DAMPING OF FILAMENT THREAD OSCILLATIONS: EFFECT OF THE SLOW CONTINUUM

The existence of the fine structure of solar filaments/prominences is clearly shown in high-resolution observations by both ground-based and satellite-on-board telescopes. These fine-structures, here called threads, have a typical width, W, and length, L, in the ranges 0.2 arcsec < W < 0.6 arcsec and 5 arcsec < L < 20 arcsec (Lin et al. 2005). Filament and prominence threads are believed to be thin magnetic flux tubes partially filled with prominence-like plasma and whose footpoints are anchored in the photosphere. Observers usually report the presence of transverse oscillations and waves in these fine structures (e.g., Lin et al. 2003, 2005, 2007; Okamoto et al. 2007), which have been interpreted in terms of magnetohydrodynamic (MHD) waves (e.g., Terradas et al. 2008). The reader is referred to Oliver & Ballester (2002), Ballester (2006), and Banerjee et al. (2007) for reviews of these events. In addition, a typical property of small-amplitude prominence oscillations observed in Doppler series is that they are quickly damped, with damping times of the order of several periods (Molowny-Horas et al. 1999; Terradas et al. 2002). Since small-amplitude oscillations are of local nature and related to periodic motions of threads, there must exist some mechanism which is able to damp thread oscillations in an efficient way.

The damping of prominence oscillations has recently been investigated in a number of papers (this topic has been reviewed by Oliver 2008). Soler et al. (2008) modeled a filament thread as a homogeneous cylinder embedded in an unbounded corona and studied the effect of nonadiabatic mechanisms (thermal conduction, radiative losses, and heating) on the wave damping. As in previous works (Carbonell et al. 2004; Soler et al. 2007), they concluded that thermal effects are only efficient in damping the slow mode, the fast kink mode being almost undamped. Subsequently, Arregui et al. (2008) used a more realistic thread model and included an inhomogeneous transition region between the cylindrical thread and the external medium. He neglected plasma pressure and adopted the so-called β = 0 approximation, where β is the ratio of the plasma pressure to the magnetic pressure. These authors found that the kink mode is resonantly coupled to Alfvén continuum modes due to the presence of the transverse inhomogeneous layer, and obtained a ratio of the damping time to the period τ_D/P ≈ 3 for typical prominence conditions. In the case of coronal loops (see, for example, recent reviews by Goossens et al. 2006; Goossens 2008), resonant absorption has been extensively investigated as a damping mechanism for the kink mode.

Although the plasma β in solar prominences is probably small, it is definitely nonzero. Hence, if gas pressure is taken into account in the model of Arregui et al. (2008), i.e., the β ≠ 0 case, it turns out that the kink mode phase speed, namely, c_k, is also within the slow (or cusp) continuum that extends between the internal, c_si, and external, c_se, sound speeds, i.e., c_si < c_k < c_se. In this case, the kink mode is not only resonantly coupled to Alfvén continuum modes but also to slow continuum modes. The frequency of the kink mode is both within the Alfvén and slow continua because of the high density and low temperature of the prominence material in comparison with the coronal values. Hence, the slow continuum damping arises as an additional mechanism to damp the kink mode in filament threads, and its efficiency needs to be assessed. Although the slow resonance has been previously investigated in the context of absorption of driven MHD waves in the solar atmosphere (e.g., Čadež et al. 1997; Erdélyi et al. 2001) and sunspots (e.g., Keppens 1996), the present work studies for the first time the effect of the slow resonance on the damping of the kink mode in filament threads. For coronal loops the kink speed is outside the slow continuum. A coronal loop is presumably hotter and denser than its surrounding corona, so the ordering of sound, Alfvén, and kink speeds is c_se < c_si < v_Ai < c_k < v_Ae. Therefore, there is no slow resonance for the kink mode in coronal loops and the present study does not apply to coronal loops.

This Letter is organized as follows. Section 2 contains a description of the model configuration and the mathematical method. The results are presented and discussed in Section 3. Finally, our conclusions are given in Section 4.

The model for the equilibrium configuration considered here is equivalent to that of Arregui et al. (2008). We model a filament thread as a straight cylinder with prominence-like conditions embedded in an unbounded corona, with a transverse inhomogeneous layer between both media (see Figure 1). We use cylindrical coordinates, namely, r, φ, and z for the radial, azimuthal, and longitudinal coordinates, respectively. We adopt the density profile by Ruderman & Roberts (2002) which only depends on the radial direction,

$$\begin{equation} \rho _{0}\left(r\right)=\left\lbrace \begin{array}{@{}clc} \rho _i,&{\rm if}&r\le a - l/2, \\ \rho _{\rm tr}\left(r\right),&{\rm if}&a-l/2<r<a+l/2,\\ \rho _e,&{\rm if}&r\ge a+l/2, \\ \end{array} \right. \end{equation} \tag{ 1 }$$

with

$$\begin{equation} \rho _{\rm tr}\left(r\right)=\frac{\rho _i}{2}\left\lbrace \left(1+\frac{\rho _e}{\rho _i}\right) - \left(1-\frac{\rho _e}{\rho _i}\right)\sin \left[\frac{\pi }{l}\left(r-a\right)\right]\right\rbrace. \end{equation} \tag{ 2 }$$

In these expressions, ρ_i is the internal density, ρ_e is the external (coronal) density, a is the tube mean radius, and l is the transitional layer width. The limits l/a = 0 and l/a = 2 correspond to a homogeneous tube and a fully inhomogeneous tube, respectively. We use the following densities: ρ_i = 5 × 10⁻¹¹ kg m⁻³ and ρ_e = 2.5 × 10⁻¹³ kg m⁻³. Therefore, the density contrast between the internal and external plasma is ρ_i/ρ_e = 200. The plasma temperature is related to the density through the usual ideal gas equation. We consider T_i = 8000 K and T_e = 10⁶ K for the internal and external temperatures, respectively. The magnetic field is taken homogeneous and orientated along the z-direction, ${\mathbf B}_0=B_0\hat{\mathbf z}$, with B₀ = 5 G everywhere. With these conditions, β ≈ 0.04.

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2.1. Analytical Dispersion Relation

We consider the linear, ideal MHD equations for the β ≠ 0 case. Since φ and z are ignorable coordinates, the perturbed quantities are put proportional to exp(iωt + imφ − ik_zz), where ω is the frequency, and m and k_z are the azimuthal and longitudinal wavenumbers, respectively. In the absence of a transitional layer, i.e., l/a = 0, the dispersion relation of magnetoacoustic waves is obtained by imposing the continuity of the radial displacement, ξ_r, and the total pressure perturbation, p_T, at r = a. This dispersion relation (Edwin & Roberts 1983) is $\mathcal {D}_m \left(\omega, k_z \right)= 0$, with

$$\begin{eqnarray} \mathcal {D}_m \left(\omega, k_z \right) &=& \rho _e \frac{m_i}{\left(\omega ^2 - k_z^2 v_{\mathrm{A}i}^2 \right)} \frac{J^{\prime }_m \left(m_i a \right)}{J_m \left(m_i a \right)} - \rho _i \frac{m_e}{\left(\omega ^2 - k_z^2 v_{\mathrm{A}e}^2 \right)} \nonumber \\ && \times\frac{K^{\prime }_m \left(m_e a \right)}{K_m \left(m_e a \right)}, \end{eqnarray} \tag{ 3 }$$

where $v_{{\rm A} i,e} = B_0/\sqrt{\mu \rho _{i,e}}$ is the internal/external Alfvén speed, and the quantities m_i and m_e are given by

$$\begin{eqnarray} && m_i^2 = \frac{\left(\omega ^2 - k_z^2 c_{\mathrm{s}i}^2 \right) \left(\omega ^2 - k_z^2 v_{\mathrm{A}i}^2 \right) }{\left(c_{\mathrm{s}i}^2 + v_{\mathrm{A}i}^2 \right) \left(c_{\mathrm{T}i}^2 k_z^2 - \omega ^2 \right) }, \nonumber \\ && m_e^2 = \frac{\left(\omega ^2 - k_z^2 c_{\mathrm{s}e}^2 \right) \left(\omega ^2 - k_z^2 v_{\mathrm{A}e}^2 \right) }{\left(c_{\mathrm{s}e}^2 + v_{\mathrm{A}e}^2 \right) \left(\omega ^2 - c_{\mathrm{T}e}^2 k_z^2 \right) }, \end{eqnarray} \tag{ 4 }$$

where $c_{{\rm T} i,e} = c_{s i,e} v_{{\rm A} i,e}\big / \sqrt{c_{s i,e}^2 + v_{{\rm A} i,e}^2 }$ is the internal/external cusp (or tube) speed.

When an inhomogeneous transitional layer is present in the model, we cannot obtain an analytical expression for the dispersion relation unless some assumptions are made. An elegant method for obtaining an analytical dispersion relation is to combine the jump conditions and the approximation of the thin boundary (TB). The jump conditions were derived by Sakurai et al. (1991a) and Goossens et al. (1995) for the driven problem and by Tirry & Goossens (1996) for the eigenvalue problem. This method was used, for example, by Sakurai et al. (1991b) for determining the absorption of sound waves in sunspots, Goossens et al. (1992) for determining surface eigenmodes in incompressible and compressible plasmas, and Van Doorsselaere et al. (2004) for kink eigenmodes in pressureless coronal loops. This method is valid if the inhomogeneous length-scale within the resonant layer is sufficiently small in comparison with the tube radius. Apart from the Alfvén resonance, Sakurai et al. (1991a) also provided jump conditions for the slow resonance, which can be applied to the present situation. Hence, the TB formalism allows us to assess the particular contribution of the slow resonance to the kink mode damping.

In our model, the magnetic field is straight and constant so that the variations of the local Alfvén frequency and the local cusp (or slow) frequency are only due to the variation of density. In that case the jump conditions at the Alfvén resonance point, namely, r = r_A, can be written as

$$\begin{equation} \left[ \xi _r \right] = -i \pi \frac{m^2 / r_{\rm A}^2}{\omega _{\rm A}^2 \left| \partial _r \rho _0 \right|_{\rm A}} p_{\rm T}, \quad \left[ p_{\rm T} \right] = 0, \quad {\rm at} \quad r = r_{\rm A}, \end{equation} \tag{ 5 }$$

where ω_A and |∂_rρ₀|_A are the Alfvén frequency and the modulus of the radial derivative of the density profile, respectively, both quantities evaluated at r = r_A. The respective jump conditions at the slow resonance point, namely r_s, are

$$\begin{eqnarray} \left[ \xi _r \right] &=& {-}i \pi \frac{k_z^2}{\omega _c^2 \left| \partial _r \rho _0 \right|_s} \left(\frac{c_{s}^2}{c_{s}^2 + v_{\mathrm{A}}^2} \right)^2 p_{\rm T}, \quad \left[ p_{\rm T} \right] = 0, \nonumber \\ && \qquad {\rm at} \quad r = r_s, \end{eqnarray} \tag{ 6 }$$

where ω_c and |∂_rρ₀|_s are the cusp frequency and the modulus of the radial derivative of the density profile, respectively, both quantities evaluated at r = r_s, while the factor $\big(\frac{c_{s}^2}{c_{s}^2 + v_{\mathrm{A}}^2} \big) = \big(\frac{\beta }{\beta + 2/\gamma } \big) \approx 0.034$ is a constant everywhere in the equilibrium, with γ = 5/3 the adiabatic index. The jump conditions (6) are independent of the azimuthal wavenumber, m, but they depend on the longitudinal wavenumber, k_z. So, for k_z = 0, jump conditions (6) become [ξ_r] = [p_T] = 0, meaning that there is no slow resonance in such a case.

Next, we use jump conditions (5) and (6) to obtain a correction to the dispersion relation (3) due to both resonances in the TB approximation,

$$\begin{equation} \mathcal {D}_m \left(\omega, k_z \right) \,{=}\, -i \pi \frac{m^2 / r_{\rm A}^2}{\omega _{\rm A}^2} \frac{\rho _i \rho _e}{\left| \partial _r \rho _0 \right|_{\rm A}}\, {-}\,i \pi \frac{k_z^2 }{\omega _c^2} \left(\frac{c_{s}^2}{c_{s}^2 + v_{\mathrm{A}}^2} \right)^2 \frac{\rho _i \rho _e}{\left| \partial _r \rho _0 \right|_s}. \end{equation} \tag{ 7 }$$

The first term on the right-hand side of Equation (7) corresponds to the effect of the Alfvén resonance, while the second term is present due to the slow resonance.

2.1.1. Expressions in the Thin Tube Approximation

Further analytical progress can be made by combining the TB and the thin tube (TT) approximations. This was done extensively by Goossens et al. (1992). We perform a asymptotic expansion of the Bessel functions present in the dispersion relation (7) by considering the long-wavelength limit, i.e., k_za ≪ 1, and only keep the lowest order, nonzero term of the expansion. Thus, the dispersion relation (7) becomes,

$$\begin{eqnarray} &&\rho _e \frac{m / a}{\left(\omega ^2 - k_z^2 v_{\mathrm{A}i}^2 \right)} + \rho _i \frac{m / a}{\left(\omega ^2 - k_z^2 v_{\mathrm{A}e}^2 \right)}+ i \pi \rho _i \rho _e \left[ \frac{m^2 / r_{\rm A}^2}{\omega _{\rm A}^2} \frac{1}{\left| \partial _r \rho _0 \right|_{\rm A}}\right. \nonumber \\ && \quad \left. +\, \frac{k_z^2 }{\omega _c^2} \left(\frac{c_{s}^2}{c_{s}^2 + v_{\mathrm{A}}^2} \right)^2 \frac{1}{\left| \partial _r \rho _0 \right|_s} \right] = 0. \end{eqnarray} \tag{ 8 }$$

Now, we write the frequency as ω = ω_R + iω_I. It follows from the resonant conditions that ω_A = ω_c = ω_R. The position of the Alfvén, r_A, and slow, r_s, resonance points can be computed by equating the real part of the kink mode frequency to the local Alfvén and slow frequencies, respectively. By this procedure, we obtain

$$\begin{equation} r_{\rm A} = a + \frac{l}{\pi } \arcsin \left[ \frac{\rho _i - \rho _e}{\rho _i + \rho _e} - \frac{2 v_{\mathrm{A}i}^2 k_z^2}{\omega _{\rm R}^2} \frac{\rho _i }{\left(\rho _i + \rho _e \right)} \right], \end{equation} \tag{ 9 }$$

for the Alfvén resonance point, and

$$\begin{equation} r_s = a + \frac{l}{\pi } \arcsin \left[ \frac{\rho _i - \rho _e}{\rho _i + \rho _e} - \frac{2 c_{\mathrm{s}i}^2 k_z^2}{\omega _{\rm R}^2} \frac{\rho _i }{\left(\rho _i + \rho _e \right)} \right], \end{equation} \tag{ 10 }$$

for the slow resonance point. In order to derive Equation (10), we have used the approximation ω²_c ≈ c²_sk²_z, which is valid for c²_s ≪ v²_A. Note that we need the value of ω_R to determine r_A and r_s. Expressions for |∂_rρ₀|_A and |∂_rρ₀|_s are obtained from the density profile (Equation (1)),

$$\begin{eqnarray} && \left| \partial _r \rho _0 \right|_{\rm A} =\left(\frac{\rho _i - \rho _e}{l} \right) \frac{\pi }{2}\cos \alpha _{\rm A},\nonumber\\ && \left| \partial _r \rho _0 \right|_s =\left(\frac{\rho _i - \rho _e}{l} \right) \frac{\pi }{2}\cos \alpha _s. \end{eqnarray} \tag{ 11 }$$

with α_A = π(r_A − a)/l and α_s = π(r_s − a)/l. We insert these expressions in Equation (8) and neglect terms with ω²_I, i.e., low-damping situation. Then we obtain an expression for the ratio ω_R/ω_I after some algebraic manipulations. Since the oscillatory period, P, and the damping time, τ_D, are related to the frequency as follows

$$\begin{equation} P = \frac{2 \pi }{\omega _{\rm R}}, \quad \tau _{\rm D} = \frac{1}{\omega _{\rm I}}, \end{equation} \tag{ 12 }$$

it is then straightforward to give an expression for τ_D/P,

$$\begin{eqnarray} \frac{\tau _{\rm D}}{P} &=& \mathcal {F} \frac{1}{(l/a)} \left(\frac{\rho _i + \rho _e}{\rho _i - \rho _e} \right) \left[ \frac{m}{\cos \alpha _{\rm A}} \right.\nonumber \\ && \left. +\,\frac{\left(k_z a \right)^2}{m} \left(\frac{c_{s}^2}{c_{s}^2 + v_{\mathrm{A}}^2} \right)^2 \frac{1}{\cos \alpha _s} \right]^{-1}, \end{eqnarray} \tag{ 13 }$$

where $\mathcal {F}$ is a numerical factor that depends on the density profile ($\mathcal {F} = 2/\pi$ in our case). As in previous expressions, the term with k_z corresponds to the contribution of the slow resonance. If this term is dropped and m = 1 and cos α_A = 1, Equation (13) is equivalent to Equation (3) of Arregui et al. (2008), which only takes the Alfvén resonance into account and was first obtained by Goossens et al. (1992) (see also equivalent expressions in Ruderman & Roberts 2002; Goossens et al. 2002).

Next, we assume r_A ≈ r_s ≈ a for simplicity, so cos α_A ≈ cos α_s ≈ 1. The ratio of the two terms in Equation (13) is

$$\begin{equation} \frac{\left(\tau _{\rm D}\right)_{\rm A}}{\left(\tau _{\rm D}\right)_s} \approx \frac{\left(k_z a \right)^2}{m^2} \left(\frac{c_{s}^2}{c_{s}^2 + v_{\mathrm{A}}^2} \right)^2, \end{equation} \tag{ 14 }$$

where (τ_D)_A and (τ_D)_s are the damping times exclusively due to the Alfvén and slow resonances, respectively. To make a simple calculation we note that the observed wavelengths of prominence oscillations correspond to 10⁻³ < k_za < 10⁻¹. For m = 1 and k_za = 10⁻² we obtain (τ_D)_A/(τ_D)_s ≈ 10⁻⁷, meaning that the Alfvén resonance is much more efficient than the slow resonance for damping the kink mode. Note that even in the extreme case of a very large β, i.e., c²_s → ∞ and so $\left(\frac{c_{s}^2}{c_{s}^2 + v_{\mathrm{A}}^2} \right) \rightarrow 1$, (τ_D)_A/(τ_D)_s ≪ 1 for typical values of k_za. By means of this simple calculation, we can anticipate that the slow resonance will be irrelevant for the kink mode damping.

2.2. Numerical Computations

We focus our investigation on the damping of the kink mode (m = 1). First, we fix the longitudinal wavenumber to k_za = 10⁻² and consider l/a = 0.2. We numerically compute (see Figure 2) the eigenfunctions: the velocity perturbation (v_r, v_φ, v_z), the magnetic field perturbation (B_1r, B_1φ, B_1z), the density perturbation (ρ₁), and the gas pressure perturbation (p₁). The behavior of v_φ and B_1r is similar to that of B_1φ and v_r, respectively, and hence they are not displayed in Figure 2. Because of the resonances the perturbations show large peaks in the transitional layer. The small panels of Figure 2 display the behavior of the eigenfunctions within the inhomogeneous layer, which allows us to ascertain the position of the peaks in more detail. The peaks of the perturbations v_r, B_1φ, and ρ₁ are related to the Alfvén resonance, while the perturbations v_z, B_1z, and p₁ are more affected by the slow resonance and their peaks appear at a different position. Moreover, we see that peaks related to the Alfvén resonance are wider than those related to the slow resonance, meaning that the slow resonance produces smaller spatial scales within the resonant layer. Considering the numerical value of ω_R, we get r_A/a ≈ 1 and r_s/a ≈ 1.08 from Equations (9) and (10), respectively. Both values are in a good agreement with the position of peaks in Figure 2.

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Next, we plot in Figure 3 the ratio of the damping time to the period, τ_D/P, as a function of k_za corresponding to the kink mode for l/a = 0.2. In this figure, we compare the numerical results with those obtained from the TB approximation. At first sight, we see that the slow resonance (dashed line in Figure 3) is much less efficient than the Alfvén resonance (symbols) in damping the kink mode. The individual contribution of each resonance in the TB approximation has been determined by solving Equation (7) and only taking the term on the right-hand side related to the desired resonance into account. For the wavenumbers relevant to prominence oscillations, 10⁻³ < k_za < 10⁻¹, the value of τ_D/P related to the slow resonance is between 4 and 8 orders of magnitude larger than the ratio obtained by the Alfvén resonance. The result of the thin tube limit (Equation (14)) agrees with this. On the other hand, the complete numerical solution (solid line) is close to the result for the Alfvén resonance. In agreement with Arregui et al. (2008), we obtain τ_D/P ≈ 3 in the relevant range of k_za. As was stated by Arregui et al. (2008, Figure 2(d)), the discrepancy between the numerical result and the TB approximation increases with l/a, the difference being around 20% for l/a = 1. However, for small, realistic values of l/a this discrepancy is less important and the TB approach is a very good approximation to the numerical result. For short wavelengths (k_za ≳ 10⁰) the value of τ_D/P increases and the efficiency of the Alfvén resonance as a damping mechanism decreases. For k_za ≈ 10² both the slow and the Alfvén resonances produce similar (and inefficient) damping times.

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In this Letter, we have studied the kink mode damping in filament threads due to resonant coupling to slow continuum modes. As far as we know, this is the first time that this phenomenon is studied in the context of solar prominences. By considering the TB approximation, we have found that, contrary to the Alfvén resonance, the slow resonance is very inefficient in damping the kink mode for typical prominence-like conditions. This conclusion also holds for fluting modes (m ⩾ 2). The very small damping due to the slow resonance is comparable to that due to thermal effects studied in previous papers (Soler et al. 2008). Therefore, we conclude that the effect of the slow resonance is not relevant to the damping of transverse thread oscillations, which is probably governed by the Alfvén resonance.

R.S., R.O., and J.L.B. acknowledge the financial support received from the Spanish MICINN and FEDER funds, and the Conselleria d'Economia, Hisenda i Innovació of the CAIB under grants AYA2006-07637 and PCTIB-2005GC3-03, respectively. R.S., R.O., and J.L.B. also want to acknowledge the International Space Science Institute teams "Coronal waves and Oscillations" and "Spectroscopy and Imaging of quiescent and eruptive solar prominences from space" for useful discussions. R.S. thanks the Conselleria d'Economia, Hisenda i Innovació for a fellowship. M.G. acknowledges support from the grant GOA/2009-009. M.G. is grateful to UIB for financial support during his stay at UIB and to J.L.B. for the hospitality at the Solar Physics Group.