Arregui et al. (2008b) considered an individual and isolated thread modeled as a cylindrical magnetic flux tube of radius in a gravity-free environment. The uniform magnetic field points along the axis of the tube (Figure 50). In the zero plasma- approximation, the thread is modeled as a density enhancement with a radial variation of density from its internal constant prominence value to the coronal constant value over a non-uniform layer of thickness . A typical value of the density contrast between the filament and coronal plasma is .

MHD waves of axisymmetric one-dimensional cylindrical flux tubes are characterized by two wavenumbers, i.e., the azimuthal wavenumber, , and the axial wavenumber, . They can have different nodes in the radial direction. Arregui et al. (2008b) concentrated their analysis on the radially and longitudinally fundamental transverse wave with azimuthal number , the kink mode. This eigenmode is consistent with the detected Doppler velocity variations (see Section 3.6.4) and their associated transverse motions, discussed in Section 4.4.1. The frequency of this mode is not influenced by the presence of a layer with small thickness, so the result of Section 4.4.1 is approximately correct; see Equation (22).

When transverse inhomogeneity is considered, the fundamental transverse kink mode resonantly couples to Alfvén waves. The consequence is the transfer of wave energy from the global transverse motion to azimuthal motions of localized nature, and thus the time damping of the kink mode (Goossens et al., 2009). Analytical expressions for the damping time scale can be obtained under the assumption that the transverse inhomogeneity length-scale is small (). This is the so-called thin boundary approximation. When the long wavelength and the thin boundary approximations are combined, the analytical expression for the damping time over period for the kink mode can be written as (see, e.g., Goossens et al., 1992, 1995; Ruderman and Roberts, 2002; Goossens et al., 2002)

Here is a numerical factor that depends on the particular variation of the density in the non-uniform layer. For a linear variation, (Hollweg and Yang, 1988; Goossens et al., 1992); for a sinusoidal variation, (Ruderman and Roberts, 2002). Consider for example as a typical density contrast and . Then, Equation (30) predicts a damping time of 6 times the oscillatory period, thus producing a time-scale compatible with observations.Quantitative parametric results for the damping of resonant kink waves in prominence threads as a function of the relevant parameters are provided by Arregui et al. (2008b). The accuracy of the analytical approximations is compared to full numerical results, beyond the long wavelength and thin boundary approximations. These results are shown in Figure 51. The damping is affected by the density contrast in the low contrast regime and rapidly decreases for increasing thread density (Figure 51a). Interestingly, stops depending on this parameter in the large contrast regime, typical of filament threads. The damping time over period is independent of the wavelength of perturbations (Figure 51b), but rapidly decreases with increasing inhomogeneity length-scale (Figure 51c).

Resonant damping in the Alfvén continuum appears to be a very efficient mechanism for the attenuation of transverse thread oscillations, especially because large density contrasts and transverse plasma inhomogeneities are combined together.

Although the plasma- in solar prominences is probably small, it is definitely non-zero. Soler et al. (2009e) showed that, in prominence plasmas, resonant damping of kink waves can additionally be produced due to the coupling to slow continuum waves. In the context of coronal loops, which are presumably hotter and denser than the surrounding corona, the ordering of sound, Alfvén and kink speeds does not allow for the simultaneous matching of the kink frequency with both Alfvén and slow continuum frequencies. Because of their relatively higher density and lower temperature conditions, this becomes possible in the case of prominence threads. Therefore, the kink mode phase speed is also within the slow (or cusp) continuum, which extends between the internal and external sound speeds, in addition to the Alfvén continuum. By considering gas pressure in the cylindrical thread model of Arregui et al. (2008b), Soler et al. (2009e) evaluated the contribution of the damping due to the slow continuum to the total resonant damping of the kink mode.

Soler et al. (2009e) used the density model of Section 5.3.1 and the plasma-. In order to obtain an analytic expression for the damping rate of the kink mode, first the long wavelength and thin boundary limits were considered. In terms of the physically relevant quantities, the damping time over the period can be cast as

Here is the same numerical factor as in Equation (30), while and , with and the Alfvén and slow resonant positions. The term with corresponds to the contribution of the slow resonance. If this term is dropped and and are taken, Equation (31) becomes Equation (30), that only takes into account the Alfvén resonance.Equation (31) can now be directly applied to measure the relative contribution of each resonance to the total damping. To do that, Soler et al. (2009e) assumed , for simplicity, so . The ratio of the two terms in Equation (31) is then

where and are the respective contributions of the Alfvén and slow resonances in Equation (31). A simple calculation shows that, for typical wavelengths of observed thread oscillations, the contribution of the slow resonance is irrelevant in front of that of the Alfvén resonance. Take for instance, and , then Equation (32) gives .This analytical predictions were further confirmed by Soler et al. (2009e) by performing numerical computations outside the thin tube and thin boundary approximations. Figure 52 shows that the slow resonance is much less efficient than the Alfvén resonance. For the wavenumbers relevant to observed prominence oscillations, the value of due to the slow resonance is between 4 and 8 orders of magnitude larger than the same ratio obtained for the Alfvén resonance. The overall conclusion by Soler et al. (2009e) is that the slow resonance is very inefficient when it comes to damping the kink mode for typical prominence conditions and in the observed wavelength range. The damping times obtained with this mechanism are comparable to those due to the thermal effects discussed in Section 5.1. Hence, resonant damping of transverse thread oscillations is governed by the Alfvén resonance.

Living Rev. Solar Phys. 9, (2012), 2
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