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Migrating Dynamo Waves and Consequences for Stellar Current Sheets

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Abstract

We study the relation between stellar dynamo-wave propagation and the structure of the stellar magnetic field. Modeling dynamo waves by the well-known Parker migratory dynamo, we vary the intensity of dynamo drivers in order to obtain activity-wave propagation toward the Equator (as in the solar-activity cycle) or towards the Poles. We match the magnetic field in the dynamo-active shell with that in the surrounding stellar material, using a simple dissipative magneto-hydrodynamic system for the transition region. Introducing a weak asymmetry between the stellar hemispheres, we study phase shifts of the dipole, quadrupole, and octupole magnetic components at various distances from the star to demonstrate that a several-percent asymmetry in dynamo drivers is sufficient to obtain a realistic relation between solar dipole and quadrupole moments. We study the behavior of the stellar current sheets and show that for the poleward-propagating activity it is substantially different from solar ones. In particular, we demonstrate conditions in which the conical current sheets propagate opposite to the solar directions.

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References

  • Akasofu, S.-I.: 1978, The interaction between a magnetized plasma flow and a magnetized celestial body: a review of magnetospheric studies. Space Sci. Rev. 21, 489. DOI. ADS.

    Article  ADS  Google Scholar 

  • Axford, W.I., Hines, C.O.: 1961, A unifying theory of high-latitude geophysical phenomena and geomagnetic storms. Can. J. Phys. 39, 1433. DOI. ADS.

    Article  MathSciNet  ADS  Google Scholar 

  • Babcock, H.W.: 1949, Three magnetic variable stars showing reversal of polarity. Publ. Astron. Soc. Pac. 61, 226. DOI. ADS.

    Article  ADS  Google Scholar 

  • Babcock, H.W.: 1954, The magnetic variable HD 188041. Astrophys. J. 120, 66. DOI. ADS.

    Article  ADS  Google Scholar 

  • Babcock, H.W.: 1956, The magnetic variable HD 71866. Astrophys. J. 124, 489. DOI. ADS.

    Article  ADS  Google Scholar 

  • Babcock, H.W.: 1958, A catalog of magnetic stars. Astrophys. J. Suppl. Ser. 3, 141. DOI. ADS.

    Article  ADS  Google Scholar 

  • Bagenal, F.: 2000, In: Murdin, P. (ed.) Planetary Magnetospheres, 2322. DOI. ADS.

    Chapter  Google Scholar 

  • Balogh, A., Erdõs, G.: 2013, The heliospheric magnetic field. Space Sci. Rev. 176, 177. DOI. ADS.

    Article  ADS  Google Scholar 

  • Balogh, A., Jokipii, J.R.: 2009, The heliospheric magnetic field and its extension to the inner heliosheath. Space Sci. Rev. 143, 85. DOI. ADS.

    Article  ADS  Google Scholar 

  • Bavassano, B., Woo, R., Bruno, R.: 1997, Heliospheric plasma sheet and coronal streamers. Geophys. Res. Lett. 24, 1655. DOI. ADS.

    Article  ADS  Google Scholar 

  • Bilous, A.V., Watts, A.L., Harding, A.K., Riley, T.E., Arzoumanian, Z., Bogdanov, S., Gendreau, K.C., Ray, P.S., Guillot, S., Ho, W.C.G., Chakrabarty, D.: 2019, A NICER view of PSR J0030+0451: evidence for a global-scale multipolar magnetic field. Astrophys. J. Lett. 887, L23. DOI. ADS.

    Article  ADS  Google Scholar 

  • Bonanno, A.: 2016, Stellar dynamo models with prominent surface toroidal fields. Astrophys. J. Lett. 833, L22.

    Article  ADS  Google Scholar 

  • Bonanno, A., Urpin, V.: 2008a, Non-axisymmetric instability of axisymmetric magnetic fields. Astron. Astrophys. 488, 1.

    Article  MATH  ADS  Google Scholar 

  • Bonanno, A., Urpin, V.: 2008b, Stability of magnetic configurations containing the toroidal and axial fields. Astron. Astrophys. 477, 35.

    Article  MATH  ADS  Google Scholar 

  • Borra, E.F., Deschatelets, D.: 2015, Measurements of stellar magnetic fields using autocorrelation of spectra. Astron. J. 150, 146. DOI. ADS.

    Article  ADS  Google Scholar 

  • Cauley, P.W., Shkolnik, E.L., Llama, J., Lanza, A.F.: 2019, Magnetic field strengths of hot Jupiters from signals of star-planet interactions. Nat. Astron. 3, 1128. DOI. ADS.

    Article  ADS  Google Scholar 

  • Christensen, U.R.: 2019, Planetary magnetic fields and dynamos. In: Read, P., et al. (eds.) Oxford Research Encyclopedia of Planetary Science, 31. DOI. ADS.

  • Crooker, N.U., Burton, M.E., Siscoe, G.L., Kahler, S.W., Gosling, J.T., Smith, E.J.: 1996, Solar wind streamer belt structure. J. Geophys. Res.: Space Phys. 101, 24331. DOI. ADS.

    Article  ADS  Google Scholar 

  • de Lima, R.C., Coelho, J.G., Pereira, J.P., Rodrigues, C.V., Rueda, J.A.: 2020, Evidence for a multipolar magnetic field in sgr j1745-2900 from x-ray light-curve analysis. Astrophys. J. 889, 165.

    Article  ADS  Google Scholar 

  • Dmitriev, A.V., Suvorova, A.V., Veselovsky, I.S., Zeldovich, M.A.: 2000, Coronal imprints in the heliospheric plasma and magnetic fields at the Earth’s orbit during the last three solar minima. Adv. Space Res. 25, 1965. DOI. ADS.

    Article  ADS  Google Scholar 

  • Eselevich, V.G., Fainshtein, V.G., Rudenko, G.V.: 1999, Study of the structure of streamer belts and chains in the solar corona. Solar Phys. 188, 277. DOI. ADS.

    Article  ADS  Google Scholar 

  • Gregory, S.G.: 2011, Equation of the field lines of an axisymmetric multipole with a source surface. Am. J. Phys. 79, 461. DOI. ADS.

    Article  ADS  Google Scholar 

  • Israelevich, P.L., Gombosi, T.I., Ershkovich, A.I., Hansen, K.C., Groth, C.P.T., DeZeeuw, D.L., Powell, K.G.: 2001, MHD simulation of the three-dimensional structure of the heliospheric current sheet. Astron. Astrophys. 376, 288. DOI. ADS.

    Article  ADS  Google Scholar 

  • Jackman, C.M., Arridge, C.S., André, N., Bagenal, F., Birn, J., Freeman, M.P., Jia, X., Kidder, A., Milan, S.E., Radioti, A., Slavin, J.A., Vogt, M.F., Volwerk, M., Walsh, A.P.: 2014, Large-scale structure and dynamics of the magnetotails of Mercury, Earth, Jupiter and Saturn. Space Sci. Rev. 182, 85. DOI. ADS.

    Article  ADS  Google Scholar 

  • Kao, M.M., Hallinan, G., Pineda, J.S., Escala, I., Burgasser, A., Bourke, S., Stevenson, D.: 2016, Auroral radio emission from late L and T dwarfs: a new constraint on dynamo theory in the substellar regime. Astrophys. J. 818, 24. DOI. ADS.

    Article  ADS  Google Scholar 

  • Katushkina, O., Izmodenov, V., Koutroumpa, D., Quémerais, E., Jian, L.K.: 2019, Unexpected behavior of the solar wind mass flux during solar maxima: two peaks at middle heliolatitudes. Solar Phys. 294, 17. DOI. ADS.

    Article  ADS  Google Scholar 

  • Khabarova, O.V., Malova, H.V., Kislov, R.A., Zelenyi, L.M., Obridko, V.N., Kharshiladze, A.F., Tokumaru, M., Sokół, J.M., Grzedzielski, S., Fujiki, K.: 2017, High-latitude conic current sheets in the solar wind. Astrophys. J. 836, 108. DOI. ADS.

    Article  ADS  Google Scholar 

  • Kislov, R.A., Khabarova, O.V., Malova, H.V.: 2019, Quasi-stationary current sheets of the solar origin in the heliosphere. Astrophys. J. 875, 28.

    Article  ADS  Google Scholar 

  • Kislyakova, K.G., Holmström, M., Lammer, H., Odert, P., Khodachenko, M.L.: 2014, Magnetic moment and plasma environment of HD 209458b as determined from Ly\(\alpha\) observations. Science 346, 981. DOI. ADS.

    Article  ADS  Google Scholar 

  • Kitchatinov, L., Khlystova, A.: 2021, Dynamo model for North-South asymmetry of solar activity. Astrophys. J. 919, 36. DOI. ADS.

    Article  ADS  Google Scholar 

  • Kleeorin, Y., Safiullin, N., Kleeorin, N., Porshnev, S., Rogachevskii, I., Sokoloff, D.: 2016, The dynamics of wolf numbers based on nonlinear dynamos with magnetic helicity: comparisons with observations. Mon. Not. Roy. Astron. Soc. 460, 3960.

    Article  ADS  Google Scholar 

  • Kochukhov, O.: 2021, Magnetic fields of M dwarfs. Astron. Astrophys. Rev. 29, 1. DOI. ADS.

    Article  ADS  Google Scholar 

  • Krause, F., Räedler, K.-H.: 1980, Mean-Field Magnetohydrodynamics and Dynamo Theory. Pergamon, Oxford. ADS.

    Google Scholar 

  • Kuzanyan, K.M., Sokoloff, D.D.: 1995, A dynamo wave in an inhomogeneous medium. Geophys. Astrophys. Fluid Dyn. 81, 113. DOI. ADS.

    Article  MathSciNet  ADS  Google Scholar 

  • Levine, R.H., Schulz, M., Frazier, E.N.: 1982, Simulation of the magnetic structure of the inner heliosphere by means of a non-spherical source surface. Solar Phys. 77, 363. DOI. ADS.

    Article  ADS  Google Scholar 

  • Linsky, J.L., Schöller, M.: 2015, Observations of strong magnetic fields in nondegenerate stars. Space Sci. Rev. 191, 27. DOI. ADS.

    Article  ADS  Google Scholar 

  • Maiewski, E., Kislov, R., Khabarova, O., Malova, H., Popov, V.Y., Petrukovich, A., Zelenyi, L.: 2020, Magnetohydrodynamic modeling of the solar wind key parameters and current sheets in the heliosphere: radial and solar cycle evolution. Astrophys. J. 892, 12.

    Article  ADS  Google Scholar 

  • Makarov, V.I., Fatianov, M.P., Sivaraman, K.R.: 1983, Poleward migration of the magnetic neutral line and the reversal of the polar fields on the sun – part one – period 1945 – 1981. Solar Phys. 85, 215. DOI. ADS.

    Article  ADS  Google Scholar 

  • Manoharan, P.K.: 2012, Three-dimensional evolution of solar wind during solar cycles 22-24. Astrophys. J. 751, 128. DOI. ADS.

    Article  ADS  Google Scholar 

  • Monsignori Fossi, B.C., Noci, G., Poletto, G.: 1992, The Ulysses space mission. Nuovo Cim. C Geophys. Space Phys. C 15, 493. DOI. ADS.

    Article  ADS  Google Scholar 

  • Moss, D., Saar, S.H., Sokoloff, D.: 2008, What can we hope to know about the symmetry properties of stellar magnetic fields? Mon. Not. Roy. Astron. Soc. 388, 416. DOI. ADS.

    Article  ADS  Google Scholar 

  • Mursula, K., Hiltula, T.: 2004, Systematically asymmetric heliospheric magnetic field: evidence for a quadrupole mode and non-axisymmetry with polarity flip-flops. Solar Phys. 224, 133. DOI. ADS.

    Article  ADS  Google Scholar 

  • Neugebauer, M., Forsyth, R.J., Galvin, A.B., Harvey, K.L., Hoeksema, J.T., Lazarus, A.J., Lepping, R.P., Linker, J.A., Mikic, Z., Steinberg, J.T., von Steiger, R., Wang, Y.-M., Wimmer-Schweingruber, R.F.: 1998, Spatial structure of the solar wind and comparisons with solar data and models. J. Geophys. Res.: Space Phys. 103, 14587. DOI. ADS.

    Article  ADS  Google Scholar 

  • Obridko, V.N., Pipin, V.V., Sokoloff, D., Shibalova, A.S.: 2021, Solar large-scale magnetic field and cycle patterns in solar dynamo. Mon. Not. Roy. Astron. Soc. 504, 4990. DOI. ADS.

    Article  ADS  Google Scholar 

  • Parker, E.N.: 1955, Hydromagnetic dynamo models. Astrophys. J. 122, 293.

    Article  MathSciNet  ADS  Google Scholar 

  • Parker, E.N.: 1961, The stellar-wind regions. Astrophys. J. 134, 20. DOI. ADS.

    Article  ADS  Google Scholar 

  • Pétri, J.: 2020, Magnetic quadri-dipolar stars rotating in vacuum. Mon. Not. Roy. Astron. Soc. 499, 4445. DOI. ADS.

    Article  ADS  Google Scholar 

  • Petrukovich, A.A., Malova, H.V., Popov, V.Y., Maiewski, E.V., Izmodenov, V.V., Katushkina, O.A., Vinogradov, A.A., Riazantseva, M., Rakhmanova, L.S., Podladchikova, T.V., Zastenker, G.N., Yermolaev, Y.I., Lodkina, I.G., Chesalin, L.S.: 2020, Modern view of the solar wind from micro to macro scales. Phys. Usp. 63, 801. DOI. ADS.

    Article  ADS  Google Scholar 

  • Pinto, R., Brun, A., Rouillard, A.: 2016, Flux-tube geometry and solar wind speed during an activity cycle. Astron. Astrophys. 592, A65.

    Article  ADS  Google Scholar 

  • Pinto, R.F., Brun, A.S., Jouve, L., Grappin, R.: 2011, Coupling the solar dynamo and the corona: wind properties, mass, and momentum losses during an activity cycle. Astrophys. J. 737, 72.

    Article  ADS  Google Scholar 

  • Pinto, R.F., Grappin, R., Velli, M., Verdini, A.: 2013, Coupling the solar surface and the corona: Coronal rotation, Alfvén wave-driven polar plumes. In: Zank, G.P., Borovsky, J., Bruno, R., Cirtain, J., Cranmer, S., Elliott, H., Giacalone, J., Gonzalez, W., Li, G., Marsch, E., Moebius, E., Pogorelov, N., Spann, J., Verkhoglyadova, O. (eds.) Solar Wind 13, CP-1539, American Institute of Physics, Melville, 58.

    Google Scholar 

  • Réville, V., Brun, A.S., Matt, S.P., Strugarek, A., Pinto, R.F.: 2015, The effect of magnetic topology on thermally driven wind: toward a general formulation of the braking law. Astrophys. J. 798, 116. DOI. ADS.

    Article  ADS  Google Scholar 

  • Richardson, J.D., Burlaga, L.F.: 2013, The solar wind in the outer heliosphere and heliosheath. Space Sci. Rev. 176, 217. DOI. ADS.

    Article  ADS  Google Scholar 

  • Robbrecht, E., Wang, Y.-M.: 2012, Determining the North-South displacement of the heliospheric current sheet from coronal streamer observations. Astrophys. J. 755, 135. DOI. ADS.

    Article  ADS  Google Scholar 

  • Roberts, D.A., Keiter, P.A., Goldstein, M.L.: 2005, Origin and dynamics of the heliospheric streamer belt and current sheet. J. Geophys. Res.: Space Phys. 110, A06102. DOI. ADS.

    Article  ADS  Google Scholar 

  • Rosenberg, R.L., Coleman, J.P.J.: 1980, Solar cycle-dependent North-South field configurations observed in solar wind interaction regions. J. Geophys. Res.: Space Phys. 85, 3021. DOI. ADS.

    Article  ADS  Google Scholar 

  • Schatten, K.H.: 1969, Coronal structure at the solar eclipse of September 22, 1968. Nature 222, 652. DOI. ADS.

    Article  ADS  Google Scholar 

  • See, V., Jardine, M., Vidotto, A.A., Donati, J.-F., Boro Saikia, S., Bouvier, J., Fares, R., Folsom, C.P., Gregory, S.G., Hussain, G., Jeffers, S.V., Marsden, S.C., Morin, J., Moutou, C., do Nascimento, J.D., Petit, P., Waite, I.A.: 2016, The connection between stellar activity cycles and magnetic field topology. Mon. Not. Roy. Astron. Soc. 462, 4442. DOI. ADS.

    Article  ADS  Google Scholar 

  • Smith, E.J.: 2001, The heliospheric current sheet. J. Geophys. Res.: Space Phys. 106, 15819. DOI. ADS.

    Article  ADS  Google Scholar 

  • Smith, E.J., Page, D.E., Wenzel, K.-P.: 1991, Ulysses: a journey above the Sun’s poles. Earth Space 4, 10. ADS.

    ADS  Google Scholar 

  • Smith, E.J., Balogh, A., Burton, M.E., Erdös, G., Forsyth, R.J.: 1995, Results of the Ulysses fast latitude scan: magnetic field observations. Geophys. Res. Lett. 22, 3325. DOI. ADS.

    Article  ADS  Google Scholar 

  • Sokoloff, D., Malova, H., Yushkov, E.: 2020, Symmetries of magnetic fields driven by spherical dynamos of exoplanets and their host stars. Symmetry 12, 12. DOI.

    Article  Google Scholar 

  • Svalgaard, L., Wilcox, J.M.: 1976, Structure of the extended solar magnetic field and the sunspot cycle variation in cosmic ray intensity. Nature 262, 766. DOI. ADS.

    Article  ADS  Google Scholar 

  • Telloni, D., Andretta, V., Antonucci, E., Bemporad, A., Capuano, G.E., Fineschi, S., Giordano, S., Habbal, S., Perrone, D., Pinto, R.F., et al.: 2021, Exploring the solar wind from its source on the corona into the inner heliosphere during the first Solar Orbiter–Parker solar probe quadrature. Astrophys. J. Lett. 920, L14.

    Article  ADS  Google Scholar 

  • Tokumaru, M., Fujiki, K., Iju, T.: 2015, North-South asymmetry in global distribution of the solar wind speed during 1985-2013. J. Geophys. Res.: Space Phys. 120, 3283. DOI. ADS.

    Article  ADS  Google Scholar 

  • Usmanov, A.V.: 1993, Interplanetary magnetic field structure and solar wind parameters as inferred from solar magnetic field observations and by using a numerical 2-D MHD model. Solar Phys. 143, 345. DOI. ADS.

    Article  ADS  Google Scholar 

  • Usmanov, A.V., Goldstein, M.L., Matthaeus, W.H.: 2014, Three-fluid, three-dimensional magnetohydrodynamic solar wind model with Eddy viscosity and turbulent resistivity. Astrophys. J. 788, 43. DOI. ADS.

    Article  ADS  Google Scholar 

  • Valdés-Galicia, J.F., Otaola, J.A.: 1996, The latitudinal structure of the heliospheric current sheet during solar activity cycles 21 and 22. J. Geophys. Res.: Space Phys. 101, 2475. DOI. ADS.

    Article  ADS  Google Scholar 

  • Vekstein, G., Priest, E.R., Amari, T.: 1991, Formation of current sheets in force-free magnetic fields. Astron. Astrophys. 243, 492. ADS.

    ADS  Google Scholar 

  • Veselovsky, I.S., Lukashenko, A.T.: 2012, Model of the magnetic field in the inner heliosphere with regard to radial field strength leveling in the solar corona. Solar Syst. Res. 46, 149. DOI. ADS.

    Article  ADS  Google Scholar 

  • Viviani, M., Warnecke, J., Käpylä, M.J., Käpylä, P.J., Olspert, N., Cole-Kodikara, E.M., Lehtinen, J.J., Brandenburg, A.: 2018, Transition from axi-to nonaxisymmetric dynamo modes in spherical convection models of solar-like stars. Astron. Astrophys. 616, A160.

    Article  ADS  Google Scholar 

  • Wang, Y.-M.: 1996, Nonradial coronal streamers. Astrophys. J. Lett. 456, L119. DOI. ADS.

    Article  ADS  Google Scholar 

  • Wang, Y.-M.: 2014, Solar cycle variation of the Sun’s low-order magnetic multipoles: heliospheric consequences. Space Sci. Rev. 186, 387. DOI. ADS.

    Article  ADS  Google Scholar 

  • Wang, Y.-M., Young, P.R., Muglach, K.: 2014, Evidence for two separate heliospheric current sheets of cylindrical shape during mid-2012. Astrophys. J. 780, 103. DOI. ADS.

    Article  ADS  Google Scholar 

  • Wang, Y.-M., Sheeley, N.R., Socker, D.G., Howard, R.A., Rich, N.B.: 2000, The dynamical nature of coronal streamers. J. Geophys. Res.: Space Phys. 105, 25133. DOI. ADS.

    Article  ADS  Google Scholar 

  • Warnecke, J., Brandenburg, A., Mitra, D.: 2011, Dynamo-driven plasmoid ejections above a spherical surface. Astron. Astrophys. 534, A11.

    Article  MATH  ADS  Google Scholar 

  • Warnecke, J., Käpylä, P.J., Käpylä, M.J., Brandenburg, A.: 2014, On the cause of solar-like equatorward migration in global convective dynamo simulations. Astrophys. J. Lett. 796, L12.

    Article  ADS  Google Scholar 

  • Warnecke, J., Käpylä, P.J., Käpylä, M.J., Brandenburg, A.: 2016, Influence of a coronal envelope as a free boundary to global convective dynamo simulations. Astron. Astrophys. 596, A115.

    Article  ADS  Google Scholar 

  • Woo, R., Armstrong, J.W., Gazis, P.R.: 1995, Doppler scintillation measurements of the heliospheric current sheet and coronal streamers close to the Sun. Space Sci. Rev. 72, 223. DOI. ADS.

    Article  ADS  Google Scholar 

  • Zeldovich, I.B., Ruzmaikin, A.A., Sokoloff, D.D.: 1983, Magnetic Fields in Astrophysics, The Fluid Mechanics of Astrophysics and Geophysics, 3. Gordon and Breach, New York. ADS.

    Google Scholar 

  • Zelenyi, L., Malova, H., Grigorenko, E., Popov, V., Delcourt, D.: 2019, Current sheets in planetary magnetospheres. Plasma Phys. Control. Fusion 61, 054002. DOI. ADS.

    Article  ADS  Google Scholar 

  • Zhilkin, A.G., Bisikalo, D.V.: 2019, On possible types of magnetospheres of hot Jupiters. Astron. Rep. 63, 550. DOI. ADS.

    Article  ADS  Google Scholar 

  • Zhukov, A.N., Saez, F., Lamy, P., Llebaria, A., Stenborg, G.: 2008, The origin of polar streamers in the solar corona. Astrophys. J. 680, 1532. DOI. ADS.

    Article  ADS  Google Scholar 

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Acknowledgments

E. Maiewski, H. Malova, V. Popov, and D. Sokoloff acknowledge the support of the Ministry of Science and Higher Education of the Russian Federation under the grant N0075-15-2020-780 (N13.1902.21.0039). The Ulysses Final Archive is available at the website ufa.esac.esa.int/ufa/#data, where the directory VHM-FGM contains the Ulysses prime resolution data of the magnetic field. The Ulysses Orbital Information (1990 – 2009) is available as follows: www.cosmos.esa.int/web/ulysses/orbit. E. Yushkov performed dynamo simulations with the financial support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement N0075-15-2019-162.

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Maiewski, E., Malova, H., Popov, V. et al. Migrating Dynamo Waves and Consequences for Stellar Current Sheets. Sol Phys 297, 150 (2022). https://doi.org/10.1007/s11207-022-02085-3

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