Mathematical modelling of pollutant dispersion in the atmosphere with allowance for the spatial--temporal variation of temperature
Received: 2026-10-14
Published: 2026-10-14
Abstract
the spatial–temporal variation of temperature. The advection–diffusion equation for the pollutant, written in conservative form with a
temperature-dependent gravitational-settling flux, is solved together with a temperature (energy) equation; the dynamic viscosity (Suther-
land’s law), the air density, the molecular diffusivity and the settling velocity are functions of the local temperature, and the atmospheric
thermal stability enters the vertical turbulent diffusion through the gradient Richardson number and a monotone stability function. The
problem is discretised by an explicit finite-volume scheme with upwind advection and a combined Courant–Friedrichs–Lewy step, point
sources are prescribed as grid-independent mass rates, and the ground boundary is treated as dry deposition with a tracked deposited mass. The scheme is validated against an analytical Gaussian solution and verified by machine-precision mass conservation and grid/time convergence. Two reproducible regimes are computed: a daytime unstable case (???????? < 0) and a night-time surface inversion (???????? > 0). Under
the inversion the vertical mixing is suppressed by more than an order of magnitude, the pollutant is confined to a shallow near-surface layer with an in-layer peak about three times the daytime value, and only about 18% of the emitted mass is ventilated out of the region against about 70% in the daytime. The example is illustrative rather than a site-specific prediction.
Keywords
List of references
-
Stull RB. An Introduction to Boundary Layer Meteorology. Kluwer Academic; 1988.
-
Byun D, Schere KL. Review of the governing equations, computational algorithms, and other components of the Models-3 Community Multiscale Air Quality (CMAQ) modeling system. Appl Mech Rev. 2006;59:51–77. https://doi.org/10.1115/1.2128636.
-
Jacobson MZ. Fundamentals of Atmospheric Modeling. 2nd ed. Cambridge University Press; 2005.
-
Zhang Y. Online-coupled meteorology and chemistry models: history, current status, and outlook. Atmos Chem Phys. 2008;8:2895–2932. https://doi.org/10.5194/acp-8-2895-2008.
-
Holmes NS, Morawska L. A review of dispersion modelling and its application to the dispersion of particles. Atmos Environ. 2006;40:5902–5928. https://doi.org/10.1016/j.atmosenv.2006.06.003.
-
Micallef A, Micallef C. The Gaussian plume model equation for atmospheric dispersion corrected for multiple reflections. Sci. 2024;6(3):49. https://doi.org/10.3390/sci6030049.
-
Kafle J, Adhikari K, Poudel E. Air pollutant dispersion using advection-diffusion equation. Nepal J Environ Sci. 2024;12(1):1–6.
-
Turner DB. Workbook of Atmospheric Dispersion Estimates: An Introduction to Dispersion Modeling. 2nd ed. CRC Press; 1994.
-
Wyngaard JC. Turbulent dispersion modeling in the boundary layer. J Atmos Sci. 1982;39(6):1342–1353.
-
Jejeniwa O, Gidey H, Appadu A. Numerical modeling of pollutant transport: results and optimal parameters. Symmetry. 2022;14:2616. https://doi.org/10.3390/sym14122616.
-
Ravshanov N, Nazarov S, Nazarov E, Boborahimov B. Development of a mathematical model of aerosol particles’ distribution process in the surface layer of the atmosphere taking into account the Earth’s heterogeneous surface. AIP Conf Proc. 2024;3004:060004.
-
Ravshanov N, Nazarov SE, Boborahimov B. Modeling the process of pollutant spread in the atmosphere with account for the capture of particles by vegetation elements. Lobachevskii J Math. 2024;45(3):1213–1226.
-
Nabieva IS. Numerical modeling of the transport and diffusion of pollutant particles taking into account airflow characteristics and temperature. Probl Comput Appl Math. 2025;1(63):27–40.
-
Ravshanov N, Nabieva IS. Mathematical modeling of the process of harmful substances dispersion in the atmosphere taking into account temperature and physical–chemical properties. Int J Theor Appl Issues Digit Technol. 2024;7(4):27–32.
-
Ravshanov N, Nabieva I, Sapaeva D, Hojiev N. Mathematical modeling of the distribution of pollutants in the atmosphere, taking into account their physical and mechanical properties. Proc SPIE. 2025;13662:136620K. https://doi.org/10.1117/12.3072577.
-
Ravshanov N, Nabieva IS, Boborakhimov BI. Software for predicting the spread of harmful substances in the atmosphere, taking into account their type and wind speed. Probl Comput Appl Math. 2025;2/2(66):54–59.
-
Boborakhimov B, Nabiyeva I, Tashtemirova N, Juraboyeva O. Mathematical modeling of the dispersion of pollutants in the atmosphere, taking into account their physico-mechanical properties and hygroscopic parameters. AIP Conf Proc. In press.
-
Ravshanov N, Boborakhimov B, Nabiyeva I. Mathematical modeling of the distribution of harmful substances in the atmosphere in variable-direction currents. E3S Web Conf. In press.
-
Seinfeld JH, Pandis SN. Atmospheric Chemistry and Physics: From Air Pollution to Climate Change. 3rd ed. John Wiley & Sons; 2016.
About the Authors
License
This work is licensed under a Creative Commons Attribution 4.0 International License.