Application of methods of mathematical physics and logical approximations to the solution of problems of distribution of temperature fields between a trend of the rising mantle-crust migrant (plume) and surrounding medium is considered. The four-dimensional problem (x, y, z, time) is reduced to the two-dimensional problem (x, time) for a particular, but the most common, case of the symmetry of the plume around the central axis and the uniform nature of its rise. A software Vladi Gear 3.0 for thermodynamic calculations in the range «plume adiabat–regional geotherms» that implements the described transformation, is developed. The analogy of the vertical thermal interaction of time layers with the internal friction (viscosity) of the medium is shown. This analogy can be used in the calculations of convective heat and mass transfer.

Vasiliev Vladimir Igorevich, candidate of geological and mineralogical sciences, research fellow. Geological Institute of Siberian Branch of Russian Acаdеmу of Sciences (GIN SB RAS). 670047, Republic of Buryatia, Ulan-Ude, Sakhyanova str., 6A, E-mail: geovladi@ginst.ru.

Zhatnuev Nikolay Sergeevich, doctor of geological and mineralogical sciences, senior researcher. Geological Institute of Siberian Branch of Russian Acаdеmу of Sciences (GIN SB RAS). 670047, Republic of Buryatia, Ulan-Ude, Sakhyanova str., 6A. E-mail: zhat@ginst.ru.

Vasilieva Eugenia Vladimirovna, candidate of geological and mineralogical sciences, research fellow. Geological Institute of Siberian Branch of Russian Acаdеmу of Sciences (GIN SB RAS). 670047, Republic of Buryatia, Ulan-Ude, Sakhyanova str., 6A. E-mail: geojenny@ginst.ru.

1. Afanasiev V.N., Burtsev S.A., Egorov K. S., Kulagin A. Yu. A cylinder in the boundary layer of a flat plate // Vestnik MGTU named after N.E. Bauman. 2011. P. 3-22.
2. Vasiliev V.I. The Object-oriented approach in computer modelling of geological phenomena and processes // Vestnik IrGSHA. 2013. V. 57. Part 1. P. 79-86.
3. Vasiliev V.I. Software product Distant 4.0 for calculation of two-dimensional distributions of physical properties of the Earth’s crust // Tectonophysics and topical issues of Earth Sciences : Proceedings of the III all-Russian tectonophysical conference. Moscow : IPE RAS, 2012. V. 2. P. 373-376.
4. Vasiliev V.I. Numerical physico-chemical model of the rise of free fluid in the subduction zone // Modern problems of Geology, Geochemistry and Geoecology of the Russian Far East. Proceedings of the II regional conference. Vladivostok : Dalnauka, 2008. P. 108-110.
5. Vasiliev V.I., Vasilieva E.V., Zhatnuev N.S. Migration of fluid-filled cracks in the lithosphere by results of tectonophysical modeling // Geodynamic evolution of the lithosphere of the Central Asian mobile belt (from the ocean to the continent) : Proceedings of the meeting. Irkutsk : IEC SB RAS, 2008. V. 6. P. 54-56.
6. Vasiliev V.I., Vasilieva E.V., Zhatnuev N.S., Sanzhiev G.D. Parameters of the formation and evolution of the mantle-crust migrant // Geoinformatika. 2019. No. 2. P. 34-42.
7. Vasiliev V.I., Zhatnuev N.S. The thermodynamic model of the evolution of a fluid-filled fracture in the zone of a plastic-brittle transition // Physicochemical and Petrophysical Studies in Earth Sciences: Materials of the X International Conference. Moscow : IGEM RAS, 2009. P. 78-82.
8. Vasiliev V.I., Zhatnuev N.S., Vasilieva E.V. A numerical model of the rise of mantle fluid in the Baikal rift zone // Petrography of igneous and metamorphic rocks: Materials of the XII All-Russian Petrographic Meeting with the participation of foreign scientists. Petrozavodsk : KSC RAS, 2015. P. 114-116.
9. Vasiliev V.I., Zhatnuev N.S., Vasilieva E.V. A numerical thermodynamic model of the lifting of fluid-filled cracks through the zone of plastic-brittle transition // Geodynamic evolution of the lithosphere of the Central Asian mobile belt (from the ocean to the continent) : Materials of the meeting. Irkutsk : IEC SB RAS, 2009. V. 7, Part 1. P. 48-50.
10. Vasiliev V.I., Zhatnuev N.S., Rychagov S.N., Vasilieva E.V., Sanzhiev G.D. Mass transfer and mineral formation in magmatogen-hydrothermal systems according to the results of numerical physical and chemical modeling // Lithosphere. 2010. No. 3. P. 145-152.
11. Vasiliev V.I., Chudnenko K.V., Zhatnuev N.S., Vasilieva E.V. Integrated computer modeling of geological objects using the example of a section of the subduction zone // Geoinformatika. 2009. No. 3. P. 15-30.
12. Vasilieva E.V., Vasiliev V.I., Zhatnuev N.S. Tectonophysical modeling of the dynamics of fluid- filled cracks in the lithosphere // Geochemistry and ore formation of radioactive, noble and rare metals in endogenous and exogenous processes : Materials of the All-Russian Conference with Foreign Participation dedicated to the 50th anniversary of the SB RAS and the 80th anniversary of Corr. RAS F.P. Krendelev. April 16-18, 2007. Ulan-Ude : Publishing House of the BSC SB RAS, 2007. V. 2. P. 122-125.
13. Vasilieva E.V., Vasiliev V.I. Evolution of fluid-filled crack in the lithosphere according to the results of tectonophysical modeling // Modern problems of geology, geochemistry and geoecology of the Russian Far East : Materials of the II regional conference. Vladivostok : Dalnauka, 2008. P. 111-113.
14. Vasilieva E.V., Vasiliev V.I., Zhatnuev N.S., Sanzhiev G.D. Tectonophysical study of the dynamics of fluid-filled cavities in a solid plastic medium // Tectonophysics and current issues of Earth sciences : Materials of the III All-Russian Tectonophysical Conference. Moscow : IPE RAS, 2012. V. 1. P. 265-268.
15. Vasilieva E.V., Vasiliev V.I., Sanzhiev G.D. The process of migration of fluid-filled cracks in the lithosphere according to the results of tectonophysical modeling // Geodynamic evolution of the lithosphere of the Central Asian mobile belt (from the ocean to the continent) : Materials of the meeting. Irkutsk : IEC SB RAS, 2010. V. 8. Part 1. P. 46-47.
16. Vasilieva E.V., Zhatnuev N.S., Vasiliev V.I., Sanzhiev G.D. The possibility of the participation of the mantle fluid in the formation of hydrotherms in the Baikal rift zone according to the results of tectonophysical and numerical physical and chemical modeling // Geology of Western Transbaikalia : Materials of the All-Russian Conference. Ulan-Ude : GIN SB RAS, 2011. P. 30-33.
17. Zhatnuev N.S. Dynamics of fractured fluid systems in the zone of plastic deformations // Lithosphere. 2006. No. 4. P. 149-158.
18. Zhatnuev N.S. Transmantle (intratelluric) fluids: a new model of plumes and plume-magmatism // Geology and Geophysics. 2016. V. 57, No. 8. P. 1445-1454.
19. Zhatnuev N.S. Transmantle fluid flows and the origin of plumes // Reports of the Russian Academy of Sciences. 2012. V. 444, No. 1. P. 50-55.
20. Zhatnuev N.S. Fractured fluid systems in the zone of plastic deformations // Reports of the Russian Academy of Sciences. 2005. V. 404, No. 3. P. 380-384.
21. Zhatnuev N.S., Vasiliev V.I., Sanzhiev G.D. Upward migration of fluids in the mantle. Conceptual, computational and analog models // Domestic Geology. 2013. No. 3. P. 24-30.
22. Karpov I.K. Physical-chemical computer simulation in geochemistry. Novosibirsk : Nauka, 1981. 247 p.
23. Kuznetsov G.V., Sheremet M.A. Difference methods for solving heat conduction problems. Tomsk : TPU, 2007. 172 p.
24. Petrishchevsky A.M. Rheological and geothermal characteristics of the Okhotsk Plume // Bulletin of the Tomsk Polytechnic University. Engineering of geological resources. 2016. V. 327, No. 2. P. 65-76.
25. Putilov K.A. Physics course. Moscow: Fizmatlitgiz, 1963. V. 1. 560 p.
26. Certificate of state registration of a computer program No. 2019613814. The Russian Federation. Vladi OverPressure – a software for parametric modeling of the formation and evolution of a mantle-crust migrant / Vasiliev V.I., Vasilieva E.V., Zhatnuev N.S. ; copyright holder GIN SB RAS (RU). – No. 2019612231 ; declared 06.03.2019 ; register. 22.03.2019 ; publ. 22.03.2019, The official bulletin «Computer programs. Database. Topologies of Integrated Circuits». No. 4.
27. Certificate of state registration of a computer program No. 2018664295. The Russian Federation. Vladi Distat – a software for calculating of two-dimensional distributions of physical fields and the viscosity of the earth’s crust / V.I. Vasiliev, N.S. Zhatnuev, E.V. Vasiliev; copyright holder GIN SB RAS (RU). – No. 2018662448 ; declared 06.11.2018 ; register. 14.11.2018 ; publ. 14.11.2018, The official bulletin «Computer programs. Database. Topologies of Integrated Circuits». No. 11.
28. Certificate of state registration of a computer program No. 2019613716. The Russian Federation. Vladi Joint – a software for statistic modeling of behavior of random uprising fluid-filled fissures (cavities) systems in plastic medium / V.I. Vasiliev ; copyright holder GIN SB RAS (RU). – No. 2019612638 ; declared 14.03.2019 ; register. 21.03.2019 ; publ. 21.03.2019, The official bulletin «Computer programs. Database. Topologies of Integrated Circuits». No. 4.
29. Certificate of state registration of a computer program No. 2019617548. The Russian Federation. Vladi DisTerm – a software for two-dimensional modeling of thermal perturbation of thermogradient medium during magmatic migrant ingress / V.I. Vasiliev; copyright holder GIN SB RAS (RU). – No. 2019616467 ; declared 03.06.2019 ; register. 17.06.2019 ; publ. 17.06.2019, The official bulletin «Computer programs. Database. Topologies of Integrated Circuits». No. 6.
30. Thermophysical properties of rocks / V.V. Babaev et al. Moscow : Nedra, 1987. 156 p.
31. Willie R.J. Petrogenesis and physics of the Earth // Evolution of igneous rocks. Moscow : Mir, 1983. P. 468-503.
32. Chudnenko K.V. Thermodynamic modeling in Geochemistry: theory, algorithms, software, applications. Novosibirsk : Academic publishing house «GEO», 2010. 287 p.
33. Berman R.G., Aranovich L.Ya. Optimized standard state and solution properties of minerals: I. Model calibration for olivine, orthopyroxene, cordierite, garnet, and ilmenite in the system FeO–MgO–CaO–Al2O3–TiO2–SiO2 // Contrib. Mineralogy and Petrology. 1996. V. 126. P. 1-24.
34. McKenzie D., Jackson J., Priestley K. Thermal structure of oceanic and continental lithosphere // Earth Planet. Sci. Lett. 2005. No. 233. P. 337-349.