Geoinformatika 2017; 4(64) : 23-27
УДК 550.425
VISCOUS DISSIPATION OF ENERGY AT THE STAGE OF Formation OF THE EARTH
Yu.V. Khachay, O.A. Hachay, A.N. Antipin
Institute of Geophysics, Ural Branch of the Russian Academy of Sciences, 100, Amundsen Str., Ekaterinburg, 620016, Russian Federation, e-mail: yu-khachay@yandex.ru; olgakhachay@yandex.ru
Purpose Modern studies have shown the importance of taking into account the contribution of internal energy sources and heat-mass transfer processes in the inner shells of the protoplanet and the irinfluence on the dynamics of the formation of the Earth in the process of its accumulation [1, 5]. Among the major contr ibutors are the potential energy accompanying the formation of acondensed body from a proto planetary cloud, the energy of naturally radioactive decay of short-l ived radioactive elements, and the energy of gravitational compression. However, attention should be paid to the fact that the formation of the Earth took place essentially in the binary system Earth-Moon. Therefore, it is important to take into account the energy release of viscous friction of the lunar tides, which change dsignificantly during the accumulation phase of the system. In [1, 5], a model for the differentiation of the substance of the solar protoplanetary cloud was proposed in the process of accumulation of terrestrial planets based on the obtained numerical results of the solut ion of the boundary value problem for a system of equations describing the balance of momentum, mas, temperature and pressure in the innerregions of the growing Earth and Moon in the process of the iraccumulation (1)—(5).
Design/ methodology/ approach In order to investigate the problem, we used the numerical solution of boundary value problems for the 3D system of differential equations. We constructed a non-uniform difference grid for time and space variables and used a standard implicit difference scheme. An iterative proces was constructed, which begins with the solution of equation (1), from which the time step is located. Then we included the solution of the problems for equation (2)—(3) with the values of the coeficients in equations (2)—(3) and (6)—(9) calculated from the values at the previous stage, and the internal iteration cycle. After the specified relative accuracy is achieved, a step is taken along the selected radius, and an external cycle is started.
Findings From the above, the folowing qualitative estimates can be obtained. Since the angular velocity of the orbital motion of the satelite is much greater than the angular velocity of the planet’s own rotation, it folows from the law of conservation of angular momentum (6) that the tidal interaction with increasing mas of the central body ensures the transfer of the orbital angular momentum of the satelite to the central masive body. Consequently, losing the orbital moment, the satelite must aproach the central body right up to the Roche orbit. The problem of the behavior of a satelite at the Roche limit requires a careful mathematical study. Due to the continuing transfer of the satelite’s orbital moment to the central body, its own angular velocity continues to grow. This should lead to an increase in the orbital angular velocity of the satelite and an increase in the radius of its orbit, which is observed up to the present time. Thus, the efect of dissipation of viscous friction must have manifested itself at the stage of formation of the outer core of the Earth. The results obtained depend on the parameters, primarily, the viscosity versus temperature and presure, the values of which are known with a high degre of uncertainty. They are to be clarified as a result of further physical-mathematical and indirect geological research.
Practical value/ implications $The results obtained will serve as a basis for constructing new systems of differential equations of Earth’s dynamics at the stage of planetary accumulation, as wel as revealing the role of the energy dissipation of viscous friction in the differentiation proceses of the mater of the interior envelopes of the Earth at the stage of its formation, which provided the features of the MHD proceses of the Earth’s dynamo and metalogeny of the Earth’s lithosphere.
Keywords: boundary problem for a system of differential equations, algorithm of numerical solution, contribution of the dissipation of tidal energy.
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