Mathematical Modeling of the Dynamics and Destruction of Magmatic Bombs Forming During the Eruption of Submarine Volcano
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Journal of Engineering Physics and Thermophysics, Vol. 93, No. 5, September, 2020
MATHEMATICAL MODELING OF THE DYNAMICS AND DESTRUCTION OF MAGMATIC BOMBS FORMING DURING THE ERUPTION OF SUBMARINE VOLCANOS V. A. Andrushchenko,a V. A. Goloveshkin,b,c I. V. Murashkin,a and N. N. Kholinb
UDC 551.217.2
Theoretical investigations into the dynamics and destruction of magmatic bombs forming during the eruption of submarine volcanos have been conducted. Continuum mechanics methods have been used for mathematical modeling of the dynamics and destruction of objects under investigation. In this case, it has been assumed that the mechanical behavior of the magmatic bomb material obeys the law of linear elastic deformation of a solid body. Keywords: deep-sea eruption, magmatic bomb, mathematical modeling, linear elasticity, destruction. Introduction. The eruption of major volcanos, as well as the earth′s collision with major extraterrestrial objects, pose a real threat to civilization [1, 2]. Whereas a great number of physicomathematical models have been constructed for cases of continental volcanic eruptions, including numerical models [3–7], the study of submarine volcanic eruptions has been dealt with in few publications [8, 9]. In the processes of an explosive-type eruption, phase transitions of magma make a determining contribution to the energy balance of all possible manifestations of a deep-sea eruption under conditions of fast decompression. They also include the processes of formation and dynamics of the crystalline phase serving as a basis for the formation of magmatic bombs representing incandescent stones as large as several meters in diameter flying out of the water into the atmosphere at high velocities (up to 200 m/s). During the eruption in the caldera of the Akademiya Nauk volcano, basaltic bombs 0.3 m in size were discharged from the bottom of Krymskoe Lake (1996) into the atmosphere for a distance of 25 km [7, 8]. This investigation deals with the study of the mechanics of a magmatic bomb′s motion from the volcanic vent to the water surface. Dynamics of a Magmatic Bomb. Consideration is given to the dynamics of a volcanic bomb′s travel from the volcanic vent to the ocean surface. Knowing the time of travel to the surface, it is possible to estimate a body′s critically stressed state under which its destruction occurs due to the nonuniformity of temperature distribution. To analyze the vertical movement of the body to the free surface of the ocean, we assume the following system of equations: mdV ρ = − mg − CS V 2 , dt 2
dH =V . dt
(1)
The solution of the system of equations (1) will be written in the form ⎛ CS ρ tan ⎜⎜ arctan V0 − 2mg ⎝ V = CS ρ 2mg
CS ρ g ⎞ t⎟ 2m ⎟⎠
,
(2)
a
Institute of Computer Aided Design of the Russian Academy of Sciences (ICAD RAS), 19/18 2nd Brestskaya Str., Moscow, 123056, Russia; email: [email protected]; bMoscow Technological University, 78 Vernadskii Ave., Moscow, 119454, Russia; email: [email protected]; cInstitute of Applied Mechanics, Russian Academy of Sciences, 4 Miusskay
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