Development of variational approach for direct and inverse problems of atmospheric hydrodynamics and chemistry

Abstract

The development of a variational approach for solving interrelated problems of atmospheric hydrodynamics and chemistry is presented. The variational approach allows expanding the class of concerned problems for a complex study of different scale physical and chemical processes using the methods of direct and inverse modeling for these purposes. A technology of constructing consistent mathematical models and methods of their numerical implementation based on the variational principle in the weak constraint state ment is described. Solutions of local and global adjoint problems are of great importance for constructing the algorithms of direct and inverse modeling. Implementing the idea of adjoint integrating factors provides unconditionally monotone and stable discrete-analytic approximations for convection-diffusion-reaction problems.

Modeling of atmospheric circulation and transport of impurities from natural and industrial sources in the Baikal region

“…Numerical implementation of models of the atmospheric dynamics and the transport of impurities is based on the variational-difference approach 12 . That ensures the energy balance of the constructed difference schemes.…”

Hypothetical scenarios of microplastics propagation in Lake Baikal

“…A detailed discussion of the approach can be found in the monograph 17 . For the purposes of this study, we follow the idea and logic of the paper 19 devoted to the similar atmospheric problems and make only a few remarks related to the lake features.…”

“…The required estimates are obtained by methods of sensitivity theory from the stationarity conditions of the functional (2) to the variations of the state function under the restriction that the state functions satisfy the equations of the model or their discrete analogues. The algorithm for obtaining estimates is described in detail in the paper 19 . Note only that the algorithms for calculating sensitivity functions involve the solutions of the corresponding adjoint problems.…”

Notes on environmental forecasting in the current conditions

“…In the case of a shortage of measurement data, it is advisable to use methods for solving inverse continuation problems, given that mathematical models in the data assimilation mode act as tools for interpolation or extrapolation of calculation data or observational data. The inverse problems of data assimilation are formulated as interrelated problems at specified intervals defined as windows of assimilation [28][29][30] . Methods of sensitivity theory work through estimates of variations of generalized functionals (or target functionals of the environment quality) to perturbations of system parameters for the entire complex of models, for individual fragments of models, etc.…”

“…Since the 70s of the last century, ICM&MG SB RAS has accumulated extensive experience in developing its own models for solving various environmental problems [26][27][28][29][30] . We are developing general approaches that allow us to work uniformly with a wide range of forward and inverse modeling problems.…”