## Abstract

The excitation and damping of transversal coronal loop oscillations is studied using one-and two-dimensional models of line-tied cylindrical loops. By solving the time-dependent magnetohydrodynamic equations it is shown how an initial disturbance generated in the solar corona induces kink mode oscillations. We investigate the effect of the disturbance on a loop with a non-uniform boundary layer. In particular, a strong damping of transversal oscillations due to resonant absorption is found, such as predicted by previous works based on normal mode analysis.

## 1. Introduction

From the theoretical point of view, the damping of transversal loop oscillations observed with Transition Region and Coronal Explorer (TRACE) has been usually studied using normal mode analysis, whereas the time-dependent problem has been mainly investigated for external driven oscillations. However, the non-driven case, more appropriate to describe the transversal oscillations, has not been investigated in much detail. The initial value problem, considered here, can help to understand how leaky and trapped modes are excited, what is the amount of energy deposited in the normal modes and what is the role of resonant absorption in damping kink mode oscillations.

## 2. Equilibrium model and wave equations

A coronal loop is modelled as a straight cylindrical tube of radius *R* and length *L*, with the magnetic field uniform and parallel to the *z*-axis. The loop density profile includes an inhomogeneous layer of width *l*,In our calculations, we set *ρ*_{i}/*ρ*_{e}=3 and *R*=0.1*L*.

To study perturbations about the equilibrium state we use the linearized magnetohydrodynamic (MHD) equations with resistivity *η* (where *R*_{m}≡1/*η* is the magnetic Reynolds number). Moreover, the low plasma-beta limit is taken. Next, a Fourier analysis in the *z*-coordinate is made and perturbations are assumed proportional to . We concentrate on the fundamental mode, with *k*_{z}=*π*/*L*, and so the photospheric line-tying effect is naturally incorporated in the model.

Now, the MHD equations in cylindrical coordinates reduce to the usual set of five coupled partial differential equations for the variables *v*_{r}, *v*_{θ}, *b*_{r}, *b*_{θ} and *b*_{z}. The independent variables are *r*, *θ* and *t*. These time-dependent equations are solved numerically with the help of the PDE2D code (Sewell 2005) for the one-dimensional problem and the CLAWPACK code (Leveque 2002) for the two-dimensional case.

## 3. One-dimensional problem

In the MHD equations, we assume that the perturbed variables behave as e^{imθ}, so that they only depend on *r* and *t*. We concentrate on the kink mode by imposing *m*=1. The initial perturbation (produced for example by a flare) is located in the coronal medium and has the following formwhere *r*_{0} is the position of the centre of the disturbance and *w* is a parameter related to its width. All other MHD variables are initially set to zero.

### (a) Homogeneous tube

We first consider *l*=0 (i.e. a homogeneous loop) and study the time evolution of the radial velocity component, *v*_{r}, at the loop centre. There are two different phases in the signal (see figure 1*a*); first comes the *impulsive leaky phase*, in which one or more leaky modes are excited (Terradas *et al*. 2005*b*). Due to its short duration and periodicity, this phase can hardly be detected with TRACE. Next comes the *stationary phase*, with the loop oscillating in the trapped kink normal mode. The amplitude of oscillation of the normal mode has a strong dependence on the location of the disturbance and for *r*_{0}≫*L* the loop is simply driven by the travelling wave passing through it (Terradas *et al*. 2005*a*).

### (b) Inhomogeneous tube

When the loop is surrounded by an inhomogeneous layer (*l*≠0), the impulsive leaky phase is practically the same as for the homogeneous case, but now the fast kink mode damps due to the resonant coupling with the Alfvén modes (figure 1*b*). In fact, the loop is resonantly damped (in an exponential fashion) almost immediately after being perturbed, so resonant absorption is quite an efficient mechanism in attenuating kink mode oscillations.

We have next studied how the damping time derived from the numerical simulations depends on the parameter *l*/*R*. We find that for the decay time agrees quite well with the analytical values given by Hollweg & Yang (1988), Goossens *et al*. (1992) and Ruderman & Roberts (2002). For , however, there is a discrepancy between the analytical and numerical values because of the thin layer approximation used in the theoretical calculations (see Van Doorsselaere *et al*. 2004 for thick layer results). Finally, the influence of the resistivity on the damping time is such that the latter is essentially independent of *R*_{m} for , in agreement with Poedts & Kerner (1990).

## 4. Two-dimensional problem

Now the assumption that perturbations have a dependence of the form e^{imθ} is dropped and the two-dimensional problem is solved in Cartesian coordinates. The initial perturbation is located in the coronal medium and has the following form

Using the same parameters as before (*r*_{0}=0.5*L*, *w*=0.2*L*, *l*/*R*=0.4), the results are very similar to those of the one-dimensional case: the loop first goes through an impulsive leaky phase, next the fundamental kink mode is excited and it is finally damped because of resonant coupling with Alfvén modes. This is just the expected behaviour since a lateral kick such as the one imposed here mostly excites the kink mode.

## Footnotes

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

- © 2005 The Royal Society