## Abstract

Resonantly damped fast kink quasi-modes are computed in fully resistive magnetohydrodynamics for fully non-uniform two-dimensional equilibrium models. The equilibrium model is a straight cylindrically symmetric flux tube with a plasma density that is non-uniform both across and along the loop. The variation of density across the loop can cover the whole loop. Our results indicate that the period and damping of coronal loop oscillations mainly depend on the density contrast and the inhomogeneity length-scale and are independent of the details of longitudinal stratification. This study extends previous studies on coronal loop oscillations, and allows for a better comparison between observations and theory.

## 1. Introduction

Transverse coronal loop oscillations triggered by explosive events, such as flares or filament eruptions, were first observed by the extreme ultraviolet telescope on board the Transition Region and Coronal Explorer spacecraft (Aschwanden *et al*. 1999; Nakariakov *et al*. 1999). These oscillations have been interpreted as magnetohydrodynamics fast kink-modes of a cylindrical coronal flux tube by Nakariakov *et al*. (1999). In most of the observed events the oscillatory amplitude has been found to be strongly damped, typically having an exponential decay time of a few oscillatory periods. The cause of the rapid damping is still a matter of considerable debate. As pointed out by Goossens *et al*. (2002), damping of quasi-mode kink oscillations by resonant absorption gives a perfect explanation of the rapid decay of the observed coronal loop oscillations, if the inhomogeneity length-scale is allowed to vary from loop to loop. Incidentally, the rapid decay of oscillations in coronal loops was predicted by Hollweg & Yang (1988), more than a decade before they were detected.

In this paper, we study the resonant damping of coronal loop oscillations in fully two-dimensional configurations, with non-uniformity of the equilibrium density in both the radial and axial directions. The equilibrium model is fully non-uniform in the radial direction and the transitional layer can cover the whole loop.

## 2. Equilibrium model

A coronal loop is modelled as a gravity-free, straight, cylindrically symmetric flux tube. In a system of cylindrical coordinates (*r*, *φ*, *z*) with the *z*-axis coinciding with the axis of the cylinder (loop), the magnetic field is pointing in the *z*-direction. As the magnetic field is straight, the classic *β*=0 approximation implies that the magnetic field is uniform, , and also that the density, *ρ*(*r*, *z*) profile can be chosen arbitrarily. Here, we consider a two-dimensional variation of plasma density of the form(2.1)where *α* is the longitudinal stratification parameter. For the radial dependence, we follow Ruderman & Roberts (2002), Van Doorsselaere *et al*. (2004), and assume a sinusoidal profile that connects the constant internal, *ρ*_{i}, and the constant external, *ρ*_{e}, values of the density along a transitional layer of length *l*.

## 3. Numerical results

The relevant parameters and the ranges of variation selected in our numerical computations are: the thickness of the inhomogeneous layer (*l*/*R*∈[0.0–2.0]), the density contrast and the longitudinal stratification parameter (*α*∈[0.0–1.0)). Figure 1*a*,*b* shows that both the period and the damping time are almost unaffected by the stratification parameter, at least for values below *α*≃0.8. A comparison between both figures shows that the period is more sensitive than the damping time, for larger values of *α*. The normalized damping rate (not shown) is completely independent of the longitudinal stratification. These figures also show that the length of the inhomogeneous layer has an important effect on both the period and damping time. In particular, the damping time is very sensitive to *l*/*R*. Figure 1*c*,*d* shows that the period and damping time are independent of *α* up to values near one. On the other hand, figure 1*c* also shows a strong dependency of period on the density contrast. For the range of variation of the density contrast considered in our computations, coronal loop oscillations are very rapidly damped in a range of 0.78–2.85 oscillatory periods.

## 4. Summary and conclusions

A parametric numerical study of the fast kink-mode frequency and damping rate has been performed for a wide range of values for several loop parameters. Our results show that the period and damping time of fast kink quasi-modes are almost independent of the longitudinal stratification parameter. The normalized damping rate is almost independent of longitudinal stratification, in agreement with Andries *et al*. (2005). The damping is strongly dependent on the thickness of the non-uniform boundary layer. Finally, the density contrast is a very important parameter, in such a way that large contrast loops get damped in less that an oscillatory period. The present results allow a better theoretical understanding of the resonant damping of coronal loop oscillations and provide us with a more accurate theoretical description for the comparison of observations and theory, opening new possibilities for coronal seismology.

## Footnotes

↵† Postdoctoral Fellow of the National Fund for Scientific Research—Flanders (Belgium; F.W.O. Vlaanderen).

One contribution of 20 to a Discussion Meeting Issue ‘MHD waves and oscillations in the solar plasma’.

- © 2005 The Royal Society