In the 'topological' (nonlocal) model of magnetic reconnection, a coronal magnetic null point is created by the superposed influence of remote photospheric sources rather than by local currents, and tiny displacements of those sources near a topologically degenerate configuration can launch a null point racing along a separator, igniting reconnection — the 'topological trigger' proposed by the Gorbachev–Somov school. This mechanism naturally explains the recently identified 'unipolar' solar flares, whose flaring arcs connect footpoints of the same magnetic polarity. However, the instability criteria that identify when a configuration is topologically unstable have been derived only for systems of at most four point-like sources in the potential-field approximation. The open problem is threefold: (i) generalize the topological instability criteria to arbitrary numbers of sources and to non-potential (force-free) fields; (ii) quantitatively determine the energy source of the unipolar microflares — direct energy input from photospheric source displacement versus release of free energy stored in a slightly non-potential field; and (iii) test the model observationally through refined identification of unipolar flares, e.g. in the He I 10830 Å chromospheric line with sub-0.2-arcsec resolution.

Background

Solar flares — sudden releases of magnetic energy in the Sun's atmosphere — are powered by magnetic reconnection, the reconfiguration of magnetic field lines in a highly conducting plasma. To understand where reconnection starts, the solar corona is often modeled in the potential-field approximation: the field above the photosphere is treated as a vacuum field generated by discrete point-like 'magnetic charges' (sources) placed on the photospheric plane, so that the field is a superposition of Coulomb-like contributions. In such a model the qualitative structure of the field is captured by its magnetic skeleton: magnetic null points (where the field vanishes), separatrix surfaces of field lines bounding distinct flux domains, and separators — the field lines lying at the intersection of two separatrices, where current sheets and reconnection preferentially occur.

A distinctive idea of the Gorbachev–Somov school (Gorbachev & Somov 1988; Somov 2008) is the 'topological trigger': for special arrangements of the photospheric sources, a tiny displacement of one source causes a topological bifurcation — a pair of null points is created or annihilated and the magnetic skeleton changes connectivity — and the newly formed null point moves rapidly along a separator, igniting fast reconnection over a large spatial scale without any local current system. The trigger conditions are formulated using index-theoretic counting arguments: the Euler–Poincaré index of a null point (the winding number of the magnetic-field direction along a small closed loop around the null, counted positive or negative according to the sense of rotation) constrains, via the Poincaré–Hopf theorem relating the sum of the indices of the equilibria of a vector field to the Euler characteristic of the underlying surface, how many nulls can exist and how they can appear or disappear when the sources move.

This theory has been fully worked out only for configurations of no more than four point sources in the potential-field approximation (Brown & Priest 1999 on topological bifurcations of three-dimensional fields; Beveridge, Priest & Brown 2002 mapping the topologies of two bipolar regions, i.e. four sources; Longcope 2005 reviewing topological methods). For many sources, Inverarity & Priest (1999) studied null-point bifurcations in particular multiple-source systems but did not obtain general criteria guaranteeing fast null-point motion, and Longcope, Brown & Priest (2003) derived only the statistical distribution of nulls above the photosphere; the connectivity theory of Longcope & Klapper (2002) goes beyond point-charge and potential approximations but does not identify unstable topological configurations. For non-potential fields, Brown & Priest (2000) showed that force-free fields — solenoidal fields with zero Lorentz force, ∇×B=αB\nabla\times\mathbf{B}=\alpha\mathbf{B}, which is the lowest-energy state above potential fields when coronal currents are present — have skeletons usually similar to potential fields for non-degenerate source configurations, but no trigger criteria exist in this class. A mathematically promising initial step toward general counting relations is the replacement of Euler–Poincaré indices by Morse–Smale indices (the Poincaré index of an equilibrium of a flow, i.e. the degree of the flow-direction map on a small sphere enclosing the equilibrium, equal to (−1)m(-1)^{m} for a hyperbolic equilibrium with an mm-dimensional unstable manifold, as studied in Morse–Smale dynamical-systems theory), used by Zhuzhoma et al. (2022) to derive universal relations between the numbers of null points and sources in branching ('anemone') flare structures.

Observationally, Dumin & Somov identified 'unipolar' flares in chromospheric movies taken in the Ca II H spectral line with the Solar Optical Telescope (SOT) aboard the Hinode satellite: short-lived flaring arcs whose footpoints lie in regions of the same magnetic polarity or in areas of negligible field, which cannot be ordinary field-line-connected loops and are instead interpreted as energy release propagating along the separator of a flipping two-dome separatrix structure (Dumin & Somov 2017, 2019, 2024). Because the potential field is the minimum-energy state for given boundary conditions, the energy budget of these microflares is itself questioned: the energy may come directly from the small photospheric displacements of the sources, or from free energy stored in a slightly non-potential (force-free) field close to the topologically unstable configuration. Higher-resolution identification is expected from the He I 10830 Å chromospheric absorption line (formed at conditions similar to Ca II H), e.g. with the New Solar Telescope at Big Bear Solar Observatory, whose ~0.16-arcsec resolution substantially exceeds Hinode/SOT's (Zeng et al. 2016 demonstrated its power on small-scale reconnection events), but sufficiently long public time series in this line do not yet exist.

Problem Statement

The problem, as posed by Dumin & Somov (2024, MNRAS 528, L15; DOI 10.1093/mnrasl/slad162), is to place the topological-trigger model of unipolar solar flares on a general footing and validate it quantitatively. Its three components are scientifically linked — they jointly test whether 'topological' (nonlocal) reconnection is a viable explanation of unipolar flares:

  1. General topological instability criteria. Derive criteria identifying topologically unstable configurations of the coronal magnetic field — configurations in which small source displacements cause a topological bifurcation with fast motion of a magnetic null point along a separator — that are valid beyond the currently solved case of at most four point-like sources in the potential-field approximation: specifically, (a) for arbitrary numbers of point sources and (b) for non-potential (force-free) fields, going beyond the point-charge and potential-field approximations in the source's sense (relaxing the Coulomb point-charge field representation, e.g. to fields anchored to discrete sources or isolated source regions of arbitrary shape, as attempted by Longcope 2001 and Longcope & Klapper 2002). The Euler–Poincaré index arguments used for the two-to-four-source cases must be generalized to the point where the results can be confronted with observations.

  2. Energy source of the microflares. Quantitatively determine, for the observed unipolar microflares, whether the released energy comes directly from the tiny displacement of the photospheric magnetic sources (potential-field energy decrease of order the relative source displacement, e.g. 5–10 per cent, in the instability region) or from the free energy stored in a slightly non-potential (force-free) field close to the topologically unstable configuration.

  3. Observational validation. Identify and characterize unipolar flares with substantially improved resolution, in particular using the He I 10830 Å chromospheric line with New Solar Telescope-class instrumentation (~0.16 arcsec) once sufficiently long time series in this line become available in the public domain (none exist at present), and test the model's predictions (near-instantaneous ignition, propagation along the flipping two-dome separator, same-polarity footpoints) against them.

The criteria of component 1 are sought (a) for arbitrary numbers of point sources, (b) for non-potential (force-free) fields, and (c) for discrete or extended source regions beyond the Coulomb point-charge representation, so that topologically unstable configurations are separated from stable configurations; the three cases are independent, and a criterion established for one of them does not by itself establish the others. In component 2 the energy attribution is sought for each unipolar microflare in the source's observational sample (Dumin & Somov 2017, 2019, 2024): whether the released energy comes from the displacement of the photospheric magnetic sources or from the free energy stored in a slightly non-potential field. In component 3 the observational validation is sought through a He I 10830 Å test specified to an explicit adequacy standard — the unipolar-flare sample it would observe, the resolution and time cadence required, the model signatures it would measure (near-instantaneous ignition, propagation along the flipping two-dome separator, same-polarity footpoints), and, for each signature, the outcome that would confirm or refute the model — the test itself being executable only once sufficiently long public time series in this line exist.

References

  1. Yurii V. Dumin, Boris V. Somov (2024), 'Unipolar solar flares as a manifestation of “topological” magnetic reconnection', Monthly Notices of the Royal Astronomical Society: Letters 528, L15–L19, DOI 10.1093/mnrasl/slad162, https://doi.org/10.1093/mnrasl/slad162 (arXiv:2309.16417)
  2. Evgeny V. Zhuzhoma, Vladislav S. Medvedev, Yurii V. Dumin, Boris V. Somov (2022), 'Topological quantification of the “anemone” (branching) solar flares', Physica D: Nonlinear Phenomena 436, 133320, DOI 10.1016/j.physd.2022.133320, https://doi.org/10.1016/j.physd.2022.133320 (arXiv:2111.06730)
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