103. Modelling multi-scale solar eruptions

Author: Petros Syntelis and Vasilis Archontis at the University of St Andrews, and Kanaris Tsinganos at the University of Athens.

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Coronal mass ejections (CMEs) are large scale eruptive events responsible for the sudden expulsion of magnetised plasma from the solar corona into the interplanetary medium. These eruptive events are commonly associated with the destabilisation of a pre-existing magnetic flux rope (MFR). Most of the time, they are also associated with the sudden release of energy in the form of a flare arcade that remains rooted in the solar surface at the location of origin of the CME. The kinetic energy content of CMEs ranges from 1026-1032 erg [1], with the bulk of the eruptive events characterized as CMEs to have energies of 1028-1031 erg.

Solar eruptive events appear also at smaller scales than traditional CMEs. Such events are usually referred to as nano-, mini-, or micro-CMEs [2-4], and they appear to be miniature versions of the larger scale CMEs, with similar morphology and dynamics such as e.g. small-scale sigmoids, eruption of small filaments and less energetic flares. Recently, solar coronal jets (Ekin~1026-1027 erg), a seemingly different energetic expulsion of plasma, have also been associated with eruptions of small scale filaments, indicating connections between jets and CMEs [5-9].

We have recently developed flux emergence numerical models for small scale eruptions and studied their formation and triggering mechanism [10]. By increasing the energy budget below the photosphere, we studied eruptions with kinetic energies spanning two orders of magnitude [11]. In this nugget, we present these models and their connection to explaining the multi-scale nature of solar eruptions.


We use the Lare3D code, which solves the 3D time-dependent, resistive, compressible MHD equations, to simulate the partial emergence of a twisted flux tube into a stratified solar atmosphere. The atmosphere consists of a sub-photospheric layer, an isothermal layer mimicking the photosphere/chromosphere, a thin layer representing the transition region and an isothermal layer mimicking the corona. The flux tube is initially placed under the photosphere. A density deficit is introduced in the flux tube, making its central part buoyant. The magnetic field of this central part partially emerges into the solar atmosphere, resulting in the formation of a bipolar region at the photosphere. The corona is initially field free, so there is no external reconnection between an atmospheric ambient field and the emerging field. We perform a parametric study by varying the energy content of the flux tube to produce eruptions of different scales.

Formation and triggering of the small scale eruptions

We first describe the eruption associated with the flux tube with the lowest energy content.
The upper part of the subphotospheric flux tube emerges above the photosphere forming a bipolar region (Fig. 1), while its axis remains below/at the photosphere. Shearing, rotational and converging motions develop self-consistently (arrows and red contours), injecting free energy into the atmosphere. A sheared arcade (blue lines, panel a) is formed near the polarity inversion line (PIL) which eventually adopts a J-shaped loops morphology (b). Reconnection between the sheared lines at a low-lying current sheet (purple isosurface) initially form long loops (c) and eventually result in a MFR that exists prior to the eruption (yellow lines, panel c).

Figure 1. (a,c) Field lines showing the initial sheared arcade (blue) and the formation of new long magnetic loops (yellow line, c). (b,d) Field lines showing J-shaped loops (blue lines) and the formation of a twisted MFR (yellow lines, d). The horizontal slice is photospheric Bz (black and white). The yellow arrows show the photospheric velocity field and the red contours show the photospheric vorticity. The purple isosurface is |J/B| [10].

The height-time profile of the MFR and the torus index at the height of the MFR is shown in Fig. 2a (upper and lower panels). The torus index (n = – dlnB/dlnz) is a metric of the decrease of the magnetic field strength with height, and a MFR is expected to be destabilized by torus instability when its located in regions of n between 1 and 2. Although the MFR is located inside a high torus-index region from the very early stages of its evolution (e.g. n > 1.5 for t > 57 min), the exponential rise associated with the eruption is not triggered by the torus instability. Only the initial accelerations of the flux rope (first vertical line and insert) could be attributed to torus instability. During the initial acceleration, the MFR moves upwards and stretched the envelope field above it (red lines, panel 2b).

The fast ejection of the MFR (second vertical line, panel 2a) occurs when the envelope field lines are stretched to the point that they reconnect underneath the MFR with other envelope field lines. Then, the MFR becomes eruptive and the reconnected flux (red lines, panel 2c) is injected towards the centre of the erupting field via a fast (right panel, 2d), hot (middle panel, 2d) and dense (left panel, 2d) reconnection flow, while a flare arcade is formed underneath the flare current sheet (cyan lines, 2c). The density and temperature profiles of the erupting field are similar to the large scale CME eruption. The kinetic energy of the eruption is 1026 erg and the physical size is ~40Mm, therefore the eruption is a small scale CME-like eruption (mini/nano-CME).

Figure 2. (a, upper) Height–time profile of the MFR. The vertical lines indicate the possible onset of torus instability (left) and explosive reconnection of envelope field lines with other envelope field lines (tether-cutting) (right). The inset shows a close-up of the height-time profile around the possible initiation of torus instability. (a, lower) The torus index measured at the MFR center. (b) Magnetic field topology before the eruption. Blue lines show J-shaped loops. Yellow lines show the MFR. Green lines show the outer envelope field lines. Red lines show the stretched envelope field lines above the MFR that are about to reconnect. The purple isosurface is a current sheet. (c) Same as (b) but during the eruption. The red lines have reconnected and have become part of the MFR. Cyan lines show the flare loops. (d) The density (left) temperature (middle) and velocity (right) during the eruption [10].

Multi-scale eruptions

To study eruptions on multiple scales, we increase the initial magnetic energy content of the subphotospheric flux tube, and increase the physical size of the numerical domain (from 653 Mm to 1533 Mm). The kinetic and the magnetic energy of the resulting eruptions against the photospheric flux are shown in Fig. 3. The energies of the eruptions follow a power law distribution. The kinetic energies are 1026 to 1028, ranging from small scale mini/nano-CME eruptions to small CMEs. Extrapolating for a typical AR flux of 1021 Mx, we predict energies of 4 × 1030 – 3 × 1033 erg, which are typical energies of CMEs.

Figure 3. Kinetic (a) and magnetic (b) energy of the eruption over the photospheric flux (diamonds). Solid lines are linear fitted lines and dashed lines are the 95% confidence level of the fit [11].


Using a self-consistent model with different initial magnetic energy contents, we study eruptions of multiple scales [10,11], with energies ranging from that of small scale eruptions and jets (1026 erg) up to small CMEs (1028 erg). A power law is predicted between the energies of the eruptions and the photospheric flux. Testing whether such scaling laws exist between the observed small scale eruption, jets and the larger scale CMEs would be a strong indication for the jet-CME connection and the multi-scale self-similar nature of eruptions.


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