Lee and Brown (2020) introduced null-like points — points on the photospheric boundary where the tangent magnetic field vanishes but the normal component does not — as a way to recover the topological structure (separatrix-like surfaces, separator-like x-lines) that is lost when discrete magnetic charge topology is replaced by continuously distributed photospheric sources. Two questions they explicitly left open remain untouched in the literature: (1) whether the separatrix-like surfaces generated by null-like points correspond to the QSL branches, spine signatures, or null halos produced by sub-photospheric nulls; and (2) whether null-like points can be located and are meaningful in non-potential regimes — force-free extrapolations and full MHD simulations — where the criterion has never been tested. The weaker, more diffuse squashing-factor signatures of null-like points noted in the paper are an obstacle to their detection.

Background

Many phenomena in the solar atmosphere — flares, jets, coronal heating — are driven by magnetic reconnection, and candidate reconnection sites are organized by the magnetic skeleton: the network of magnetic null points (points where the field B=0\mathbf{B}=0), the separatrix surfaces that fan out from them, and the separator field lines where two separatrices intersect. Classically this skeleton is computed with magnetic charge topology (MCT), in which photospheric flux concentrations are approximated by discrete point sources. Real solar magnetic fields, however, as measured by magnetograms and as produced by numerical simulations, are continuously distributed over the photospheric plane (z=0z=0). Lee and Brown (2020) showed that the transition from discrete to continuous sources has a topological cost: when two sources of the same polarity overlap, the null point between them disappears from the z=0z=0 plane, and with it the separatrix wall and the separator (an x-line, a key candidate reconnection site) that it anchored, leaving only a separatrix dome.

To recover such lost structure, Lee and Brown (2020) introduced the null-like point (NLP): a point on the boundary where the field component parallel to the boundary vanishes, B∥=0B_{\parallel}=0, while the perpendicular component is allowed to remain nonzero, B⊥≠0B_{\perp}\ne 0 — a null of the tangent field rather than of the full field. Tracing fan-like field lines from an NLP defines separatrix-like surfaces, whose intersections give separator-like lines that retain the x-line structure of true separators. Using the squashing factor QQ — a standard measure of how rapidly the field-line mapping between the two footpoints of a field line changes across the boundary (Titov, Hornig and Démoulin 2002; Titov 2007) — Lee and Brown showed that these separatrix-like surfaces are a subclass of quasi-separatrix layer (QSL), a thin region of high QQ where field lines change connectivity rapidly and reconnection can proceed without true null points (Priest and Démoulin 1995); field lines crossing a QSL undergo slip-running reconnection (Aulanier et al. 2006), a continuous apparent motion of the footpoints rather than a discrete jump.

Two prior results frame the unresolved questions. First, Restante, Aulanier and Parnell (2009) studied how skeletons degrade into QSLs when the magnetic sources are placed below the photospheric plane: they found QSLs associated with the spine field lines of sub-photospheric nulls, a second category (termed branches) located along the photospheric endpoints of fan field lines from sub-photospheric sources, and a null halo — a diffuse region of raised QQ above a sub-surface null. Second, all of Lee and Brown's examples use potential (current-free) fields, although the NLP criterion involves only the boundary field and is in principle model-agnostic. Within Lee and Brown's own Q-maps, the QQ signature of an NLP is visibly weaker and more diffuse than that of a true null footprint, so reliable detectability of NLPs is not established even in the potential case. Lee and Brown observed that pinning down this association — whether NLP-generated separatrix-like surfaces correspond to the branches, spine signatures, or null halos, and whether the NLPs lie along the x-line structure propagating from a sub-surface null — would involve a comparison of the same topological setup realized both ways: with sources placed below the photospheric plane so that a sub-photospheric null is present, and with continuous sources on the plane whose tangent field contains an NLP.

Problem Statement

Establish null-like points as a well-founded and generally applicable extension of the coronal magnetic skeleton, by answering the two linked questions left open by Lee and Brown (2020):

  1. Correspondence with sub-photospheric-null topology. Determine whether the NLP-origin separatrix-like surfaces and their QQ-map footprints correspond to the branches, the spine-line signatures, or the null halos that Restante, Aulanier and Parnell (2009) attribute to sub-surface nulls, and whether the NLPs themselves lie along the x-line structure that propagates from a sub-photospheric null point.

  2. Generalization beyond potential fields. Implement and validate NLP detection in non-potential regimes — force-free field extrapolations and full magnetohydrodynamic simulations — establishing whether NLPs exist in such fields, how they behave under non-potential currents and dynamical evolution, and which topological features (separatrix-like QSLs, separator-like x-lines) they reveal there. Establishing NLP existence and detectability is a prerequisite for the behavioural and topological-feature results.

Behind both questions stands a detectability obstacle noted in the source: the QQ signatures of NLPs are weaker and more diffuse than null footprints, and the reasons for these diminished signatures are not established. Resolving the correspondence and generalization questions determines whether NLP-generated QSLs matter for reconnection — for instance whether crossing them drives slip-running reconnection — but the reconnection physics itself is a downstream implication, not part of what must be established here.

References

  1. Daniel T. Lee, Daniel S. Brown (2020), Topology of Coronal Magnetic Fields: Extending the Magnetic Skeleton Using Null-like Points, Solar Physics 295(12), article 168. DOI 10.1007/s11207-020-01729-6. https://doi.org/10.1007/s11207-020-01729-6 (preprint arXiv:2011.10272)
  2. A. L. Restante, G. Aulanier, C. E. Parnell (2009), How skeletons turn into quasi-separatrix layers in source models, Astronomy & Astrophysics 508, 433–443. DOI 10.1051/0004-6361/200912664. https://doi.org/10.1051/0004-6361/200912664
  3. David I. Pontin, Eric R. Priest (2022), Magnetic reconnection: MHD theory and modelling, Living Reviews in Solar Physics 19, article 1. DOI 10.1007/s41116-022-00032-9. https://doi.org/10.1007/s41116-022-00032-9
  4. E. R. Priest, P. Démoulin (1995), Three-dimensional magnetic reconnection without null points: 1. Basic theory of magnetic flipping, Journal of Geophysical Research: Space Physics 100, 23443–23463. DOI 10.1029/95JA02740. https://doi.org/10.1029/95JA02740
  5. Vyacheslav S. Titov, Gunnar Hornig, Pascal Démoulin (2002), Theory of magnetic connectivity in the solar corona, Journal of Geophysical Research: Space Physics 107(A8), 1164. DOI 10.1029/2001JA000278. https://doi.org/10.1029/2001JA000278
  6. V. S. Titov (2007), Generalized Squashing Factors for Covariant Description of Magnetic Connectivity in the Solar Corona, The Astrophysical Journal 660, 863–873. DOI 10.1086/512671. https://doi.org/10.1086/512671
  7. G. Aulanier, E. Pariat, P. Démoulin, C. R. DeVore (2006), Slip-Running Reconnection in Quasi-Separatrix Layers, Solar Physics 238, 347–376. DOI 10.1007/s11207-006-0230-2. https://doi.org/10.1007/s11207-006-0230-2