Dynamically changing bottom relief modeling based on spatially inhomogeneous 3D mathematical model of wave hydrodynamics

. The article is devoted to the construction and adaptation to natural and climatic conditions and geographical features of precision mathematical models of hydrodynamics of wave processes and relief formation. The incompleteness of data is characteristic of the new tasks considered in the article, the solution of this problem is carried out by involving remote sensing and cadastral survey data. It is essential to include in this complex algorithm for constructing raster models of dynamically changing bottom relief based on known cadastral survey data and remote sensing data. The software package described in this paper makes it possible to reduce the forecast time of hydrophysics processes, including dangerous and catastrophic phenomena. On the basis of the developed software package, prognostic calculations of the processes of coastal erosion and changes in the bottom relief were performed. Modeling of the state of a water body takes into account the characteristic features of natural processes, among which it is worth noting the spatial and temporal variability of relief formation.


Introduction
Currently, the trends of combining modern mathematical apparatus and computing technologies to solve practical problems in various areas of human-environment interaction under natural and man-made impacts are actively manifested. One of the most important tasks is to combine all these factors into a single computational process based on large-scale physical modeling [1][2][3][4]. Thus, the task of constructing and researching a spatial-threedimensional model of wave hydrodynamics designed to simulate hydrodynamic processes in the presence of multi-scale turbulent exchange is relevant. Forecasting the state of shallow water bodies in emergency situations caused by human activity or natural and climatic disasters and phenomena is based on mathematical modeling. When modeling the state of shallow water bodies, it should take into account such features of the water body as: hydrodynamic regimes, climatic factors and the geometry of the reservoir and its coastal area. Assessment of the situation, development and adoption of measures for the rational use and conservation of water resources are an important and ur-gent problem, while in the conditions of modern reality, undesirable and inevitable violations of the natural environment and the natural balance of the ecosystem of these water bodies are increasingly observed. A serious problem is the change in the bottom relief, destruction of the shores, pollution of the aquatic environment. Despite conducting a wide range of studies focused on the problem under consideration, they do not fully reflect the totality of various factors and processes: hydrodynamic, hydrobiological, meteorological and anthropogenic [5][6]. This indicates the need for a systematic analysis of the problem and the construction of an interconnected set of models, high-performance algorithms and programs.
Run-up fluctuations in the level of the Azov Sea lead to problems with navigation, destruction of the shores, redistribution of bottom sediments, flooding of coastal areas. As the analysis of remote sensing data has shown, they represent a fundamental source for the observation and prediction of similar phenomena. In November 2019, a strong easterly wind blew over the Azov Sea (up to 20 m/s), which in the last decade of November led to wind-driven water runoff and drying of the Taganrog Bay (Fig. 1). According to observations, the bay became shallow catastrophically, due to the shallowing of shipping channels, navigation was completely stopped. In addition, a strong wind lifted the sand of the shallow bottom of the sea into the air and carried it over considerable distances. A similar situation was repeated in December 2022.
Data from remote sensing of the Earth made it possible to see the dynamics of this phenomenon in radar images (Fig. 1). In the images, water runoff is visible due to the fact that the drying zone looks less rough and is displayed in tones from dark gray to black. On the contrary, the sea surface under the influence of the blowing strong wind is covered with intense wind waves and is displayed in lighter tones. Analysis of a series of images showed that the water from the bay began to recede on November 21, maximum drying was recorded on November 22-23, and on November 24-25, the water began to return to the Taganrog Bay [9][10][11].
The incompleteness of data, or their absence, is characteristic of the new tasks considered in the article, therefore, there is a problem of calibration and verification of mathematical models, the solution of which is feasible by involving remote sensing and cadastral survey data.
The article describes the construction and adaptation to natural and climatic conditions and geographical features of precision mathematical models of hydrodynamics of wave processes and relief formation depending on the expected scenarios of changes in weather and climatic conditions. It is essential to include in this complex algorithm for constructing raster models of dynamically changing bottom relief based on known cadastral survey data and remote sensing data. The software package described in this paper makes it possible to assimilate space sensing data and reduce the forecast time for hydrophysics processes, including dangerous and catastrophic phenomena. On the basis of the developed software package, prognostic calculations of the processes of coastal erosion and changes in the bottom relief were performed.

2
Materials and methods

Raster model of the Azov Sea bottom construction on the basis of geodetic survey data of the bottom level
The Taganrog Cadastre Bureau conducted a geodetic survey of the bottom level of the Taganrog Bay in the Pushkin Embankment area at low tide. Figure 2 shows a fragment of the area where the geodetic survey was carried out, crosses mark the points at which the depth used to create the raster model is measured (Fig. 2). The level of zero height corresponds to zero in the coordinate system MSK-61 zone 1. Based on the geodetic survey data, a raster model of the bottom of the Azov Sea section was constructed, which was used as the geometry of the computational domain during modeling (Fig. 3).

Spatially inhomogeneous three-dimensional mathematical model of wave hydrodynamics
Spatially inhomogeneous 3D wave hydrodynamics model includes: -Navier-Stokes motion equations: -continuity equation: is the velocity vector of the water flow of a shallow water body;  is the density of the aquatic environment; P is the hydrodynamic pressure; g is the gravitational acceleration; , are coefficients of turbulent exchange in the horizontal and vertical directions [7].

The discrete model of hydrodynamics
The main requirement for a discrete model is the implementation of conservation laws that are valid in the initial physical and mathematical formulation of the problem, the so-called principle of conservativeness. The Navier-Stokes equations for an incompressible fluid are the laws of conservation of mass and momentum written in differential form. There is no equation for kinetic energy, and the balance of kinetic energy is a consequence of the laws of conservation of mass and momentum. In the difference case, the fulfillment of the laws of conservation of mass and momentum is achieved by approximating the divergent form of the original differential equations by the integro-interpolation method. Schemes that are stable at small values of the Reynolds number and unstable at large ones are especially For the numerical realization the uniform grid is introduced: l are the boundaries along the parallelepiped [8].
The variant of method of correction to pressure in the case of a variable density will take form: , ,    For the Navier-Stokes equations in the velocity-pressure variables, numerical methods that preserve kinetic energy are naturally constructed on spaced grids (Fig. 4-6). The figures show the Arakawa C grid indicating the distribution of prognostic and diagnostic variables in slices of the Ox, Oy, and Oz planes with indexing. Each velocity and pressure component is set on its own grid, the convective terms are approximated by central differences, and the scheme has a second order of accuracy in space. Comparison of algorithms on spaced and combined grids showed that conservation laws are better performed on spaced grids. Another advantage of circuits on spaced grids is the connection between speed and pressure, which does not give non-physical, grid oscillations characteristic of circuits on combined grids.
In LES approach, it is assumed that Kolmogorov scales are not resolved. As a result, numerical sampling acts as a filter. LES models are based on the gradient transport hypothesis, which suggests that turbulent transport acts similarly to molecular transport in the sense that quantities are carried downwards by allowed gradients. When replaced back into filtered equations, the gradient transfer model LES take exactly the same form as the terms of molecular transfer, but with the replacement of constant molecular transfer coefficients with turbulent equivalents. Consequently, when the code is executed in LES mode, the set of equations remains the same, but all variables alternate as the corresponding filtered version and turbulent transfer coefficients are used.
During sampling, projections were used taking into account the influence of cartographic coefficients, which are defined as the ratio of the distance in computational space to the corresponding distance on the earth's surface. When LES models are used, the cell volume is used to calculate the vortex viscosity. The volume of the cell must be adjusted using cartographic coefficients.

Results and discussions
As an example of using the developed software package, the problem of calculating the hydrodynamic effect of waves on the bottom of the Taganrog Bay in the Pushkin Embankment area was solved, the modeling area has dimensions of 100 by 50 m and a depth of 1.8 m. The calculations used a grid of 100×200×40 calculation nodes, the time step was 0.01 seconds.

Summary
The article solves the problem of constructing and researching a spatial-three-dimensional model of wave hydrodynamics designed for modeling hydrodynamic processes in the presence of multi-scale turbulent exchange, based on the coordination of analytical, numerical, experimental approaches and field data. The article describes the construction and adaptation to climatic conditions and geographical features of precision mathematical models of hydrodynamics of wave processes and relief formation. It is essential to include in this complex algorithm for constructing raster models of dynamically changing bottom relief based on known cadastral survey data and remote sensing data. A raster model of a section of the bottom of the Azov Sea is constructed. The described software package, e, makes it possible to reduce the forecast time of hydrophysics processes, including dangerous and catastrophic phenomena. On its basis, prognostic calculations of the processes of coastal erosion and changes in the bottom relief were performed. Modeling of the state of a water body takes into account the characteristic features of natural processes, among which it is worth noting the spatial and temporal variability of relief formation and sedimentation, and, as a consequence, changes in the coastline.