The influence of longitudinal density variation in coronal loops on the eigenfunctions of kink-oscillation overtones

Abstract

Context. As coronal loops are spatially at least partially resolved in the longitudinal direction, attempts have been made to use the longitudinal profiles of the oscillation amplitudes as a seismological tool. Aims. We aim to derive simple formulae to assess which oscillation modes and which quantities of the oscillation (displacement or compression) are most prone to modifications induced by stratification of the equilibrium density along the loop. We furthermore clarify and quantify that the potential of such a method could be enhanced if observational profiles of the compression in the oscillations could be determined. Methods. By means of a linear expansion in the longitudinal stratification along with the "thin tube" approximation, the modifications to the eigenfunctions are calculated analytically. The results are validated by direct numerical computations. Results. Higher axial overtones are found to be more affected by equilibrium stratification and hence would provide a much better tool if observed. For the k − 1th overtone the compression is found to be around (k + 2) 2 /k 2 times more sensitive to longitudinal density variation than the displacement. While the linear formulae do give a good indication of the strength of the effects of longitudinal density stratification, the numerical computations indicate that the corrections to the approximate analytical results are significant and cannot be neglected under the expected coronal conditions.

Torsional Alfvén waves in stratified and expanding magnetic flux tubes

“…It shows that as a result of density stratification, the anti-nodes of the first-overtone eigenfunction shift away from the loop apex (z = 0). This result was also obtained by Andries et al (2009) for kink modes. Figure 7 clears also that in the presence of magnetic field expansion, the anti-nodes shift towards the loop apex which is in agreement with the result obtained by Verth and Erdélyi (2008) for the kink body modes.…”

“…For instance, it was pointed out by Verth (2007) that if the magnetic field becomes weaker towards the loop apex, a useful magneto-seismological signature is the anti-node of the 1st harmonic since it shifts towards the loop apex. Also Andries et al (2009) used the anti-node shift in the presence of longitudinal density stratification as signature of the eigenmode modifications. They showed that the anti-node shift of the eigenfunctions of the higher kink- Fig.…”

Resonant MHD Waves in the Solar Atmosphere

“…Hollweg and Yang 1988;Ruderman and Roberts 2002;Van Doorsselaere et al 2004), while axial stratification affects the periods of oscillation (see e.g. Andries et al 2005aAndries et al , 2005bAndries et al , 2009aAndries et al , 2009bArregui et al 2005;Goossens et al 2006;Dymova and Ruderman 2005, 2006a, 2006bMcEwan et al 2006McEwan et al , 2008Ruderman et al 2008).…”

Coronal Seismology by Means of Kink Oscillation Overtones

“…E.g., L = 100 Mm and H = 50 Mm causes a shift of 5.6 Mm, well within the resolution of TRACE (714 km). This was taken up further by Andries et al (2009) anticipating possible detection of higher overtones in the future. Carrying out a linear expansion of the eigenfunctions in the stratification parameter α (as had done earlier for the frequency), an approximation is found for the lth Fourier contribution in the kth mode with k and l arbitrary.…”

“…Another interesting development concerns the use of the longitudinal eigenfunctions as seismological tools Safari et al 2007;Andries et al 2009). Although its application is subject to considerably more observational uncertainty, it is most certainly an avenue that must be considered.…”