Mathematical modelling of proton migration in Earth mantle

Vadim Bobrovskiy^1, 2 ; Juan Galvis³ ; Alexey Kaplin² ; Alexander Sinitsyn³ ; Marco Tognoli⁴ ; Paolo Trucco¹

¹ Department of Management, Economics and Industrial Engineering, Politecnico di Milano, Milan 20156, Italy
² Cosmetecor UK, London, England W1H 1PJ, UK
³ Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
⁴ Energy Engineering Department, Politecnico di Milano, Milan 20156, Italy

Mathematical modelling of natural phenomena, Tome 17 (2022), article no. 14.

Résumé

In the study, we address the mathematical problem of proton migration in the Earth’s mantle and suggest a prototype for exploring the Earth’s interior to map the effects of superionic proton conduction. The problem can be mathematically solved by deriving the self-consistent electromagnetic field potential U(x, t) and then reconstructing the distribution function f(x,v,t). Reducing the Vlasov-Maxwell system of equations to non-linear sh-Gordon hyperbolic and transport equations, the propagation of a non-linear wavefront within the domain and transport of the boundary conditions in the form of a non-linear wave are examined. By computing a 3D model and through Fourier-analysis, the spatial and electrical characteristics of potential U(x, t) are investigated. The numerical results are compared to the Fourier transformed quantities of the potential (V) obtained through field observations of the electric potential (Kuznetsov method). The non-stationary solutions for the forced oscillation of two-component system, and therefore, the oscillatory strengths of two types of charged particles can be usefully addressed by the proposed mathematical model. Moreover, the model, along with data analysis of the electric potential observations and probabilistic seismic hazard maps, can be used to develop an advanced seismic risk metric.

Détail

DOI : 10.1051/mmnp/2022018

Bibliographie

[1] R. Adams and J. Fournier, Sobolev Spaces, Academic Press, New York (2003).

[2] M.S. Alnœs, J. Blechta The FEniCS Project Version 1.5 Arch. Numer. Softw 2015 100

[3] W. Bangerth, R. Hartmann, G. Kanschat Deal. II—a general-purpose object-oriented finite element library ACM Trans. Math. Softw 2007 24 es

[4] V. Bobrovskiy, Software and hardware of international spatially distributed monitoring network for investigation local and global effects prior to the strong earthquakes. Ph.D. thesis. Geophysics. (in Russian), Russian State University for Geological Prospecting, Moscow (2016).

[5] V. Bobrovskiy and D. Kuznetsov, Seismic global conception on the example of strongest earthquakes with M8+ occurred in 2001-2015 (in Russian). Scientific World, Moscow (2016).

[6] V. Bobrovskiy, F. Stoppa, L. Nicoli, Y. Losyeva Nonstationary electrical activity in the tectonosphere-atmosphere interface retrieving by multielectrode sensors: case study of three major earthquakes in Central Italy with M 6 Earth Sci. Inf 2017 269 285

[7] P. Braasch, Semilineare elliptische Differentialgleichungen und das Vlasov-Maxwell-System. Utz, Verlag Wiss. (1997).

[8] J.W. Cooley, J.W. Tukey An algorithm for the machine calculation of complex Fourier series Math. Comput 1965 297 301

[9] L. Fortuin The wave equation in a medium with a time-dependent boundary J. Acoust. Soc. Am 1973 1683 1685

[10] H. Gao, C. Liu, J. Shi, S. Pan, T. Huang, X. Lu, H.-T. Wang, D. Xing, J. Sun Superionic silica-water and silica-hydrogen compounds in the deep interiors of Uranus and Neptune Phys. Rev. Lett 2022 035702

[11] D. Gayer, C. O’Sullivan et al., FreeCAD visualization of realistic 3D physical optics beams within a CAD system-model, in vol. 9914 of Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VIII. International Society for Optics and Photonics (2016) 99142Y.

[12] C. Geuzaine, J.-F. Remacle Gmsh: a 3-D finite element mesh generator with built-in pre-and post-processing facilities Int. J. Numer. Methods Eng 2009 1309 1331

[13] M. Hanfland, R.J. Hemley, H.-K. Mao Novel infrared vibron absorption in solid hydrogen at megabar pressures Phys. Rev. Lett 1993 3760 3763

[14] Y. He Superionic iron alloys and their seismic velocities in Earth’s inner core Nature 2022 258 262

[15] R.J. Hemley, Z.G. Soos, M. Hanfland, H.-K. Mao Charge-transfer states in dense hydrogen Nature 1994 384 387

[16] M. Hou, Y. He, B. Jang, S. Sun, Y. Zhuang, L. Deng, R. Tang, J. Chen, F. Ke, Y. Meng, V. Prakapenka, B. Chen, J. Shim, J. Liu, D. Kim, Q. Hu, C. Pickard, R.J. Needs, H-K. Mao Superionic iron oxide—hydroxide in Earth’s deep mantle Nat. Geosci 2021 174 178

[17] W. Kauzmann, Quantum Chemistry. Academic, New York (1957).

[18] D. Kuznetsov, Practice of short-term prediction of earthquakes: astro-, cosmo-geophysical impulses of Vernadsky-Vlasov-Vorob jev-Prigozhin on the vertical sequence of underground electrodes at PedInstitute fault at the magnetic meridian of Petropavlovsk-Kamchatsky. All-Union Institute for science and technical information (VINITI), Moscow (1991) 1–9.

[19] V. Larin, Hypothesis of a primordially hydride earth. Moscow, Izdatel’stvo Nedra (1980).

[20] K. Lee A mixed problem for hyperbolic equations with time-dependent domain J. Math. Anal. Appl 1966 471 495

[21] A. Logg, K.-A. Mardal, G.N. Wells et al., Automated Solution of Differential Equations by the Finite Element Method. Springer (2012).

[22] M. Millot, S. Hamel, J. Rygg, P.M. Celliers, G.W. Collins, F. Coppari, D.E. Fratanduono, R. Jeanloz, D.C. Swift, J.H. Eggert Experimental evidence for superionic water ice using shock compression Nat. Phys 2018 297 302

[23] M.N. Nabighian, Electromagnetic methods in applied geophysics. Soc. of Exploration Geophysicists, Tulsa, Okla (1993).

[24] M. Nishi, T. Irifune, J. Tsuchiya, Y. Tange, Y. Nishihara, K. Fujino, Y. Higo Stability of hydrous silicate at high pressures and water transport to the deep lower mantle Nat. Geosci 2014 224 227

[25] C. Pao, Nonlinear Parabolic and Elliptic Equations. Springer (1992).

[26] G.A. Rudykh, N.A. Sidorov, A.V. Sinitsyn Nonstationary solutions of the two-particle Vlasov—Maxwell system Dokl. Math 1989 700 701

[27] A. Sinitsyn, V. Vedenyapin and E. Dulov, Kinetic Boltzmann, Vlasov and related equations. Elsevier (2011).

[28] A.H. Squillacote, J. Ahrens, C. Law, B. Geveci, K. Moreland and B. King, Vol. 366 of The paraview guide. Kitware Clifton Park, NY (2007).

[29] E. Sugimura, T. Komabayashi, K. Ohta, K. Hirose, Y. Ohishi, L. Dubrovinsky Experimental evidence of superionic conduction in H2O ice J. Chew,. Phys 2012 194505

[30] V. Surkov and M. Hayakawa, Ultra and Extremely Low Frequency Electromagnetic Fields. Springer, Tokyo, Japan (2014).

[31] S. Uyeda, T. Nagao, Y. Orihara, T. Yamaguchi, I. Takahashi Geoelectric potential changes: possible precursors to earthquakes in Japan Proc Natl Acad Sci USA 2000 4561 4566

[32] P.P. Varostos, K. Alexopoulos Physical properties of the variation of the electric fields of the Earth preceding earthquakes Tectonophysics 1984 73 98

[33] V. Vernadsky Gas exchange in earth’s crust Proc. St. Petersburg Royal Acad. Sci 1912 141 162

[34] A.A. Vlasov, Many-particle theory and its application to plasma. Gordon and Breach (1961).