Influence of the principle of minimum potential energy on the distribution of density and gravitational energy of the Earth for the reference model PREM
From the point of view of modeling the internal structure of the Earth, its figure and evolution play an important role, which to one degree or another are associated with gravitational energy and the principle of its minimization. The realization of the minimum principle of potential for models of the distribution of the Earth’s density is the key in studies on the detection of inhomogeneous mass distribution. Achieving the minimum gravitational energy of the Earth is equivalent to the approximation of the internal structure to the hydrostatic state, and this condition is achieved due to variations in density. Therefore, for correct geophysical interpretation of gravimetric data, it is necessary to align the PREM (Preliminary Reference Earth Model) with harmonic coefficients of geopotential and minimum functional condition that determines gravitational energy, and only on this basis to estimate variations in density and tectonosphere. An algorithm for representing a piecewise continuous density distribution function in a spherical PREM model by Legendre polynomials is proposed in the paper to calculate the density, potential and energy distribution in an ellipsoidal planet using an additional condition — the minimum of gravitational energy. The use of such an algorithm made it possible to transform the spherically symmetric PREM model to a hydrostatically balanced state. It turned out that in the model obtained, the excess potential energy is concentrated in the inner and outer core of the Earth, and also insignificantly in the planet’s crust. The total energy E for the PREM reference model, which is subdivided into ellipsoidal layers, is 2.3364∙1024 erg, and in the modified PREM model after its correction for the hydrostatic component, it is 2.2828∙1024 erg. Estimation of density redistribution and identification of areas of their greatest change provide a mechanism for explaining the dynamic processes in the middle of the Earth.
Barkin, Yu.V. (2011). Century variations of the Earth’s figure in the modern era. In The current state of Earth sciences (pp. 183—187). Moscow: Ed. of the Faculty of Geology, Moscow State University (in Russian).
Bullen, K.E. (1978). Density of the Earth. Moscow: Mir, 442 p. (in Russian).
Vikulin, A.V. (2004). Introduction to the physics of the Earth. Petropavlovsk-Kamchatsky: KGPU publishing house, 240 p. (in Russian).
Gutenberg, B. (1963). Physics of the Earth’s Interior. Moscow: Inostrlit, 264 p. (in Russian).
Zharkov, V.N., Trubitsyn, V.P. & Samsonenko, L.V. (1971). Physics of the Earth and planets. Figures and internal structure. Moscow: Nauka, 384 p. (in Russian).
Kozlenko, V.G. (1984). System interpretation of geophysical fields. Kiev: Naukova Dumka, 220 p. (in Russian).
Kozlenko, V.G., Starostenko, V.I., Meshcheryakov, G.A. & Deyneka, Yu.P. (1979). Building a density model of the Earth from gravimetric data. Geofizicheskiy Zhurnal, 1(3), 3—21 (in Russian).
Kondratyev, B.P. (2007). Potential theory. New methods and problems with solutions. Moscow: Mir, 512 p. (in Russian).
Kuznetsov, V.V. (2005). The principle of minimizing the gravitational energy of the Earth and the mechanisms of its implementation. Vestnik Otdeleniya nauk o Zemle RAN, (1), 1—27 (in Russian).
Lavrentev, M.M. (1962). On some non-correct tasks in Mathematical Physics. Novosibirsk: Nauka, 92 p. (in Russian).
Magnitskiy, V.A. (1965). Internal structure and physics of the Earth. Moscow: Nedra, 380 p. (in Russian).
Marchenko, A.N. & Zayats, A.S. (2008). Estimation of the potential gravitational energy of the Earth based on reference density models. Geodynamika, (1), 5—24. https://doi.org/10.23939/jgd2008.01.005 (in Ukrainian).
Mashimov, M.M. (1982). Planetary theory of Geodesy. Moscow: Nedra, 261 p. (in Russian).
Meshcheryakov, G.A. (1991). Problems of the theory of potential and the generalized Earth. Moscow: Nauka, 216 p. (in Russian).
Moritz, G. (1994). Figure of the Earth: Theoretical Geodesy and the Internal Structure of the Earth. Kiev: Publ. House of the National Academy of Sciences of Ukraine, 240 p. (in Russian).
Muratov, R.Z. (1976). Potentials of an ellipsoid. Moscow: Atomizdat, 144 p. (in Russian).
Newton, I. (1989). Mathematical Principles of Natural Philosophy. Moscow: Nauka, 690 p. (in Russian).
Stacey, F. (1972). Physics of the Earth. Moscow: Mir, 344 p. (in Russian).
Tikhonov, A.N. (1943). On the stability of inverse problems. Doklady AN SSSR, 39(5), 195—198 (in Russian).
Fys, M.M., Brydun, A.M. & Yurkiv, M.I. (2019). Researching the influence of the mass distribution inhomogeneity of the ellipsoidal planet’s interior on its stokes constants. Geodynamika, (1), 17—27. https://doi.org/10.23939/jgd2019.01.017 (in Ukrainian).
Fys, M., & Nikulishin, V. (2011). Analysis of the efficiency of the Earth figurine on the internal structure of the PREM model. Geodynamika, (1), 17—21 (in Ukrainian).
Fys, M.M., Nikulishin, V.I., & Ozimblovskyy, R.M. (2010). Use of Legendre polynomials to approximate one-dimensional mass density distributions of planets and study their convergence. Heodeziya, kartohrafiya i aerofotoznimannya, (73), 3—6 (in Ukrainian).
Tserklevych, A.L. (2005). Method of building a model of the distribution of the Earth’s tectonosphere density, consistent with the gravitational field and other geological-geophysical information. Geofizicheskiy Zhurnal, 27(2), 309—314 (in Russian).
Tserklevych, A.L., Shylo, Y.O. & Shylo, O.M. (2019). Earth’s figure changes — geodynamic factor of stressed-deformed lithosphere state. Geodynamika, (1), 28—42. https://doi.org/10.23939/jgd2019.01.028 (in Ukrainian).
Chandrasekhar, S. (1973). Ellipsoidal figures of equilibrium. Moscow: Mir, 288 p. (in Russian).
Shen, E.L. (1977). On the variation approach to the construction of the density model of the Earth. Geofizicheskiy sbornik AN USSR, (80), 44—77 (in Russian).
Dzewonski, A. & Anderson, D. (1981). Preliminary reference Earth model. Physics of the Earth and Planet Interiors, 25, 297—356. https://doi.org/10.1016/0031-9201(81)90046-7.
Shen, W., Shen, Z., Sun, R. & Barkin, Yu. (2015). Evidences of the expanding Earth from space-geodetic data over solid land and sea level rise in recent two decades. Geodesy and Geodynamics, 6(4), 248—252. https://doi.org/10.1016/j.geog.2015.05.006.
Tserklevych, A.L., Zayats, O.S. & Shylo, Y.O. (2017). Dynamics of the Earth shape transformation. Kinematics and Physics of Celestial Bodies, 33(3), 130—141. https://doi.org/10.3103/S0884591317030060.
How to Cite
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).