Construction of a consistent differential formulation of the adjointproblem for the passive impurity transfer model

Abstract

When implementing numerical algorithms for identifying the power of pollution sources in the passive impurity transfer model based on measurement data, the question naturally arises of constructing conjugate statements consistent with the main task. Such a matching can be considered from the perspective of a differential formulation, as well as from the point of view of discretization of the problem in its numerical implementation. This paper discusses various aspects of such alignment for the model of transport in the Sea of Azov and a similar model for the Black Sea. Ocean dynamics models are nonlinear in nature. When solving the problem of assimilation of measurement data in hydrodynamic models, linearization is most often performed over a certain time interval (assimilation interval). In this paper, a model of passive impurity transfer is considered, i.e., it does not affect the dynamic processes in the liquid itself. Such models are often used in solving environmental problems. The impurity transfer model is linear. The quadratic prediction quality functional is convex, i.e. it has one extremum. Using a linear model as links to minimize such a functional does not change its convexity, which leads to reliable operation of the procedure for searching for the extremum of the functional. Such a search is performed using an appropriate iterative process. The equations of the transfer model used may have different forms depending on the problem being solved and the region of its application. For example, a $\sigma$-coordinate model is used to solve a problem in the waters of the Sea of Azov. The results of the work can be useful in solving various environmental problems in the process of studying the effects of anthropogenic pollution sources in the waters of the Azov and Black Seas.