109. Kink oscillations of sigmoid coronal loops

Author: Norbert Magyar and Valery M. Nakariakov at the University of Warwick.

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In solar physics, coronal loops have long been in the spotlight. As building blocks of the closed solar corona, understanding their structure and evolution is akin to understanding the coronal heating problem. Moreover, oscillations detected in loops can serve as natural diagnostic probes of their physical properties through coronal seismology, a field in which the properties of coronal plasmas are inferred from observed wave properties and wave theory. Since the first observation of kink oscillations of coronal loops and the first application of coronal seismology [2], models have continuously improved to account for effects such as loop curvature, density stratification, loop cross-sectional variations, cooling, elliptic cross-sections, and so on. A frequently observed property of loops is non-planarity, i.e. exhibiting a helical or sigmoid shape [3]. The effects of coronal loop helicity on their standing kink oscillations were previously investigated analytically, but only through their effect on the local stratified equilibrium density [4]. Here, for the first time, we simulate kink oscillations of sigmoid coronal loops and investigate the ability of coronal seismology to assess the sigmoidity.

Numerical model

Our 3D numerical model consists of a background coronal plasma in a hydrostatic equilibrium in which we embed a coronal loop of higher density. The magnetic field is a force-free dipole with constant ⍺ parameter, adapted from [5]. The ⍺ parameter controls the helicity of the field lines (according to ∇ X B = ⍺B ). We add a higher density loop by tracing a single magnetic field line, and then using it as a central axis to construct a tube. The origin of this single field line, which varies depending on ⍺, is chosen in order to maximise the sigmoidity of the resulting loop while keeping it in the simulation domain. See Figure 1 for an example of a sigmoid flux tube.

Figure 1.Movie showing the magnetic field lines of the employed force-free magnetic field. Towards the end of the movie, a selected flux tube is shown, based on which a denser loop is constructed. The colour represents magnetic field intensity in user units.

The pulse is a horizontally polarised velocity perturbation varying sinusoidally along the loop, which aims to excite a fundamental standing kink mode. However, note that this initial perturbation probably does not coincide with the eigenfunction of the fundamental kink, which is not known. Therefore, while preferentially exciting the fundamental kink mode, other modes are also excited to a small degree, including leaky waves. The resulting kink oscillation of the loop is shown in Figure 2.

Figure 2.Movies showing the simulated kink oscillation of a sigmoid coronal loop, viewed approximately along the loop axis (left frame), and perpendicular to the solar surface (right frame). The colour represents synthetic 171 Å intensity in user units.

The ideal MHD equations are solved in a 3D rectangular domain using MPI-AMRVAC 2.0 [6], with a finite-volume approach. We applied a splitting strategy for the magnetic field, with the time-independent force-free magnetic field considered as a background field. Thus we only solve for the (nonlinear) perturbed magnetic field components. For this, we used the newly-implemented HLLD solver adapted for magnetic field decomposition described in [7].


Analysis of the oscillation properties is based on synthetic 171 Å intensity and Doppler shift images with the lines of sight corresponding to the coordinate axes. After measuring the oscillation properties, we proceed to infer magnetic field estimates seismologically by calculating the theoretical kink period. We do this using the WKB approximation, as the kink speed varies along the loop. From this the magnetic field intensity is determined using the measured oscillation period, loop length, and estimations of the density.

The results are shown in Figure 3. We have considered a range of an order of magnitude for the precision to which the average internal density can be determined, while the density ratio (internal to external density) is taken to range from 1.5 to 10. In the simulation, the average density ratio is close to 2. Here we assume that the measurements of the length of the loop and of the oscillation period are exact.

Figure 3. Plot showing the actual minimum, near the apex (blue square), maximum, near the footpoints (yellow square), average (green diamond), and seismologically estimated (red circle) magnitude of the magnetic field inside the oscillating loop, for different values of the ⍺ parameter. The seismologically estimated value uses the average value of internal density and density ratio. Values shown are for the single traced field line used to construct the loop. The average is taken over the full loop length. The error bar extends to the highest/lowest estimate resulting from the range of values considered for internal density and the density ratio. Note that the ordinate axis is logarithmic.

For the simulation with no sigmoidity, despite the measured and theoretically calculated periods being close to each other, the seismologically predicted magnetic field value is lower than the average value. This can be understood in the following way: as the displacement amplitude of the fundamental mode has a maximum near the apex, the oscillation period is more sensitive to the weaker magnetic field near the apex rather than near footpoints. With increasing sigmoidity however, the predicted magnetic field shows an increasing trend with respect to the average value. This observation might allow for the seismological determination of the sigmoidity of a coronal loop, if some other method to determine the average magnetic field is available, such as force-free extrapolations. In this sense, the free magnetic energy in a coronal loop could be estimated seismologically.


We propose that the dependence of the magnetic field estimate on the loop sigmoidity could be exploited seismologically in order to measure the non-potentiality, i.e. the free magnetic energy in coronal loops. However, for this method to work, the determination of the average magnetic field along the loop is needed, as well as an accurate measurement of the density along the loop. The external/internal density ratio only weakly impacts the results. On the other hand, we demonstrated the robustness of the seismological method, even when applied to non-planar or sigmoid coronal loops. For all values of sigmoidity considered, the estimation of the magnetic field is within the extremal magnetic field values measured in the loop, despite considering an order of magnitude accuracy for the average density determination.


  • [1] Magyar, N. & Nakariakov, V. M., ApJL 894 L23 (2020)
  • [2] Nakariakov, V. M. & Ofman, L., A&A, 372, L53 (2001)
  • [3] Aschwanden, M. J. et al., ApJ, 756, 124 (2012)
  • [4] Ruderman, M. S. & Scott, A., A&A, 529, A33 (2011)
  • [5] Cuperman, S. et al., A&A, 216, 265 (1989)
  • [6] Xia, C. et al., ApJS, 234, 30 (2018)
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