Mathematical modeling the density of sea water in the deep pond

Taganrog, Russia The paper covers the mathematical model of hydrodynamics of a deep water reservoir-the sea of Japan, taking into account the complex geometry of the shoreline and the bottom, friction on the bottom and wind currents, evaporation, deviation of the pressure field from the hydrostatic approximation, water density as a function of the spatial distribution of temperature and salinity, is proposed and investigated. The models of observations for the functions included in the model problem, including the dependence of the water density distribution on the spatial distribution of its temperature, salinity and ionic composition, are determined and investigated. The application of these models is based on electrical conductivity and temperature, which makes them as accurate as possible in

where t is the time; is the angular velocity of the earth's rotation;  is the angle between the angular velocity of the Earth's rotation and the vertical;  is the water density; p is the pressure increase over the hydrostatic pressure of the undisturbed liquid;  is the value of the gravitational potential;  , are the diffusion coefficients in horizontal and vertical directions, respectively [7];

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on the side surface  and bottom where n is the vector of the external normal to the surface ; v  is the he density of the terrigenous suspension, which appears as a result of erosion of the banks and river flow; where 0 u , 0 v , 0 w , 0 T , 0 S are given functions. Model (1)-(9) takes into account the motion of the water flow, microturbulent diffusion, the Coriolis force, the spatial distribution of salinity and TEM-temperature. In the majority of works on Oceanology and thermohydrodynamics the linear variant of water density distribution (LV) depending only on temperature changes is considered: where 1028   ;  is the coefficient expressing the dependence of water density changes on temperature changes T (Table 1). This dependence is the maximum permissible simplification, which significantly reduces the accuracy of the models using it. A more accurate model of observations, based on the equation of state (ES-80), used to determine the density of sea water [8], has the form:

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; (17) (20) is the average modulus of elasticity; n  is the density of the standard medium-oceanic clean water; The observation model (11)-(24) is applicable in the following ranges: However, it should be noted that for the application of the model (11)-(24) it is necessary to have average values of temperature and salinity in the nodes of the measuring grid, which does not allow instantaneous calculations.
In this regard, this paper proposes to modify the system (11)-(24), adding the equation of the form: and replacing equation (11) with equation: where  s the coefficient obtained experimentally; 1 0, l are given factors. This modification (ES -18) not only improves the accuracy of predictive modeling, but also allows to use as input data not the temperature and salinity deviations from the mean values, but the data measured in the grid nodes at a specific time. This circumstance allows to optimize the performance of software systems using dependence (11)-(24), and to shorten the program, which is numerically realized in] the developed hydrodynamic model. On the coastline of the sea of Japan, there are 18 coastal hydrological stations, forming a constantly growing database «UISGO», the use of which allows for calibration and comparative analysis of models of observations of the species (10), (11) and (26).
The spatial distribution of water temperature on the surface of the sea of Japan throughout the year is characterized by the following large-scale regularity: the maximum values of water temperature in the active layer of the water body are typical for the southern part of the sea. This is mainly due to two reasons, one of which is the inflow of modified subtropical waters from the Pacific ocean, and the second is the geographical location of the region and the associated increased solar radiation (compared to the more Northern parts of the sea). The minimum values of water temperature in the active layer of the reservoir are typical for the Northern and North-Western parts of the sea. The warming effect of subtropical waters of Pacific origin is practically not affected. In winter, the lowest values of air temperature are observed over these waters, which leads to intensive cooling of the surface layer of water. In spring and summer, the Northern and North-Western parts of the sea are characterized by weather conditions with high frequency of fogs, which leads to the weakening of solar radiation penetrating into the surface layer of water. All considered factors lead to the fact that in the Northern and North-Western parts of the sea the lowest heat storage of the surface water layer is formed. Figure 2 shows the average monthly temperature values for 2017 in the mountain zone from 0 to 100 meters for the sea of Japan. There are two types of spatial distribution of salinity in the layer of 0-100 meters in the waters of the sea of Japan during the year. The first type of salinity distribution manifests itself from January to June. During this period of the year, maximum values are observed in the southern and Eastern parts of the sea. The main feature of the second type of salinity distribution, which is allocated fro m July to December, is the presence of maximum values in the Central part of the sea. As you get closer to the coast the salinity is lowered. Starting from the 200 m horizon, seasonal changes in salinity become unreliable. Therefore, for these horizons, only the average annual salinity distribution in the sea of Japan is considered. In the layer of 600-800 m, it is very difficult to distinguish the spatial laws of the distribution of salinity of the waters of the sea of Japan. Limits spatial changes in salinity become comparable to the point-ness of registrations salinity. Thus, until the mid-1970s, salinity was determined by the method of chlorine titration, the accuracy of which did not exceed 0.02‰. From the mid-70s to the end of the 80s to determine the salinity, mainly used electric meters with an accuracy of up to 0.005‰. Only in the last decade of the 20th century, water salinity is really determined to an accuracy of 0.001-0.003‰. Average long-term data show that in the deep and bottom layers of the sea of Japan variations of salinity are in the range of 34,050-34,075‰. In the fields of salinity is quite difficult to identify regional patterns. This is mainly due to the lack of accuracy of the predominant number of salinity definitions currently available at great depths of the sea. However, as follows from viscotec governmental registrations of salinity made in recent years, the salinity of the deep sea-the deep waters of the sea of Japan lies in a fairly narrow range (34,065 -34,069‰). Figure 3 shows the average monthly salinity values for 2017 on the horizon from 0 to 100 meters for the sea of Japan. The criterion for checking the adequacy of the considered models of observations VI-da (10), (11) and (26) where nat k  are the full-scale measurements of water density values; k  шы density value calculated using models (10), (11), (26).
For the comparative analysis of the observation models, the full-scale values of the seawater density in the nodes of the computational grid for the high seas on the horizons from 0 to 100 meters, taken from the database «UISGO» (Fig. 7). It is obvious that the universal model of the species (10) is significantly inferior to the exact news of the model of the species (11), which in turn is a rough approximation to the developed model of observations of the species (26).
Conclusion. The mathematical model of hydrodynamics of a deep water reservoir-the sea of Japan, taking into account the complex geometry of the shoreline and the bottom, friction on the bottom and wind currents, evaporation, deviation of the pressure field from the hydrostatic approximation, water density as a function of the spatial distribution of temperature and salinity, is proposed and investigated. The models of observations for the functions included in the model problem, including the dependence of the water density distribution on the spatial distribution of its temperature, salinity and ionic composition, are determined and investigated. The application of these models is based on electrical conductivity and temperature, which makes them as accurate as possible in different conditions.
Taking into account the dependence of the density of sea water on the spatial distribution of its temperature, salinity and ionic composition increases the accuracy of mathematical modeling of the processes of hydrodynamics of the reservoir. Modeling of water flow movement in deep-water reservoirs taking into account variable density of sea water allows to improve accuracy of calculations of TRANS-loads for vessels, to calculate moments of formation of natural catastrophes, such as floods, coast abrasion, eutrophication, Zamora, and in real time to carry out calculations for allocation of the zones polluted following catastrophes of technogenic character, for example, emergency oil spills.

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In the future, it is possible to develop and implement an algorithm for the PLIS of a modified alternating triangular method for solving the grid equations arising from the discretization of the hydrodynamic model problem [10][11][12][13][14][15][16][17]. In order to improve the efficiency of the computational method for the implementation of the hydrodynamic model, algorithms and library elements focused on multiprocessor computing systems will be developed in the future.