3.1 TRACE observations

On 14 July 1998, the imaging telescope on board the Transition Region and Coronal Explorer (TRACE) registered, in both 171 Å and 195 Å lines, spatially resolved decaying oscillating displacements of coronal loops in the active region AR 8270 (Aschwanden et al., 1999

; Nakariakov et al., 1999

; Schrijver et al., 1999 ). These oscillations happened shortly after a solar flare and, most probably, were generated by the flare. The mechanism of the excitation remains hidden, but it can be connected with a blast wave generated in the flare epicentre. Some of the loops seem to be more responsive to the oscillation than others and it could probably be connected with the magnetic topology of the active regions (Schrijver and Brown, 2000 ). Oscillations of different loops were not synchronised in phase. The highest amplitude was seen near the loop apices. Since this discovery, the loop oscillations were subject to an extensive observational study and the results are summarised in Schrijver et al. (2002 ) and Aschwanden et al. (2002

). In particular, it was found that the kink oscillations do not always have a simple form of a global (or principle) mode and there can be higher spatial harmonics observed.

Anyway, the oscillation examined by Nakariakov et al. (1999 ) may be considered as a typical example of kink oscillations of coronal loops. The analysis of the loop displacement shows that the oscillation is almost harmonic with the period of about $P = 256 s$ (the frequency about $4 mHz$ ). Figure 10 shows the temporal evolution of the displacement at the loop apex. About three periods of oscillation were observed. Displacement amplitudes are several $Mm$ for the distance between the loop footpoints estimated to be about $2L/p = 83 Mm$ . The displacement amplitude is several times larger than the loop cross-section radius, which was observed to be about $2a = 1 Mm$ . The oscillation shows evidence of strong damping. Simultaneously, similar quasi-periodic oscillations were observed in several other loops at the distance of several $Mm$ to $60- 70 Mm$ from the flare epicentre (Aschwanden et al., 1999 ). All these observational findings suggested the oscillations, at least observed in this event, to be interpreted as a kink global standing mode of the loop.

Figure 10:

The temporal evolution of the loop displacement as an average coordinate of the loop position for four neighbouring, perpendicular cuts through the loop apex (diamonds), with error bars ( $± 0.5$ pixels), starting at 13:13:51 UT on 14 July 1998. The solid curve is a best fit of the function $A sin(wt + f) exp(- ct)$ with $A = 2030 ± 580 km$ , $-1 w = 1.47 ± 0.05 rad min$ , and $-1 c = 0.069 ± 0.013 min$ , corresponding to a period and e-folding decay time of $P = 4.3± 0.9 min$ and $t = 14.5 ± 2.7 min$ , respectively (from Nakariakov et al., 1999

Taking the observed period $P$ and loop length $L$ , and applying that the wavelength of a global standing mode is double the length of the loop, one can estimate the phase speed required as

According to the theory of MHD modes of a magnetic cylinder, discussed in Section 2, the fast kink magnetoacoustic modes of a magnetic cylinder do not have dispersive cut-offs and exist for all wavenumbers. In all cases, the wavelength of the observed kink oscillations is much longer than the loop cross-section diameter (e.g., the width of the oscillating loop observed on 14 July 1998 is about $1 Mm$ , while the loop length may be estimated as $261 Mm$ for the distance between the footpoints of about $83 Mm$ ; Nakariakov and Ofman, 2001 ). In this limit, the phase speed of fast kink modes waves approaches the kink speed $CK$ given by Equation (8 ).

Terradas and Ofman (2004 ) pointed out intensity variations localised at the tops of some large-amplitude oscillating loops observed with TRACE in the flaring event on 14 July 1998. As no noticeable changes of the plasma temperature were found at those regions, the intensity variations were interpreted as density variations, approximately in the range $14% -52%$ . The amplitude could possibly be even higher if the filling factor was less than 1. In the analysed loop, the projected maximum amplitude of the oscillations was about $-1 90 km s$ . In the loops oscillating with smaller amplitudes this effect is not detected, indicating its nonlinear nature.

The majority of kink oscillation events corresponds to the horizontal perturbations of coronal loops, which do not change the length of the loop. Recently, Wang and Solanki (2004 ) found an example of vertically polarised kink oscillations in TRACE 195 Å data. The oscillation period was $3.9 min$ , the displacement amplitude was about $8 Mm$ and the decay time was $11.9 min$ . The main difference of this polarisation from the horizontal one is that in the curved loop the vertically polarised kink mode changes the length of the loop. Consequently, as the mass in the loop should be conserved, this mode can have a significant compressible component. The fractional intensity perturbations associated with this mode were estimated as

where $Dr$ is the density perturbation and $DL$ is the perturbation of the loop length. Here it is assumed that the oscillation does not deform the loop cross-section, i.e., $Da/a « DL/L$ . An alternative is associated with bulk field-aligned flows through the loop footpoints, which do not have observational confirmation.

Another possible polarisation of kink oscillations is when the loop moves in the same plane and its apex displaces horizontally. This mode has not been identified yet.