5.3 Resonant damping of infinitely long thread oscillations

The phenomenon of resonant wave damping in non-uniform plasmas has been extensively studied in connection to wave-based coronal heating mechanisms (Ionson, 1978) and the damping of transverse coronal loop oscillations (Hollweg and Yang, 1988Jump To The Next Citation Point; Ruderman and Roberts, 2002Jump To The Next Citation Point; Goossens et al., 2002Jump To The Next Citation Point, 2010). The mechanism relies in the energy transfer from the transverse kink mode to small scale Alfvén waves because of the plasma inhomogeneity at the transverse spatial scales of the structures. This idea was put forward by Arregui et al. (2008bJump To The Next Citation Point), whose analysis is restricted to the damping of kink oscillations due to the resonant coupling to Alfvén waves in a pressureless (zero plasma-β) plasma. It was extended to the case in which both the Alfvén and the slow resonances are present by Soler et al. (2009eJump To The Next Citation Point). Here we discuss the main results from these works, whose aim is to assess the damping properties of resonant absorption. For this reason, the considered configurations are based on the infinitely long thread model of Figure 31View Image.

5.3.1 Resonant damping in the Alfvén continuum

Arregui et al. (2008bJump To The Next Citation Point) considered an individual and isolated thread modeled as a cylindrical magnetic flux tube of radius a in a gravity-free environment. The uniform magnetic field points along the axis of the tube (Figure 50View Image). In the zero plasma-β approximation, the thread is modeled as a density enhancement with a radial variation of density from its internal constant prominence value ρ p to the coronal constant value ρc over a non-uniform layer of thickness l. A typical value of the density contrast between the filament and coronal plasma is ζ = ρp∕ρc = 200.

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Figure 50: Model used by Arregui et al. (2008bJump To The Next Citation Point) to represent a radially non-uniform filament fine structure of mean radius a and transverse inhomogeneity length scale l.

MHD waves of axisymmetric one-dimensional cylindrical flux tubes are characterized by two wavenumbers, i.e., the azimuthal wavenumber, m, and the axial wavenumber, kz. They can have different nodes in the radial direction. Arregui et al. (2008bJump To The Next Citation Point) concentrated their analysis on the radially and longitudinally fundamental transverse wave with azimuthal number m = 1, 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 (22View Equation).

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 (l∕a ≪ 1). 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., 1992Jump To The Next Citation Point, 1995; Ruderman and Roberts, 2002Jump To The Next Citation Point; Goossens et al., 2002Jump To The Next Citation Point)

τd = F a- ζ-+-1. (30 ) P l ζ − 1
Here F is a numerical factor that depends on the particular variation of the density in the non-uniform layer. For a linear variation, 2 F = 4∕π (Hollweg and Yang, 1988; Goossens et al., 1992); for a sinusoidal variation, F = 2∕π (Ruderman and Roberts, 2002). Consider for example ζ = 200 as a typical density contrast and l∕a = 0.1. Then, Equation (30View Equation) 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. (2008bJump To The Next Citation Point). 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 51View Image. The damping is affected by the density contrast in the low contrast regime and τd∕P rapidly decreases for increasing thread density (Figure 51View Imagea). Interestingly, τd∕P 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 51View Imageb), but rapidly decreases with increasing inhomogeneity length-scale (Figure 51View Imagec).

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.

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Figure 51: Wave damping by Alfvén resonant absorption in an infinitely long prominence thread. Damping time over period for fast kink waves in filament threads with radius a = 100 km. (a) As a function of the density contrast, with l∕a = 0.2 and for two wavelengths. (b) As a function of the wavelength, with l∕a = 0.2, for two density contrasts. (c) As a function of the transverse inhomogeneity length-scale, for two combinations of wavelength and density contrast. In all plots solid lines correspond to analytical solutions given by Equation (30View Equation), with F = 2∕π (from Arregui et al., 2008bJump To The Next Citation Point).

5.3.2 Resonant damping in the slow continuum

Although the plasma-β in solar prominences is probably small, it is definitely non-zero. Soler et al. (2009eJump To The Next Citation Point) 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. (2008bJump To The Next Citation Point), Soler et al. (2009eJump To The Next Citation Point) evaluated the contribution of the damping due to the slow continuum to the total resonant damping of the kink mode.

Soler et al. (2009eJump To The Next Citation Point) used the density model of Section 5.3.1 and the plasma-β ≃ 0.04. 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

( ) ⌊ ( )2 ⌋− 1 τd a- ζ +-1- ⌈--m---- (kza)2 --c2s---- --1---⌉ P = F l ζ − 1 cos αA + m c2s + v2A cos αS . (31 )
Here F is the same numerical factor as in Equation (30View Equation), while αA = π(rA − a)∕l and αS = π(rS − a)∕l, with rA and rS the Alfvén and slow resonant positions. The term with kz corresponds to the contribution of the slow resonance. If this term is dropped and m = 1 and cos αA = 1 are taken, Equation (31View Equation) becomes Equation (30View Equation), that only takes into account the Alfvén resonance.

Equation (31View Equation) can now be directly applied to measure the relative contribution of each resonance to the total damping. To do that, Soler et al. (2009eJump To The Next Citation Point) assumed rA ≃ rS ≃ a, for simplicity, so cos αA ≃ cos αS ≃ 1. The ratio of the two terms in Equation (31View Equation) is then

τ (k a)2( c2 )2 -dA-≃ --z--- ----s-2- , (32 ) τdS m2 c2s + vA
where τ dA and τ dS are the respective contributions of the Alfvén and slow resonances in Equation (31View Equation). 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, m = 1 and kza = 10−2, then Equation (32View Equation) gives τdA ∕τdS ≃ 10−7.
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Figure 52: Wave damping by Alfvén and slow resonances in an infinitely long prominence thread. Kink mode ratio of the damping time to the period, τ ∕P d, as a function of the dimensionless wavenumber, kza, for l∕a = 0.2. The solid line is the full numerical solution. The symbols and the dashed line are the results of the thin boundary approximation for the Alfvén and slow resonances, i.e., the two terms in Equation (31View Equation). The shaded region represents the range of typically observed values for the wavelengths in prominence oscillations (from Soler et al., 2009eJump To The Next Citation Point).

This analytical predictions were further confirmed by Soler et al. (2009eJump To The Next Citation Point) by performing numerical computations outside the thin tube and thin boundary approximations. Figure 52View Image shows that the slow resonance is much less efficient than the Alfvén resonance. For the wavenumbers relevant to observed prominence oscillations, the value of τd∕P 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.

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