Abstract
Abstract
This study develops a two-dimensional, two-layer tempered time-fractional diffusion–reaction model for pollutant transport in a thermally stratified atmosphere. The model represents a temperature inversion through a lower turbulent layer and an upper stable layer, with continuity of pollutant concentration and diffusive flux imposed at the layer interface. An exact analytical solution is derived using eigenfunction expansion and the Laplace transform and is expressed in terms of coupled spatial eigenfunctions and tempered Mittag–Leffler functions. The existence, uniqueness, and Hadamard well-posedness of the model are established. Numerical analysis demonstrates that the fractional order strongly influences pollutant trapping and vertical redistribution, with smaller fractional orders producing greater pollutant accumulation beneath the inversion layer. The tempering parameter primarily controls concentration attenuation while preserving the characteristic layered spatial structure. Limiting-case analysis recovers both the classical diffusion–reaction equation and the standard fractional diffusion–reaction equation, confirming the consistency of the formulation. The framework is further illustrated using an idealized representation of the Kathmandu Valley, Nepal, where the results demonstrate the effects of anomalous diffusion and memory attenuation on pollutant retention under inversion conditions. The proposed model provides an analytical framework for investigating pollutant transport in vertically heterogeneous atmospheric environments and offers a basis for further data-driven and observational studies.
Mathematical Subject Classification (MSC): 35R11, 35K57, 26A33, 76R50.