Nonlinear problem of temperature distribution inside the Earth

• Main Author: Davidovich, Mihail Vladimirovich
• Format: Article
• Language: English
• Published: Saratov State University 2020-04-01
• Series: Известия высших учебных заведений: Прикладная нелинейная динамика
• Subjects: radiant thermal conductivity; earth heat balance; stefan–boltzmann law; blackness coefficient; earth’s core temperature
• Online Access: https://andjournal.sgu.ru/sites/andjournal.sgu.ru/files/text-pdf/2020/04/davidovich.pdf
• ISSN: 0869-6632; 2542-1905

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.