Mathematical modelling of pollutant dispersion in the atmosphere with allowance for the spatial–temporal variation of temperature

A three-dimensional model is developed for the dispersion of harmful substances in the atmospheric surface layer with allowance for 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 (Sutherland’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 and a night-time surface inversion. 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.