Abstract
A flux rope is a domain where a twisted magnetic field [B] is concentrated; it can be described as the core of a singularity of the outer field or the outer vector potential [A] (Kleman and Robbins in Solar Phys. 289, 1173, 2014). This latter case, occurring when the outer field is vanishing, is mathematically analysed for a straight infinite rope. Concepts from condensed-matter physics defect theory are used: the flux [Φ], measured as ∮ C A⋅ds along any loop [C] surrounding the rope, is a topological constant of the theory. A flux rope with a small outer magnetic field can be treated as a perturbation of the above. This theoretical framework allows for the use of classical configurations inside the core, e.g. the linear force-free field (LFFF) Lundquist model or the nonlinear (NLFFF) Gold–Hoyle model, but restricts the number of stable solutions: they are quantised into strata of increasing energies (an infinite number of strata in the first case, only one stratum in the second case); each stratum is defined by a number 2πζ=b/r 0, where b is the periodicity along the axis of the rope and r 0 is its radius, and the rope is made of a continuous set of stable states. We also analyse the merging of identical flux ropes (belonging to the same stratum), with conservation of the relative magnetic helicity: they merge into a unique rope of the first stratum, with a considerable release of energy.
These results might apply to ropes that nucleate in the convection zone and the photosphere, where the magnetic field outside the ropes is weak, and less accurately to the magnetic clouds (MCs) into which they evolve after coronal mass ejections (CMEs) are triggered. However, the lowest LFFF stratum and the unique NLFFF stratum, which numerically come close to each other in this analysis, match the spacecraft data remarkably well.
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Notes
A right-handed helix carries a positive pitch, a left-handed helix a negative pitch, by convention; this choice implies the use of a right-handed coordinate frame.
We denote any quantity f evaluated at a critical point as \(f|_{r_{0}}\). For example, \(\frac {\partial D}{\partial\eta}|_{r_{0}}=0\).
Use in sequence Equation 11.3.31 (μ=−ν=1) and Equation 9.1.27 (ν=1) from Abramowicz and Stegun (1970).
We give in Equation (29) a closed expression of the relative helicity for a cylindrical tube, which is made possible by its quantised stratum character.
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Acknowledgements
I am indebted to J.M. Robbins for long-lasting fruitful exchanges on the space physics of electromagnetism. I thank P. Démoulin for pointing out to me the possible relevance of the flux tube singularity theory to magnetic clouds, and for useful correspondence. Both made relevant remarks on the text. I am grateful to J.-L. Le Mouël, who read the manuscript with the utmost care and with whom I have had fruitful discussions. I am also very grateful to the anonymous referee for very valuable remarks. This is IPGP contribution No. 3382.
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Kleman, M. Flux Ropes as Singularities of the Vector Potential. Sol Phys 290, 707–725 (2015). https://doi.org/10.1007/s11207-015-0647-6
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DOI: https://doi.org/10.1007/s11207-015-0647-6