ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Davidovich M. V. Nonlinear problem of temperature distribution inside the Earth. Izvestiya VUZ. Applied Nonlinear Dynamics, 2020, vol. 28, iss. 2, pp. 140-157. DOI: 10.18500/0869-6632-2020-28-2-140-157

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Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
537.525.5, 536.244
DOI: 
10.18500/0869-6632-2020-28-2-140-157

Nonlinear problem of temperature distribution inside the Earth

Autors: 
Davidovich Mihail Vladimirovich, Saratov State University
Abstract: 

The purpose of this research is to obtain a nonlinear heat conduction equation based on the Stefan–Boltzmann law and energy balance to study the temperature distribution inside the Earth, taking into account usual conductivity and radiant heat conductivity. Resulting equation with a fourth-degree nonlinearity allows to consider the heat transfer between a layer of matter and the environment. Methods to obtain the equation are based on the energy conservation law, taking into account the Kirchhoff’s law and introducing the variation of the blackness coefficient. Results are presented as a solution to the stationary problem of heat distributed along the Earth’s radius, wherefore the obtained equation has been applied. Therefore, thermal balance of the Earth has been considered, stationary functionals for temperature distribution have been obtained, whereof the temperature in the center of the Earth has been estimated. This estimate is based on the study of the transparency of the surface layer and the atmosphere for IR and optical range radiation. It is shown that the temperature is affected by transparency of a small surface layer and the value of variation of the blackness coefficient. It is also shown that the temperature in the center correlates well with the recent data from indirect experiments which is about 6000...6500 K. Temperature distribution along the radius is obtained. In a large area of central values, the temperature changes very slightly. Main temperature variation occurs at a layer of about 600 km above the surface. Discussion. Estimates without taking into account radiative transfer lead to a higher temperature in the center. It should be noted that the considered model is approximate. It uses spherical symmetry and does not consider heat transfer due to convection in the liquid internal region, which leads to a stronger temperature equalizing. A more accurate model requires specifying the spatial distributions of the coefficients included in the equation.

 

Acknowledgements. The work was supported by Russian Science Foundation, project no. 16-19-10033.

Key words: 
radiant thermal conductivity
Earth heat balance
Stefan–Boltzmann law
blackness coefficient
Earth’s core temperature
Reference: 
  1. Jacobs J.A. The Earth’s inner core. Nature, 1953, vol. 172, no. 4372, p. 297.
  2. Tikhonov A.N., Samarskii A.A. Equations of Mathematical Physics. Moscow: Nauka, 1977. 736 p. (in Russian).
  3. Tikhonov A.N. On the influence of radioactive decay on the temperature of the Earth crust. Proc. of USSR Academy of Sciences, Mathematics and Nature Sciences, 1937, p. 431 (in Russian).
  4. Anzellini S. Science. AAAS, 2013, vol. 340, no. 6136, p. 464.
  5. Patankar S. Numerical Methods for Solving Problems of Heat Transfer and Fluid Dynamics. Translation from English edited by V.D. Vilensky. Moscow: Energoatomizdat, 1984. 512 p. (Patankar, Suhas V. Numerical Heat Transfer and Fluid Flow. Washington: New York: Hemisphere; McGraw-Hill, cop. 1980.)
  6. Shashkov A.G., Bubnov V.A., Yanovsky S.Yu. Wave Phenomena of Thermal Conductivity. Moscow: URSS, 2004. 298 p. (in Russian). 
  7. Dulnev G.N., Parfenov V.G., Sigalov A.V. Application of Computers for Solving Heat Transfer Problems. Moscow: Higher school, 1990. 207 p. (in Russian).
  8. Kutateladze S.S. Fundamentals of the Theory of Heat Transfer. M: Atomizdat, 1979, 416 p. (in Russian).
  9. Sveshnikov A.G., Bogolyubov A.N., Kravtsov V.V. Lectures on Mathematical Physics: Studies. benefit. Moscow: Moscow State University, 1993. 352 p. (in Russian).
  10. Unsold A. Physics of Stellar Atmosphere. 2ed. Berlin: Springer, 1955.
  11. Ambartsumyan V.A., Mustel E.R., Severniy A.B., Sobolev V.V. Theoretical Astrophysics. Ed. V.A. Ambartsumyan. Moscow: Gostekhizdat, 1952. 635 p. (in Russian).
  12. Chandrasekhar S. Radiative Transfer. 2ed. Dover Publications Inc. 1960.
  13. Ivanov V.V. Radiation Transfer and Spectra of Celestial Bodies, Moscow: Nauka, 1969. 472 p. (in Russian).
  14. Sparrow E.M., Sess R.D. Heat transfer by radiation. Leningrad: Trans. from English, 1970. 295 p. (Sparrow E.M. and Cess R.D., 1978, Radiative Heat Transfer, McGraw-Hill. Google Scholar. Tien, C.-L., 1988).
  15. Adrianov V.N. Fundamentals of Radiation and Complex Heat Transfer. Moscow: Energia, 1972. (in Russian).
  16. Muchnik G.F., Rubashov I.B. Methods of Heat Transfer Theory. Part 2. Thermal Radiation. Moscow: Higher School, 1974. 270 p. (in Russian).
  17. Siegel R., Howell J. Thermal Radiation Heat Transfer. McGraw-Hill Book Company, New York Google Scholar, 1972. (Siegel, Robert; Howell, John R. Thermal radiation heat transfer New Jourk a. o., 1972).
  18. Adzerikho K.S. Lectures on the Theory of Transfer of Radiant Energy. Ed. M.A. Elyashevich. Minsk: BSU publishing house, 1975. 192 p. (in Russian).
  19. Ozisik M.N. Radiative Transfer and Interactions with Conduction and Convection. C. John Wiley, 1973.
  20. Apresyan L.A., Kravtsov Yu.A. Theory of Radiation Transfer: Statistical and Wave Aspects, Moscow: Nauka, 1983, 216 p. (in Russian).
  21. Nagirner D.I. Lectures on the Theory of Radiation Transfer: Studies benefit. SPb.: Publishing house SPb. Univ., 2001. 284 p. (in Russian).
  22. Zeldowich Ya.B., Raizer Yu.P. Physics of Shock Waves and High-temperature Hydrodynamics Phenomena. Moscow: Nauka, 1966. 688 p. (in Russian).
  23. Laktionov A.G. Thermal Balance of the Earth. Vol. 5. Physical Encyclopedia / Under the editorship of A.M. Prokhorov. M.: Great Russian encyclopedia, 1998, p. 75 (in Russian).
  24. Trenberth K.E., Fasullo J.T., Kiehl J. Bulletin of the American Meteorological Society, 2009, vol. 90, no. 3, p. 311.
  25. Pollack H.N., Hurter S.J., Johnson J.R. Reviews of Geophysics, 1993, vol. 31, no. 3, p. 267.
  26. Lyubimova E.A. Heat Flows from the Crust and Mantle of Continents / In Tectonosphere of the Earth. M.: Nauka, 1978. p. 327 (in Russian).
  27. Handbook of Physical Constants of Rocks. Ed. by S. Clarke, Jr. M.: Mir, 1969. 544 p. (in Russian).
  28. Magnitsky V.A. Fundamentals of Earth physics. M.: Geodesizdat, 1953. 289 p.
  29. Earth’s crust and upper mantle. Edited By P. HART. M.: Mir, 1972. 640 p.
  30. General Geophysics. Under the editorship of V.A. Magnitsky. M.: Publishing house of Moscow University, 1995. 317 p. (in Russian). 
  31. Van Schmus W.R. Natural radioactivity of the crust and mantle / AGU Handbook of Physical Constants. Edited by T.J. Ahrens. Washington: Am. Geophys., Un., 1994. V. 1. P. 283.
  32. Carison R.W. Planetary science: A new recipe for Earth formation. Nature, 2015, vol. 520, p. 298.
  33. Davidovich M.V., Kornev I.A., Timofeev A.I. Nonlinear dynamics of heat transfer in cylindrical and spherical structures. In: Questions of applied physics. Intercollegiate scientific collection. Saratov, 2015, pp. 93–98. (in Russian).
Received: 
17.03.2019
Accepted: 
13.01.2020
Published: 
30.04.2020
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