The Parker problem: existence of smooth force-free fields and coronal heating
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REVIEW ARTICLE
The Parker problem: existence of smooth force-free fields and coronal heating David I. Pontin1,2
•
Gunnar Hornig2
Received: 21 January 2020 / Accepted: 8 July 2020 Ó The Author(s) 2020
Abstract Parker (Astrophys J 174:499, 1972) put forward a hypothesis regarding the fundamental nature of equilibrium magnetic fields in astrophysical plasmas. He proposed that if an equilibrium magnetic field is subjected to an arbitrary, small perturbation, then—under ideal plasma dynamics—the resulting magnetic field will in general not relax towards a smooth equilibrium, but rather, towards a state containing tangential magnetic field discontinuities. Even at astrophysical plasma parameters, as the singular state is approached dissipation must eventually become important, leading to the onset of rapid magnetic reconnection and energy dissipation. This topological dissipation mechanism remains a matter of debate, and is a key ingredient in the nanoflare model for coronal heating. We review the various theoretical and computational approaches that have sought to prove or disprove Parker’s hypothesis. We describe the hypothesis in the context of coronal heating, and discuss different approaches that have been taken to investigating whether braiding of magnetic field lines is responsible for maintaining the observed coronal temperatures. We discuss the many advances that have been made, and highlight outstanding open questions. Keywords Sun: corona Sun: magnetic fields Magnetic reconnection Magnetohydrodynamics
& David I. Pontin [email protected] 1
School of Mathematical and Physics Sciences, University of Newcastle, Callaghan, NSW 2308, Australia
2
School of Science and Engineering, University of Dundee, Dundee DD1 4HN, UK
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D. I. Pontin, G. Hornig
Contents 1 2 3 4
Introduction............................................................................................................................... Statement of the Parker hypothesis ......................................................................................... Energy injection ....................................................................................................................... Existence and structure of magnetohydrostatic equilibria: theory ........................................... 4.1 Force-free equilibria: general properties........................................................................... 4.2 Linear perturbative analyses.............................................................................................. 4.3 Arguments against the existence of smooth equilibria .................................................... 4.3.1 Parker’s optical analogy.................................................................................... 4.3.2 Work of Low et al. on ‘‘topologically untwisted’’ fields ................................ 4.3.3 Uniqueness of equilibria in reduced MHD ...................................................... 4.4
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