Abstract

In this work, we derive magnetic toroids from surface magnetograms by employing a novel optimization method, based on the trust region reflective algorithm. The toroids obtained in this way are combinations of Fourier modes (amplitudes and phases) with low longitudinal wavenumbers. The optimizati... Show moreIn this work, we derive magnetic toroids from surface magnetograms by employing a novel optimization method, based on the trust region reflective algorithm. The toroids obtained in this way are combinations of Fourier modes (amplitudes and phases) with low longitudinal wavenumbers. The optimization also estimates the latitudinal width of the toroids. We validate the method using synthetic data, generated as random numbers along a specified toroid. We compute the shapes and latitudinal widths of the toroids via magnetograms, generally requiring several m's to minimize residuals. A threshold field strength is chosen to include all active regions in the magnetograms for toroid derivation, while avoiding non-contributing weaker fields. Higher thresholds yield narrower toroids, with an m = 1 dominant pattern. We determine the spatiotemporal evolution of toroids by optimally weighting the amplitudes and phases of each Fourier mode for a sequence of five Carrington Rotations (CRs) to achieve the best amplitude and phases for the middle CR in the sequence. Taking more than five causes "smearing" or degradation of the toroid structure. While this method applies no matter the depth at which the toroids actually reside inside the Sun, by comparing their global shape and width with analogous patterns derived from magnetohydrodynamic (MHD) tachocline shallow water model simulations, we infer that their origin is at/near the convection zone base. By analyzing the "Halloween" storms as an example, we describe features of toroids that may have caused the series of space weather events in 2003 October-November. Calculations of toroids for several sunspot cycles will enable us to find similarities/differences in toroids for different major space weather events. Show less