Introduction. The use of two-dimensional (2D) hydrodynamic models is relevant, despite the development of numerical methods of marine hydrodynamics focused on the use of three-dimensional spatial models. This is due to the modelling of hydrodynamic processes in shallow and coastal systems in solving practically important problems of predicting the transport of pollutants in suspended and dissolved forms. Evaporation for the Southern of Russia marine coastal systems (the Azov Sea, the Northern Caspian, etc.), and even more so in the coastal areas of the Red Sea, is a significant factor that affects not only the balance of water masses, but also makes changes in the momentum of the system and the distribution of the velocity vector of the aquatic environment. This effect is significant for coastal currents and shallow-water systems.

Materials and Methods. The traditional method of converting the terms of the Navier-Stokes equations containing differentiation by horizontal spatial variables was used, involving the rearrangement of differentiation operations by horizontal spatial coordinates and integration by vertical coordinate when constructing a spatially two-dimensional model of hydrodynamics of marine coastal systems when integrated by vertical coordinate. This made it possible to avoid the appearance of non-physical sources of energy and momentum in the spatially two-dimensional model, which can be essential in traditional 2D models with significant depth differences characteristic of coastal systems. The implementation of the analogue of the law of conservation of the total mechanical energy of the system for the constructed 2D model is investigated.

Results. Using the correct transformation of the 3D model (integration of the Navier-Stokes equations and continuity along a vertical coordinate, taking into account evaporation from a free surface), spatially two-dimensional models of hydrodynamics are constructed, for which the basic conservation laws, including mass and total mechanical energy of the system, are fulfilled. The implementation of an analogue of the law of conservation of total mechanical energy for various types of boundary conditions, including at the bottom, is investigated. The evaporation from the free surface is correctly accounted for not only in the continuity equation, but also in the equations of motion taking into account wind and waves.

Discussion and Conclusion. 2D model of hydrodynamics has been constructed and studied, taking into account evaporation not only in the mass balance equation (continuity), but also in the Navier-Stokes equations of motion. The proposed model can be used for predictive modelling of hydrophysical processes, including the spread of pollutants in the aquatic environment of coastal systems and shallow reservoirs in relation to marine systems such as the Sea of Azov, the Northern Caspian Sea, coastal areas of the Red Sea, etc. Spatially two-dimensional models of marine hydrodynamics, without replacing three-dimensional models, can serve as a model basis for operational forecasting of situations in coastal systems and shallow-water objects using computing systems with relatively low performance and a moderate amount of RAM (5–10 Tflops, 2–4 TB, respectively).

Introduction. Mathematical modelling of the aerodynamics of mountain gorges is an actual means of studying possible man-made emissions in various meteorological conditions that increase the transfer of pollutants in the direction of densely populated areas. Aerodynamics and climatic conditions are unique for various mountain gorges. This requires a separate study for each specific case. The paper studies the distribution of dust aerosol from the Unal tailings dump, located near the village of Verkhny Unal (Alagir Gorge, RNO-Alania, RF), with south and south-easterly winds. With these wind directions, the dust of the tailings dump is carried by air currents in the north direction, towards Alagir. The aim of the study is to obtain a forecast for the surface concentration of dust with an increased content of lead, zinc and other elements near densely populated areas of the flat part of RNO-Alania.

Materials and Methods. The model takes into account the terrain, surface wind roses and dust deposition processes. The calculations were carried out for the case of neutral stratification and without taking into account the influence of seasonal factors using a mathematical model previously published by the authors.

Results. The model prediction of the dust concentration distribution obtained from calculations is shown. The frequencies and amplitudes of oscillations of unsteady jet streams in the cross section of the Alagir gorge are analyzed. Based on the data of satellite sensing of the Earth’s atmosphere, the frequency of winds leading to the transfer of dust in the direction of densely populated areas is estimated.

Discussion and Conclusion. The Unal tailings dump is a source of pollutants and over many years of its existence, soil contamination can be significant. Field studies of the soil in the Alagir area and, possibly, measures for its reclamation are necessary.

Introduction. Mathematical modelling of currents is an urgent research topic in the field of hydrodynamics and oceanography. Despite ongoing research in the field of developing accurate and efficient numerical methods for solving Navier-Stokes equations that take into account vortex viscosity, the problems of accurate prediction and control of turbulence remain unresolved. The influence of nonlinear effects in vortex viscosity models on the accuracy of forecasts and their applicability to various flow conditions also remains relevant. The aim of the study is to study the influence of linearized and quadratic bottom friction and two turbulence models on the numerical solution of stationary and non-stationary periodic flows. Special emphasis is placed on comparing numerical results with analytical solutions within the framework of using various models of bottom friction.

Materials and Methods. The computational models used in this study are based on a simplified two-dimensional wave model and full three-dimensional Navier-Stokes equations. The classical model of shallow water motion and the 2D model without taking into account dynamic changes in the geometry of the reservoir surface are derived from a system of equations for a spatially inhomogeneous three-dimensional mathematical model of wave hydrodynamics of a shallow reservoir. Analytical solutions were found by linearization of the equations, which obviously has its limitations. A distinction is made between two types of nonlinear effects – nonlinearities caused by higher-order terms in the equations of motion, i. e. terms of advective acceleration and friction, and nonlinear effects caused by geometric nonlinearities, this is due, for example, to different water depths and reservoir widths, which will be important when modelling a real sea.

Results. The results of modeling stationary and non-stationary periodic flows in a schematized rectangular basin using linearized bottom friction are presented. The influence of linearization on the numerical solution is investigated in comparison with analytical profiles using models calculating bottom friction in a quadratic formulation. In combination with quadratic bottom friction, two turbulence models are studied: the constant vortex viscosity and the Prandtl mixing length model. The results obtained as a result of three-dimensional modelling are compared with the results of two-dimensional modeling and analytical solutions averaged in depth.

Discussion and Conclusion. New approaches to modelling and studying flows with variable vortex viscosity are proposed, including analysis of the influence of linearization and the use of various turbulence models. For the linearized and quadratic formulations of bottom friction, it is proved that the numerical results for the case of stationary flow show great similarity with analytical solutions, since the surface height is much less than the water depth and advection can be neglected. The numerical results for the unsteady flow also show a good agreement with the theory. Unlike analytical solutions, numerical modelling has minor deviations in the long run. The study of flows, within the framework of using various turbulence models, will make it possible to take into account the influence of nonlinear effects in vortex viscosity models on the accuracy of forecasts and their applicability to various flow conditions. The results obtained make it possible to better understand and describe the physical processes occurring in shallow waters. This opens up new possibilities for applying mathematical modelling to predict and analyze the impact of human activities on the marine environment and to solve other problems in the field of oceanology and geophysics.

Introduction. The negative consequences that may arise due to an accidental oil spill are difficult to account for, since they disrupt many natural processes and relationships within the ecosystem of the reservoir. After an oil spill, a dense layer of oil film forms on the water surface quite quickly, preventing access to air and light (after a spill of one ton of oil, an oil slick about 10 mm thick forms on the surface of the reservoir after 10 minutes). As a result, the fauna and flora of the reservoir suffer. If the accident occurred in the coastal zone near a populated area, then the toxic effect is enhanced, because petroleum products in combination with various pollutants of human origin can form dangerous compounds. For high-risk areas (the main routes of transportation of petroleum products, places of their bunkering and unloading, etc.), it is necessary to predict various scenarios for the spread and transformation of oil pollution, taking into account their multifractional composition, turbulent diffusion and advective transport, destruction under the influence of natural factors. The aim of the work is to build a linearized non-stationary spatially heterogeneous mathematical model of transport and transformation of oil pollution, taking into account the above factors.

Materials and Methods. The oil that has entered the aquatic environment is represented as a surface and suspended substance in the water column. Oil is subject to a variety of transformation processes: advection, gravitational spreading, emulsification, dispersion, dissolution, biodegradation, etc. The study of these processes and their forecasting, as a rule, requires the development of mathematical and software. In mathematical and numerical modeling, one should start from the system of Navier-Stokes equations and continuity equations, as well as introduce additional physical tolerances of the flow geometry, acceptable and justified in each case, as shown by world experience and objective analysis of the physical picture of processes. Mathematical modeling of the oil distribution process in coastal marine systems has been performed.

Results. Mathematical oil distribution model has been created, taking into account its multifractional composition. It is assumed that oil fractions can be in water in dissolved or undissolved states. The modeling takes into account such physical characteristics of particles as density, acceleration of gravity, molar mass, etc. After the linearization of the problem under consideration, difference schemes using extended uniform grids were constructed.

Discussion and Conclusion. Pollution caused by an oil spill in the aquatic environment occurs very quickly and is often very destructive. An important factor will be prompt response, which plays a crucial role in minimizing its negative consequences. Modeling of the oil spill process can be useful for determining the location and condition of oil at sea, conducting a risk analysis of the spread of the substance and developing measures to localize and eliminate pollution.

Introduction. Pollution of shallow waters is becoming an increasingly serious problem. It is important to study the mechanisms of pollution distribution in them to protect and restore such vulnerable ecosystems, it is necessary to develop strategies for the development of sustainable and environmentally friendly use of natural resources, minimizing the negative impact on the environment. Part of this work is the construction of a mathematical model for the spread of pollutants (in particular, phosphates) in shallow reservoirs. The aim of the work is to construct scenarios for changes in the concentration of phosphates at various parameters of the model.

Materials and Methods. The phosphate transport mathematical model in a shallow reservoir is described, implemented using a modified alternating triangular iterative method to solving grid equations.

Results. The developed mathematical model is numerically implemented in the form of a software module. This model is an important tool for assessing and predicting the various pollution sources impact to the water quality of ecosystems such as lakes and reservoirs.

Discussion and Conclusion. The resulting model can be used to analyze various pollution scenarios, for example, to determine optimal waste management strategies and prevent pollution of water resources. In addition, the software module developed by the authors allows you to simulate the process of the phosphates concentration changing and can be useful for conducting scientific and engineering research in the aquatic ecology field and developing effective methods for adapting hydrobiocenosis to changes in the aquatic ecosystem.

Introduction. Coastal systems of Southern Russia are constantly exposed to biotic, abiotic and anthropogenic factors. In this regard, there is a need to develop non-stationary spatially inhomogeneous interconnected mathematical models that make it possible to reproduce various scenarios for the dynamics of biological and geochemical processes in coastal systems. There is also the problem of the practical use of mathematical modelling, namely its equipping with real input data (boundary, initial conditions, information about source functions). An operational source of field information can be data received from artificial Earth satellites. Therefore, the problem arises of identifying phytoplankton populations in images of reservoirs, which, as a rule, have a spotty structure, with low image contrast relative to the background, as well as determining the boundaries of their location.

Materials and Methods. This work is based on the correct application of modern mathematical analysis methods, mathematical physics and functional analysis, the theory of difference schemes, as well as methods for solving grid equations. Biogeochemical processes are described based on convection-diffusion-reaction equations. Linearization of the constructed model is carried out on a time grid with step τ. A method for recognizing the boundaries of spotted structures is being developed based on Earth remote sensing data. A combination of methods is considered as image processing algorithms: local binary patterns (LBP) and a two-layer neural network.

Results. The developed software-algorithmic tools for space image recognition are presented, based on a combination of methods — local binary patterns (LBP) and neural network technologies, focused on the subsequent input of the obtained initial conditions for the problem of phytoplankton dynamics into a mathematical model. Regarding the necessary mathematical model, a continuous linearized model has been proposed and studied, and on its basis a linearized discrete model of biogeochemical cycles in coastal systems, for which practically acceptable time step values have been established for numerical (predictive) modelling of problems of the dynamics of planktonic populations and biogeochemical cycles, including in the event of death phenomena, which makes it possible to reduce the time of operational forecasting. At the same time, for the constructed discrete model, properties that are practically significant for discrete models are guaranteed to be satisfied: stability, monotonicity and convergence of the difference scheme, which is important for reliable forecasts of adverse and dangerous phenomena.

In the process of work, referring to satellite images, which make it possible to obtain the state of coastal systems with high accuracy, initial conditions are entered into the mathematical (computer) model. The model analyzes satellite image data and determines levels of “pollution”, the formation of extinction zones and other factors that may threaten nature.

Discussion and Conclusion. Discussion and conclusions. Using this model, it is possible to predict possible changes in coastal ecosystems and develop strategies to protect them. The results obtained make it possible to significantly reduce the time of forecast calculations (by 20−30 %) and increase the likelihood of early detection of unfavorable and dangerous phenomena, such as intense “blooming” of the aquatic environment and the formation of extinction zones in coastal systems.