Abstract
We discuss application of contemporary methods of the theory of dynamical systems with regular and chaotic hyperbolic dynamics to investigation of topological structure of magnetic fields in conducting media. For substantial classes of magnetic fields, we consider well-known physical models allowing us to reduce investigation of such fields to study of vector fields and Morse–Smale diffeomorphisms as well as diffeomorphisms with nontrivial basic sets satisfying the A axiom introduced by Smale. For the point–charge magnetic field model, we consider the problem of the separator playing an important role in the reconnection processes and investigate relations between its singularities. We consider the class of magnetic fields in the solar corona and solve the problem of topological equivalency of fields in this class. We develop a topological modification of the Zeldovich funicular model of the nondissipative cinematic dynamo, constructing a hyperbolic diffeomorphism with chaotic dynamics that is conservative in the neighborhood of its transitive invariant set.
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References
H. Alfven, “On sunspots and the solar cycle,” Arc. F. Mat. Ast. Fys., 29A, 1–17 (1943).
H. Alfven, “Electric currents in cosmic plasmas,” Rev. Geophys. Space Phys., 15, 271 (1977).
H. Alfven and C.-G. Fälthammar, Cosmical Electrodynamics: Fundamental principles, Clarendon, Oxford (1963).
D. V. Anosov and V. V. Solodov, “Hyperbolic sets,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fundam. Napravl., 66, 12–99 (1991).
V. I. Arnol’d and B. A. Khesin, Topological Methods in Hydrodynamics [in Russian], MTsNMO, Moscow (2007).
P. Baum and A. Bratenahl, “Flux linkages of bipolar sunspot groups: a computer study,” Solar Phys., 67, 245–258 (1980).
C. Beveridge, E. R. Priest, and D. S. Brown, “Magnetic topologies due to two bipolar regions,” Solar Phys., 209, No. 2, 333–347 (2002).
C. Beveridge, E. R. Priest, and D. S. Brown, “Magnetic topologies in the solar corona due to four discrete photospheric flux regions,” Geophys. Astrophys. Fluid Dyn., 98, No. 5, 429–445 (2004).
C. Bonatti, V. Grines, V. Medvedev, and E. Pecou, “Three-dimensional manifolds admitting Morse-Smale diffeomorphisms without heteroclinic curves,” Topol. Appl., 117, 335–344 (2002).
H. Bothe, “The ambient structure of expanding attractors, II. Solenoids in 3-manifolds,” Math. Nachr., 112, 69–102 (1983).
D. S. Brown and E. R. Priest, “The topological behaviour of 3D null points in the Sun’s corona,” Astron. Astrophys., 367, 339 (2001).
S. Childress and A. D. Gilbert, Stretch, Twist, Fold: The Fast Dynamo, Springer, Berlin–Heidelberg–N.Y. (1995).
R. M. Close, C. E. Parnell, and E. R. Priest, “Domain structures in complex 3D magnetic fields,” Geophys. Astrophys. Fluid Dyn., 99, No. 6, 513–534 (2005).
T. G. Cowling, Magnetohydrodynamics, Interscience, New York (1956).
G. Duvaut and J. L. Lions, “Inéquations en thermoélasticité et magnétohydrodynamique,” Arch. Ration. Mech. Anal., 46, 241–279 (1972).
W. M. Elsässer, “Magnetohydrodynamics,” Am. J. Phys., 23, 590 (1955).
W. M. Elsässer, “Magnetohydrodynamics,” Usp. Fiz. Nauk, 64, No. 3, 529–588 (1958).
A. T. Fomenko, Differential Geometry and Topology, Plenum Press, N.Y.–London (1987).
V. S. Gorbachev, S. R. Kel’ner, B. V. Somov, and A. S. Shvarts, “New topological approach to the problem of trigger for solar flares,” Astron. Zh., 65, 601–612 (1988).
V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, and S. Kh. Zinina, “Heteroclinic curves of Morse–Smale diffeomorphisms and separators in the plasma magnetic field,” Nelin. Dinam., 10, 427–438 (2014).
V. Grines, T. Medvedev, and O. Pochinka, Dynamical Systems on 2- and 3-Manifolds, Springer, Berlin (2016).
V. Grines, T. Medvedev, O. Pochinka, and E. Zhuzhoma, “On heteroclinic separators of magnetic fields in electrically conducting fluids,” Phys. D. Nonlin. Phenom., 294, 1–5 (2015).
V. Z. Grines and O. V. Pochinka, Introduction to Topological Classification of Cascades on Manifolds of Dimension Two and Three [in Russian], Moscow–Izhevsk (2011).
V. Z. Grines and O. V. Pochinka, “Morse–Smale cascades on 3-manifolds,” Russ. Math. Surv., 68, No. 1, 117–173 (2013).
V. Z. Grines and O. V. Pochinka, “Morse–Smale cascades on 3-manifolds,” Usp. Mat. Nauk, 68, No. 1, 129–188 (2013).
V. Grines and O. Pochinka, “Topological classification of global magnetic fields in the solar corona,” Dyn. Syst., 33, No. 3, 536–546 (2018).
V. Z. Grines, E. V. Zhuzhoma, and V. S. Medvedev, “New relations for flows and Morse–Smale diffeomorphisms,” Dokl. RAN, 382, No. 6, 730–733 (2002).
V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, and O. V. Pochinka, “Global attractor and repeller of Morse–Smale diffeomorphisms,” Tr. MIAN, 271, 111–133 (2010).
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge–N.Y. (1995).
A. Katok and B. Hasselblatt, Introduction to the Theory of Dynamical Systems [in Russian], Faktorial, Moscow (1999).
I. Klapper and L.-S. Young, “Rigorous bounds of the fast dynamo growth rate involving topological entropy,” Commun. Math. Phys., 173, 623–646 (1995).
L. D. Landau and E. M. Lifshits, Theoretical Physics in 10 Volumes. Vol. VIII. Continuum Electrodynamics [in Russian], Fizmatlit, Moscow (2005).
D. W. Longcope, “Topological and current ribbons: a model for current, reconnection anf flaring in a complex, evolving corona,” Solar Phys., 169, 91–121 (1996).
R. C. Maclean, C. Beveridge, G. Hornig, and E. R. Priest, “Coronal magnetic topologies in a spherical geometry, I. Two bipolar flux sources,” Solar Phys., 235, No. 1-2, 259–280 (2006).
R. Maclean, C. Beveridge, D. Longcope, D. Brown, and E. Priest, “A topological analysis of the magnetic breakout model for an eruptive solar flare,” Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci., 461, 2099 (2005).
R. Maclean, C. Beveridge, and E. Priest, “Coronal magnetic topologies in a spherical geometry, II. Four balanced flux sources,” Solar Phys., 238, 13–27 (2006).
R. C. Maclean and E. R. Priest, “Topological aspects of global magnetic field behaviour in the solar corona,” Solar Phys., 243, No. 2, 171–191 (2007).
H. K. Moffatt, Magnetic Field Generation in Electrically Conducting Fields, Cambridge University Press, Cambridge (1978).
H. K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids [Russian translation], Mir, Moscow (1980).
S. A. Molchanov, A. A. Ruzmaykin, and D. D. Sokolov, “Kinematic dynamo in random flow,” Usp. Fiz. Nauk, 145, 593–628 (1985).
M. M. Molodenskiy and S. I. Syrovatskiy, “Magnetic fields of active areas and their null points,” Astron. Zh., 54, 1293–1304 (1977).
Z. Nitecki, Differential Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, M.I.T. Press, Cambridge–London (1971).
A. V. Oreshina, I. V. Oreshina, and B. V. Somov, “Magnetic-topology evolution in NOAA AR 10501 on 2003 November 18,” Astron. Astrophys., 538, 138 (2012).
E. N. Parker, “Hydromagnetic dynamo models,” Astrophys. J., 122, 293–314 (1955).
H. Poincaré, “Sur les courbes définies par une équation différentielle, III,” J. Math. Pures Appl., 4, No. 1, 167–244 (1882).
E. R. Priest, Solar Magnetohydrodynamics, Springer, Dordrecht (1982).
E. Priest, T. Bungey, and V. Titov, “The 3D topology and interaction of complex magnetic flux systems,” Geophys. Astrophys. Fluid Dyn., 84, 127–163 (1997).
E. Priest and T. Forbes, Magnetic Reconnection: MHD Theory and Applications, Cambridge Univ. Press, New York (2000).
E. Priest and T. Forbes, Magnetic Reconnection: MHD Theory and Applications, FML, Moscow (2005).
E. Priest and C. Schriver, “Aspects of three-dimensional magnetic reconnection,” Solar Phys., 190, 1–24 (1999).
E. R. Priest and V. S. Titov, “Magnetic reconnection at three-dimensional null points,” Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 354, 2951–2992 (1996).
Shao Shu-Guang, Wang Shu, Xu Wen-Qing, and Ge. Yu-Li, “On the local C1,α solution of ideal magnetohydrodynamical equations,” Discrete Contin. Dyn. Syst., 37, No. 4, 2103–2118 (2007).
B. V. Somov, Plasma Astrophysics, Part II: Reconnection and Flares, Springer, N.Y. (2013).
S. Smale, “Diffeomorphisms with many periodic points,” Mathematica, 11, No. 4, 88–106 (1967).
S. Smale, “Differentiable dynamical systems,” Bull. Am. Math. Soc., 73, 741–817 (1967).
D. D. Sokolov, “Problems of magnetic dynamo,” Usp. Fiz. Nauk, 185, 643–648 (2015).
D. D. Sokolov, R. A. Stepanov, and P. G. Frik, “Dynamo: from astrophysic models to laboratory experiment,” Usp. Fiz. Nauk, 184, 313–335 (2014).
P. A. Sweet, “The production of high energy particles in solar flares,” Nuovo Cimento Suppl., 8, Ser. X, 188–196 (1958).
S. I. Syrovatskiy, “Magnetohydrodynamics,” Usp. Fiz. Nauk, 62, No. 7, 247–303 (1957).
S. I. Vaynshteyn and Ya. B. Zel’dovich, “On genesis of magnetic fields in astrophysics (Turbulent mechanisms “dynamo”),” Usp. Fiz. Nauk, 106, 431–457 (1972).
Ya. B. Zel’dovich and A. A. Ruzmaykin, “Hydromagnetic dynamo as a source of planetary, solar, and galactic magnetism,” Usp. Fiz. Nauk, 152, 263–284 (1987).
E. V. Zhuzhoma and N. V. Isaenkova, “On zero-measure solenoidal basic sets,” Mat. Sb., 202, No. 3, 47–68 (2011).
E. V. Zhuzhoma, N. V. Isaenkova, and V. S. Medvedev, “On topological structure of magnetic field of regions of the photosphere,” Nelin. Dinam., 13, No. 3, 399–412 (2017).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 63, No. 3, Differential and Functional Differential Equations, 2017.
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Grines, V.Z., Zhuzhoma, E.V. & Pochinka, O.V. Dynamical Systems and Topology of Magnetic Fields in Conducting Medium. J Math Sci 253, 676–691 (2021). https://doi.org/10.1007/s10958-021-05261-1
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DOI: https://doi.org/10.1007/s10958-021-05261-1