Abstract
We discuss the model for the evolution of an active region (AR) in which the graph constructed from the singular points of the magnetic field codes the AR magnetic patterns. The AR dynamic scenarios are mapped by the discrete structure of the network formed from the maxima, minima, and saddle points of the magnetic field. The AR dynamics leads to a rearrangement of the graph, graph dynamics. We discuss two graphs. The first of them is the graph of a Morse–Smale complex. It is a cellular gradient model of the magnetic field each cell of which contains a maximum, a minimum, and two saddle points. The Morse–Smale complex admits a simplification (editing) with prescribed detail that preserves the field topology. The second graph codes the AR dynamics simultaneously on different scales, in the so-called scale-space. This space is formed by a sequence of convolutions of the original magnetogram with a Gaussian kernel, so that the blurring scale is its additional coordinate. Non-Morse singular points, with a degenerate Hessian and Laplacian, are considered in each scale-space layer. The curves connecting these points in different scales form the critical paths whose vertices are called top points. The graph constructed from these points codes the structure of the degenerate AR field structures on different scales. We propose an efficient method of calculating the critical paths using a Jacobi set. The pre-flare regimes are believed to be associated with a significant change in the field topology, which controls the graph dynamics. Consequently, they must be accompanied by noticeable variations in the spectrum of eigenvalues for the discrete Laplacian of the graph (Kirchhoff–Laplace matrix). As an example, we present the evolution of the spectra for these graphs constructed from the magnetograms of the flaring AR 12673. The SDO/HMI magnetograms of the AR for the scalar line-of-sight (LOS) component served as the data. The possible connection of large variations in the spectrum with succeeding X flares is discussed.
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Notes
Thus, the ‘‘predictions’’ of the events that have already occurred in the AR history.
Simplex stands for simplest. The 0-dimensional or 0-simplexes are simply points, 1-simplexes are edges.
Homotopy is a family of mappings of two spaces that are continuous in parameter, but not necessarily unique. In our example these are the intersecting disks and the edge to which they shrink.
A one-to-one differentiable mapping of two spaces, the inverse of which is also smooth, supplied with the structure of a one-parameter group relative to time.
Interestingly, the complexes themselves may be considered as discrete multivalued dynamical systems and can be used to analyze the critical image regions (Allili et al. 2007).
Imagine a horizontal plane crossing two hills separated by a saddle. There is only one connectivity component in the section below the saddle. There are already two when crossing the saddle and above.
Another way to prove the equivalence of (1) and (2) is to write the solution (2) via the Green function using the Fourier transform.
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Alekseev, V.V., Makarenko, N.G. & Knyazeva, I.S. Graph Dynamics of Solar Active Regions: Morse–Smale Complexes and Multiscale Graphs of Magnetograms. Astron. Lett. 46, 488–500 (2020). https://doi.org/10.1134/S1063773720070014
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DOI: https://doi.org/10.1134/S1063773720070014