Resumen
The advection-diffusion equation is used for describing the dispersion of pollutants in a limited area. Methods for preventing dangerous levels of pollutants in ecologically important zones are suggested. The methods are based on the control of emission rates of sources and use the direct and adjoint estimates of the average pollution concentration in the zones. While the direct estimates use solutions of the pollution transport problem and permit to study the ecological situation in the whole domain, the adjoint estimates allow getting information only in the selected zones of the domain. The adjoint estimates are obtained with solutions to the adjoint problem and depend explicitly on the positions of the sources and their emission rates, and on the initial distribution of pollutants in the region. In each such estimate, the adjoint problem solution serves as the influence function that shows the quantitative contribution of every source into the pollution of the corresponding zone. This makes the adjoint estimates very efficient tools in the study of the model response to changes in emission rates and initial conditions, as well as in the development of control strategies. Both non-optimal (sufficient) and optimal control strategies are suggested. Each strategy consists in reducing the emission rates of sources, and defines maximum allowable intensity (in case of optimal control), or sufficient intensity (in case of sufficient control) of each source to avoid violations of hygiene standards. Such criteria are designed taking into account dynamic conditions in the atmosphere or ocean (sea), that is, the processes of propagation, dispersion and transformation of pollutants, as well as the number of sources to control,their locations and the sanitary norms. The control methods developed are illustrated with simple examples using two-dimensional dispersion models. However, these methods can also be applied to three-dimensional models. As an example, in the last part of the article, a three-dimensional model of dispersion is considered. In addition, to expand the scope of application of the methods of control of the intensity of sources, the optimal control strategies are applied to a source that emits a chemical substance to clean aquatic systems contaminated with biofilms (remediation) or oil (bioremediation).