50. 3D Visualisation of the Eigenmodes of a Straight Magnetic Flux Tube

Author: Ioannis Giagkiozis, Viktor Fedun, Robertus Erdélyi, and Gary Verth at the Solar Physics and Space Plasma Research Centre (SP2RC), Solar WAve Theory Group (SWAT) and Space Systems Laboratory (SSL) at the University of Sheffield

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A magnetised plasma supports a wealth of waves. In the magnetohydrodynamic (MHD) approximation in a homogeneous plasma the eigenmodes are commonly referred to as the Alfvén [1], slow and fast magnetoacoustic modes. However, it is interesting to note that the behaviour of these waves heavily depends on the magnetic field geometry and the background plasma parameters. A prevalent magnetic configuration in the solar atmosphere is the “magnetic flux tube”, occurring both in curved coronal loops and in almost straight versions, in fibrils, spicules, mottles etc. An idealisation of a flux tube is a straight magnetic cylinder. In [2] such a model was considered for photospheric and coronal regions, under the assumption of constant equilibrium plasma parameters (i.e. plasma density, pressure and magnetic field strength) inside and outside the flux tube. In this nugget we show 3D visualizations of the analytic solutions initially derived by Edwin and Roberts [2] for a straight magetic flux tube. The remaining visualizations can be accessed in on our website and YouTube channel [3,7].

Visualisations of MHD Eigenmodes for a Magnetic Flux Tube

Although the eigenmodes explored in [2] for a straight magnetic cylinder are still predominantly referred to as the Alfvén, slow and fast magnetoacoustic modes there are some interesting new physical features introduced. One key difference in comparison to the eigenmodes in a uniform unbounded plasma is that the magnetoacoustic waves in a magnetic flux tube are further classified as surface and body modes. The surface modes are localised near the boundary with the velocity amplitude falling off rapidly towards the centre and similarly in the exterior of the tube (evanescent waves). The body modes exhibit an oscillatory character inside the flux tube while outside the waves are evanescent.

Here we would like to introduce an extensive set of eigenmode visualisations illustrating the 3D behaviour of the Alfvén, slow and fast magnetoacoustic modes based on the solutions of the dispersion equation originally derived by [2]. We consider the azimuthal wavenumbers n=0 to n=6 and the first 3 radial harmonics (or branches) of the fast and slow waves. Additionally we show the Alfvén mode for n=0 and n=1. For the magnetoacoustic modes the azimuthal wavenumbers n=0, and n=1 correspond to the sausage and kink modes respectively, while the remaining correspond to the first up to fifth fluting modes.

Figure 1. In all figures, the coordinate unit vectors can be seen in the lower left corner and are encoded as red, greed and blue corresponding to the x-,y- and z-coordinates respectively. Alfvén mode for azimuthal wavenumbers, n=1 and n=-1.

A sample of the visualisations of the MHD eignemodes, from different perspectives, can be seen in Figure 1 to 3. The magnetic field is represented by the thin red lines, and the white arrows depict the velocity vector field. The field lines for a quarter of the flux tube have been removed to show the corresponding perturbation of the plasma density and velocity field. Additionally, a density perturbation higher than the background equilibrium density is coloured red while a lower density is coloured blue. Further details on the parameters used to produce these visualisations, as well as the relevant dispersion diagrams, are available in [3]. In Figure 1 we show the Alfvén mode for azimuthal wavenumbers, n=1,-1. Notice that the different magnetic surfaces for the Alfvén mode in this magnetic configuration are decoupled.

Figure 2. Vertical slice of a magnetic flux tube for the fast sausage mode (azimuthal wavenumber, n=0) and the first radial harmonic, otherwise referred to as first branch.

In Figure 2 we present three periods of the fast sausage mode, see the dispersion diagram in [4]. The fast kink mode is shown in Figure 3, also see [5] for the dispersion diagram. For slow modes the dominant velocity component is field-aligned, while for the fast modes the largest velocity component is in the perpendicular direction. The only components of the velocity field for the Alfvén mode are perpendicular to the magnetic field lines within the magnetic surface Figure 1.

Figure 3.Horizontal slice of a magnetic flux tube for the fast kink mode (azimuthal wavenumber, n=1) and the first radial harmonic.


Better understanding of MHD modes of relevant magnetic configurations can aid solar physicists in interpreting both imaging and spectroscopic data. It is noteworthy that 2D visualisations of a number of solutions presented here [3], have been previously made available by E. Verwichte [6]. All movies (more than 300 in total) for the fast and slow magnetoacoustic modes for azimuthal wavenumbers n=0 to n=6 and the first 3 radial harmonics as well as the Alfvén modes for n=0 and n=1, are available for viewing and download at [3]. Future updates and additions will be made available at [7]. Lastly, we would like to mention that all movies in [3] have been generated using VAPOR [8], and, we wish to acknowledge the people behind its development for kindly sharing their work openly with the community.