RESONANTLY DAMPED KINK MAGNETOHYDRODYNAMIC WAVES IN A PARTIALLY IONIZED FILAMENT THREAD

We model a filament thread as an infinite and straight cylindrical magnetic flux tube of radius a surrounded by an unbounded coronal environment. Cylindrical coordinates are used, namely r, φ, and z, for the radial, azimuthal, and longitudinal directions, respectively. In the following expressions, subscript 0 indicates equilibrium quantities while we use subscripts f and c to explicitly denote filament and coronal quantities, respectively. The magnetic field is homogeneous and orientated along the cylinder axis, ${\mathit \bf {B}}_0 = B_0 \hat{e}_z$, with B₀ = 5 G constant everywhere. We adopt the one-fluid approximation and consider a hydrogen plasma composed of ions (protons), electrons, and neutral atoms. Since we are interested in kink MHD waves supported by the filament thread body, we also assume the β = 0 approximation, which neglects gas pressure effects. With such assumptions (see details in Forteza et al. 2007 and Paper II), the plasma properties are characterized by two quantities: the fluid density, ρ₀, and the ionization fraction, $\tilde{\mu }_0$, which gives us information about the plasma degree of ionization. The allowed values of $\tilde{\mu }_0$ range between $\tilde{\mu }_0 = 0.5$ for a fully ionized plasma and $\tilde{\mu }_0 = 1$ for a neutral plasma.

The density profile assumed here only depends on the radial direction and is the same as in Paper I (which was adopted after Ruderman & Roberts 2002), namely,

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

with

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

where we take ρ_f = 5 × 10⁻¹¹ kg m⁻³ and ρ_c = 2.5 × 10⁻¹³ kg m⁻³, the density contrast between the filament and coronal plasma being ρ_f/ρ_c = 200. In these expressions, a is the thread mean radius and l is the transitional layer width. The limit cases l/a = 0 and l/a = 2 correspond to a homogeneous thread without transitional layer and a fully inhomogeneous thread, respectively. We also take the same functional dependence for the ionization fraction profile,

$$\begin{equation} \tilde{\mu }_{0}\left(r\right)=\left\lbrace \begin{array}{@{}l@{}l@{}l} \tilde{\mu }_f,& \quad {\rm if} &\quad r\le a - l/2, \\ \tilde{\mu }_{\rm tr}(r), &\quad {\rm if} &\quad a-l/2<r<a+l/2,\\ \tilde{\mu }_c, &\quad {\rm if} &\quad r\ge a+l/2, \\ \end{array} \right. \end{equation} \tag{ 3 }$$

with

$$\begin{equation} \tilde{\mu }_{\rm tr}\left(r\right)=\frac{\tilde{\mu }_f}{2}\left\lbrace \left(1+\frac{\tilde{\mu }_c}{\tilde{\mu }_f}\right) - \left(1-\frac{\tilde{\mu }_c}{\tilde{\mu }_f}\right)\sin \left[\frac{\pi }{l}\left(r-a\right)\right]\right\rbrace, \end{equation} \tag{ 4 }$$

where the filament ionization fraction, $\tilde{\mu }_f$, is considered a free parameter and the corona is assumed to be fully ionized, so $\tilde{\mu }_c = 0.5$. Recent studies by Gouttebroze & Labrosse (2009) on radiative transfer in cylindrical threads suggest (see their Figure 1(b)) an almost constant ionization degree in the core of the thread surrounded by a transitional zone where the plasma ionization degree abruptly increases toward fully ionized coronal conditions. Our ionization fraction profile (Equations (3) and (4)) attempts to represent such a behavior.

We consider the basic MHD equations for a partially ionized plasma in the one-fluid approach derived by Forteza et al. (2007; see also Braginskii 1965). Since we are dealing with small-amplitude perturbations, we restrict ourselves to linear perturbations. After removing gas pressure terms by setting β = 0, the relevant equations for our investigation are the momentum and the induction equations, whose linearized version in MKS units are

$$\begin{equation} \rho _0 \frac{\partial {\mathit {\bf v}}_1}{\partial t} = \frac{1}{\mu } \left[ \left(\nabla \times {\mathit {\bf B}}_1 \right) \times {\mathit {\bf B}}_0 \right], \end{equation}\vspace*{-8pt} \tag{ 5 }$$

$$\begin{eqnarray} \hspace{-2pt}\frac{\partial {\mathit {\bf B}}_1}{\partial t} &=& \nabla \times \left({\mathit {\bf v}}_1 \times {\mathit {\bf B}}_0\right) - \nabla \times \left(\eta \nabla \times {\mathit {\bf B}}_1 \right) + \nabla \times \left\lbrace \eta _{\rm A} \left[ \left(\nabla \times {\mathit {\bf B}}_1 \right)\right.\right.\nonumber\\ &&\left.\left.\times {\mathit {\bf B}}_0 \right] \times {\mathit {\bf B}}_0 \right\rbrace - \nabla \times \left[ \eta _{\rm H} \left(\nabla \times\;\, {\mathit {\bf B}}_1 \right) \times {\mathit {\bf B}}_0 \right],\!\! \end{eqnarray} \tag{ 6 }$$

along with the condition ∇ · B₁ = 0. In these equations, B₁ = (B_r, B_φ, B_z), v₁ = (v_r, v_φ, v_z) are the components of the magnetic field and velocity perturbations, respectively, while μ = 4π × 10⁻⁷ N A⁻² is the vacuum magnetic permeability. Quantities η, η_A, and η_H in Equation (6) are the coefficients of ohmic, ambipolar, and Hall's magnetic diffusion. Note that these nonideal terms of the induction equation appear due to collisions between the different plasma species. In particular, ohmic diffusion is mainly governed by electron–ion collisions and ambipolar diffusion is mostly caused by ion–neutral collisions. Hall's effect is enhanced by ion–neutral collisions since they tend to decouple ions from the magnetic field while electrons remain able to drift with the magnetic field (Pandey & Wardle 2008). The ambipolar diffusivity is commonly expressed in terms of the Cowling's coefficient, η_C, as follows:

$$\begin{equation} \eta _{\rm A} = \frac{\eta _{\rm C}- \eta }{B_0^2}. \end{equation} \tag{ 7 }$$

As explained in Paper II, η and η_C correspond to the magnetic diffusivities longitudinal and perpendicular to magnetic field lines, respectively. Due to the presence of neutrals, η_C ≫ η, meaning that perpendicular magnetic diffusion is much more efficient than longitudinal magnetic diffusion in a partially ionized plasma. In a fully ionized plasma η_C = η, so magnetic diffusion is then isotropic. Since η, η_C, and η_H are functions of the plasma physical conditions (see, e.g., Braginskii 1965; Khodachenko et al. 2004; Leake et al. 2005; Leake & Arber 2006; Pandey & Wardle 2008; Paper II for expressions), their values in our equilibrium depend on the radial direction. It is convenient for our following analysis to write ohmic, Cowling's, and Hall's diffusivities in a dimensionless form,

$$\begin{equation} \tilde{\eta }= \frac{\eta }{v_{\mathrm{A}f}a}, \qquad {\tilde{\eta }_{\rm C}}= \frac{\eta _{\rm C}}{v_{\mathrm{A}f}a}, \qquad \tilde{\eta }_{\rm H}= \frac{\eta _{\rm H}B_0}{v_{\mathrm{A}f}a}, \end{equation} \tag{ 8 }$$

where the tilde denotes the dimensionless quantity and $v_{\mathrm{A}f}= B_0 / \sqrt{ \mu _0 \rho _f}$ is the filament Alfvén speed. Note that $\tilde{\eta }^{-1}$ is usually called the magnetic Reynolds number. Figure 1 displays the radial profiles of $\tilde{\eta },{\tilde{\eta }_{\rm C}}$, and $\tilde{\eta }_{\rm H}$ according to the equilibrium properties. The figure focuses on the transitional layer, where the dimensionless coefficients vary by several orders of magnitude from internal to external values.

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Since φ and z are ignorable coordinates, perturbations are put proportional to exp(imφ + ik_zz − iωt), where ω is the oscillatory frequency, and m and k_z are the azimuthal and longitudinal wavenumbers, respectively. Then, Equations (5)–(6) become,

$$\begin{equation} \omega v_r = - \frac{v_{\mathrm{A}}^2}{B_0} (k_z B_r + i B^{\prime }_z), \end{equation} \tag{ 9 }$$

$$\begin{equation} \omega v_\varphi = \frac{v_{\mathrm{A}}^2}{B_0} \left(\frac{m}{r} B_z - k_z B_\varphi \right), \end{equation}\vspace*{-8pt} \tag{ 10 }$$

$$\begin{equation} \omega v_z = 0, \end{equation}\vspace*{-8pt} \tag{ 11 }$$

$$\begin{eqnarray} \omega B_r &=& -\!B_0 k_z v_r + \eta \left(\frac{m}{r} B^{\prime }_\varphi + \frac{m}{r^2} B_\varphi - i \frac{m^2}{r^2} B_r \right) \nonumber\\ &&+\, \eta _{\rm C}\left(k_z B^{\prime }_z - i k_z^2 B_r \right) + \eta _{\rm H}B_0 \left(i k_z \frac{m}{r} B_z - i k_z^2 B_\varphi \right),\nonumber\\ \end{eqnarray}\vspace*{-8pt} \tag{ 12 }$$

$$\begin{eqnarray} \omega B_\varphi\,{=}&&-\!B_0 k_z v_\varphi + \eta \left(i B^{\prime \prime }_\varphi + i \frac{1}{r} B^{\prime }_\varphi - i \frac{1}{r^2} B_\varphi + \frac{m}{r} B^{\prime }_r - \frac{m}{r^2} B_r \right)\nonumber \\ &&\,+\,\eta ^{\prime } \left(i B^{\prime }_\varphi + i \frac{1}{r} B_\varphi + \frac{m}{r} B_r \right) + \eta _{\rm C}\left(i k_z \frac{m}{r} B_z -i k_z^2 B_\varphi\vphantom{i k_z \frac{m}{r} B_z}\right)\nonumber\\ &&\, +\, \eta _{\rm H}B_0 \left(i k_z^2 B_r - k_z B^{\prime }_z \right), \end{eqnarray}\vspace*{-8pt} \tag{ 13 }$$

$$\begin{eqnarray} \omega B_z\,{=}&&-\!B_0 \left(i v^{\prime }_r + i \frac{1}{r} v_r - \frac{m}{r} v_\varphi \right)\,{+}\,\eta_{\rm C}\left(i B^{\prime \prime }_z + i \frac{1}{r} B^{\prime }_z\right.\nonumber\\ &&\left.{-}\,i \frac{m^2}{r^2} B_z + k_z B^{\prime }_r + k_z \frac{1}{r} B_r + i k_z \frac{m}{r} B_\varphi \right)\nonumber\\ &&\;{+}\,\eta _{\rm C}^{\prime }(k_z B_r + i B^{\prime }_z) + \eta _{\rm H}B_0 \left(k_z B^{\prime }_\varphi + k_z \frac{1}{r} B_\varphi - k_z \frac{m}{r} B_r \right)\nonumber\\ &&\;{+}\,\eta _{\rm H}^{\prime } B_0 \left(k_z B_\varphi - \frac{m}{r} B_z \right), \end{eqnarray} \tag{ 14 }$$

where the prime denotes the derivative with respect to r. From Equation (11) we get v_z = 0, so no longitudinal displacements are allowed in the β = 0 approximation. These equations form an eigenvalue problem, with ω being the eigenvalue and (v_r, v_φ, B_r, B_φ, B_z) the eigenvector. In the present work, we study the time damping of kink waves, so we assume m = 1 and take a real and positive k_z. A complex frequency, ω = ω_R + iω_I, is therefore expected, damped solutions corresponding to ω_I < 0. The period, P, and damping time, τ_D, are related to the real and imaginary parts of the frequency as follows:

$$\begin{equation} P = \frac{2 \pi }{\omega _{\rm R}}, \qquad \tau _{\rm D} = \frac{1}{\left|\omega _{\rm I}\right|}. \end{equation} \tag{ 15 }$$

2.2.1. Importance of Hall's Term

In Paper II, we showed that ohmic diffusion can be neglected in front of Cowling's diffusion for large k_za, while ohmic diffusion dominates for small values of k_za. In the present work, we have also included Hall's diffusion. Here, we compare the importance of Hall's term with the other nonideal terms in the induction equation. To achieve this, we first have to assess the typical length scales of the dissipative terms of the induction equation. We note that longitudinal derivatives only appear in the terms with ${\tilde{\eta }_{\rm C}}$ and $\tilde{\eta }_{\rm H}$ of Equations (12)–(14), while the terms with $\tilde{\eta }$ only contain radial and azimuthal derivatives. Therefore, the typical length scale of both Cowling's and Hall's terms is the longitudinal wavelength, λ_z, which can be expressed in terms of the longitudinal wavenumber, λ_z = 2πk⁻¹_z. On the other hand, the typical length scale of the ohmic term is the filament thread mean radius, a. By taking into account these relevant length scales, we define the dimensionless number $\mathcal {H}$ as the magnitude of Hall's term with respect to that of the ohmic term,

$$\begin{equation} \mathcal {H} = \frac{\left| \nabla \times \left[ \eta _{\rm H} \left(\nabla \times {\mathit {\bf B}}_1 \right) \times {\mathit {\bf B}}_0 \right] \right|}{\left| \nabla \times \left(\eta \nabla \times {\mathit {\bf B}}_1 \right) \right|} \sim \left(\frac{k_z a}{2 \pi } \right)^2 \frac{\tilde{\eta }_{\rm H}}{\tilde{\eta }}. \end{equation} \tag{ 16 }$$

From Equation (16), one can compute the value of k_za for which Hall's term becomes more important than the Ohm's term by setting $\mathcal {H} \approx 1$. So, one obtains

$$\begin{equation} \left(k_z a \right)_{\rm H} \approx 2 \pi \sqrt{\frac{\tilde{\eta }}{\tilde{\eta }_{\rm H}}}. \end{equation} \tag{ 17 }$$

We can compare this transitional wavenumber with that obtained in Section 2.3 of Paper II that determines when Cowling's diffusion begins to dominate over ohmic diffusion,

$$\begin{equation} \left(k_z a \right)_{\rm C} \approx 2 \pi \sqrt{\frac{\tilde{\eta }}{{\tilde{\eta }_{\rm C}}}}. \end{equation} \tag{ 18 }$$

Since in our equilibrium ${\tilde{\eta }_{\rm C}}> \tilde{\eta }_{\rm H}$, one gets (k_za)_C < (k_za)_H, meaning than Cowling's diffusion becomes dominant over ohmic diffusion for smaller wavenumbers than those needed for Hall's term to become relevant. Next, we have to know whether Cowling's diffusion or Hall's diffusion dominate for k_za beyond both transitional values. To do so, we define the dimensionless number $\mathcal {H}_{\rm C}$ as the magnitude of Hall's term with respect to that of the ambipolar term,

$$\begin{equation} \mathcal {H_{\rm C}} = \frac{\left| \nabla \times \left[ \eta _{\rm H} \left(\nabla \times {\mathit {\bf B}}_1 \right) \times {\mathit {\bf B}}_0 \right] \right|}{\left| \nabla \times \left\lbrace \eta _{\rm A} \left[ \left(\nabla \times {\mathit {\bf B}}_1 \right) \times {\mathit {\bf B}}_0 \right] \times {\mathit {\bf B}}_0 \right\rbrace \right|} \sim \frac{\tilde{\eta }_{\rm H}}{{\tilde{\eta }_{\rm C}}}. \end{equation} \tag{ 19 }$$

Considering again that ${\tilde{\eta }_{\rm C}}> \tilde{\eta }_{\rm H}$, the condition $\mathcal {H_{\rm C}} <1$ is always satisfied, therefore Cowling's diffusion is always more important than Hall's diffusion. By means of this simple dimensional analysis, we expect the presence of Hall's term to have a minor effect on the results.

Some analytical progress can be performed when no transitional layer is present, i.e., l/a = 0. We showed in Paper II that it is possible to give an analytical dispersion relation when the terms with η and η_H are dropped from the induction equation. In such a situation, the induction equation (Equation (6)) can be written in a compact form as follows:

$$\begin{equation} \frac{\partial {\mathit {\bf B}}_1}{\partial t} = \frac{\Gamma ^2_{\rm A} }{v_{\mathrm{A}}^2} \nabla \times \left({\mathit {\bf v}}_1 \times {\mathit {\bf B}}_0\right), \end{equation} \tag{ 20 }$$

where Γ²_A ≡ v²_A − iωη_C is the modified Alfvén speed squared (Forteza et al. 2008). Next, we follow the standard procedure (see, e.g., Edwin & Roberts 1983; Goossens et al. 2009) and obtain the dispersion relation of trapped waves by imposing that the radial displacement, ξ_r = iv_r/ω, and the total pressure perturbation, p_T = B₀B_z/μ, are both continuous at the thread boundary, r = a, and that perturbations must vanish at infinity. So, the dispersion relation governing transverse oscillations is $\mathcal {D}_m \left(\omega, k_z \right) = 0$, with

$$\begin{eqnarray} \mathcal {D}_m \left(\omega, k_z \right) =&& \frac{n_c}{\rho _c \left(\omega ^2 - k_z^2 \Gamma _{{\rm A}c}^2 \right)} \frac{K^{\prime }_m \left(n_c a \right)}{K_m \left(n_c a \right)}\nonumber\\ &&{-}\,\frac{m_f}{\rho _f \big(\omega ^2 - k_z^2 \Gamma _{{\rm A}f}^2 \big)} \frac{J^{\prime }_m (m_f a)}{J_m (m_f a)}, \end{eqnarray} \tag{ 21 }$$

where J_m and K_m are the Bessel function and the modified Bessel function of the first kind (Abramowitz & Stegun 1972), respectively, and the quantities m_f and n_c are given by

$$\begin{equation} m_f^2 = \frac{\big(\omega ^2 - k_z^2 \Gamma _{{\rm A}f}^2 \big)}{\Gamma _{{\rm A}f}^2}, \qquad n_c^2 = \frac{\left(k_z^2 \Gamma _{{\rm A}c}^2 - \omega ^2 \right)}{\Gamma _{{\rm A}c}^2}. \end{equation} \tag{ 22 }$$

Note that Equation (21) is valid for any value of m.

Next, our aim is to extend this analytical analysis to the case l/a ≠ 0. When an inhomogeneous transitional layer is present in the equilibrium, the kink mode is resonantly coupled to Alfvén continuum modes. The radial position where the Alfvén resonance takes place, r_A, can be computed by setting the kink mode frequency, ω_k, equal to the local Alfvén frequency, ω_A = v_Ak_z. The expression of r_A corresponding to our density profile was obtained in Paper I (Equation (9)),¹

$$\begin{equation} r_{\rm A} = a + \frac{l}{\pi } \arcsin \left[ \frac{\rho _f + \rho _c}{\rho _f - \rho _c} - \frac{2 v_{\mathrm{A}f}^2 k_z^2}{\omega _k^2} \frac{\rho _f }{(\rho _f - \rho _c)} \right]. \end{equation} \tag{ 23 }$$

The ideal MHD equations are singular at r = r_A. This singularity is removed if dissipative effects, such as magnetic diffusion or viscosity, are considered in a region around the resonance point, i.e., the dissipative layer. A method to obtain an analytical dispersion relation in the presence of an inhomogeneous transitional layer is to combine the jump conditions at the resonance point with the so-called thin boundary (TB) approximation, which was first used by Hollweg & Yang (1988). The jump conditions were first derived by Sakurai et al. (1991a) and Goossens et al. (1995) for the driven problem, and later by Tirry & Goossens (1996) for the eigenvalue problem. They have been used in a number of papers in the context of MHD waves in the solar atmosphere (e.g., Sakurai et al. 1991b; Goossens et al. 1992; Keppens et al. 1994; Erdélyi et al. 1995; Stenuit et al. 1998; Andries et al. 2000; Van Doorsselaere et al. 2004; Paper I among other works). An important result for the present investigation was obtained by Goossens et al. (1995), who proved that the jump conditions derived by Sakurai et al. (1991a) for Alfvén resonances in ideal MHD remain valid in dissipative MHD. This allows us to apply the jump conditions derived by Sakurai et al. (1991a) to our case. The assumptions behind the TB approximation and its applications have been recently reviewed by Goossens (2008). In short, the main assumption of the TB approximation is that there is a region around the dissipative layer where both ideal and dissipative MHD applies. In our equilibrium, we can assume that the thickness of the dissipative layer, namely δ_A, roughly coincides with the width of the inhomogeneous transitional layer. This condition is approximately verified for thin layers, i.e., l/a ≪ 1. So, we can simply connect analytically the perturbations from the homogeneous part of the tube to those of the external medium by means of the jump conditions and avoid the numerical integration of the dissipative MHD equations across the inhomogeneous transitional layer.

The jump conditions for the radial displacement and the total pressure perturbation provided by Sakurai et al. (1991a) in the case of a straight magnetic field are

$$\begin{equation} \left[ \left[ \xi _r \right]\right] = - i \pi \frac{m^2 \big / r_{\rm A}^2}{\left| \rho _0 \Delta \right|_{r_{\rm A}}} p_{\rm T}, \qquad \left[ \left[ p_{\rm T} \right] \right]= 0, \quad \hbox{at} \quad r= r_{\rm A}, \end{equation} \tag{ 24 }$$

where [[X]] = X_c − X_f stands for the jump of the quantity X, and $\Delta = \frac{d}{{d}r} (\omega ^2 - \omega _{\rm A}^2)$. By considering that in our model the magnetic field is straight and constant so that the variations of the local Alfvén frequency are only due to the variation of the equilibrium density, we can write $\left| \rho _0 \Delta \right|_{r_{\rm A}} = \omega _{\rm A}^2 \left| \partial _r \rho _0 \right|_{r_{\rm A}}$. In addition, from the resonance condition we have ω²_A = ω²_k. Applying the jump conditions (Equation (24)), we arrive at the dispersion relation in the TB approximation,

$$\begin{equation} \mathcal {D}_m \left(\omega, k_z \right) = -i \pi \frac{m^2 \big/ r_{\rm A}^2}{\omega _k^2 \left| \partial _r \rho _0 \right|_{r_{\rm A}}}, \end{equation} \tag{ 25 }$$

with $\mathcal {D}_m \left(\omega, k_z \right)$ defined in Equation (21). Note that in order to solve Equation (25) we need the value of the kink frequency, ω_k. For this reason, we use a two-step procedure. First, we solve the dispersion relation for the case l/a = 0 (Equation (21)) and obtain ω_k. Next, we assume that the real part of the frequency is approximately the same when the inhomogeneous transitional layer is included, allowing us to determine r_A from Equation (23) and therefore $\left| \partial _r \rho _0 \right|_{r_{\rm A}}$. Finally, we solve the complete dispersion relation (Equation (25)) with these parameters.

2.3.1. Expressions in the Thin Tube Limit

Equation (25) is a transcendental equation that has to be solved numerically. As in Paper I, it is possible to go further analytically by considering the thin tube (TT) approximation, i.e., k_za ≪ 1. The combination of both the TB and TT approximations has been applied in several works (Goossens et al. 1992, 2002, 2009; Ruderman & Roberts 2002). In the TT limit, we approximate the kink mode frequency as

$$\begin{equation} \omega _k \approx \sqrt{\frac{2}{ 1 + \rho _c / \rho _f }} v_{\mathrm{A}f}k_z. \end{equation} \tag{ 26 }$$

We put this expression in Equation (23) and obtain that r_A ≈ a. Next, we perform a first order, asymptotic expansion for small arguments of the Bessel functions of Equation (25). The dispersion relation then becomes

$$\begin{eqnarray} &&\hspace{-25pt}\rho _f \left(\omega ^2 - k_z^2 \Gamma _{{\rm A}f}^2 \right) + \rho _c \left(\omega ^2 - k_z^2 \Gamma _{{\rm A}c}^2 \right)\nonumber\\ &&= i \pi \left(\frac{m}{a} \right) \frac{\rho _f \rho _c}{\left| \partial _r \rho _0 \right|_{r_{\rm A}}} \frac{\big(\omega ^2 - k_z^2 \Gamma _{{\rm A}f}^2 \big) \left(\omega ^2 - k_z^2 \Gamma _{{\rm A}c}^2 \right)}{\omega _k^2 }. \end{eqnarray} \tag{ 27 }$$

We now write the frequency as ω = ω_k + iω_I, and the modified Alfvén speed squared is approximated by Γ²_A ≈ v²_A − iω_kη_C. We insert these expressions in Equation (27) and neglect terms with ω²_I and ω_Ik²_z. It is straightforward to obtain an expression for the ratio ω_I/ω_k,

$$\begin{eqnarray} \frac{\omega _{\rm I}}{\omega _k} =&& -\!\frac{\pi }{2} \left(\frac{m}{a} \right) \frac{\rho _f \rho _c}{(\rho _f + \rho _c)} \frac{1}{ \left| \partial _r \rho _0 \right|_{r_{\rm A}}}\nonumber\\ &&\times\left[ \frac{1}{4} \frac{(\rho _f - \rho _c)^2}{\rho _f \rho _c} + \left(\frac{1}{v_{\mathrm{A}f}^2} + \frac{1}{v_{\mathrm{A}c}^2} \right) \frac{{\eta _{\rm C}}_f {\eta _{\rm C}}_c}{2} k_z^2 \right]\nonumber \\ &&-\frac{1}{2} \frac{(\rho _f {\eta _{\rm C}}_f + \rho _c {\eta _{\rm C}}_c)k_z}{\sqrt{(\rho _f + \rho _c) \big(\rho _f v_{\mathrm{A}f}^2+ \rho _c v_{\mathrm{A}c}^2\big)}}. \end{eqnarray} \tag{ 28 }$$

The first term in Equation (28) owes its existence to the factor in the dispersion relation related to the TB approximation and represents the contribution of resonant absorption. For long wavelengths, the factor (1/v²_Af + 1/v²_Ac)η_{C f}η_{C c}k²_z/2 can be neglected, so the term related to the resonant damping is independent of the value of Cowling's diffusivity and, therefore, of the ionization degree. Then, one can see that in the TT case the term of Equation (28) due to the resonant damping takes the same form as in a fully ionized plasma, which was previously obtained by, e.g., Goossens et al. (1992, Equation (77)) and Ruderman & Roberts (2002, Equation (56)). On the other hand, the second term in Equation (28) is related to the damping by Cowling's diffusion and is also present in the case l/a = 0. This term is proportional to k_z, so we also expect it to be of a minor influence in the TT regime.

Next, we take into account that for the present sinusoidal density profile, $\left| \partial _r \rho _0 \right|_{r_{\rm A}} \approx \pi (\rho _f - \rho _c) / 2 l$. We insert this expression in Equation (28) and then use it to give a relation for the ratio of the damping time to the period,

$$\begin{equation} \frac{\tau _{\mathrm{D}}}{P} = \frac{2}{\pi } \left[ m \left(\frac{l}{a} \right) \left(\frac{\rho _f - \rho _c}{\rho _f + \rho _c } \right) + \frac{2 (\rho _f \tilde{\eta }_{{\rm C}f}+ \rho _c \tilde{\eta }_{{\rm C}c})k_z a}{\sqrt{2 \rho _f (\rho _f + \rho _c) }} \right]^{-1}, \end{equation} \tag{ 29 }$$

where we have used Equation (8) to rewrite Cowling's diffusivities in their dimensionless form. To perform a simple application, we compute τ_D/P from Equation (29) in the case m = 1, k_za = 10⁻², and l/a = 0.2, resulting in τ_D/P ≈ 3.18 for a fully ionized thread $(\tilde{\mu }_f = 0.5)$, and τ_D/P ≈ 3.16 for an almost neutral thread $(\tilde{\mu }_f = 0.95)$. We note that the obtained damping times are consistent with the observations. Furthermore, the ratio τ_D/P depends very slightly on the ionization degree, suggesting that resonant absorption dominates over Cowling's diffusion. To check this last statement, we compute the ratio of the two terms on the right-hand side of Equation (29), which allows us to compare the damping times exclusively due to resonant absorption, (τ_D)_RA, and Cowling's diffusion, (τ_D)_C,

$$\begin{equation} \frac{(\tau _{\mathrm{D}})_{\rm RA}}{(\tau _{\mathrm{D}})_{\rm C}} = \sqrt{\frac{2 (\rho _f + \rho _c)}{\rho _f}} \left(\frac{\rho _f \tilde{\eta }_{{\rm C}f}+ \rho _c \tilde{\eta }_{{\rm C}c}}{\rho _f - \rho _c} \right) \frac{k_z a}{m \left(l/a \right)}. \end{equation} \tag{ 30 }$$

This last expression can be further simplified by considering that in filament threads ρ_f ≫ ρ_c and $\tilde{\eta }_{{\rm C}f}\gg \tilde{\eta }_{{\rm C}c}$, so that

$$\begin{equation} \frac{\left(\tau _{\mathrm{D}}\right)_{\rm RA}}{\left(\tau _{\mathrm{D}}\right)_{\rm C}} \approx \sqrt{2} \tilde{\eta }_{{\rm C}f}\frac{k_z a}{m \left(l/a \right)}. \end{equation} \tag{ 31 }$$

Thus, we see that the efficiency of Cowling's diffusion with respect to that of resonant absorption increases with k_za and $\tilde{\mu }_f$ (through $\tilde{\eta }_{{\rm C}f}$). Considering the same parameters as before, one obtains (τ_D)_RA/(τ_D)_C ≈ 2 × 10⁻⁸ for $\tilde{\mu }_f = 0.5$, and (τ_D)_RA/(τ_D)_C ≈ 6 × 10⁻³ for $\tilde{\mu }_f = 0.95$, meaning that resonant absorption is much more efficient than Cowling's diffusion. From Equation (31) it is also possible to give an estimation of the wavenumber for which Cowling's diffusion becomes more important than resonant absorption by setting (τ_D)_RA/(τ_D)_C ≈ 1. So, one gets

$$\begin{equation} k_z a \approx \frac{m \left(l/a \right)}{\sqrt{2} \tilde{\eta }_{{\rm C}f}}. \end{equation} \tag{ 32 }$$

Considering again the same parameters, Equation (32) gives k_za ≈ 5 × 10⁵ for $\tilde{\mu }_f = 0.5$, and k_za ≈ 1.7 for $\tilde{\mu }_f = 0.95$. One has to bear in mind that Equation (31) is valid only for k_za ≪ 1, so we expect resonant absorption to be the dominant damping mechanism in the TT regime even for an almost neutral filament plasma. We will verify the analytical estimations of the present section by means of numerical computations.

To numerically solve the full eigenvalue problem (Equations (9)–(14)) we use the PDE2D code based on finite elements (Sewell 2005). An explanation of the numerical method can be found in Terradas et al. (2006), who used the PDE2D code to investigate the process of resonant absorption in coronal loops. We follow the same procedure as in Papers I and II. The integration of Equations (9)–(14) is performed from the cylinder axis, r = 0, to the edge of the numerical domain, r = r_max, where all perturbations vanish since the evanescent condition is imposed in the coronal medium, i.e., we restrict ourselves to trapped modes. The boundary conditions at r = 0 are imposed by symmetry arguments. To obtain a good convergence of the solution and to avoid numerical errors, we need to locate the edge of the numerical domain far enough from the filament thread mean radius to satisfy the evanescent condition. We have considered r_max = 100a. We use a nonuniform grid with a large density of grid points within the inhomogeneous transitional layer in order to correctly describe the small spatial scales that develop due to the Alfvén resonance.

First, we start with the case of a homogeneous filament thread without transitional layer, i.e., l/a = 0. This is the configuration studied in Paper II with the addition of Hall's term in the induction equation. Hence, we briefly discuss these results here and refer the reader to Paper II for a more complete description. Figure 2(a) displays τ_D/P versus k_za for different ionization degrees. In agreement with Paper II, we obtain that τ_D/P has a maximum which corresponds to 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 position of this transitional wavenumber is well approximated by Equation (18). As expected, the approximate solution obtained by solving Equation (21) for $\tilde{\mu }= 0.8$ (symbols in Figure 2(a)) agrees well with the numerical solution in the range of k_za 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 of filament oscillations (shaded region), τ_D/P is between 1 and 2 orders of magnitude larger than the observed values, which implies that neither ohmic nor Cowling's diffusion provide realistic damping times compatible with those observed.

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On the other hand, the kink mode exists as a propagating wave for k_za in the range k^{c −}_za < k_za < k^{c +}_za, where k^{c −}_z and k^{c +}_z are the critical wavenumbers described by Equation (38) of Paper II. With the notation adopted in the present work, these critical wavenumbers can be expressed in dimensionless form as follows:

$$\begin{equation} k_z^{c-} a \approx \frac{\tilde{\eta }}{2} \left(k_\perp a \right)^2, \qquad k_z^{c+} a \approx \frac{2}{{\tilde{\eta }_{\rm C}}}, \end{equation} \tag{ 33 }$$

where k_⊥ is the perpendicular wavenumber to the magnetic field lines and contains the effect of the model geometry (see details in Paper II). For our present cylindrical filament thread, this geometry-related factor can be written as (k_⊥a)² ≈ (k_ra)² + (k_φa)², where k_r and k_φ are the radial and azimuthal wavenumbers, respectively. We approximate k²_r by its expression in the ideal case,

$$\begin{equation} k_r^2 \approx \frac{\omega _k^2 - k_z^2 v_{\mathrm{A}f}^2}{v_{\mathrm{A}f}^2}, \end{equation} \tag{ 34 }$$

which for k_za ≪ 1 and ρ_f ≫ ρ_c simplifies to k²_r ≈ k²_z. On the other hand, the azimuthal wavenumber can be expressed as k²_φ ≈ m²/a². In the long-wavelength case, the azimuthal wavenumber dominates over the radial wavenumber, so we get (k_⊥a)² ≈ m². Next, we plot in Figure 2(b) the ratio of the damping time to the period as a function of k_za for $\tilde{\mu }_f = 0.8$ considering three different values of the filament thread radius. We see that the smaller the thread radius, the smaller τ_D/P and so the more attenuated the kink wave. In addition, the critical wavenumbers are shifted when a changes, the range of k_za for which the kink wave propagates being wider for thick threads than for thin threads. However, one must bear in mind that the observed width of filament threads is in the range 02–06 (Lin 2004), and therefore a ranges from 75 km to 375 km, approximately. The critical wavenumbers are far from the relevant values of k_za for the observed thread widths, and so they should not affect the kink wave propagation in realistic filament threads.

Hereafter, we include an inhomogeneous transitional layer in the model. This causes the Alfvén continuum to be present and so the kink mode is now resonantly coupled to continuum modes. In Figure 3(a), we assess the effect of the transitional layer on the ratio τ_D/P. For a fixed $\tilde{\mu }_f$, we compare the results corresponding to several values of the transitional layer width. Some relevant differences appear with respect to the case l/a = 0. First, we see that τ_D/P is dramatically reduced for intermediate values of k_za including the region of typically observed wavelengths. In this region, the ratio τ_D/P becomes smaller as l/a is increased, a behavior consistent with damping by resonant absorption. This result is confirmed by means of Figures 4 and 5, which display the kink mode eigenfunctions close to the resonance point given by Equation (23). We see that, with the exception of B_z, which is directly proportional to the total pressure perturbation, the other perturbations show small spatial-scale oscillations of large amplitude around the resonance point, i.e., within the so-called dissipative layer. The azimuthal components of both the velocity and magnetic field perturbations have the largest amplitudes because of the coupling to torsional Alfvén continuum modes. The thickness of the dissipative layer, δ_A, is given by Equation (5) of Sakurai et al. (1991a),

$$\begin{equation} \delta _{\rm A} = \left[ \left| \frac{\omega }{\Delta } \right| \eta \right]^{1/3}_{r_{\rm A}}. \end{equation} \tag{ 35 }$$

By considering again that in our model the variations of the local Alfvén frequency are only due to the variation of the equilibrium density, and approximating the resonant point by r_A ≈ a, we rewrite Equation (35) as follows:

$$\begin{equation} \delta _{\rm A} = \left[\frac{l}{\pi } \left(\frac{\rho _f + \rho _c}{\rho _f - \rho _c} \right) \eta \right]^{1/3}_{r_{\rm A}}. \end{equation} \tag{ 36 }$$

We see that δ_A ∼ l^1/3, so the larger the transitional region width, the larger the thickness of the dissipative layer. This is consistent with Figures 4 and 5, in which the small spatial-scale oscillations due to the resonance spread over a wider region for l/a = 0.5 (Figure 5) than for l/a = 0.2 (Figure 4).

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Turning back again to Figure 3(a), we see that the ratio τ_D/P for large k_za is independent of the transitional layer width and coincides with the solution in the absence of transitional layer. The cause of this behavior is that perturbations are essentially confined within the homogeneous part of the thread for large k_za, and therefore the kink mode is mainly governed by the internal plasma conditions. On the other hand, the solution for small k_za is completely different when l/a ≠ 0. We note that the inclusion of the inhomogeneous transitional layer (i.e., for l/a ≠ 0) removes the smaller critical wavenumber, k^{c −}_z, and consequently the kink mode exists for very small values of k_za.

We study in Figure 3(b) the dependence of τ_D/P on the ionization degree for a fixed l/a, and it turns out that the ionization degree is only relevant for large k_za, where the ratio τ_D/P significantly depends of $\tilde{\mu }_f$. Figure 6 allows us to shed light on this result. There, we assess the ranges of k_za where Cowling's and Hall's diffusion dominate. As expected, we find that Hall's diffusion is irrelevant in the whole studied range of k_za, while Cowling's diffusion, caused by the presence of neutrals, dominates the damping for large k_za. 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 was analytically predicted in Section 2.3.1. On the contrary, we find that ohmic diffusion dominates for very small k_za. It is important noting that although the kink mode is still resonantly coupled to Alfvén continuum modes, the damping time related to Ohm's dissipation becomes smaller than that due to resonant absorption, meaning that the kink wave is mainly damped by ohmic diffusion for very small k_za. Since η is almost independent on the ionization degree, the ratio τ_D/P in the region of small k_za slightly depends on the value of $\tilde{\mu }_f$.

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Finally, we compare the full numerical solution with that obtained by solving the analytical dispersion relation in the TB approximation (Equation (25)). For the case l/a = 0.2, and $\tilde{\mu }_f = 0.8$, we plot by means of symbols in Figure 3(a) the result obtained from Equation (25). We can see a very good agreement between the approximation and the numerical result (solid line) for k_za ≳ 10⁻⁴, while both solutions do not agree for k_za ≲ 10⁻⁴, which corresponds to the range of k_za for which ohmic diffusion dominates. Equation (25) was derived by taking into account the effect of resonant absorption and Cowling's diffusion, but the influence of ohmic diffusion on the damping is not included. For this reason, the approximate solution does not correctly describe the kink mode damping for very small k_za, which corresponds to extremely long and unrealistic wavelengths, while it successfully agrees with the full numerical solution for realistic wavelengths and larger values of k_za.