Some mathematical problems of atmospheric electricity

A. V. Kalinin ; A. A. Tyukhtina

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Winter Mathematical School "Modern Methods of Function Theory and Related Problems", Voronezh, January 28 - February 2, 2021, Part 2, Tome 207 (2022), pp. 48-60.

Résumé

In this paper, we discuss various formulations of mathematical problems arising in the description of the global electric circuit in the Earth's atmosphere. We consider initial-boundary-value problems for the nonstationary system of Maxwell equations, the system of Maxwell equations in the nonrelativistic electric approximation, and for the system of Maxwell equations in the quasistationary approximation generalizing the nonrelativistic electric and magnetic approximations.

Détail

Keywords: atmospheric electricity, global electric circuit, system of Maxwell equations, quasi-stationary approximation.

Bibliographie

[1] Galanin M. P., Popov Yu. P., Kvazistatsionarnye elektromagnitnye polya v neodnorodnykh sredakh, Fizmatlit, M., 1995

[2] Dyuvo G., Lions Zh.-L., Neravenstva v mekhanike i fizike, Nauka, M., 1980

[3] Zhidkov A. A., Kalinin A. V. Korrektnost odnoi matematicheskoi zadachi atmosfernogo elektrichestva, Vestn. NNGU im. N. I. Lobachevskogo., 2009, no. 4, 123–129

[4] Kalinin A. V., Slyunyaev N. N., Mareev E. A., Zhidkov A. A., “Statsionarnye i nestatsionarnye modeli globalnoi elektricheskoi tsepi: korrektnost, analiticheskie sootnosheniya, chislennaya realizatsiya”, Izv. RAN. Fiz. atmosf. okeana., 50:3 (2014), 355–364

[5] Kalinin A. V., Sumin M. I., Tyukhtina A. A., “Ustoichivye sekventsialnye printsipy Lagranzha v obratnoi zadache finalnogo nablyudeniya dlya sistemy uravnenii Maksvella v kvazistatsionarnom magnitnom priblizhenii”, Differ. uravn., 52:5 (2016), 608–624 | Zbl

[6] Kalinin A. V., Sumin M. I., Tyukhtina A. A., “Ob obratnykh zadachakh finalnogo nablyudeniya dlya sistemy uravnenii Maksvella v kvazistatsionarnom magnitnom priblizhenii i ustoichivykh sekventsialnykh printsipakh Lagranzha dlya ikh resheniya”, Zh. vychisl. mat. mat. fiz., 57:2 (2017), 18–40

[7] Kalinin A. V., Tyukhtina A. A., “Kvazistatsionarnye elektromagnitnye polya v neodnorodnykh sredakh s neprovodyaschimi i slaboprovodyaschimi vklyucheniyami”, Zh. Srednevolzh. mat. o-va., 18:4 (2016), 119–133

[8] Kalinin A. V., Tyukhtina A. A., “Priblizhenie Darvina dlya sistemy uravnenii Maksvella v neodnorodnykh provodyaschikh sredakh”, Zh. vychisl. mat. mat. fiz., 60:8 (2020), 1408–1421 | Zbl

[9] Kalinin A. V., Tyukhtina A. A., Lavrova S. R., “Neklassicheskie zadachi v modelyakh globalnoi elektricheskoi tsepi”, Tr. Mezhdunar. konf. «Aktualnye problemy vychislitelnoi i prikladnoi matematiki». Marchukovskie nauchnye chteniya–2019 (Novosibirsk, 1-5 iyulya 2019 g.), IPTs NGU, Novosibirsk, 2019, 203–209 | MR

[10] Landau L. D., Lifshits E. M., Teoreticheskaya fizika. Tom 8. Elektrodinamika sploshnykh sred, Nauka, M., 1982 | MR

[11] Mareev E. A., “Dostizheniya i perspektivy issledovanii globalnoi elektricheskoi tsepi”, Usp. fiz. nauk., 180:5 (2010), 527–534 | DOI

[12] Temam R., Uravneniya Nave—Stoksa. Teoriya i chislennyi analiz, Mir, M., 1981 | MR

[13] Tolmachev V. V., Golovin A. M., Potapov V. S., Termodinamika i elektrodinamika sploshnoi sredy, Izd-vo MGU, M., 1988

[14] Tamm I. E., Osnovy teorii elektrichestva, Nauka, M., 1989

[15] Alonso Rodriguez A., Valli A., Eddy Current Approximation of Maxwell Equations. Theory, Algorithms, and Applications, Spriner-Verlag, Milan, 2010 | MR | Zbl

[16] Ammari H., Buffa A., Nedelec J.-C., “A justification of eddy currents model for the Maxwell equations”, SIAM J. Appl. Math., 60:5 (2000), 1805–1823 | DOI | MR | Zbl

[17] Anisimov S. V., Mareev E. A., “Geofizicheskie issledovaniya globalnoi elektricheskoi tsepi”, Fizika Zemli., 2008, no. 10, 8–18

[18] Bayona V., Flyer N., Lucas G. M., Baumgaertner A. J. G., “A 3-D RBF-FD solver for modeling the atmospheric global electric circuit with topography (GEC-RBFFD v1.0)”, Geosci. Model Dev., 8:10 (2015), 3007–3020 | DOI

[19] Boström R., Fahleson U., “Vertical propagation of time-dependent electric fields in the atmosphere and ionosphere”, Electrical Processes in Atmospheres, eds. Dolezalek H., Reiter R., Steinkopff, 1977, 529-535. | MR

[20] Degond P., Raviart P.-A., “An analysis of the Darwin model of approximation to Maxwell's equations”, Forum Math., 4 (1992), 13–44 | DOI | MR | Zbl

[21] Evtushenko A., Kuterin F., Svechnikova E., “A plasmachemical axially symmetric self-consistent model of daytime sprite”, Atmos. Chem. Phys., 2020.

[22] Girault V., Raviart P., Finite element methods for Navier–Stokes equations, Springler-Verlag, N.Y., 1986 | MR | Zbl

[23] Jansky J., Pasko V. P., “Charge balance and ionospheric potential dynamics in timedependent global electric circuit model”, J. Geophys. Res. Space Phys., 229:12 (2014), 10184–10203

[24] Kawashima S., Shizuta Y., “Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, II”, Proc. Jpn. Acad. Ser. A., 62:5 (1986), 181–184 | DOI | MR | Zbl

[25] Kalinin A. V., Slyunyaev N. N., “Initial-boundary value problems for the equations of the global atmospheric electric circuit”, J. Math. Anal. Appl., 450:1 (2017), 112–136 | DOI | MR | Zbl

[26] Kalinin A. V., Tyukhtina A. A., “$L_p$-estimates for scalar products of vector fields and their application to electromagnetic theory problems”, Math. Meth. Appl. Sci., 41:18 (2018), 9283–9292 | DOI | MR | Zbl

[27] Kolmbauer M., Existence and Uniqueness of Eddy Current Problems in Bounded and Unbounded Domains, Inst. Comput. Math. J. Kepler Univ., Linz, Austria, 2011

[28] Larsson J., “Electromagnetics from a quasistatic perspective”, Am. J. Phys., 75:3 (2007), 230–239 | DOI

[29] Liu C., Williams E. R., Zipser E. J., Burns G. Diurnal variation of global thunderstorms and electrified shower clouds and their contribution to the global electrical circuit, J. Atmos. Sci., 67:2 (2010), 309–323 | DOI

[30] Mach D. M., Blakeslee R. J., Bateman M. G., “Global electric circuit implications of combined aircraft storm electric current measurements and satellite-based diurnal lightning statistics”, J. Geophys. Res., 116:D5 (2011), D05201

[31] Mareev E. A., Yashunin S. A., Davydenko S. S., et al., “On the role of transient currents in the global electric circuit”, Geophys. Res. Lett., 35:15 (2008), L15810 | DOI

[32] Markson R. The global circuit intensity: its measurement and variation over the last 50 years, Bull. Am. Meteor. Soc., 88:2 (2007), 223–241 | DOI

[33] Pasko V. P., Inan U. S., Bell T. F., Taranenko Y. N., “Sprites produced by quasi-electrostatic heating and ionization in the lower ionosphere”, J. Geophys. Res., 102:A3 (1997), 4529–4561 | DOI

[34] Raviart P.-A., Sonnendrücker E., “Approximate models for the Maxwell equations”, J. Comput. Appl. Math., 63 (1994), 69–81 | DOI | MR

[35] Raviart P.-A., Sonnendrücker E., “A hierarchy of approximate models for the Maxwell equations”, Numer. Math., 73 (1996), 329–372 | DOI | MR | Zbl

[36] Rycroft M. J., Harrison R. G., Nicoll K. A., Mareev E. A., “An overview of Earth's global electric circuit and atmospheric conductivity”, Space Sci. Rev., 137:1-4 (2008), 83–105 | DOI

[37] Rycroft M. J., Harrison R. G., “Electromagnetic atmosphere plasma coupling: the global atmospheric electric circuit”, Space Sci. Rev., 168:1-4 (2011), 363–384 | DOI

[38] Shalimov S. L., Bösinger T., “An alternative explanation for the ultra-slow tail of sprite-associated lightning discharges”, J. Atm. Solar-Terrestr. Phys., 68 (2006), 814–820 | DOI

[39] Tinsley B. A., “The global atmospheric electric circuit and its effects on cloud microphysics”, Rep. Progr. Phys., 71:6 (2008), 066801 | DOI

[40] Weitzner H., Lawson W. S., “Boundary conditions for the Darwin model”, Phys. Fluids B., l (1989), 1953–1957 | DOI | MR

[41] Williams E. R., “The global electrical circuit: a review”, Atmos. Res., 91:2-4 (2009), 140–152 | DOI

[42] Williams E., Mareev E., “Recent progress on the global electrical circuit”, Atmos. Res., 135–136 (2014), 208–227 | DOI

[43] Wilson T. R., “Investigations on lightning discharges and on the electric field of thunderstorms”, Phil. Trans. Roy. Soc. London. Ser. A., 221 (1921), 73–115 | DOI

[44] Wilson T. R., “The electric field of a thundercloud and some of its effects”, Proc. Phys. Soc. London., 37 (1924), 32D–37D | DOI

[45] Yashunin S. A., Mareev E. A., Rakov V. A., “Are lightning M components capable of initiating sprites and sprite halos?”, J. Geophys. Res., 112 (2007), D10109

[46] Zhou L., Tinsley B. A., “Global circuit model with clouds”, J. Atmos. Sci., 67:4 (2010), 1143–1156 | DOI