Modeling the impacts of chemical substances and time delay to mitigate regional atmospheric pollutants and enhance rainfall

Abstract

Rainfall, a crucial process of the hydrological cycle, involves the condensation of atmospheric cloud droplets into raindrops that fall on the Earth’s surface, providing essentials for human well-being and ecosystem. Research studies show that the condensation–nucleation process for forming raindrops is reduced due to atmospheric pollutants. In this scenario, introducing chemical substances may effectively mitigate regional atmospheric pollution, and reduced atmospheric pollution may lead to adequate rainfall. In the present research work, we analyze rainfall dynamics using a modeling approach with the incorporation of a time lag involved between measuring the data for atmospheric pollution and introducing chemical substances in the regional atmosphere. Here, we assume the formation rate of cloud droplets as a decreasing function of atmospheric pollutants. It is also assumed that introducing chemical substances reduces regional atmospheric pollution. Involving time delay as a bifurcation parameter, we analyze the stability, direction, and period of the bifurcating periodic solutions arising through Hopf bifurcation. Along with this, the presented numerical simulations corroborate the analytical results of our mathematical model. The modeling study reveals that the use of chemical substances in proportion to the concentration of atmospheric pollutants measured at time ($t-\tau$) becomes crucial to mitigate the atmospheric pollutants because as time delay exceeds a threshold value, the system loses its stability and undergoes Hopf bifurcation.