5.5 Resonant damping in partially ionized finite length threads

The results described in Sections 5.3, 5.4.1, and 5.4.2 indicate that, because of the coupling of kink waves to Alfvén waves, resonant absorption constitutes a very plausible mechanism for the explanation of the observed spatial and time decay of transverse oscillations. The main limitation of these studies is that they adopt a one-dimensional density model that might not be appropriate in view of the longitudinal structuring of prominence threads. This led Soler et al. (2010aJump To The Next Citation Point) to investigate the time damping properties of two-dimensional thread models, that is, with density inhomogeneity across the thread and along the magnetic tube in which it is contained. In this study, resonant absorption and damping by partial ionization effects were considered simultaneously.

Soler et al. (2010aJump To The Next Citation Point) (see Figure 59View Image) 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 Lp. 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 + L e and − L e, one has + − Le = L − L e − Lp, with L 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. (2002Jump To The Next Citation Point), discussed in Section 4.5 (see Figure 41View Image), are the ability to model non-centered threads, the inclusion of a non-uniform transverse layer and partial ionization of the thread plasma.

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Figure 59: Model used by Soler et al. (2010aJump To The Next Citation Point) to represent a finite length thread. A partially filled magnetic flux tube, with length L and radius a, is considered. The tube ends are fixed by two rigid walls representing the solar photosphere. The tube is composed of a dense region of length Lp surrounded by two much less dense zones corresponding to the evacuated parts of the tube. In the prominence region a transversely inhomogeneous layer of length l is considered. The plasma in the prominence region is assumed to be partially ionized with an arbitrary ionization degree &tidle;μ p. Both the evacuated part and the corona are taken to be fully ionized.

First, damping by Cowling’s diffusion alone is considered by setting l = 0. 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

( )1∕2 ∘ --(-------)---- τd≈ -1- ρp-+-ρc -1-- 2 1 − Lp- Lp-, (34 ) P 2π ρp &tidle;ηCp L L
with η&tidle;Cp = ηC∕vApa the filament Cowling’s diffusivity in dimensionless form. For ρp∕ ρc = 200, Lp ∕L = 0.1 and L = 105 km, Equation (34View Equation) gives τd∕P ≈ 5 × 103 for &tidle;μp = 0.8 and τd∕P ≈ 150 for &tidle;μp = 0.99. Therefore, in a transversally homogeneous thread, an almost neutral prominence plasma is needed, i.e., &tidle;μp ≈ 1, for the damping due to Cowling’s diffusion to be efficient. Although the precise ionization degree is unknown, such large values of &tidle;μp are probably unrealistic in the context of prominences.

Next, Soler et al. (2010aJump To The Next Citation Point) considered l∕a ⁄= 0, 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

⌊ ⌋−1 ( ) ( ) ( )1∕2 τd ≈ 2-||m l- ρp-−-ρc + η&tidle; --ρp--- ∘-----4--------|| . (35 ) P π ⌈ a ρp + ρc Cp ρp + ρc ( Lp) Lp ⌉ 2 1 − L L

As a numerical example, in the case m = 1, Lp ∕L = 0.1, L = 107 m and l∕a = 0.2, the damping ratio is τ ∕P ≈ 3.18 d for a fully ionized thread (&tidle;μ = 0.5 p) and τ ∕P ≈ 3.16 d for an almost neutral thread (&tidle;μp = 0.95). 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 τd∕P depends only very slightly on the ionization degree and the second term on the right-hand side of Equation (35View Equation) can in principle be neglected.

When the prominence region is not at the center of the tube, and assuming l = 0, an approximate expression for the damping ratio is

( )1 ∕2 ┌│ --[-----------------------]---- τd -1- ρp-+-ρc -1-│∘ ( Lp) Lp- L-−e L+e Lp- P ≈ 2 π ρ &tidle;η 2 1 − L L + 4 L2 L . (36 ) p Cp
Taking the limits L−e → 0 or L+e → 0 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 (− + Le = Le).

Soler et al. (2010aJump To The Next Citation Point) find that for l ⁄= 0 and under the thin tube and thin boundary approximations, the period and damping time by resonant absorption have the same dependence on L −e and L+e. 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.

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Figure 60: Results for the thread model in Figure 59View Image without transverse transitional layer and for the prominence thread located at the central part of the magnetic tube. (a) Period, P, of the fundamental kink mode in units of the internal Alfvén travel time, τ Ap, as a function of L ∕L p. The horizontal dotted line corresponds to the period of the kink mode in a homogeneous prominence cylinder. The symbols are the analytic solution (Equation (24) in Soler et al., 2010aJump To The Next Citation Point). (b) Damping time, τd, in units of the internal Alfvén travel time, τAp, as a function of Lp∕L. The different lines denote μ&tidle;p = 0.5 (dotted), 0.6 (dashed), 0.8 (solid), and 0.95 (dash-dotted). The symbols are the analytic approximation for &tidle;μp = 0.8 (Equation (27) in Soler et al., 2010aJump To The Next Citation Point). (c) τd∕P versus Lp ∕L. The line styles have the same meaning as in panel (b) and the symbols are the approximation given by Equation (34View Equation) (from Soler et al., 2010aJump To The Next Citation Point).

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

In the case without transverse transitional layer, l∕a = 0, damping is only due to Cowling’s diffusion. Figure 60View Imagea displays the period as a function of Lp∕L for different values of the ionization degree in the prominence region, whereas Figure 60View Imageb 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 Lp ∕L → 1. 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 &tidle;μp it slightly increases as Lp∕L 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 Lp∕L, i.e., Lp ∕L ≤ 0.4, whereas the approximate expressions diverge from the actual solution when the prominence region occupies most of the magnetic tube. Figure 60View Imagec displays τ ∕P d versus L ∕L p. The numerical solution shows little dependence on Lp ∕L, while the analytical approximation (Equation [34View Equation]) diverges from the numerical value in the limit of large Lp ∕L. Given the obtained large values of τd∕P, Soler et al. (2010aJump To The Next Citation Point) 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. (2010aJump To The Next Citation Point) included resonant damping. The period and the damping time of the fundamental kink mode were computed as a function of the different parameters, namely &tidle;μp, l∕a, and Lp ∕L. Regarding the period, Soler et al. (2010aJump To The Next Citation Point) found that both its value and its dependence on Lp ∕L are the ones plotted in Figure 60View Imagea because the period is almost independent &tidle;μ p and l∕a. For a fixed ionization degree of &tidle;μ = 0.8 p, the damping time decreases with l∕a. The approximate analytical estimate of the damping time is in good agreement with the full solution for Lp ∕L below 0.4. In order to assess the efficiency of resonant damping, Figure 61View Imageb displays the corresponding values of τd∕P. In comparison with the damping ratio by Cowling’s diffusion (see Figure 60View Imagec), much smaller values of τd∕P 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. (2010aJump To The Next Citation Point), 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.

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Figure 61: Results for a thread configuration with a transverse transitional layer and for the prominence thread located at the central part of the magnetic tube. (a) τd in units of the internal Alfvén travel time, τ Ap, and (b) τ ∕P d as a function of L ∕L p. The different lines in both panels denote l∕a = 0.05 (dotted), 0.1 (dashed), 0.2 (solid), and 0.4 (dash-dotted). The symbols in panels (a) and (b) correspond to the analytic approximations with l∕a = 0.2, Equations (34) and (32) in Soler et al. (2010aJump To The Next Citation Point).

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.


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