Generalized 3D Model of Crosswind Concentrations and Deposition in the Atmospheric Boundary Layer
Abstract
:1. Introduction
2. Analytical Solutions to the Atmospheric Pollutant Equation
- represents the substantial derivative, capturing the rate of change in pollutant concentration from the perspective of a moving air parcel, embodying both the temporal and spatial variations due to airflow.
- stands for the total concentration of pollutants, which includes both the average concentration and the transient fluctuations attributable to atmospheric turbulence.
- denotes the total wind velocity vector represented by , where U indicates the east–west component, V the north–south component, and W the vertical component, encompassing both the average wind speed and its instantaneous fluctuations.
- indicates the partial derivative with respect to time, denoting how the concentration of pollutants changes at a fixed point as time progresses.
- is the gradient operator, employed to calculate the spatial gradient of the pollutant’s concentration.
- describes the divergence of the wind velocity and pollutant concentration product, representing the net movement of pollutants carried by the wind across the spatial domain.
- S is the source term within the equation, quantifying the addition or removal of pollutants from the atmosphere via various processes such as industrial emissions, chemical reactions, or their deposition and uptake by terrestrial surfaces and vegetation.
- (i)
- The apex of the mixing or inversion layer functions as an impervious boundary preventing the diffusion of contaminants. Consequently, a reflective flux boundary condition is postulated at the height of the inversion layer, articulated mathematically as
- (ii)
- Addressing the scenario where a pollutant is partially absorbed by the ground surface, Calder [15] delineates an applicable boundary condition in flux terms. It is posited that the ground surface may facilitate the partial removal or absorption of the pollutant via deposition, characterized by a deposition velocity :
- (iii)
- The lateral diffusive flux diminishes with increasing distance from the source, approaching zero as y extends towards the lateral boundary , formalized by
- The system reaches a state of equilibrium dispersion ();
- Contributions from and are omitted, given that the x-axis aligns with the mean wind direction, rendering the w and v components of wind velocity minor;
- Turbulent diffusion along the mean wind direction is disregarded in comparison to advective transport, implying that
- Consider as the solution for of the system , and define as follows:We will establish that satisfies the requirements of system .
- To improve solution regularity, we apply mollifiers , as referenced in [16], over , which are defined as follows:
- We differentiate with respect to x, leading to
- Integrating the advection and diffusion terms, we findThis confirms that corresponds to the system when n approaches infinity, thus completing the proof of the equivalence between the systems and .
- The concentration at the end of a sub-layer must equal the concentration at the beginning of the subsequent sub-layer to ensure the continuity of concentration across sub-layer interfaces, as denoted by the equation
- The concentration flux, which is the product of diffusivity () and the concentration gradient with respect to the vertical axis (z), must also be continuous across the interfaces between sub-layers. This continuity is captured by the equation
- The coefficient is determined by the following relationship:
- The coefficient is determined by the following relationship:
- The norm is defined as follows:
- The parameter is characterized by the equation
- We begin by applying the Fourier transform with respect to the lateral coordinate y, as presented in Equation (16), to the problem. The resulting equation is
- We calculate the derivative of with respect to x, obtaining
- We multiply both sides of Equation (17) by the exponential term . Given that and remain constant within each sub-layer, this simplifies to
- We apply the same method to the boundary conditions, expressing them in terms of to ensure consistency with the transformed system.
- Applying Lemma 1, we transform the problem into a formulation that depends on .
- Representing as a series, we express it in a separated-variables form
- In verifying that the functions and satisfy their respective differential equations, we must therefore confirm that satisfies the equation
- Determining the specific forms of and ensures that they are, respectively, expressed as
- The continuity condition between sub-layers necessitates a recursion of the eigenvalues and the eigenfunctions for . This process begins by determining the eigenvalues associated with the first layer, , using the conditions related to . The resulting expressions are enumerated below:
- –
- The function is expressed as
- –
- The norm of is given by
The recursive generation of eigenvalues for the intermediate and final layers depends on the known values of the previous layers, in accordance with their respective conditions. This approach makes it possible to construct the associated eigenfunctions, succinctly maintaining continuity between the layers. For further details, please refer to Appendix A.1. - The Sturm–Liouville theorem [19] allows for the expanded representation of as
- Applying the inverse Fourier transform to from Equation (16), and considering that the inverse transform of
- 1.
- The Single-Layer and Two-Dimensional Solution:
- 2.
- The Multi-Layer and Three-Dimensional Solution:
- represents the average wind speed over the height of the mixing layer, calculated as
- denotes the average eddy diffusivity over the same height, expressed as
- Substitute Single-Layer Values into the CIC Equation—begin by inserting the conditions of a single-layer setup into the original CIC formula from Equation (27):
- Apply Limits as Approaches 0 and Extends to —evaluate the limits to reflect the shift to a single-layer framework that spans the entire mixing height :
3. Turbulent Parameters in Pollutant Dispersion Modeling
- (a)
- Parameterizing the Wind Speed Profile, .The wind speed profile is principally derived using surface layer theory enhanced by Monin–Obukhov (M-O) similarity theory [26]. While traditionally limited to the surface layer, extending the M-O scaling across the entire atmospheric boundary layer (ABL) involves maintaining a constant wind speed profile above this initial layer, ensuring the model’s extended applicability and continuity.Gryning et al. [27] have refined this approach to create a comprehensive wind speed profile applicable over the entire ABL, which is particularly suited for homogeneous terrains. Their advanced model employs a local mixing length composed of three component scales, leading to a robust formulation that incorporates adjustments for atmospheric stability throughout the entire ABL:In this formulation, represents the Coriolis parameter, is the von Karman constant, and and are resistance law functions dependent on the stability parameter . Additionally, Gryning et al. [27] proposed an empirical formula for that enhances the model’s ability to adapt to various atmospheric stability conditions:
- (b)
- Parameterizing the Diffusivity Profiles
- (b.1)
- Vertical Diffusivity:In atmospheric dispersion models, the parameterization of eddy diffusivity, , significantly influences the model’s accuracy and performance. Traditionally, has been modeled as solely a function of the height above the ground, z. However, recent advancements have extended this parameterization to include the downwind distance x, recognizing its impact on near-source dispersion dynamics.An innovative approach by Mooney and Wilson [28] introduced a composite model for , blending both spatial variables into a single expression:The parameterization of at the source height is given byMangia et al. [29] further refined based on local similarity and statistical diffusion theories, resulting in the following relation:
- (b.2)
- Horizontal Diffusivity:In the realm of atmospheric dispersion modeling, lateral eddy diffusivity plays a crucial role in determining the spread of pollutants in the crosswind direction. This diffusivity is fundamentally expressed through a relationship that ties it to the wind speed profile and the variance in lateral dispersion:This parameterization, based on the foundational work cited in Huang’s theory [31], encapsulates the dynamic interactions between wind speed and atmospheric turbulence. By integrating the vertical wind profile with the derivative of the squared standard deviation with respect to x, the model effectively captures how lateral diffusion varies with both altitude and distance from the source.
4. Results: Data Analysis and Model Validation
Results and Discussions
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
ABL | Atmospheric boundary layer |
CIC | Crosswind-integrated concentration |
Appendix A. Construction of the Solution Associated with the First Sub-Layer and an Eigenvalue Analysis of the Problem (uid56)
Appendix A.1. Construction of the Solution Associated with the First Sub-Layer of the Problem (uid56)
Appendix A.2. Eigenvalue Analysis of the Problem (uid56)
Appendix B. Derivation of Specific Solutions for cH
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Run | Date | Distance | ||||||
---|---|---|---|---|---|---|---|---|
(DD/MM/YY) | () | () | () | () | () | () | () | |
800 | 3.73 | 2.24 | 4.21 | |||||
1 | 18 May 1983 | 165 | 0.40 | 325 | 1600 | 2.14 | 0.98 | 4.05 |
3200 | 1.30 | 0.57 | 3.65 | |||||
800 | 12.90 | 7.47 | 1.93 | |||||
2 | 26 May 1983 | 44 | 0.26 | 135 | 1600 | 9.08 | 3.25 | 1.80 |
3200 | 7.22 | 2.31 | 1.74 | |||||
800 | 5.91 | 3.06 | 3.14 | |||||
3 | 05 June 1983 | 77 | 0.27 | 182 | 1600 | 3.31 | 1.32 | 3.02 |
3200 | 1.79 | 0.66 | 2.84 | |||||
800 | 20.10 | 8.04 | 1.75 | |||||
4 | 12 June 1983 | 34 | 0.20 | 104 | 1600 | 13.10 | 4.26 | 1.62 |
3200 | 9.15 | 3.14 | 1.31 | |||||
800 | 10.50 | 5.25 | 1.56 | |||||
5 | 24 June 1983 | 59 | 0.26 | 157 | 1600 | 8.61 | 3.38 | 1.47 |
3200 | 6.64 | 2.92 | 1.14 | |||||
800 | 13.40 | 7.23 | 1.17 | |||||
6 | 27 June 1983 | 71 | 0.30 | 185 | 1600 | 6.15 | 2.52 | 1.15 |
3200 | 3.11 | 1.25 | 1.10 |
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Farhane, M.; Souhar, O. Generalized 3D Model of Crosswind Concentrations and Deposition in the Atmospheric Boundary Layer. Atmosphere 2024, 15, 1054. https://doi.org/10.3390/atmos15091054
Farhane M, Souhar O. Generalized 3D Model of Crosswind Concentrations and Deposition in the Atmospheric Boundary Layer. Atmosphere. 2024; 15(9):1054. https://doi.org/10.3390/atmos15091054
Chicago/Turabian StyleFarhane, Mehdi, and Otmane Souhar. 2024. "Generalized 3D Model of Crosswind Concentrations and Deposition in the Atmospheric Boundary Layer" Atmosphere 15, no. 9: 1054. https://doi.org/10.3390/atmos15091054