Graph Dynamics of Solar Active Regions: Morse–Smale Complexes and Multiscale Graphs of Magnetograms

Vladislav V. Alekseev, N. G. Makarenko, Irina Knyazeva

2020

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DOI 10.1134/s1063773720070014
Data confidence 70 medium
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Abstract

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.