Zhuzhoma, Medvedev, Dumin and Somov (Physica D 436 (2022) 133320) used Morse–Smale theory to derive universal topological inequalities constraining the numbers of magnetic null points and heteroclinic separators in the branching ("anemone") configurations of coronal magnetic fields modeled by effective point-like magnetic charges. Their theorems apply only to positively or negatively unbalanced charge groups, are purely topological (position-independent), and rely on the potential (current-free) point-charge field satisfying the Morse–Smale hypotheses. Two extensions remain open beyond these hypotheses: (i) counting inequalities and separator-existence statements for balanced (zero net flux) charge groups; and (ii) position-dependent criteria or constructive location bounds relating null and separator positions to the geometric arrangement and magnitudes of the charges. No resolution of either direction is known in the literature, and the adjacent equilibrium-counting problem for balanced point charges remains open. The problem is open.

Background

A solar flare is a sudden release of magnetic energy in the Sun's atmosphere, believed to be driven by magnetic reconnection, the reorganization of magnetic field lines that occurs in current sheets. Three-dimensional reconnection tends to be concentrated near magnetic null points, locations where the magnetic field B\mathbf{B} vanishes, and along separators, field lines that connect two null points and form the transversal intersection of the fan surfaces (the two-dimensional separatrix surfaces emanating from nulls). Because the solar corona is nearly current-free outside reconnection regions, the magnetic field there is often approximated as potential (∇×B=0\nabla\times\mathbf{B}=0) and generated by effective point-like magnetic charges: monopole-like sources and sinks representing the footpoints of concentrated magnetic flux tubes at the photosphere. "Anemone" flares are branching configurations in which multiple null points are connected to each other and to the magnetic sources by separators, producing several emissive loops or jets from one reconnection region.

The topological approach to flares asks what constraints the arrangement of sources imposes on the numbers and locations of nulls and separators, since these determine where reconnection can occur. Gorbachev, Kel'ner, Somov and Shvarts (Soviet Astronomy 32 (1988) 308) derived position-dependent criteria for the existence of null points for four arbitrarily located charges and discovered the "topological trigger": tiny source displacements that create a new null point high above the source plane. Longcope and collaborators developed a general theory of connectivity and current sheets anchored to point sources using Euler-characteristic (Poincaré–Hopf) counting (Longcope 2001; Longcope and Klapper 2002, ApJ 579:468), and Grines, Medvedev, Pochinka and Zhuzhoma (Physica D 294 (2015) 1) proved separator-existence theorems for special plasma geometries. Observationally, Dumin and Somov (Solar Physics 295 (2020) 92) showed that chromospheric anemone microflares with more than four footpoints require clusters of nulls, and often develop in regions of unbalanced magnetic polarity.

Zhuzhoma, Medvedev, Dumin and Somov (2022) imported Morse–Smale theory from dynamical systems. A Morse–Smale vector field is one whose equilibrium states are all hyperbolic (linearization has no eigenvalues on the imaginary axis), whose stable and unstable manifolds intersect transversally, and whose trajectories have no periodic orbits, so every orbit limits to equilibria. Treating the magnetic field lines as trajectories (the path variable playing the role of time), charges and nulls become equilibria: a positive charge is a source, a negative charge a sink, and a generic null a saddle whose Morse index (dimension of its unstable manifold) is 1 ("positive" null) or 2 ("negative" null). A group of charges is called positively (negatively) unbalanced if it can be enclosed in a ball whose boundary the field crosses outward (inward) everywhere. Gluing a compensating sink ball to the source region yields a Morse–Smale field on the three-sphere S3S^3, where the Poincaré–Hopf relation 1+N−−S++S−−N+=01 + N^- - S^+ + S^- - N^+ = 0 holds (N±N^\pm = numbers of positive/negative charges, S±S^\pm = numbers of positive/negative nulls). Theorem 1 of that paper: a positively unbalanced group with l≥2l \ge 2 positive charges (and any number of negative charges) contains at least l−1l-1 negative nulls; if exactly l−1l-1 nulls exist, all are negative, no separators exist, and the field structure is unique up to topological equivalence. Corollaries give the mirrored statements for negatively unbalanced groups and dominant-charge configurations, and configurations with prescribed null counts but no separators are constructed. These are the first counting theorems for an arbitrary number of charges, but they hold only under three restrictive hypotheses, and the paper's own conclusions acknowledge that the formulas (unlike the four-charge results of Gorbachev et al.) specify no relation between charge positions and null locations, and that the unbalanced-group requirement narrows applicability, while the entire derivation sits inside the potential point-charge Morse–Smale idealization described in the paper's Section 1.1.

Problem Statement

Extend the Morse–Smale-based topological quantification of null points and separators in point-charge models of anemone-type solar magnetic configurations beyond the hypotheses under which it is currently proven, in the two directions identified above:

(1) Balanced (zero net flux) groups. Prove (or disprove) analogues of the counting inequalities and separator-existence statements for groups of effective point magnetic charges with zero net flux, where the unbalanced-group hypothesis fails and the current continuation-to-S3S^3 construction with a single compensating source or sink is unavailable: what are the sharp lower bounds on the numbers of positive and negative nulls and on separator existence for such groups, as functions of the numbers and types of charges?

(2) Position-dependent criteria. Derive relations between the geometric positions and magnitudes of the charges and the existence and spatial location of the null points and separators for an arbitrary number of irregularly placed charges, extending the position-dependent four-charge criteria of Gorbachev et al. (1988) — including constructive criteria or bounds locating nulls and separators in space from the charge data, rather than only position-independent counts.

Both directions are required for a complete resolution, although each is an independently attackable sub-question. These two extensions share one scientific objective: a topological theory of null-point and separator counts and locations valid for the charge configurations actually realized in complex solar active regions, which is what the 2022 results — restricted to unbalanced groups and to position-independent statements within the potential point-charge Morse–Smale setting — do not yet provide.

References

  1. E. V. Zhuzhoma, V. S. Medvedev, Y. V. Dumin, B. V. Somov, Topological quantification of the "anemone" (branching) solar flares, Physica D 436 (2022) 133320, DOI 10.1016/j.physd.2022.133320, https://doi.org/10.1016/j.physd.2022.133320 (arXiv:2111.06730)
  2. Y. V. Dumin, B. V. Somov, Unipolar solar flares as a manifestation of 'topological' magnetic reconnection, Monthly Notices of the Royal Astronomical Society: Letters 528 (1) (2023) L15–L19, DOI 10.1093/mnrasl/slad162, https://doi.org/10.1093/mnrasl/slad162 (arXiv:2309.16417)
  3. Y. V. Dumin, B. V. Somov, Topological ignition of the stealth coronal mass ejections (2025), arXiv:2504.05876, https://arxiv.org/abs/2504.05876
  4. Y. V. Dumin, B. V. Somov, New types of the chromospheric anemone microflares: case study, Solar Physics 295 (2020) 92, DOI 10.1007/s11207-020-01662-8, https://doi.org/10.1007/s11207-020-01662-8
  5. V. S. Gorbachev, S. R. Kel'ner, B. V. Somov, A. S. Shvarts, A new topological approach to the question of the trigger for solar flares, Soviet Astronomy 32 (1988) 308, NASA ADS bibcode 1988SvA....32..308G, https://ui.adsabs.harvard.edu/abs/1988SvA....32..308G (no DOI registered)
  6. D. W. Longcope, I. Klapper, A general theory of connectivity and current sheets in coronal magnetic fields anchored to discrete sources, The Astrophysical Journal 579 (2002) 468–481, DOI 10.1086/342750, https://doi.org/10.1086/342750
  7. D. W. Longcope, Separator current sheets: generic features in minimum-energy magnetic fields subject to flux constraints, Physics of Plasmas 8 (2001) 5277–5290, DOI 10.1063/1.1418431, https://doi.org/10.1063/1.1418431
  8. V. Grines, T. Medvedev, O. Pochinka, E. Zhuzhoma, On heteroclinic separators of magnetic fields in electrically conducting fluids, Physica D 294 (2015) 1–5, DOI 10.1016/j.physd.2014.11.004, https://doi.org/10.1016/j.physd.2014.11.004
  9. H. Edelsbrunner, C. Fillmore, G. Oliveira, Counting equilibria of the electrostatic potential (2025), arXiv:2501.05315, https://arxiv.org/abs/2501.05315
  10. P. Arathoon, G. Ball, M. D. Kvalheim, The Maxwell conjecture is false (2026), arXiv:2607.27197, https://arxiv.org/abs/2607.27197
  11. R. L. Edgar, S. Régnier, Properties of magnetic null points associated with X-class flares during solar cycle 24, Monthly Notices of the Royal Astronomical Society 532 (1) (2024) 755–762, DOI 10.1093/mnras/stae1470, https://doi.org/10.1093/mnras/stae1470