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Joint instability and abrupt nonlinear transitions in a differentially rotating plasma

Part of: Focus on Plasma Astrophysics

Published online by Cambridge University Press:  18 February 2019

A. Plummer ,
J. B. Marston Open the ORCID record for J. B. Marston [Opens in a new window]  and
S. M. Tobias
A. Plummer
Affiliation:
Department of Physics, Harvard University, Cambridge, MA 02138, USA
J. B. Marston*
Affiliation:
Department of Physics, Brown University, Providence, RI 02912, USA
S. M. Tobias
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
*
†Email address for correspondence: [email protected]
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Abstract

Global magnetohydrodynamic (MHD) instabilities are investigated in a computationally tractable two-dimensional model of the solar tachocline. The model’s differential rotation yields stability in the absence of a magnetic field, but if a magnetic field is present, a joint instability is observed. We analyse the nonlinear development of the instability via fully nonlinear direct numerical simulation, the generalized quasi-linear approximation (GQL) and direct statistical simulation (DSS) based upon low-order expansion in equal-time cumulants. As the magnetic diffusivity is decreased, the nonlinear development of the instability becomes more complicated until eventually a set of parameters is identified that produces a previously unidentified long-term cycle in which energy is transformed from kinetic energy to magnetic energy and back. We find that the periodic transitions, which mimic some aspects of solar variability – for example, the quasiperiodic seasonal exchange of energy between toroidal field and waves or eddies – are unable to be reproduced when eddy-scattering processes are excluded from the model.

Keywords

astrophysical plasmasplasma dynamicsplasma nonlinear phenomena
Type
Research Article
Information
Journal of Plasma Physics , Volume 85 , Issue 1 , February 2019 , 905850113
Copyright
© Cambridge University Press 2019 

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References

Allawala, A., Tobias, S. M. & Marston, J. B.2017 Dimensional reduction of direct statistical simulation. arXiv:1708.07805v1.Google Scholar
Bakas, N. A. & Ioannou, P. J. 2013 Emergence of large scale structure in barotropic $\unicode[STIX]{x1D6FD}$ -plane turbulence. Phys. Rev. Lett. 110 (22), 224501.Google Scholar
Bakas, N. A. & Ioannou, P. J. 2014 A theory for the emergence of coherent structures in beta-plane turbulence. J. Fluid Mech. 740, 312–341.Google Scholar
Balbus, S. A. & Hawley, J. F. 1991 A powerful local shear instability in weakly magnetized disks. I – Linear analysis. II – Nonlinear evolution. Astrophys. J. 376, 214–233.Google Scholar
Cally, P. S. 2001 Nonlinear evolution of 2D tachocline instabilities. Solar Phys. 199 (2), 231–249.Google Scholar
Cally, P. S., Dikpati, M. & Gilman, P. A. 2003 Clamshell and tipping instabilities in a two-dimensional magnetohydrodynamic tachocline. Astrophys. J. 582 (2), 1190.Google Scholar
Cally, P. S., Dikpati, M. & Gilman, P. A. 2008 Three-dimensional magneto-shear instabilities in the solar tachocline – II. Axisymmetric case. Mon. Not. R. Astron. Soc. 391, 891–900.Google Scholar
Chandrasekhar, S. 1960 The stability of non-dissipative Couette flow in hydromagnetics. Proc. Natl Acad. Sci. USA 46, 253–257.Google Scholar
Charbonneau, P., Christensen-Dalsgaard, J., Henning, R., Larsen, R. M., Schou, J., Thompson, M. J. & Tomczyk, S. 1999 Helioseismic constraints on the structure of the solar tachocline. Astrophys. J. 527 (1), 445–460.Google Scholar
Child, A., Hollerbach, R., Marston, B. & Tobias, S. 2016 Generalised quasilinear approximation of the helical magnetorotational instability. J. Plasma Phys. 82 (03), 905820302.Google Scholar
Constantinou, N. C. & Parker, J. B. 2018 Magnetic suppression of zonal flows on a beta plane. Astrophys. J. 863 (1), 1–13.Google Scholar
Cowley, S. C., Wilson, H., Hurricane, O. & Fong, B. 2003 Explosive instabilities: from solar flares to edge localized modes in tokamaks. Plasma Phys. Control. Fusion 45 (12A), A31.Google Scholar
Dikpati, M., Cally, P. S. & Gilman, P. A. 2004 Linear analysis and nonlinear evolution of two-dimensional global magnetohydrodynamic instabilities in a diffusive tachocline. Astrophys. J. 610 (1), 597.Google Scholar
Dikpati, M., Cally, P. S., McIntosh, S. W. & Heifetz, E. 2017 The origin of the ‘seasons’ in space weather. Sci. Rep. 7 (1), 14750.Google Scholar
Dikpati, M. & Gilman, P. A. 2001 Flux-transport dynamos with $\unicode[STIX]{x1D6FC}$ -effect from global instability of tachocline differential rotation: a solution for magnetic parity selection in the sun. Astrophys. J. 559, 428–442.Google Scholar
Dritschel, D. G., Qi, W. & Marston, J. B. 2015 On the late-time behaviour of a bounded, inviscid two-dimensional flow. J. Fluid Mech. 783, 1–22.Google Scholar
Furth, H. P., Killeen, J. & Rosenbluth, M. N. 1963 Finite-resistivity instabilities of a sheet pinch. Phys. Fluids 6, 459–484.Google Scholar
Gilman, P. A. & Dikpati, M. 2002 Analysis of instability of latitudinal differential rotation and toroidal field in the solar tachocline using a magnetohydrodynamic shallow-water model. I. Instability for broad toroidal field profiles. Astrophys. J. 576 (2), 1031–1047.Google Scholar
Gilman, P. A. & Fox, P. A. 1997 Joint instability of latitudinal differential rotation and toroidal magnetic fields below the solar convection zone. Astrophys. J. 484 (1), 439.Google Scholar
Herring, J. R. 1963 Investigation of problems in thermal convection. J. Atmos. Sci. 20 (4), 325–338.Google Scholar
Hollerbach, R. & Cally, P. S. 2009 Nonlinear evolution of axisymmetric twisted flux tubes in the solar tachocline. Solar Phys. 260, 251–260.Google Scholar
Malkus, W. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225 (1161), 196–212.Google Scholar
Marston, J. B., Chini, G. P. & Tobias, S. M. 2016 Generalized quasilinear approximation: application to zonal jets. Phys. Rev. Lett. 116 (21), 214501.Google Scholar
Marston, J. B., Qi, W. & Tobias, S. M. 2019 Direct statistical simulation of a jet. In Zonal Jets: Phenomenology, Genesis, and Physics, Cambridge University Press.Google Scholar
Miesch, M., Matthaeus, W., Brandenburg, A., Petrosyan, A., Pouquet, A., Cambon, C., Jenko, F., Uzdensky, D., Stone, J., Tobias, S. et al. 2015 Large-eddy simulations of magnetohydrodynamic turbulence in heliophysics and astrophysics. Space Sci. Rev. 194, 97–137.Google Scholar
Miesch, M. S. 2007 Sustained magnetoshear instabilities in the solar tachocline. Astrophys. J. 658 (2), L131–L134.Google Scholar
Ogilvie, G. I. & Pringle, J. E. 1996 The non-axisymmetric instability of a cylindrical shear flow containing an azimuthal magnetic field. Mon. Not. R. Astron. Soc. 279, 152–164.Google Scholar
Pitts, E. & Tayler, R. J. 1985 The adiabatic stability of stars containing magnetic fields. IV – The influence of rotation. Mon. Not. R. Astron. Soc. 216, 139–154.Google Scholar
Seshasayanan, K., Dallas, V. & Alexakis, A. 2017 The onset of turbulent rotating dynamos at the low magnetic prandtl number limit. J. Fluid Mech. 822, R3.Google Scholar
Spiegel, E. A. & Zahn, J.-P. 1992 The solar tachocline. Astron. Astrophys. 265, 106–114.Google Scholar
Spruit, H. C. 1999 Differential rotation and magnetic fields in stellar interiors. Astron. Astrophys. 349, 189–202.Google Scholar
Tayler, R. J. 1973 The adiabatic stability of stars containing magnetic fields-I. Toroidal fields. Mon. Not. R. Astron. Soc. 161, 365.Google Scholar
Tayler, R. J. 1980 The adiabatic stability of stars containing magnetic fields. IV – Mixed poloidal and toroidal fields. Mon. Not. R. Astron. Soc. 191, 151–163.Google Scholar
Taylor, J. B. 1968 Plasma containment and stability theory. Proc. R. Soc. Lond. A 304, 335–360.Google Scholar
Taylor, J. B. & Newton, S. L. 2015 Special topics in plasma confinement. J. Plasma Phys. 81, 205810501.Google Scholar
Tobias, S. M. 2005 The solar tachocline: formation, stability and its role in the solar dynamo. In Fluid Dynamics and Dynamos in Astrophysics and Geophysics (ed. Soward, A. M., Jones, C. A., Hughes, D. W. & Weiss, N. O.), p. 193. CRC Press.Google Scholar
Tobias, S. M., Dagon, K. & Marston, J. B. 2011 Astrophysical fluid dynamics via direct statistical simulation. Astrophys. J. 727 (2), 127.Google Scholar
Tobias, S. M. & Marston, J. B. 2013 Direct statistical simulation of out-of-equilibrium jets. Phys. Rev. Lett. 110 (10), 104502.Google Scholar
Tobias, S. M. & Marston, J. B. 2016 Three-dimensional rotating Couette flow via the generalised quasilinear approximation. J. Fluid Mech. 810, 412–428.Google Scholar
Vedenov, A. A., Velikhov, E. P. & Sagdeev, R. Z. 1962 Quasilinear theory of plasma oscillations. Nucl. Fusion Suppl. Part 2, 465–475.Google Scholar
Velikhov, E. P. 1959 Stability of an ideally conducting liquid flowing between rotating cylinders in a magnetic field. Zhur. Eksptl’. i Teoret. Fiz. 36.Google Scholar
Watson, M. 1980 Shear instability of differential rotation in stars. Geophys. Astrophys. Fluid Dyn. 16 (1), 285–298.Google Scholar
Wilks, T. M., Garofalo, A. M., Diamond, P. H., Guo, Z. B., Hughes, J. W., Burrell, K. H. & Chen, X. 2018 Scaling trends of the critical $E\times B$ shear for edge harmonic oscillation onset in DIII-D quiescent H-mode plasmas. Nucl. Fusion 58, 112002.Google Scholar
Williams, P. D. 2009 A proposed modification to the Robert–Asselin time filter. Mon. Weath. Rev. 137 (8), 2538–2546.Google Scholar