3.4 Decay of the oscillations

The physical mechanism responsible for the quick decay of the oscillations is under intensive discussion. The direct dissipation caused by viscosity or resistivity, considering classical values of the coronal viscosity and resistivity, cannot explain the observed decay times (see the discussion in Roberts, 2000

). Ofman et al. (1994 ) numerically established the scaling law

which connects the decay time $t$ of the oscillation, the Reynolds number $Re$ ( $= LCA0/n$ ) associated with the shear viscosity $n$ , and the oscillation period $P$ of a fundamental mode with wavelength $2L$ . In this study, the decay mechanism was resonant absorption (see Section 2.2.1) of a collective kink mode and the authors restricted their attention to linear perturbations only. Applying this law, Nakariakov et al. (1999

) suggested that the decay of kink oscillations might be explained by resonant absorption with enhanced shear viscosity. It should be pointed out that this result is not in contradiction to common sense, as enhanced dissipation actually means that the coefficient of shear viscosity in the Braginskii viscosity tensor is comparable to the bulk viscosity coefficient. If so, this must be connected with coronal micro-turbulence. Such a situation is not unusual in astrophysics (e.g., the invoking of turbulent viscosity in the understanding of the physical properties of accretion disks) and is often seen in laboratory plasmas. The basic idea behind resonant absorption is that the wave energy of the damped global fast kink mode is converted to a localised Alfvén mode through resonant coupling. This coupling occurs in a shell in the loop boundary where the kink mode frequency, which is always between the internal and external Alfvén frequencies, matches the local Alfvén frequency. The combination of these two modes, through resonant coupling, is also called a quasi-mode. In the absence of an equilibrium flow this mode damps independently of dissipation. For a thin, weakly dissipative loop with a thin boundary layer between $a - l$ and $a$ of the form $r(r) = [(r0 + re)- (r0- re) cos((r - a)p/l)]/2$ , the global kink wave has a decay time (which may be derived from Equation (13

)) (Ruderman and Roberts, 2002 )

where $l$ is the width of the loop boundary, which is assumed to be thin compared with the loop width $2a$ , and $P$ is the mode period $2L/CK$ . This expression does not contain the viscosity coefficient, and the authors concluded that the observed decay time was not connected with dissipation. However, the observed decay times could be explained if $l = 0.23a$ . Sharper profiles give longer damping times; shallower profiles lead to global motions that are rapidly damped. Goossens et al. (2002a ) examined the observations further, using the selection of eleven loops presented by Ofman and Aschwanden (2002

), and concluded that the resonant absorption decay Equation (34

) was able to reproduce the observed decay times provided the inhomogeneity scale $l$ as a fraction of tube radius $a$ ranged in value from $l/a = 0.16$ to $l/a = 0.49$ . This result would violate, though, the assumption $l « a$ . But Van Doorsselaere et al. (2004 ) showed numerically that, even for this range of values of $l/a$ , the analytical Equation (34

) remains valid. Aschwanden et al. (2003 ) compared the theoretically predicted values of the density ratio $re/r0$ with observations and concluded that they are in poor agreement, which was attributed to the narrowness of the TRACE 171 Å temperature bandpass.

In addition, Equation (34 ) may give unrealistically short decay times comparable with the oscillation period. It is not clear how the phenomenon of resonant absorption can happen in those cases when the decay time is comparable with the oscillation period and, consequently, the oscillation is not harmonic, making the resonance impossible. In any case, resonant absorption remains a plausible interpretation of the decay.

Another alternative interpretation is connected with the effect of phase-mixing (Roberts, 2000 ; Ofman and Aschwanden, 2002 ) (see Section 2.2.2). The decay of a standing Alfvén waves by phase-mixing, given by Equation (17 ), becomes after assuming that $dCA/dx ~~ CA/l$ :

In principle $Re$ is the shear Reynolds number, $Re = LCA/n$ , with $n$ the kinematic shear viscosity coefficient. If we take the theoretical values for $n$ , then the decay time due to phase-mixing is orders of magnitude longer than the observed decay. Ofman and Aschwanden (2002

) claimed that, due to micro-turbulence or kinetic processes modifying the velocity distribution of ions, $n$ could be of the same order as the bulk viscosity coefficient, practically returning back to the idea suggested by Nakariakov et al. (1999

). Hence, decay due to phase-mixing can be of the same order as the observed decay. We would like to emphasise that in the case discussed here, the phase mixing mechanism does not involve the torsional modes, but is connected with kink perturbations of neighbouring loops.

A different approach to the problem is based on the leakage of wave energy from the loop. Cally (1986 , 2003 ) showed that the fast kink mode is almost identical to a fast leaky kink mode. In the long wavelength limit both modes propagate at the kink speed. In the limit of the plasma- $b$ tending to zero, the amplitude of the leaky kink mode decreases as it radiates into the coronal environment with a decay time (Cally, 2003 , note a factor $2$ error in his expression)

which, unless the loop width $2a$ is considerably larger than the observed loop emission width, is too long to explain the observed decay, especially for large loops (see Figure 13

). For example for a loop of length $200 Mm$ , $a$ needs to be in the range of $10 -20 Mm$ .

The wave energy may also leak through its footpoints. Roberts (2000 ) concluded from the theoretical study by Berghmans and de Bruyne (1995 ) that this type of leakage is insufficient to explain the observed decay. De Pontieu et al. (2001 ), though, argued that this study did not take the chromosphere into account. The amplitude decay of an Alfvén wave leaking into a chromosphere with density scale-height $h$ , typically $150- 200 km$ , is (Hollweg, 1984 )

which is about five times longer than the observed decay (see Figure 13

). This was confirmed by the numerical study by Ofman (2002 ), who also pointed out some inconsistencies in the work by De Pontieu et al. (2001 ). A similar study of footpoint leakage of fast kink modes themselves has not yet been undertaken but it should take into account the stratification of the external medium and the magnetic field divergence with height. Both effects are expected to increase the efficiency of the reflection of kink modes at the footpoints and, consequently, reduce the leakage.

Ofman and Aschwanden (2002 ) suggested that observationally determined scaling laws, connecting the decay times with oscillation periods and lengths of oscillating loops, may provide some information allowing us to distinguish between the interpretations discussed above. Indeed different decay mechanisms give different scaling laws: coronal leakage ( $t oc L2P$ ), footpoint leakage ( $t oc LP$ ), resonant absorption ( $t oc P$ ), and phase-mixing ( $2/3 t oc P$ ). By using the fact that the period is proportional to the loop length, one can be eliminated in favour of the other.

Ofman and Aschwanden (2002 ) determined scaling laws from observational decay times measured by Nakariakov et al. (1999 ) and Aschwanden et al. (2002 ). In particular they found that $t oc P 1.17± 0.34$ . They concluded that the mechanism of phase-mixing agrees best with the scaling laws. They, though, assumed that the length $l$ in Equation (35 ) is proportional to the length of the loop to obtain a modified scaling law for phase-mixing:

There is indeed a good correspondence between the theoretical and observational power law index. However, it is not clear why $l$ should be proportional to $L$ . If the original scaling law for phase-mixing is considered, then the difference is larger than one standard deviation. Also, the mechanism of resonant absorption, which has a power law index of 1 falls within one standard deviation of the observationally determined power law index and can, therefore, not be excluded. The mechanisms of coronal and footpoint leakage have power law indices of $3$ and $2$ , respectively, which is clearly not consistent with observations.

Figure 13 shows the observationally determined decay times as a function of wavelength and period and uses measurements from Nakariakov et al. (1999 ), Aschwanden et al. (2002 ), Wang and Solanki (2004 ), and Verwichte et al. (2004 ), which adds two more measurements compared with the study of Ofman and Aschwanden (2002 ). We have to bear in mind, though, that the measurement from Wang and Solanki (2004 ) refers to a vertically polarised oscillation and that the measurement from Verwichte et al. (2004 ) is an averaged value from seven damped kink oscillations in a prominence driven loop arcade. The data points are fitted by power laws which have the dependencies $0.70±0.32 t oc c$ and $1.12± 0.36 t oc P$ . The dependence on the wavelength seems to favour phase-mixing, without excluding resonant absorption, and the dependence on period seems to favour resonant absorption but is also consistent with the modified scaling law for phase-mixing. In any case, more examples are needed as the data are too scattered.

Figure 13:

Observationally determined loop oscillation decay times as a function of wavelength (left) and period (right) from: Nakariakov et al. (1999 ) and Aschwanden et al. (2002 ) ( $<>$ ), Wang and Solanki (2004 ) ( $*$ ), and Verwichte et al. (2004

) ( $o$ ). The solid lines are a best fit to the observations, corresponding to $t oc c0.70±0.32$ and $t oc P 1.12±0.36$ . The dashed and dash-dotted lines are the best fits using the models of resonant absorption and phase-mixing, respectively. The vertically and diagonally shaded region correspond to theoretical decay times due footpoint leakage of Alfvén waves (Hollweg, 1984 ) and coronal leakage of a fast leaky kink mode (Cally, 2003 ), respectively, for a range of realistic coronal values for $CA = 500 -2000 km s-1$ , $a = 1 -8 Mm$ , and $r /r = 5- 100 e 0$ .

Verwichte et al. (2004 ) studied TRACE observations of 15 April 2001 of kink oscillations in a post-flare loop arcade. The loops of this arcade were made to oscillate by the actions of a nearby prominence eruption. The oscillation signatures of nine loops were determined as a function of distance along each loop. They found that the displacement amplitude decreases with distance from the loop top as expected for a fundamental mode. However, two of the loops showed two, simultaneous oscillation modes with the longest period, roughly twice the shortest period. Also, the displacement amplitude of the shortest period oscillation increases with distance from the loop top, which indicates that it is an harmonic mode. The measured decay times of the loop oscillations shows the same dependence on the oscillation period as for previous observations, but with a clear bias towards longer decay times. There are several possible explanations for this bias. Firstly, this study considered an oscillating arcade of post-flare loops, while previous observations are concerned with more isolated active region loops. The decay times may be influenced by the different structuring of post-flare loops or by interactions between neighbouring loops in the arcade. Secondly, the arcade oscillation is driven by a nearby prominence eruption that may have given multiple impulses to the arcade. If a time of $5- 10 min$ is subtracted from all the decay times, then they fall in the range of decay times of the earlier studies. The relationship between the decay time and the period, though, would then be much steeper.

Further development of this discussion on the decay of kink oscillations is connected with the improvement of the observational statistics and high-resolution 3D numerical modelling of oscillating loops. In particular, an important issue is whether the loop cross-section remains unperturbed during kink oscillations. Also, the question of the excitation of these modes, connected with the observational fact that the kink oscillations are a rather rare phenomenon, is still open.