Effect of coronal structure on loop oscillations: exponential profiles

Aims. The role of longitudinal structuring of the surrounding corona on the modes of oscillation of a coronal magnetic flux tube was studied in Donnelly et al. (2006) for a piecewise uniform profile. Here we investigate whether a more realistic continuous exponential profile changes the conclusions drawn from that paper.

Methods. A partial differential equation is derived for the total pressure perturbation of the fast modes, which is then decomposed by separation of variables. The longitudinal part is solved numerically, obtaining a dispersion relation. These results are supported by an analytical investigation in terms of Bessel functions of purely imaginary order.

Results. Structure in the interior of the loop shifts the frequencies of the modes (and may trap higher harmonics), an effect which can be understood by taking an averaged profile with a suitable weight. Structure in the environment modifies only slightly the frequencies, but displaces the cutoff frequency. The shift due to the structure in the fundamental period is small, but the ratio between the periods of the fundamental mode and its harmonics can be used to probe the structure.

Conclusions. The results support our previous study in a more realistic, continuously varying profile and provide limits to the conclusions drawn in coronal seismology if an unstructured loop is used. Also, the ratio between the period of the fundamental kink (even) mode and its first (odd) harmonic is proven as an extra seismological tool for coronal loops.