Soler et al. (2010a) (see Figure 59) modeled a prominence fine structure as a straight cylindrical magnetic tube only partially filled with the cold and dense material. The length of the dense part is . The thread may either occupy the centre of the magnetic tube or be displaced, so that the lengths of both evacuated parts of the tube are different. By denoting the lengths of the right and left hand-side evacuated regions as and , one has , with the full length of the tube. Just like in the works discussed in Section 5.4, the prominence plasma is partially ionized and a transverse inhomogeneous transitional layer is included between the prominence thread and the coronal medium. Ion-neutral collisions and resonant absorption are the considered damping mechanisms. The main model improvements in comparison to the thread model by Díaz et al. (2002), discussed in Section 4.5 (see Figure 41), are the ability to model non-centered threads, the inclusion of a non-uniform transverse layer and partial ionization of the thread plasma.

First, damping by Cowling’s diffusion alone is considered by setting . When the thread is located in the center of the tube the ratio of the damping time to the period is given by the approximate expression

with the filament Cowling’s diffusivity in dimensionless form. For , and , Equation (34) gives for and for . Therefore, in a transversally homogeneous thread, an almost neutral prominence plasma is needed, i.e., , for the damping due to Cowling’s diffusion to be efficient. Although the precise ionization degree is unknown, such large values of are probably unrealistic in the context of prominences.Next, Soler et al. (2010a) considered , so that both resonant absorption and Cowling’s diffusion can cause wave damping. An approximate expression for the damping ratio for a sinusoidal density variation in the transitional layer is

As a numerical example, in the case , , and , the damping ratio is for a fully ionized thread () and for an almost neutral thread (). Note that the obtained damping times are consistent with the observations. Moreover, as seen in Section 5.4.1, the contribution of resonant absorption to the damping is much more important than that of Cowling’s diffusion, so the ratio depends only very slightly on the ionization degree and the second term on the right-hand side of Equation (35) can in principle be neglected.

When the prominence region is not at the center of the tube, and assuming , an approximate expression for the damping ratio is

Taking the limits or in this expression, it can be shown that the minimum value of the damping ratio by Cowling’s diffusion takes place when the prominence region is located at the magnetic tube center ().Soler et al. (2010a) find that for and under the thin tube and thin boundary approximations, the period and damping time by resonant absorption have the same dependence on and . This means that for resonant absorption the damping ratio does not depend on these quantities. Since resonant damping dominates over Cowling’s diffusion, this leads to the conclusion that when considering both damping mechanisms, the damping ratio will be almost unaffected by the position of the prominence region within the fine structure.

The accuracy of the above analytical solutions can be assessed by numerically solving the general dispersion relation derived by Soler et al. (2010a). Here we only show the results obtained by Soler et al. (2010a) for the case in which the prominence thread is centered in the tube.

In the case without transverse transitional layer, , damping is only due to Cowling’s diffusion. Figure 60a displays the period as a function of for different values of the ionization degree in the prominence region, whereas Figure 60b shows the corresponding values of the damping time. As can be seen, the period increases when the length of the thread is increased and tends to the value for a homogeneous prominence cylinder when . In addition, the period is independent of the ionization degree. On the contrary, the damping time strongly depends on the ionization degree, and for a fixed it slightly increases as becomes larger. In all solutions, the analytical expressions for the period and the damping time are in agreement with the solution of the full dispersion relation for realistic, small values of , i.e., , whereas the approximate expressions diverge from the actual solution when the prominence region occupies most of the magnetic tube. Figure 60c displays versus . The numerical solution shows little dependence on , while the analytical approximation (Equation [34]) diverges from the numerical value in the limit of large . Given the obtained large values of , Soler et al. (2010a) concluded that the efficiency of the damping due to Cowling’s diffusion in a partially filled flux tube does not improve with respect to the longitudinally homogeneous tube case shown in Section 5.4.1.

Next, Soler et al. (2010a) included resonant damping. The period and the damping time of the fundamental kink mode were computed as a function of the different parameters, namely , , and . Regarding the period, Soler et al. (2010a) found that both its value and its dependence on are the ones plotted in Figure 60a because the period is almost independent and . For a fixed ionization degree of , the damping time decreases with . The approximate analytical estimate of the damping time is in good agreement with the full solution for below 0.4. In order to assess the efficiency of resonant damping, Figure 61b displays the corresponding values of . In comparison with the damping ratio by Cowling’s diffusion (see Figure 60c), much smaller values of are now obtained. The damping ratio is almost independent of the length of the thread. This is because, under the assumptions made by Soler et al. (2010a), the dependence of the period and damping time on the length of the thread is the same. Overall, a very good agreement is obtained between the numerical result and the analytical approximation, even for large values of the length of the thread.

In summary, the dominant damping mechanism is resonant absorption, which provides damping ratios in agreement with the observations, whereas ion-neutral collisions are irrelevant for the damping. The values of the damping ratio are independent of both the prominence thread length and its position within the magnetic tube, and coincide with the values for a tube fully filled with the prominence plasma. A recent study that further analyses resonant damping of thread oscillations in two-dimensional equilibrium models can be found in Arregui et al. (2011). These authors additionally analyzed the influence of the density in the evacuated part of the thread. This quantity is seen to influence periods and damping times, but has little influence on the damping rate of transverse thread oscillations. The implications of some of these results for the determination of physical properties in transversely oscillating threads are discussed in Section 6.

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