Bifurcation scenarios for separators and transitions between topological classes of global coronal magnetic fields in the point-charge model

In the point-charge (magnetic charge topology) model of the global solar coronal magnetic field, Grines and Pochinka (2018) obtained, for a fixed configuration, a complete set of criteria for the existence of separators — field lines lying on intersections of the fan surfaces of two null points — and a classification of such fields up to topological equivalence, with a complete invariant given by the genus of an associated level surface and the coherent-equivalence class of a self-indexing energy function. The dynamic counterpart remains open: describing the transition scenarios between topologically inequivalent domain configurations — the bifurcations governing the appearance and disappearance of separators and the restructuring of the flux-domain decomposition — that are invoked to explain flare- and prominence-related magnetic reconnection. A systematic, invariant-based bifurcation theory of this classification, valid for arbitrary charge configurations, remains open.

Background

The solar corona — the hot outer atmosphere of the Sun — is threaded by a magnetic field whose structure largely determines coronal activity. Of particular importance are solar flares (sudden releases of magnetic energy) and prominences (dense, cool plasma structures suspended in the corona), both of which are believed to be powered by magnetic reconnection: the reorganization of magnetic field-line connectivity that occurs when field regions of different directions are brought together in a highly conducting plasma. Since the 1980s, a productive way to study this reorganization has been the topological approach to solar flares (surveyed in Grines, Zhuzhoma and Pochinka, 2017/2021): the coronal field is idealized as potential, $B=-\nabla\Phi$, generated by point "magnetic charges" of positive and negative polarity placed on the photosphere (the Sun's visible surface), each charge abstracting a compact region where magnetic flux threads the surface. Although magnetic monopoles do not exist in nature, this magnetic charge topology model reproduces the large-scale connectivity of the corona and is the standard setting in which flare-related topological questions are posed.

In this model the field has isolated null points — points where $B=0$. A null is a saddle-type equilibrium of the associated vector field: it has a one-dimensional invariant manifold called the spine (a single field line through the null) and a two-dimensional invariant manifold called the fan (a separatrix surface of field lines). Physically, a null can be positive or negative according to the sign of the product of the eigenvalues of the field's Jacobian at the null. When the fan surfaces of two nulls of opposite sign intersect, the intersection is typically a field line connecting the two nulls, called a separator. Separators are the natural sites of magnetic reconnection and of the current sheets that release flare energy, so their number and locations carry the essential information about where and how the corona can reconfigure. The fan surfaces partition the corona into connected flux domains ("domes"), regions of field lines of common connectivity, and a change in this partition — typically through the birth or death of a separator, accompanied by reconnection — changes the global structure of the corona.

Grines and Pochinka (2018) formalized this model geometrically, following earlier spherical-geometry studies of the St Andrews school (Beveridge, Priest and Brown, 2002; Maclean and Priest, 2007). The field is defined on the corona $\{w>0\}$ of the three-sphere $S^3=\{(x,y,z,w)\in\mathbb{R}^4:\,x^2+y^2+z^2+w^2=1\}$, with the photosphere the equatorial two-sphere $P=\{w=0\}$ carrying $k$ point charges and the field extended symmetrically to the mirror corona $\{w<0\}$; the potential $\Phi$ is a Morse function (a smooth function whose critical points are all nondegenerate), and the field is assumed to be in general position ("quasi-typical": hyperbolic nulls and charges, transverse intersections of fans, no spine lying in a fan). In this setting they proved, first, criteria for separator existence in terms of the types of the nulls — for instance, the fan of every coronal or vertical null contains at least one separator, while the fan of a horizontal (photospheric) null contains a separator exactly when the closure of its photospheric trace limits on more than one charge — and, second, a classification of such fields up to topological equivalence (a homeomorphism of the punctured sphere taking field lines to field lines with orientation). The complete invariant is remarkably concrete: writing $m$ for the number of nulls, the quantity $g=(m-k+2)/2$ is a nonnegative integer, the level surface $\Sigma=\varphi^{-1}(3/2)$ of the self-indexing energy function $\varphi$ (a standard auxiliary Morse function whose critical points coincide with those of the potential and whose value at each critical point equals its Morse index) is an orientable surface of genus $g$, at least $2g$ nulls have fans containing separators, and two fields are topologically equivalent if and only if their energy functions are "coherently equivalent": there is an orientation-preserving homeomorphism $H$ with $\varphi' \circ H = \varphi$ that carries the decoration of $\Sigma$ by the positive and negative null fans to the corresponding decoration of the level surface $\Sigma'$ of $\varphi'$ (Definition 4.2 of the authors' survey). This is the classical machinery of Morse–Smale dynamical systems (flows whose nonwandering set consists of finitely many hyperbolic equilibria, of which the point-charge field is a gradient example) applied to the coronal field.

The classification is a fixed-time, static theory. The physics that motivated it, however, is dynamic: as photospheric flux sources emerge, move, and decay, the charge configuration changes, and the field passes through bifurcations in which separators appear and disappear and the dome decomposition is restructured — precisely the topological changes accompanying flares and prominence eruptions, in which magnetic energy is released at separators by reconnection. The solar-physics literature had studied such transitions only computationally and case by case: Beveridge, Priest and Brown (2002) produced the bifurcation diagram for two bipolar regions (four charges), identifying four stable topological states connected by a global separator bifurcation, a local double-separator bifurcation, and separatrix- and spine-quasi-bifurcations; Brown and Priest (1999) analyzed how separators arise and are destroyed for sources in a plane; Maclean and Priest (2007) exhibited seventeen topological changes in a six-source model of the global field reversal over the solar cycle; Haynes, Parnell, Galsgaard and Priest (2007) tracked the bifurcations of the magnetic skeleton (the web of nulls, spines, fans and separators) through a three-dimensional magnetohydrodynamic numerical experiment; and Platten, Parnell, Haynes, Priest and Mackay (2014) followed the global coronal topology through a solar cycle. What does not yet exist is a systematic, invariant-based theory of these transitions valid for arbitrary charge configurations, building on criteria and classification of the Grines–Pochinka type: which transitions between topological equivalence classes are possible, through which generic mechanisms they proceed, and how the genus and the decorated surface evolve across them.

Problem Statement

In the point-charge model of the global coronal magnetic field — the potential field $B=-\nabla\Phi$ on the corona $\{w>0\}$ of $S^3$ whose photosphere $P=\{w=0\}$ carries $k$ point magnetic charges, with Morse potential $\Phi$ and in the quasi-typical general position of Grines and Pochinka (2018) — develop the bifurcation theory of the topological classification: describe the transition scenarios, as the charge configuration (number, positions and fluxes of the charges) varies, between topologically inequivalent configurations of the coronal flux domains into which the null fans divide the corona. Concretely: (i) characterize the mechanisms of appearance and disappearance of separators (field lines on transverse intersections of the fans of two nulls of opposite sign), including the local bifurcations of nulls, spines and fans through which they proceed; (ii) determine which transitions between topological equivalence classes are possible, and which are generic in the charge-parameter space, and how the complete invariants of the classification — the genus $g=(m-k+2)/2$ of the level surface $\Sigma=\varphi^{-1}(3/2)$ and the coherent-equivalence class of the self-indexing energy function $\varphi$, where $m$ is the number of nulls — change across each transition.

References

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