, the same formulations as those of Yamada and Mellor (1975) and Liu and Leung (1998) are adopted for the turbulent length scales in neutral and stable stratifications, and those suggested by Enger (1986) are used in unstable stratification. These formulations are summarized below.
i1520-0450-40-1-92-e11a

The height of the convective boundary layer zi is defined as the height where wθ has its minimum value (Enger 1986). The stability is determined by the Monin–Obukhov length scale L defined in Barthelmie et al. (1993), which is simplified from five to three classes as follows:

  1. stable stratification: 0 m ⩽ L < 1000 m,

  2. neutral stratification: L ⩽ −1000 m or L ≥ 1000 m, and

  3. unstable stratification: −1000 m ⩽ L < 0 m.

Model coefficients

The procedures for determining the model coefficients in (4), (9), and (10) generally follow those of Andrén (1990) and Enger (1986) with only minor changes. After Andrén (1990), the diffusive length scale constant d and the dissipation closure constant Cϵ in (4) are equal to ⅓ and 21.3, respectively, in the current study.

In determining the empirical constant α1, (9) is simplified by assuming that the underlying surface is flat and homogeneous. Furthermore, the concentration field is assumed to be stationary and horizontally homogeneous. The equation is simplified to (13) under neutral conditions (Nieuwstadt and Ulden 1977) as follows:
i1520-0450-40-1-92-e13
According to the measurements in the neutral surface layer, q2 = 9u2τ and λ = κz (Enger 1990). The results to be discussed in section 6 show that ww1.2u2τ in the neutral surface layer. Inserting the above expressions into (13), α1 is found to be 0.4.

A value of ⅓ is assigned to α2 according to Enger (1990). The diffusion length scales a3 and a4 in (9) and (10) have been used with the same value. These length scales used in Enger (1990) and Pai and Tsang (1991) are ⅓ and 0.3, respectively. After some sensitivity tests, the difference is found to be insignificant and thus a3 (=a4) is assigned to be ⅓.

The value of α3 used in Sun and Chang (1986) and Enger (1990) is ≈0.13; a value of 0.45 is adopted by Pai and Tsang (1991). According to the tests carried out by Enger (1983b), 0.1 ⩽ α3 ⩽ 0.3. We have performed some sensitivity tests and found that α3 = 0.13 gave satisfactory agreement with measurements; hence, this value is adopted for the current study.

The computational domain and grid system

As shown in Fig. 1, a rectangular computational domain is used in the current calculation in which the lengths in the streamwise (x), lateral (y), and vertical (z) directions are equal to 40, 3.6, and 3.0 km, respectively. These lengths are large enough for the mesoscale perturbation to be well within the model domain and also for applying the Neumann boundary conditions along the boundary of the model. The pollutant dispersion is assumed to be symmetric along the centerline plane of the pollutant source (y = 0) to economize the computer resources.

For the one-dimensional atmospheric boundary layer simulation, 30 grid points are used in the vertical plane. A log-linearly spaced grid is adopted to improve the accuracy of the numerical model. Details of the grid generation can be found elsewhere (Liu and Leung 1998).

The same vertical grid is used for the three-dimensional pollutant dispersion simulation in which the numbers of grid points used in the X and Y directions are 30 and 25, respectively. The horizontal grid is a variable resolution one so as to obtain a more refined solution near the pollutant point source. The grids in the X–Y, X–Z, and Y–Z planes are shown in Fig. 2.

Initial and boundary conditions

The initial and boundary conditions used in this atmospheric simulation are the same as those adopted in the case-2 study of Liu and Leung (1998). A diurnal cycle is simulated to produce different atmospheric stability classes by varying the ground-surface potential temperature. It consists of 24 simulated hours starting at midnight. The roughness parameter z0 and the geostrophic wind components Ug and Vg are adopted to be 0.05 m, 18.0 m s−1, and 0.0 m s−1, respectively.

The boundary conditions used in the pollutant dispersion module are shown in Fig. 1. Gaussian distribution of pollutant concentration was used on the first grid points downstream of the source. The point source should not overlap with any grid point in the computation domain, because c = ∞ at the source. The computational domain is assumed to be free from pollutants initially.

Numerical method

In this study, the gradient of the calculated pollutant concentration is very large near the point source; thus, a variable-resolution mesh as shown in Fig. 2 is adopted to obtain local refinement of the solution near the point source. The finite-element method is used to discretize the continuous mathematical model because of its flexibility in handling the variable-resolution mesh.

The detailed finite-element method for solving the governing equations of the current one-dimensional atmospheric boundary layer model can be found in Liu and Leung (1998) and will not be repeated here. This section only describes in detail the finite-element method adopted for solving the pollutant dispersion model.

The eight-nodes brick element and the linear basis interpolation function are used to approximate the dependent variable ψ as follows:
ψx, y, z, tNx, y, zt
The weighting function W(x, y, z), the shape function N(x, y, z), and (14) are substituted into (5), (9), and (10) to obtain the weak formulation of the mathematical model. This formulation is then rearranged to obtain the global semidiscrete weighted residual approximation, which, written in matrix notation, is
MΨ̇KΨF
In establishing the finite element formulation discussed above for computation, it is more convenient to divide the global computation domain Ω into nele number of nonoverlapping elements Ωe. Each element constructs its semidiscrete weighted residual approximation in Ωe as follows:
MeΨ̇KeΨFe
where
i1520-0450-40-1-92-e16b
The global semidiscrete weighted residual approximation [(15)] can then be established as the sum of the contribution from (16) (assembling). The matrices Me, Ke, and vector Fe depend on the differential equations to be solved. In the calculation of c, the matrices and vector are given by
i1520-0450-40-1-92-e17a
The symbol ;p0 represents the average of the dependent variable ψ at t − Δt within an element Ωe, that is,
ψ̂NξηζΨtt
where ξ, η, and ζ are the normalized coordinates in an element Ωe and lie between −1 and 1. The mass matrix Me is the same as (17b) for solving the differential equations (9) and (10). In the calculation of uic, the stiffness matrix and force vector are given by
i1520-0450-40-1-92-e19a
Last, in the calculation of cθ, the stiffness matrix and force vector are given by
i1520-0450-40-1-92-e20a
It is worth mentioning that the second-order derivative terms in (9) and (10) have been integrated by parts that lead to surface integral terms. In addition, one of the derivatives is moved from the shape functions to the weighting functions. The boundary conditions for the second-order turbulent-fluxes are either zero Neumann or Dirichlet ones. Hence, the surface integral terms are equal to zero and are not included in the above finite-element formulations for simplicity. For calculating Me, Ke, and Fe in (17), (19), and (20), a linear basis interpolation function is used, which is defined in terms of the normalized coordinates in each element Ωe as
i1520-0450-40-1-92-e21
Because the convection terms dominate the transport process in the current problem, the Galerkin finite-element method becomes unstable when the local Peclet number γe is greater than 2. To avoid any instabilities caused by convection domination, the Petrov–Galerkin finite-element method (Hughes and Brooks 1982; Zienkiewicz and Taylor 1989) is used instead of the Galerkin finite-element method. The implementation of the Petrov–Galerkin finite-element method is equivalent to the use of a modified weighting function Wi, which is written in tensor notation as
i1520-0450-40-1-92-e22
When applying the Petrov–Galerkin finite-element method discussed above in diffusion-dominated problems, (uekuek)1/2 ≈ 0, γe ≈ 0, αe ≈ 0, and WN. Therefore, the implementation of the Petrov–Galerkin finite-element method in diffusion problems is equivalent to the use of the Galerkin finite-element method.
The calculations of Me, Ke, and Fe involve some volume integral terms in each element Ωe. Because the shape function N and the weighting function W are defined in the normalized coordinates (ξ, η, ζ), integrations of (17), (19), and (20) are transformed from the global coordinates (x, y, z) to the normalized ones (ξ, η, ζ) by using the isoparametric finite-element method. Details of this finite-element method can be found in Zienkiewicz and Taylor (1989). Equations (17), (19), and (20) are generally transformed as
i1520-0450-40-1-92-e23
Here Me, Ke, and Fe in each element Ωe are transformed to the form of (23) and integrated numerically by the Gaussian quadrature approximation. They are then assembled as the global semidiscrete weighted residual approximation using (16). Afterward, the temporal derivative is integrated by using the Crank–Nicolson scheme and is rearranged as the following system of linear equations:
i1520-0450-40-1-92-e24
The above system of linear equations is solved by the successive-overrelaxation method after applying the initial and boundary conditions at each time step.

Results and discussion

The simulations consist of two parts as according to the modules of the model. The first part simulates the vertical profiles of turbulence for different stabilities, and the second part simulates the pollutant dispersion at different emission elevations under a convective boundary layer.

Turbulence simulation

To investigate the performance of the model, it is tested against several experimental observations. Liu and Leung (1998) tested the model against the results of the Wangara (Yamada and Mellor 1975) and convective water channel (Willis and Deardorff 1974) experiments. The simulated mean velocity and potential temperature showed general agreement with those experimental results.

Apart from these parameters, the turbulence is also an essential factor that affects the pollutant dispersion. Thus, the predicted turbulence structure is compared with the measurements conducted by Izumi and Caughey (1976), Caughey and Palmer (1979), Caughey et al. (1979), Brost et al. (1982), Nieuwstadt (1984), and Grant (1986). These measurements provide datasets in neutral, stable, and unstable atmospheric boundary layers over flat and horizontally homogeneous terrain, thus representing a wide spectrum of atmospheric conditions.

Neutral atmospheric boundary layer

The results of the 24-h simulation as mentioned in section 4 are summarized in Table 1, which indicates 15 h of stable stratification, 8 h of unstable stratification, and 1 h of neutral stratification. A neutrally stratified case is observed at 1700 LT. Note that this experiment is not an exact neutral atmospheric simulation, because the Monin–Obukhov length scale L is −1103 m instead of negative infinity. An ideal neutral atmospheric boundary layer dataset is rarely available for comparison, because the atmosphere is usually affected by the weak heat flux on the ground surface and stable stratification above the boundary layer.

The computed vertical profiles of the normalized velocity variances uu/u2τ, υυ/u2τ and ww/u2τ are shown in Fig. 3. Field measurements obtained by Brost et al. (1982) and Grant (1986) are also shown for comparison and provide velocity variances for a nearly neutral atmosphere.

Good agreement between the experimental observations and computed parameters is found except in some underprediction of uu/u2τ and υυ/u2τ at higher levels. This underprediction is mainly due to the use of the averaged empirical constant term Cϵ (dissipation relation in surface layer) obtained from laboratory and field measurements for the current calculation. The empirical constant determined from laboratory measurements is smaller than that measured in the atmospheric surface layer (Andrén 1990). The normalized velocity variances decrease with increasing altitude and have maximum values (4.2, 3.1, and 1.2 for uu/u2τ, υυ/u2τ, and ww/u2τ, respectively) near the ground surface. The computed results of ww/u2τ in a neutrally stratified atmospheric surface layer are useful in deriving the empirical constant α1 used in (13).

Stable atmospheric boundary layer

According to Table 1, a stably stratified atmosphere is formed after sunset and continues until sunrise the next morning. The two most stable cases among the simulated hours are at 0700 (L = 68 m) and 1900 LT (L = 73 m), which are presented here.

Similar to the case of neutral stability, the normalized velocity variances decrease with increasing altitudes and have maximum values near the ground surface (Fig. 4). These maxima are located at an altitude of approximately 0.05 zh, where zh is the height of the atmospheric boundary layer. By definition, zh is the level at which the vertical heat flux has decreased to 5% of its surface value. The computed altitudes of the maximum velocity variances are different from those of the field measurements that show maxima on the ground surface. This difference is mainly caused by the use of an underpredicted Cϵ similar to that of the case of neutral stratification discussed previously. Nevertheless, the horizontal and vertical velocity variances at 1900 are within the scattered range of the data, but slight underprediction is observed at 0700. The vertical profiles of heat flux show good agreement for both simulated periods.

Unstable atmospheric boundary layer

Convective atmospheric boundary layers are found during the daytime from 0900 to 1600 LT with a wide range of Monin–Obukhov length scale (Table 1). The most unstable cases occurred at 1300 and 1400 and have been adopted for the current comparison. The simulated turbulence field at 1500 will also be used for calculating the pollutant dispersion to be discussed in the next section.

The vertical profiles of the normalized horizontal velocity variance (uu + υυ)/2w2, vertical velocity variance ww/w2, vertical heat flux wθ/ wθO, and temperature covariance θθ/T2* are shown in Fig. 5. The calculated normalized horizontal velocity variances show nearly constant values between 0.3 and 0.5 within the mixed layer, values that matched with the convective channel experimental results obtained by Willis and Deardorff (1974). On the other hand, the normalized vertical velocity variances show greater variations from 0.2 on the ground surface to 0.6 at 0.3 ⩽ z/zi ⩽ 0.5 and decrease again thereafter. This observation agreed well with that obtained by Willis and Deardorff (1974). The maximum value of ww/w2* (0.5–0.6) also matched with that simulated by Enger (1986), which, however, occurs at a lower altitude (z/zi = 0.3).

The normalized vertical heat fluxes decrease linearly with altitudes for z/zi < 0.9, and are slightly higher than those of the measured data of Caughey and Palmer (1979) but in line with those obtained by Willis and Deardorff (1974) and Enger (1986). These values become fairly constant above the convective boundary layer.

The calculated temperature covariance agreed well with the experimental observations for z/zi < 0.6 (Fig. 5). However, greater variations can be observed at higher altitudes (z/zi > 0.7). The temperature covariance decreases rapidly with altitude and reaches a minimum at z/zi ≈ 0.8–0.9. This trend then reverses completely above the mixed layer. Enger (1986) showed that better simulation could be obtained by calculating the temperature covariance prognostically. The diagnostic temperature covariance is a simplified expression of the prognostic one by assuming negligible time derivative and diffusion terms. Hence, the observed discrepancy is mainly caused by the negligence of the diffusion terms. Prognostic calculation will increase the computation load tremendously without significantly improving the accuracy. Thus, the temperature covariance is calculated diagnostically in this study. Note that the irregularity of velocity and temperature variance for z/zi > 1 is caused by the rapid change of the calculated mean variables between the mixing layer and the upper stable layer.

Dispersion simulation

The second-order closure dispersion model is tested with the use of the wind, potential temperature, and background turbulence fields generated at 1500 LT as described in the previous section. It is a simulation under a convective atmospheric boundary layer of parameters as listed in Table 1 and normalized mean wind Um/w∗ ≈ 5.7. Willis and Deardorff (1976, 1981) studied the pollutant dispersion with midlevel and ground-level pollutant releases in a convective water channel, and results of three-dimensional pollutant distribution were obtained. On the other hand, some of the experimental measurements and numerical models only considered two-dimensional crosswind-integrated pollutant distribution in the vertical (X–Z) plane. Typical examples are Sun and Chang (1986) and Pai and Tsang (1991) who studied pollutant dispersion over flat terrain under a convective boundary layer using two-dimensional models. The same turbulent length-scale formulations are used in their studies. The former uses a finite difference approach, and the latter uses a finite-element approach. As such, we calculated the pollutant concentrations in both two-dimensional and three-dimensional spaces to facilitate comparisons with these experimental and computational data. For the two-dimensional comparisons, the modeled concentration was integrated along the y direction to obtain the crosswind-integrated pollutant concentration
i1520-0450-40-1-92-e25
The upper limit of the integration in (25) is ymax, because the horizontal domain is not extended to infinity in this study. Therefore, the calculated cy would be smaller than the actual value.

To facilitate comparisons with experimental data, simulations have been conducted at four different emission elevations, corresponding to zs = 0.067, 0.25, 0.5, and 0.75 times zi. These simulations are discussed in the following sections.

Emission height zs = 0.067zi

Figure 6 shows the normalized centerline concentration contours in the X–Z plane for a ground-level release (zs = 0.067zi). The plume rises once it is released into the convective boundary layer and becomes level with Z = 0.9 at a downwind distance X ≈ 1.5. The agreement between the experimental observation (Willis and Deardorff 1976) and the numerical simulation is in general good, except that the calculated plume ascends higher than the observed one (Z = 0.65). Both results show nearly uniform pollutant concentration in the mixing layer for downwind distance X > 3.0.

The shapes of the crosswind-integrated pollutant concentration Cy contours resemble those of the centerline concentration and agree well with experimental results (Willis and Deardorff 1976) except for a slight underestimation of the local maximum concentration of the plume (Fig. 7). The limited lateral domain used in (25) is one of the reasons for this undersimulation, because the experiments were performed in a confined water channel.

Figure 8 shows the Cy results from the current model and from Pai and Tsang (1991). Both models used the same turbulent length-scale formulation in neutral and stable stratifications; Pai and Tsang (1991) used the following expression for unstable stratification:
i1520-0450-40-1-92-e26
The turbulent length scale used in this case is smaller than that of Pai and Tsang (1991). This difference results in greater dissipation, which leads to smaller magnitude and gradient of the simulated turbulent fluxes. Hence, the spatial variation of mean pollutant concentration is smaller than that of Pai and Tsang (1991). The lateral domain in calculating the crosswind-integrated properties of Pai and Tsang (1991) is infinity, but the current crosswind pollutant concentration is integrated from the three-dimensional results within a limited lateral domain. The difference of the lateral domains also affects the results calculated. Although different turbulent length-scale formulations are used, both models are able to simulate the plume ascent feature and the positions of local minimum and maximum Cy (at X ≈ 1.5). The altitude of the local maximum obtained by Pai and Tsang (1991), the current model, and Willis and Deardorff (1976) are Z = 1.0 (Cy = 1.42), Z = 0.75 (Cy = 1.2), and Z = 0.65 (Cy = 1.25), respectively. Therefore, the current model results compare more favorably with the observations of Willis and Deardorff (1976) than with the model results of Pai and Tsang (1991).

Emission height zs = 0.5zi

The pollutant discharged at higher altitudes has different characteristics from that of the ground-level emission. Figure 9 shows the centerline pollutant concentration contour in the X–Z plane for an emission height of 0.5zi. Instead of rising, initially the simulated plume falls to the ground surface, producing a local maximum at X ≈ 1.0 (C = 1.5). This phenomenon can be observed from the experimental results of Willis and Deardorff (1981), which, however, have a higher concentration (C = 3.0). This difference may be caused by the assumption here of neutrally stratified Gaussian pollutant concentration distribution as the boundary conditions near the point source. The effect of stratification (cθ) on the pollutant dispersion is not included in the pollutant source boundary condition. Thus, the plume only descends gradually to the ground surface instead of a more rapid descent under normal convective condition.

The calculated plume path shows clearly that, after impinging on the ground, the plume rebounds and rises in the mixing layer (Fig. 9). Because the local maximum on the ground surface is much smaller than the measured value, the contours that show the rebound of the plume have a lower concentration than the corresponding measured one.

Although the plume mainly descends toward the ground surface, part of the pollutant spreads and rises into the upper mixing layer once it is released into the atmosphere. This plume development feature has been simulated by the current model. However, the magnitude and location of the local minimum cannot be simulated accurately.

Figure 10 shows the ground-surface pollutant concentration distribution in the horizontal (X–Y) plane. Generally, the calculated concentrations are comparable to the observed values except for the deviation in the location of the local maximum as discussed previously. The ground-surface pollutant distribution generally follows a Gaussian distribution. Thus, the pollutant dispersion simulation can be simplified from a three-dimensional model to a two-dimensional one by considering the crosswind-integrated pollutant concentration. The cy values in the vertical plane can be calculated by the two-dimensional models similar to those used by Sun and Chang (1986) or Pai and Tsang (1991). The ground-surface pollutant concentration is then calculated by assuming a Gaussian distribution. However, attention should be paid to the choice of suitable horizontal dispersion coefficient, which might be site specific.

Figure 11 shows the distribution of Cy in the X–Z plane. It agrees well with experimental measurements (Willis and Deardorff 1981). The calculated local maximum (Cy = 1.4) on the ground surface (X ≈ 1.1) is comparable to the observed value (Cy = 1.8, X ≈ 0.8). In addition, it is observed that the phenomenon of plume rise to the upper mixing layer and the occurrence of local minimum near X = 1 and Z = 0.5 are simulated accurately by the current calculation.

Figure 12 shows the Cy values obtained by the current model and by Pai and Tsang (1991). As can be observed, the simulated plume path, location of the local maximum and minimum, and the rise of the plume into the upper mixing layer are all comparable to those of Pai and Tsang (1991). An improvement in the simulation accuracy is observed for the current model when the results are compared with the experimental results of Willis and Deardorff (1981).

According to the distribution of C and Cy obtained, the agreement between the calculated and observed C values is in general fair (see Fig. 9), but much better agreement between the calculated and observed crosswind pollutant concentration can be found (Figs. 11 and 12). Further works, such as increasing the calculation resolution, could be conducted to improve the performance of the model in this aspect.

Emission height zs = 0.25zi

Figure 13 shows the distribution of Cy calculated by the current model and that of Sun and Chang (1986) for an emission height of zs = 0.25zi. The simulated plume first descends after being emitted into the atmosphere. This descent brings the plume down to the ground surface at X ≈ 0.5 with a concentration Cy = 2.4. This local maximum agrees well with that calculated by Sun and Chang (1986). After the impingment, the plume rebounds and rises into the mixing layer. A local maximum (Cy = 1.15) caused by the rebound occurs at X ≈ 1.7 and Z ≈ 0.7, which is also comparable to that obtained by Sun and Chang (1986) (Cy = 1.21 at X ≈ 2.0 and Z ≈ 1.0).

Emission height zs = 0.75zi

Figure 14 shows the distribution of Cy calculated by the current model and that of Sun and Chang (1986) for an emission height zs = 0.75zi. The agreement between them is in general good. Unlike the previous three cases, the current calculation does not show spread and ascent of the pollutant into the upper atmosphere, because wθ is smaller at the upper part of the mixing layer. The plume descends once it is released into the atmosphere and reaches the ground surface at X ≈ 1.8 of Cy = 1.2, which is comparable to those calculated by Sun and Chang (1986) (X ≈ 1.7 and Cy = 1.35). According to the distribution of Cy calculated at various emission heights, it is found that the downwind distance of the maximum ground-level concentration from the pollutant point source is greater for higher elevation of the pollutant source. However, the maximum ground-level values of Cy are greater for lower source elevation. This result is in line with usual observations during pollutant measurements.

Comparison of other dispersion characteristics

Apart from the comparisons of the pollutant distribution by the contour plots (Figs. 6–14), several parameters relating to the dispersion characteristics are discussed in this section. These parameters are stated and defined as follows:
i1520-0450-40-1-92-e27

Figure 15 shows the normalized mean concentration height for different emission heights. For the cases of emission height zs/zi = 0.067 and 0.25, the pollutant is carried aloft once it is emitted into the convective boundary layer. These plume ascents are followed by a slight overshoot at X = 1.3 and 1.6 that matches with the experimental observations (Willis and Deardorff 1976, 1978). The calculated plume path for pollutant emitted at zs/zi = 0.067 reaches a maximum altitude at X ≈ 1.8 (z/zi ≈ 0.55); for the case of zs/zi = 0.25 it occurs at a greater downstream distance X ≈ 2.1 (z/zi ≈ 0.51). The plume paths of both cases finally descend toward their final equilibrium heights. Unlike the above two low-level releases, pollutant emitted at zs/zi = 0.5 and 0.75 descends initially instead of rising upward once it is emitted into the convective boundary layer. The calculated minimum plume heights are z/zi ≈ 0.45 (at X ≈ 1.0) and 0.43 (at X ≈ 1.9) for zs/zi = 0.5 and 0.75, respectively. Similar to the other two cases discussed above, the plume paths finally reach their equilibrium heights. The final equilibrium heights of the four cases are similar and are approximately equal to 0.5zi (at X > 3.2), indicating that pollutant is finally well mixed in the convective boundary layer.

Figure 16 shows the horizontal (σy) and vertical (σz) dispersion coefficients at different emission heights. Though the concentration contours (Figs. 6–8) and mean concentration heights (Fig. 15) show reasonably good agreement between the calculated and experimental results, undersimulation, especially on σz, can be observed for pollutant emitted at zs/zi = 0.067 and 0.25. The current model simulates the plume centerlines well as shown in Fig. 15; hence, the dispersion coefficients calculated by (28) and (29) solely represent the pollutant dispersion from the plume centerline. On the other hand, conventional Gaussian plume models assume pollutant plume traveling horizontally in a streamwise direction, and the crosswind pollutant transport is modeled by employing empirical pollutant dispersion coefficients. Therefore, the crosswind pollutant transported by both diffusion and plume rise/descent is included in the empirical dispersion coefficients, which are obviously greater than the current calculated values. The above explanation is further confirmed by the dispersion coefficients shown in Fig. 16 in which the measured vertical dispersion coefficients are greater than those of our numerical ones. Comparatively, the discrepancy between the horizontal dispersion coefficients is not so large, because the contribution from plume rise/descent is zero in the horizontal direction. As shown in Fig. 15, the plume rise/descent is less significant at higher pollutant emission elevations, resulting in better agreements between the measured and numerical vertical dispersion coefficients. Despite the above undersimulation, the calculated dispersion coefficients for other emission heights show good agreement with experimental measurements. For zs/zi = 0.25 the calculated σy agrees well with the experimental measurements, and the final equilibrium value of σz is undersimulated slightly. Excellent agreements can be found throughout the range of simulation for the case of zs/zi = 0.5. In this case, maximum and minimum σz occur at X = 0.8 and 1.6, respectively, which agree well with those of the experimental measurements.

Conclusions

A practical three-dimensional numerical model based on a second-order closure scheme has been developed to simulate the wind field and pollutant dispersion from an elevated point source. It consists of an atmospheric boundary layer module and a pollutant dispersion module that is able to simulate the mean flow, turbulence parameters, and pollutant dispersion under different atmospheric stratifications. The mathematical model is discretized into a variable-resolution mesh by using an isoparametric finite-element method that can refine resolution in regions of strong gradient (e.g., near pollutant point sources). Validity of the model is tested by simulating the atmospheric boundary layer structure and pollutant dispersion in a horizontally homogeneous atmosphere. Some non-Gaussian plume characteristics are simulated, and they compared well with experimental measurements as well as with other numerical results. This study provides an alternative method to study the interesting phenomenon of air pollutant dispersion, overcoming the weaknesses of conventional Gaussian plume and k-theory dispersion models by calculating explicitly the turbulent mass fluxes. Moreover, it provides a consistent approach for air-quality studies by considering the atmospheric flow and pollutant dispersion simultaneously.

In this study, the one-dimensional version of the atmospheric boundary layer module is used to simulate the vertical profiles of mean velocity, potential temperature, and turbulence parameters under neutral, stable, and unstable atmospheric boundary layers by simulating a hypothetical diurnal cycle. The mean velocity, potential temperature, and turbulent kinetic energy are calculated prognostically; all other turbulence parameters are calculated diagnostically by the Yamada–Mellor 2.5-level turbulence closure scheme. The modeled vertical profiles of turbulence are similar to field measurements under different stability classes. It is worth mentioning that the calculated horizontal and vertical velocity variances are slightly undersimulated in neutral and stable stratifications, mainly caused by employing a smaller empirical dissipation closure constant. The calculated temperature covariance shows a suppression near the top of an unstable atmospheric boundary layer, which is the main deficiency of the adopted diagnostic turbulence closure scheme. However, the accuracy is not affected much by this suppression, because temperature covariance is not involved directly in the pollutant dispersion calculation. Moreover, the effect of the suppression is insignificant on other calculated parameters, as indicated by their well-simulated results in both atmospheric boundary layer and pollutant dispersion simulations.

The three-dimensional version of the dispersion module calculates both the mean concentration and the second-order moments prognostically. The dispersion of pollutant emitted from a point source at different altitudes (zs = 0.067, 0.25, 0.5, and 0.75 times zi) and under a convective boundary layer is simulated. Unlike results from traditional Gaussian plume and k-theory dispersion models, several non-Gaussian plume behaviors such as descent and rise are simulated that are comparable to the experimental observations. Because these non-Gaussian plume behaviors significantly affect the plume trajectories, their effect should be considered in pollutant dispersion analyses. The current three-dimensional simulations demonstrate the applicability of a second-order dispersion model, although it might be computationally expensive for some routine predictions. A simplified two-dimensional second-order dispersion model has been suggested that simulates the vertical non-Gaussian plume features and calculates the pollutant concentration by assuming horizontal Gaussian pollutant distribution. This model is also validated using the mean concentration height and the pollutant dispersion coefficients, which are comparable to other experimental measurements.

This study calculates the horizontally homogeneous atmospheric turbulence in different stabilities, which is helpful to the analysis of the simulated pollutant dispersion. In addition, simulations of the dispersion of pollutant emitted from a point source into a convective atmospheric boundary layer are performed, and some interesting plume behaviors are obtained. This study not only enhances our knowledge in pollutant dispersion but also suggests a practical approach for solving realistic and hypothetical air dispersion problems.

Acknowledgments

The authors acknowledge the Hong Kong Research Grant Council for supporting this project and the Computer Centre of the University of Hong Kong for their help in using the supercomputer.

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APPENDIX

Nomenclature

  • a3, a4   diffusion length scale, =⅓ and ⅓, respectively

  • C   normalized pollutant concentration, C(X, Y, Z) = c(x, y, z)z2iUm/Q

  • Cy   normalized crosswind-integrated pollutant concentration, C(X, Z) = cy(x, z)ziUm/Q

  • Cϵ   dissipation closure constant, =21.3

  • c   ensemble-averaged pollutant concentration

  • cy   ensemble-averaged and crosswind-integrated pollutant concentration, cy(x, z) = −∞c(x, y, z) dy

  • cθ   concentration–temperature covariance

  • D/Dt   substantial derivative, =∂/∂t + ui(∂/∂xi)

  • d   empirical diffusive length scale constant, =⅓

  • F = {F}   global force vector

  • Fe = {Fe}   force vector in an element Ωe

  • F   a function of space

  • f   Coriolis parameter, =2ω sinϕ, where ϕ is the latitude and ω is the earth rotation speed

  • g   gravity acceleration

  • gi   gravity acceleration vector, =(0, 0, −g)

  • he   characteristic mesh length in an element Ωe

  • ke   artificial viscosity in an element Ωe

  • km, kh   vertical eddy coefficients for momentum and heat

  • L   Monin–Obukhov length scale, =−θu3τ/κgwθ0

  • K = [K]   global stiffness matrix

  • Ke = [Ke]   stiffness matrix in an element Ωe

  • M = [M]   global mass matrix

  • Me = [Me]   mass matrix in an element Ωe

  • N = {N}T   linear isoparametric shape function, ={N1 N2 N3 N4 N5 N6 N7 N8}

  • nele   number of elements Ωe divided in the global domain Ω

  • P   pressure

  • p′   fluctuating pressure

  • Q   pollutant emission rate

  • q2   two times the turbulent kinetic energy, =uiui

  • T∗   convective temperature scale, =wθ0/w

  • t   time

  • Ug, Vg   geostrophic wind speed components in x and y directions, respectively

  • Um   mean wind speed in the convective boundary layer

  • u, υ   ensemble-averaged velocity components in x and y directions, respectively

  • uei   ensemble-averaged velocity in i direction in an element Ωe

  • ui   ensemble-averaged velocity in i direction

  • uic   turbulent mass flux in tensor notation

  • uiuj   Reynolds stress in tensor notation

  • uw, υw   turbulent momentum fluxes

  • uτ   friction velocity on the ground surface

  • wθ   turbulent vertical heat flux

  • wθ0   turbulent vertical heat flux on the ground surface

  • W = {W}   weighting function, ={W1 W2 W3 W4 W5 W6 W7 W8}T

  • w∗   convective velocity scale, =(ziwθg/θ)1/3

  • X, Y, Z   normalized spatial coordinate in x, y, and z directions, respectively, X = (w∗/Um)(x/zi), Y = y/zi and Z = z/zi

  • x, y, z   spatial coordinates in streamwise, lateral, and vertical directions, respectively

  • xi   Cartesian coordinate in tensor notation

  • xmax, ymax, zmax   size of the computational domain in x, y, and z directions, respectively, xmax = 40 000 m, ymax = 3600 m, and zmax = 3000 m

  • z   mean concentration height

  • z0   roughness length parameter

  • zh   depth of the neutral and stable atmospheric boundary layer

  • zi   depth of the convective boundary layer

  • zs   pollutant emission height

  • αe   parameter used for calculation of the weighting function W = coth(γe/2) − 2/γe

  • α1, α2, α3   empirical constant, =0.4, ⅓, and 0.1333, respectively

  • β   coefficient of thermal expansion ≈ 1/θ

  • Δt   time incremental interval

  • Δz   vertical grid interval

  • δij   Kronecker delta, =1 (i = j); =0 (ij)

  • γe   local Peclet number, =(ueiuei)1/2he/ke

  • κ   von Kármán constant, =0.35

  • λ   turbulent length scale

  • Ω   global computation domain

  • Ωe   computation domain in an element

  • Ψ = {ψ}   nodal value of the dependent variable

  • Ψ̇   time derivative of Ψ

  • ρ   air density

  • σy, σz   horizontal and vertical dispersion coefficients

  • θ   ensemble averaged potential temperature

  • θθ   potential temperature covariance

  • θ∗   scaling temperature, =wθ0/uτ

  • ξ, η, ζ   normalized spatial coordinates in an element Ωe

  • ξi, ηi, ζi   normalized nodal coordinates of the node point i in an element Ωe

Fig. 1.
Fig. 1.

Computation domain and the boundary conditions.

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 2.
Fig. 2.

Distribution of the grids in X–Y, X–Z, and Y–Z planes.

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 3.
Fig. 3.

Normalized velocity variances profiles under neutral atmospheric boundary layer Solid line is current study at 1700 LT; plus signs are Brost et al. (1982); circles are Grant (1986).

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 4.
Fig. 4.

Normalized velocity variances and heat flux profiles under stable atmospheric boundary layer. Dotted lines are 0700 and solid lines are 1900 LT of current study; circles are Caughey et al. (1979); plus signs are Nieuwstadt (1984).

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 5.
Fig. 5.

Normalized velocity variances and heat flux profiles under unstable atmospheric boundary layer. Solid line is 1300 and dashed line is 1400 LT of current study; circle is Caughey and Palmer (1979); filled diamond, filled square, filled circle, and filled triangle denote 5–8 Jul, respectively, of Minnesota data (Izumi and Caughey 1976) and Ashchurch data (Caughey and Palmer 1979); dash–dotted line and dash–double-dotted line denote cases S1 and S2 of Willis and Deardorff (1974); plus sign is 1200, times sign is 1300, open diamond is 1400, open triangle is 1500, and open square is 1600 LT of Enger (1986).

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 6.
Fig. 6.

Normalized centerline pollutant concentrations [C(X, 0, Z)] in the X–Z plane for zs = 0.067zi: (a) current model and (b) Willis and Deardorff (1976).

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 7.
Fig. 7.

Normalized crosswind-integrated pollutant concentrations (Cy) in the X–Z plane for zs = 0.067zi: (a) current model and (b) Willis and Deardorff (1976).

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 8.
Fig. 8.

Same as Fig. 7, but for (a) current model and (b) Pai and Tsang (1991).

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 9.
Fig. 9.

Normalized centerline pollutant concentrations in the X–Z plane for zs = 0.5zi: (a) current model and (b) Willis and Deardorff (1981).

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 10.
Fig. 10.

Normalized ground-surface pollutant concentrations in the X–Y plane for zs = 0.5zi: (a) current model and (b) Willis and Deardorff (1981).

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 11.
Fig. 11.

Normalized crosswind-integrated pollutant concentrations (Cy) in the X–Z plane for zs = 0.5zi: (a) current model and (b) Willis and Deardorff (1981).

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 12.
Fig. 12.

Same as Fig. 11, but for (a) current model and (b) Pai and Tsang (1991).

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 13.
Fig. 13.

Normalized crosswind-integrated pollutant concentrations (Cy) in the X–Z plane for zs = 0.25zi: (a) current model and (b) Sun and Chang (1986).

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 14.
Fig. 14.

Same as Fig. 13, but for zs = 0.75zi.

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 15.
Fig. 15.

Normalized concentration height for different emission heights zs/zi. Solid line is 0.067, dashed line is 0.25, dotted line is 0.5, and dash–dotted line is 0.75 of current study; solid circle is 0.067 for Willis and Deardorff (1976); solid square is 0.25 for Willis and Deardorff (1978); solid triangle is 0.5 for Willis and Deardorff (1981).

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Fig. 16.
Fig. 16.

Mean dimensionless variance for different emission heights zs/zi: (a) 0.067 (b) 0.25, and (c) 0.5 m. Solid line is horizontal variance and dashed line is vertical variance of current study; filled circle is horizontal variance and filled square is vertical variance for data reproduced from Willis and Deardorff (1976, 1978, 1981).

Citation: Journal of Applied Meteorology 40, 1; 10.1175/1520-0450(2001)040<0092:TADSUA>2.0.CO;2

Table 1.

Results of the 24-h diurnal cycle simulation.

Table 1.
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