Nonlinear dynamo theory

M. I. Kopp ; A. V. Tur ; V. V. Yanovskii

Russian journal of nonlinear dynamics, Tome 11 (2015) no. 2, pp. 241-266.

Résumé

Using the asymptotic method of multiple scales construct nonlinear theory of large-scale structures in stratified conducting medium in the presence of small-scale oscillations of the velocity field and magnetic fields. Such small-scale stationary oscillations are generated by small external sources at low Reynolds numbers. The nonlinear system of equations describing the evolution of largescale structure of the velocity field and the magnetic fields are obtained. The linear stage of evolution leads to the well known instability. We study the equations of non-linear instability and its stationary solutions.

Détail

Keywords: stratified conducting medium, nonlinear system of equations, instability, multiscale method.
Mots-clés : large scale structures

Bibliographie

[1] Petviashvili V. I., Pokhotelov O. A., Uedinennye vikhri v plazme i atmosfere, Energoatomizdat, Moskva, 1989, 200 pp.

[2] Aburdzhaniya G. D., Samoorganizatsiya nelineinykh vikhrevykh struktur i vikhrevoi turbulentnosti v dispergiruyuschikh sredakh, KomKniga, Moskva, 2006, 328 pp.

[3] Kolesnichenko A. V., Marov M. Ya., Turbulentnost i samoorganizatsiya: Problemy modelirovaniya kosmicheskikh i prirodnykh sred, Binom, Moskva, 2009, 648 pp.

[4] Onischenko O. G., Pokhotelov O. A., Astafeva N. M., “Generatsiya krupnomasshtabnykh vikhrei i zonalnykh vetrov v atmosferakh planet”, UFN, 178:6 (2008), 605–618

[5] Sommeria J., Meyers S. P., Swinney H. L., “Laboratory simulation of Jupiters Great Red Spot”, Nature, 331 (1988), 689–693

[6] Lakhin V. P., Neustoichivosti i volny vo vraschayuscheisya plazme i turbulentnaya generatsiya regulyarnykh struktur, Dis. na soisk. uch. stepeni dokt. fiz-mat. nauk, Natsionalnyi issledovatelskii tsentr, Kurchatovskii institut, Moskva, 2013, 257 pp.

[7] Gershuni G. Z., Zhukhovitskii E. M., Konvektivnaya ustoichivost neszhimaemoi zhidkosti, Nauka, Moskva, 1972, 392 pp.

[8] Kuklin V. M., “Rol pogloscheniya i dissipatsii energii v formirovanii prostranstvennykh nelineinykh struktur v neravnovesnykh sredakh”, Ukr. fiz. zhurnal, 1 (2004), 49–81

[9] Shmerlin B. Ya., Kalashnik M. V., “Konvektivnaya neustoichivost Releya v prisutstvii fazovykh perekhodov vlagi: Formirovanie krupnomasshtabnykh vikhrei i oblachnykh struktur”, UFN, 183:5 (2013), 497–510

[10] Moiseev S. S., Rutkevich P. B., Tur A. V., Yanovskii V. V., “Vikhrevoe dinamo v konvektivnoi srede so spiralnoi turbulentnostyu”, ZhETF, 94:2 (1988), 144–153

[11] Lupyan E. A., Mazurov A. A., Rutkevich P. B., Tur A. V., “Generatsiya krupnomasshtabnykh vikhrei pod deistviem spiralnoi turbulentnosti konvektivnoi prirody”, ZhETF, 102:5 (1992), 1540–1552

[12] Moiseev S. S., Oganyan K. R., Rutkevich P. B., Tur A. V., Khomenko G. A., Yanovskii V. V., “Vikhrevoe dinamo v spiralnoi turbulentnosti”, Integriruemost i kineticheskie uravneniya dlya solitonov: Sb. nauch. tr., ed. V. G. Baryakhtar, Naukova dumka, Kiev, 1990, 280–382

[13] Levina G. V., Moiseev S. S., Rutkevitch P. B., “Hydrodynamic alpha-effect in a convective system”, Nonlinear instability, chaos and turbulence: Vol. 2, Adv. Fluid Mech., 25, eds. L. Debnath, D. N. Riahi, WIT Press, Southampton, 2000, 111–162

[14] Zimin V. D., Levina G. V., Moiseev S. S., Tur A. V., “Vozniknovenie krupnomasshtabnykh struktur pri turbulentnoi konvektsii v podogrevaemom snizu vraschayuschemsya sloe”, Dokl. AN SSSR, 309 (1989), 88–84

[15] Frishe U., She Z. S., Sulem P. L., “Large scale flow driven by the anisotropic kinetic alpha effect”, Phys. D, 28:3 (1987), 382–392

[16] Druzhinin O. A., Khomenko G. A., “Nelineinaya teoriya gidrodinamicheskogo alfa-effekta v szhimaemoi srede i obratnyi kaskad energii”, Tr. Mezhdunarodn. konf. «Nelineinye i turbulentnye protsessy v fizike»: T. 2, Naukova dumka, Kiev, 1989, 83–86

[17] Rutkevitch P. B., Sagdeev R. Z., Tur A. V., Yanovsky V. V., “Nonlinear dynamic theory of the $\alpha$-effect in compressible fluid”, Proc. of the 4th Internat. Workshop on Nonlinear and Turbulent Processes in Physics: Vol. 2 (Kiev, USSR, October 9–22, 1989), eds. V. G. Baryakhtar, V. M. Chernousenko, N. S. Erokhin, World Sci., Singapore, 1990, 172–175

[18] Tur A. V., Yanovsky V. V., Large-scale instability in hydrodynamics with stable temperature stratification driven by small-scale helical force, 2012, arXiv: 1204.5024v1

[19] Tur A. V., Yanovsky V. V., “Non linear vortex structure in stratified driven by small-scale helical forse”, Open J. Fluid Dynam., 3:2 (2013), 64–74

[20] Arnold V. I., Khesin B. A., Topologicheskie metody v gidrodinamike, MTsNMO, Moskva, 2007, 392 pp.

[21] Zeldovich Ya. B., Ruzmaikin A. A., Sokolov D. D., Magnitnye polya v astrofizike, RKhD, Institut kompyuternykh issledovanii, Moskva – Izhevsk, 2006, 384 pp.

[22] Arnold V. I., Zeldovich Ya. B., Ruzmaikin A. A., Sokolov D. D., “Magnitnoe pole v dvizhuscheisya provodyaschei zhidkosti”, UMN, 35:5 (1981), 220–221

[23] Arnold V. I., “Evolyutsiya magnitnogo polya pod deistviem perenosa i diffuzii”, UMN, 38:2 (1983), 225–227

[24] Zeldovich Ya. B., Ruzmaikin A. A., “Problemy dinamo v astrofizike”, Vspyshki na zvezdakh (sverkhnovye, rentgenovskie istochniki, Solntse), Itogi nauki i tekhniki. Ser. Astronomiya, 21, ed. R. A. Syunyaev, VINITI, Moskva, 1982, 151–187

[25] Kopp M. I., Tur A. V., Yanovskii V. V., “Krupnomasshtabnaya konvektivnaya neustoichivost v elektroprovodyaschei srede s melkomasshtabnoi spiralnoi turbulentnostyu”, ZhETF, 147:4 (2015), 846–866

[26] Moffat G., Vozbuzhdenie magnitnogo polya v provodyaschei srede, Mir, Moskva, 1980, 285 pp.

[27] Parker Yu., Besedy ob elektricheskikh i magnitnykh polyakh v kosmose, RKhD, Institut kompyuternykh issledovanii, Moskva – Izhevsk, 2010, 207 pp.

[28] Sokolov D. D., Stepanov R. A., Frik P. G., “Dinamo: Na puti ot astrofizicheskikh modelei k laboratornomu eksperimentu”, UFN, 184:3 (2014), 318–335

[29] Busse F. H., “Generation of planetary by convection”, Phys. Earth Planet. In., 12 (1976), 350–358

[30] Zang K. K., Busse F. H., “Generation of magnetic fields by convection in a rotating spherical fluid shell of infinite Prandtl number”, Phys. Earth Planet. In., 59 (1990), 202–222

[31] Jones C. A., “Convection-driven geodynamo models”, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 358:1768 (2000), 873–897

[32] Glatzmaier G. A., Roberts P. H., “Simulating the geodynamo”, Contemp. Phys., 38:4 (1997), 269–288

[33] Dolginov A. Z., Urpin V. A., “Termomagnitnaya neustoichivost neodnorodnoi plazmy”, ZhETF, 75:5 (1979), 1921–1932

[34] Vainshtein S. I., Zeldovich Ya. B., Ruzmaikin A. A., Turbulentnoe dinamo v astrofizike, Nauka, Moskva, 1980, 354 pp.

[35] Vainshtein S. I., Magnitnye polya v kosmose, Nauka, Moskva, 1983, 237 pp.

[36] Montgomery D., Chen H., “Turbulent amplification of large-scale magnetic fields”, Plasma Phys. Control. Fusion, 26:10 (1984), 1199–1210

[37] Brandt P. N., Scharmert G. B., Ferguson S., Shine R. A., Tarbell T. D., Title A. M., “Vortex flow in the solar photosphere”, Nature, 335 (1988), 238–240