Fast kink modes of longitudinally stratified coronal loops

Abstract

Aims. We investigate the standing kink modes of a cylindrical model of coronal loops. The density is stratified along the loop axis and changes discontinuously at the surface of the cylinder. The periods and mode profiles are studies with their deviation from those of the unstratified loops. The aim is to extract information on the density scale heights prevailing in the solar corona. Methods. The problem is reduced to solving a single second-order partial differential equation for δB z (r, z), the longitudinal component of the Eulerian perturbations of the magnetic field. This equation, in turn, is separated into two second-order ordinary, differential equations in r and z that are, however, connected through a dispersion relation between the frequencies and the longitudinal wave numbers. In the thin tube approximation, the eigensolutions are obtained by a perturbation technique, where the perturbation parameter is the density stratification parameter. Otherwise the problem is solved numerically. Results. 1) On functional dependencies of the dispersion relation the radial wave number is independent of the longitudinal stratification. 2) We verify the earlier computational finding that the first overtone frequencies increase with increasing stratification and the observational finding (from analysis of TRACE data) that the ratio of the first to the fundamental overtone frequency decreases with increasing stratification. The method we use to arrive at these conclusions, however, is more analytical than computational, and yet our numerical results agree with the earlier results.3) The mode profiles depart from the sinusoidal mode profiles of the unstratified loops. This departure and its dependence on the scale height is obtained, and might serve to determine scale heights once high resolution data become available.

Application of the Rayleigh-Ritz variational technique for coronal loop oscillations

“…For typical loop lengths, 100 − 400Mm, the density scale heights fall in the range of 13 − 108Mm. These results are in agreement with Verwichte et al (2004), McEwan et al (2006), and Safari et al (2007). The application of variational method in real model of coronal loop are implementing in our group.…”

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

“…It was strongly emphasised by and Verth et al (2007) that the longitudinal spatial resolution could possibly allow for the estimation of loop model parameters based on the deviations in the eigenfunctions rather than the eigenfrequencies. The effect of longitudinal density stratification on the transversal displacement amplitude was however found to be not particularly strong for the fundamental oscillation Safari et al 2007), and due to several uncertainties in the observations (projection effects, localisation of the loop footpoints), probably not detectable at present with reasonable confidence. For the first overtone however it was found that the shift of the antinodes towards the footpoints is much more pronounced and could possibly be detectable.…”

“…The derivation starts from a tube of arbitrary thickness but eventually specializes to a "thin tube" to obtain concise formulae. A similar formula (for the displacement only) was derived by Safari et al (2007) starting directly from the "thin tube" formula by Dymova & Ruderman (2005). They used it to validate the results about the modifications to the displacement profiles of the fundamental and its first harmonic by and Verth et al (2007).…”

Sausage oscillations in loops with a non-uniform cross-section

“…They found that stratification changed the amplitude profile from one of constant longitudinal density, but that the effect was too subtle for such observational signatures to be detectable by EUV imagers. Safari et al (2007) also considered standing kink modes of longitudinally stratified coronal loops. They found changes in amplitude profiles of a few percent, consistent with Erdélyi & Verth (2007), and showed that the predicted departure of P 1 /2P 2 from unity was in good agreement with the observational data of Verwichte et al (2004).…”