5.4 Resonant damping in partially ionized infinitely long threads

5.4.1 Temporal damping

Damping by resonant absorption in a partially ionized prominence plasma was studied by Soler et al. (2009dJump To The Next Citation Point), who integrated both mechanisms in a non-uniform cylindrical prominence thread model in order to assess their combined effects. Partial ionization is relevant for the damping of short wavelength fast waves (Forteza et al., 2007), while resonant damping of kink waves is efficient whenever a transverse density inhomogeneity is present. The question arises on whether partial ionization affects the mechanism of resonant absorption and vice versa.

The model adopted by Soler et al. (2009dJump To The Next Citation Point) has the magnetic and density structuring of the models used in Section 5.3 (see Figure 50View Image), but the plasma properties are also characterized by the ionization fraction, &tidle;μ. The radial behaviour of the ionization fraction in threads is unknown, so Soler et al. (2009dJump To The Next Citation Point) assumed a one-dimensional transverse profile akin to the one employed to model the equilibrium density. The thread ionization fraction, μ&tidle;p, is considered a free parameter and the corona is assumed to be fully ionized, so &tidle;μc = 0.5. The non-uniform transitional layer of thickness l therefore connects two plasmas with densities ρp and ρc and ionization degrees &tidle;μp and &tidle;μc. Soler et al. (2009dJump To The Next Citation Point) used the one-fluid approximation and, for simplicity, the β = 0 limit, which excludes slow waves. The quantities η, η C and η H are here functions of the radial direction.

Soler et al. (2009dJump To The Next Citation Point) first considered resonant damping in combination with Cowling’s diffusion and excluded Hall’s dissipation. They derived the following approximate expression for the damping ratio over the period, under the thin boundary approximation,

⌊ ( ) ( ) ⌋−1 τd = 2-⌈m l- ρp −-ρc + 2-(ρ∘p&tidle;ηCp-+-ρc&tidle;ηCc)kza-⌉ , (33 ) P π a ρp + ρc 2ρp (ρp + ρc)
with &tidle;ηCc,p = ηC∕vAc,pa the coronal and prominence Cowling’s diffusivities in dimensionless form. Notice that Equation (33View Equation) reduces to Equation (30View Equation) in a fully ionized plasma, and is in agreement with Equation (31View Equation), in which the slow resonance is additionally included. In this expression, the term due to resonant damping is independent of the value of Cowling’s diffusivity and, therefore, of the ionization degree. The second term, related to the damping by Cowling’s diffusion, is proportional to kz, so its influence in the long-wavelength limit is expected to be small. Soler et al. (2009dJump To The Next Citation Point) performed a simple calculation by considering m = 1, kza = 10−2 and l∕a = 0.2. This results in 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). Thus, this approximate expression suggests that the ratio τd∕P depends very slightly on the ionization degree, suggesting that resonant absorption dominates over Cowling’s diffusion.

The analytical estimates described above can be verified and extended by numerically solving the full eigenvalue problem. This approach allowed Soler et al. (2009dJump To The Next Citation Point) to additionally include Hall’s diffusion in addition to ohmic and Cowling’s dissipation. In their study, these authors first considered a configuration with an abrupt density variation across the thread boundary (that is, l = 0), which prevents resonant absorption from working. Next, they included the thin transitional layer between the thread and the corona, so that both resonant absorption and ion-neutral effects are at work.

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Figure 53: Wave damping by ion-neutral effects in an infinitely long cylindrical prominence thread. Ratio of the damping time to the period of the kink mode as a function of kza for a thread without transitional layer, i.e., l∕a = 0. (a) Results for a = 100 km and different ionization degrees: &tidle;μp = 0.5 (dotted line), &tidle;μp = 0.6 (dashed line), &tidle;μp = 0.8 (solid line), and &tidle;μp = 0.95 (dash-dotted line). Symbols are the approximate solution, given by Equation (33View Equation), for &tidle;μp = 0.8. (b) Results for &tidle;μ = 0.8 p and different thread widths: a = 100 km (solid line), a = 50 km (dotted line) and a = 200 km (dashed line). The shaded zone corresponds to the range of typically observed wavelengths of prominence oscillations (from Soler et al., 2009dJump To The Next Citation Point).

For a homogeneous thread (l∕a = 0), Soler et al. (2009dJump To The Next Citation Point) computed the damping rate for different ionization degrees (see Figure 53View Image). In agreement with the results displayed for the kink mode in Figure 49View Image, τd∕P has a maximum at the transition between the ohmic-dominated regime, which is almost independent of the ionization degree, to the region where Cowling’s diffusion is more relevant and the ionization degree has a significant influence. The approximate analytical solution for a given value of &tidle;μp agrees very well with the numerical solution in the region where Cowling’s diffusion dominates, while it significantly diverges from the numerical solution in the region where ohmic diffusion is relevant. Within the range of typically reported wavelengths, τd∕P is between 1 and 2 orders of magnitude larger than the measured values, so neither ohmic nor Cowling’s diffusion can account for the observed damping time.

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Figure 54: Wave damping by resonant absorption and ion-neutral effects in an infinitely long cylindrical prominence thread. Ratio of the damping time to the period of the kink mode as a function of kza for a thread with an inhomogeneous transitional layer. (a) Results for μ&tidle;p = 0.8 and different transitional layer widths: l∕a = 0 (dotted line), l∕a = 0.1 (dashed line), l∕a = 0.2 (solid line), and l∕a = 0.4 (dash-dotted line). Symbols are the solution in the thin boundary approximation (Equation [33View Equation]) for l∕a = 0.2. (b) Results for l∕a = 0.2 and different ionization degrees: &tidle;μp = 0.5 (dotted line), &tidle;μp = 0.6 (dashed line), &tidle;μp = 0.8 (solid line), and &tidle;μp = 0.95 (dash-dotted line). In both panels a = 100 km (from Soler et al., 2009dJump To The Next Citation Point).

For the inhomogeneous thread case (l∕a ⁄= 0), Figure 54View Imagea displays some relevant differences. First, the damping time is dramatically reduced for intermediate values of kza, which include the region of typically observed wavelengths. In this region, the ratio τd∕P becomes smaller as l∕a is increased, a behaviour consistent with damping by resonant absorption. The inclusion of the inhomogeneous transitional layer removes the smaller critical wavenumber and consequently the kink mode exists for very small values of kza. Figure 54View Imagea also shows a very good agreement between the numerical and the approximate solutions, this one given by Equation (33View Equation), for wavenumbers above kza ∼ 10−4, and a poor agreement in the range for which ohmic diffusion dominates, below kza ∼ 10−4. To understand this behaviour one has to bear in mind that the analytic approximate solution includes the effects of resonant absorption and Cowling’s diffusion, but not the influence of ohmic diffusion. Such as shown in Figure 54View Imageb, the ionization degree is only relevant for large wavenumbers, where the damping rate significantly depends on the ionization fraction through ohmic diffusion.

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Figure 55: Ratio of the damping time to the period of the kink mode as a function of k a z in an infinitely long thread with a = 100 km and l∕a = 0.2. The different line styles represent the results for a partially ionized thread with &tidle;μp = 0.8 and considering all the terms in the induction equation (solid line), for a partially ionized thread with &tidle;μp = 0.8 and neglecting Hall’s term (symbols) and for a fully ionized thread (dotted line) (from Soler et al., 2009dJump To The Next Citation Point).

Figure 55View Image displays the ranges of kza for which Cowling’s and Hall’s diffusion dominate. Hall’s diffusion is irrelevant in the whole range of kza studied by Soler et al. (2009dJump To The Next Citation Point), while Cowling’s diffusion dominates the damping for large kza. In the whole range of relevant wavelengths, resonant absorption is the most efficient damping mechanism and the damping time is independent of the ionization degree, as predicted by the analytical result (Equation [33View Equation]). On the contrary, ohmic diffusion dominates for very small kza. In this region, the damping time related to Ohm’s dissipation becomes even shorter than that due to resonant absorption, which means that the kink wave is mainly damped by ohmic diffusion.

5.4.2 Spatial damping

Motivated by the spatially damped propagating waves observed by Terradas et al. (2002) (see Section 3.6.3), the spatial damping of linear non-adiabatic magnetohydrodynamic waves in a homogenous, unbounded, magnetized, and fully ionized plasma was studied by Carbonell et al. (2006Jump To The Next Citation Point). The spatial damping in a flowing partially ionized plasma has been studied by Carbonell et al. (2010Jump To The Next Citation Point). Carbonell et al. (2006) found that the thermal (fast) wave shows the strongest (weakest) spatial damping. For periods longer than 1 s the spatial damping of magnetoacoustic waves is dominated by radiation, while at shorter periods the spatial damping is dominated by thermal conduction. Therefore, radiative effects on linear magnetoacoustic slow waves can be a viable mechanism for the spatial damping of short period prominence oscillations, while thermal conduction does not play any role. On the other hand, Carbonell et al. (2010) found that in the presence of a background flow, new strongly damped fast and Alfvén waves appear whose features depend on the joint action of flow and resistivity. The damping lengths of adiabatic fast and slow waves are strongly affected by partial ionization, which also modifies the ratio between damping lengths and wavelengths. For non-adiabatic slow waves, the unfolding in both wavelength and damping length induced by the flow allows efficient damping for periods compatible with those observed in prominence oscillations. In the case of non-adiabatic slow waves and within the range of periods of interest for prominence oscillations, the joint effect of both flow and partial ionization leads to efficient spatial damping of oscillations. For fast and Alfvén waves, the most efficient damping occurs at very short periods not compatible with those observed in prominence oscillations.

Using the same equilibrium model as in Soler et al. (2009d) (see Figure 50View Image), whose results have been presented in Section 5.4.1, Soler et al. (2011Jump To The Next Citation Point) investigated the spatial damping of propagating kink MHD waves in transversely non-uniform and partially ionized prominence threads. The damping mechanisms are resonant absorption and ion-neutral collisions (Cowling’s diffusion). In the absence of transitional layer, i.e., when the damping is due to Cowling’s diffusion exclusively, the non-dimensional wavelength, the damping length, LD, and the ratio of the damping length to the wavelength are displayed in Figure 56View Image. Regarding the wavelength, we see that the effect of Cowling’s diffusion is only relevant for periods much shorter than those observed (1 – 10 min, corresponding to 40 ≤ P ∕τ ≤ 400 Ap, with τ = a∕v Ap Ap the thread Alfvén travel time). On the other hand, an almost neutral plasma, i.e., &tidle;μp → 1, has to be considered to obtain an efficient damping and to achieve small values of the damping ratio within the relevant range of periods. Such very large values of &tidle;μp are probably unrealistic (Labrosse et al., 2010).

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Figure 56: Spatial damping of kink waves due to ion-neutral effects in an infinitely long prominence thread. Results for the kink mode spatial damping in the case l∕a = 0: (a) λ∕a, (b) LD∕a, and (c) L ∕λ D versus P ∕τ Ap for &tidle;μ p = 0.5, 0.6, 0.8, and 0.95. Symbols in panels (a), (b), and (c) correspond to the analytical solution given by Equations (12), (13), and (14) in Soler et al. (2011Jump To The Next Citation Point) in the thin tube approximation, while the horizontal dotted line in panel (c) corresponds to the limit of LD ∕λ for high frequencies. The shaded area denotes the range of observed periods of thread oscillations.

For resonantly damped modes, Figure 57View Image shows the results for different values of the thickness of the layer and fixed ionization degree. Figure 58View Image displays the results for different values of the ionization degree and a fixed transverse inhomogeneity length scale. Since the wavelength is not affected by the value of l∕a and has the same behaviour as in Figure 56View Imagea, both Figures 57View Image and 58View Image focus on LD ∕a and LD ∕λ. Depending on the period, two different behaviours of the solutions are obtained. For short periods, the damping length is independent of the layer thickness and is governed by the value of the ionization degree. On the contrary, for large periods, the damping length depends on the value of l∕a, but is independent of the ionization degree. This result indicates that resonant absorption dominates the damping for large periods, whereas Cowling’s diffusion is more relevant for short period oscillations. In addition, we can observe that the approximate transitional period for which the damping length by Cowling’s diffusion becomes shorter than that due to resonant absorption is much lower than the typically observed periods. This shows that resonant absorption is the dominant damping mechanism in the relevant range. The analytical approximation for the damping ratio obtained by Soler et al. (2011) provides an accurate description of the kink mode spatial damping in the relevant range of periods, such as shown by the diamonds in Figures 56View Image and 57View Image.

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Figure 57: Results for the kink mode spatial damping in an infinitely long prominence thread, in the case l∕a ⁄= 0: (a) LD ∕a and (b) LD ∕λ versus P ∕τAp for l∕a = 0.05, 0.1, 0.2, and 0.4, with &tidle;μp = 0.8. Symbols in panel (b) correspond to the analytical solution in the thin tube approximation, while the vertical dotted line is the approximate transitional period for l∕a = 0.1. The shaded area denotes the range of observed periods of thread oscillations.
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Figure 58: Results for the kink mode spatial damping in an infinitely long prominence thread, in the case l∕a ⁄= 0: (a) LD ∕a and (b) LD ∕λ versus P ∕τAp for &tidle;μp = 0.5, 0.6, 0.8, and 0.95, with l∕a = 0.2. Symbols in panel (b) correspond to the analytical solution in the thin tube approximation. The shaded area denotes the range of observed periods of thread oscillations.

For typically reported periods of thread oscillations, resonant absorption is an efficient mechanism for the kink mode spatial damping, while ion-neutral collisions have a minor role. Cowling’s diffusion dominates both the propagation and damping for periods much shorter than those observed, while resonant absorption could explain the observed spatial damping of kink waves in prominence threads.


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