The Thermal Instability of Solar Prominence Threads

THE THERMAL INSTABILITY OF SOLAR PROMINENCE THREADS

R. Soler1,2, J. L. Ballester2, and M. Goossens1 1 Centre for Plasma Astrophysics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium; [email protected] 2 Departament de F´ısica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain Received 2010 November 16; accepted 2011 February 10; published 2011 March 21

ABSTRACT The fine structure of solar prominences and filaments appears as thin and long threads in high-resolution images. In Hα observations of filaments, some threads can be observed for only 5–20 minutes before they seem to fade and eventually disappear, suggesting that these threads may have very short lifetimes. The presence of an instability might be the cause of this quick disappearance. Here, we study the thermal instability of prominence threads as an explanation of their sudden disappearance from Hα observations. We model a prominence thread as a magnetic tube with prominence conditions embedded in a coronal environment. We assume a variation of the physical properties in the transverse direction so that the temperature and density continuously change from internal to external values in an inhomogeneous transitional layer representing the particular prominence–corona transition region (PCTR) of the thread. We use the nonadiabatic and resistive magnetohydrodynamic equations, which include terms due to thermal conduction parallel and perpendicular to the magnetic field, radiative losses, heating, and magnetic diffusion. We combine both analytical and numerical methods to study linear perturbations from the equilibrium state, focusing on unstable thermal solutions. We find that thermal modes are unstable in the PCTR for temperatures higher than 80,000 K, approximately. These modes are related to temperature disturbances that can lead to changes in the equilibrium due to rapid plasma heating or cooling. For typical prominence parameters, the instability timescale is of the order of a few minutes and is independent of the form of the temperature profile within the PCTR of the thread. This result indicates that thermal instability may play an important role for the short lifetimes of threads in the observations. Key words: instabilities – magnetic fields – magnetohydrodynamics (MHD) – Sun: corona – Sun: filaments, prominences

1. INTRODUCTION Theoretically, the fine structures have been modeled as magnetic flux tubes anchored in the solar photosphere (e.g., Solar prominences and filaments are large-scale magnetic Ballester & Priest 1989; Rempel et al. 1999), which are structures of the solar corona. The main issues regarding the only partially filled with the cool (∼104 K) filament material, physics, dynamics, and modeling of these coronal inhabitants while the rest of the tube is occupied by hot coronal plasma. have been recently reviewed by Labrosse et al. (2010) and Therefore, the magnetic field is oriented along the axis of the Mackay et al. (2010). High-resolution observations reveal that fine structure. This model is conceptually in agreement with the prominences and filaments are formed by long (5–20) and thin idea that the dense prominence material is trapped in dips near (0.2–0.6) fine structures, usually called threads. Although the the apex of a magnetic arcade connecting two photospheric existence of the fine structure of prominences was discovered a regions of opposite magnetic polarity. The dips are supposed long time ago (e.g., Menzel & Wolbach 1960; Engvold 1976), to correspond to the observed threads, which are piled up to its properties and dynamics could only be studied in more detail form the prominence body. It has also been suggested from with recent high-resolution observations. The fine structures differential emission measure studies that each thread might show up as dark ribbons in Hα images of filaments on the solar be surrounded by its own prominence–corona transition region disk from the Swedish Solar Telescope (e.g., Lin 2004;Lin (PCTR) where the plasma physical properties would abruptly et al. 2007, 2008, 2009) and as bright features in observations of vary from prominence to coronal conditions (Cirigliano et al. prominences in the solar limb from the Solar Optical Telescope 2004). aboard the Hinode satellite (e.g., Okamoto et al. 2007; Berger Prominence threads are highly dynamic (see, e.g., Heinzel et al. 2008; Chae et al. 2008; Ning et al. 2009; Schmieder et al. 2007; Engvold 2008). For example, transverse thread oscil- 2010). Statistical studies show that the orientation of threads lations and propagating waves along the threads seem to be with respect to the filament long axis can significantly vary ubiquitous in prominences, which have been interpreted in terms within the same filament (Lin 2004), with 20◦ a mean value of magnetohydrodynamic (MHD) waves (see the reviews by typically reported. Vertical threads are more commonly seen Ballester 2006;Oliver2009; Arregui & Ballester 2010). Mass in quiescent prominences (e.g., Berger et al. 2008; Chae et al. flows along threads, with typical flow velocities of less than 2008) whereas horizontal threads are usually observed in active 30 km s−1, have also been frequently reported (e.g., Zirker et al. region prominences (e.g., Okamoto et al. 2007). Schmieder et al. 1994, 1998; Lin et al. 2003; Chae et al. 2008). An interest- (2010) recently pointed out that vertical threads might actually ing property of the observations is the apparent short lifetime of be a pileup of horizontal threads which seem to be vertical some threads when they are observed in Hα sequences (e.g., Lin structures when projected on the plane of the sky. However, 2004; Lin et al. 2005, 2009). Typically, the threads can be fol- this is still a matter of controversy. Since threads are observed lowed for only 5–20 minutes before they seem to fade with time in both spines and barbs, it is believed that they are the basic and eventually disappear. The cause of this quick disappearance building blocks of prominences and filaments (Engvold 2004). is unknown, although several explanations have been proposed.

1 The Astrophysical Journal, 731:39 (13pp), 2011 April 10 Soler, Ballester, & Goossens A possible explanation is related to the presence of flows and thermal conduction may be responsible for the formation of mass motions (e.g., Chae et al. 2008; Chae 2010; Schmieder prominence fine structure. In subsequent works, Ireland et al. et al. 2010), which may trigger a Kelvin–Helmholtz instability (1992, 1998) investigated the influence of magnetic diffusion on (KHI). Recently, Soler et al. (2010b) investigated the KHI in the thermal continuum. These authors concluded that magnetic magnetic flux tubes due to shear flows generated by transverse diffusion plays a similar role to that of perpendicular thermal motions of the tube, while Zaqarashvili et al. (2010) studied the conduction as both mechanisms replace the continuum by dis- KHI in twisted tubes due to longitudinal flows. When the results crete modes. of both papers are applied to prominences, one obtains that nei- In the present work, we follow the method of Van der ther the observed transverse velocity amplitudes of oscillating Linden et al. (1991) and Van der Linden & Goossens (1991) threads nor the longitudinal flow velocities are large enough to and investigate the thermal stability of an inhomogeneous trigger a KHI on timescales consistent with the apparent lasting prominence thread model. The equilibrium model adopted here time of the threads. On the other hand, a different explanation is a cylindrical magnetic tube, representing a prominence thread, was suggested by Lin (2004) and Lin et al. (2005). According embedded in a coronal environment. The plasma physical to these authors, rapid cooling or heating may cause the maxi- properties, i.e., temperature, density, etc., are inhomogeneous mum of the plasma emission to fall outside the bandpass of the in the transverse direction and vary from prominence to PCTR filter, and so the thread would become invisible in Hα images. and coronal values. In this investigation, we assume that the A mechanism that could lead to a rapid heating or cooling of the equilibrium is uniform in the longitudinal direction. The effect of prominence material is a thermal instability. Here, we explore longitudinal plasma inhomogeneity is relegated to a forthcoming this possibility by studying the thermal instability of prominence study. We use the nonadiabatic and resistive MHD equations threads. and superimpose linear perturbations on the equilibrium state. Thermal or condensation modes have been extensively inves- Parallel and perpendicular thermal conduction, radiative losses, tigated in homogeneous plasmas (e.g., Parker 1953; Field 1965; heating, and magnetic diffusion are the nonideal effects included Heyvaerts 1974). A relevant work in the context of prominences in our equations. We combine both analytical methods and was performed by Carbonell et al. (2004) who studied the ther- numerical computations to study the properties of the discrete mal mode in homogeneous plasmas with prominence, PCTR, thermal continuum modes, focusing on the unstable modes with and coronal conditions, considering parallel thermal conduction largest growth rates. to magnetic field lines and the optically thin radiative loss func- This paper is organized as follows. Section 2 contains the tion of Hildner (1974). Carbonell et al. (2004) obtained that, basic equations and a description of the equilibrium configu- for long wavelengths, the thermal mode is unstable for PCTR ration. Then, Sections 3–5 contain a theoretical study of the temperatures since thermal conduction is not efficient enough properties of the thermal instability. The stability of the thermal to stabilize the thermal disturbance. In the case of an inhomoge- continuum in the absence of perpendicular thermal conduction neous plasma, thermal modes were studied in detail in the works and magnetic diffusion is discussed in Section 3 by following by Van der Linden et al. (1991), Van der Linden & Goossens the analysis of Van der Linden et al. (1991), while the unstable (1991), and Van der Linden (1993). As the present investigation thermal modes are investigated in Section 4 when both perpen- is based on and inspired by these previous works, we summarize dicular thermal conduction and magnetic diffusion are included. their relevant results in the following paragraph. This will also A parametric study of the growth rate of the most unstable solu- enable us to clearly define the advancement made in the present tion is performed in Section 5. Then, we discuss in Section 6 the paper compared with these previous investigations. physical implication of our theoretical results for the stability Van der Linden et al. (1991) studied the linear nonadiabatic and lifetime of prominence threads. Finally, the summary of our MHD spectrum of a magnetic cylinder taking into account the results is given in Section 7. effect of plasma inhomogeneity in the transverse direction. They included radiative losses, heating, and thermal conduction par- 2. BASIC EQUATIONS AND EQUILIBRIUM allel to the magnetic field lines but neglected perpendicular thermal conduction. These authors pointed out the important The basic MHD equations governing the dynamics of a result that, along with the classical Alfven´ and slow continua, nonadiabatic and resistive plasma are there exists an additional thermal continuum, which can be un- Dρ stable depending on the physical properties of the equilibrium. + ρ∇·v = 0, (1) Subsequently, Van der Linden & Goossens (1991) investigated Dt the effect on the thermal continuum of thermal conduction perpendicular to the magnetic field lines and showed that perpen- Dv 1 ρ =−∇p + (∇×B) × B, (2) dicular thermal conduction replaces the continuum by a dense Dt μ set of discrete quasi-continuum modes. These quasi-continuum modes retain the basic stability properties and growth rates of ∂B the continuum and are confined within thin conductive layers =∇×(v × B) −∇×(η∇×B) , (3) around the position of the continuum singularities. The width ∂t of the conductive layer, and so the spatial scale of the quasi- continuum modes, depends on the value of the perpendicular Dp γp Dρ − + (γ − 1) [ρL(T,ρ) −∇·(κ ·∇T ) thermal conductivity. Van der Linden & Goossens (1991) and Dt ρ Dt Van der Linden (1993) applied these results to the context of −η|∇ × B|2 = 0, (4) solar prominences and showed that the spatial scales related to the most unstable quasi-continuum modes of the spectrum are consistent with the size of the prominence threads reported from T high-resolution observations. This suggests that perpendicular p = ρR , (5) μ˜

2 The Astrophysical Journal, 731:39 (13pp), 2011 April 10 Soler, Ballester, & Goossens

Table 1 Values in MKS Units of the Parameters in the Radiative Loss Function (Equation (8)) Corresponding to Several Temperature Regimes

Regime Temperature Range χ ∗ α Prominence-1.1 T 15×103 K1.76 × 10−13 7.4 Prominence-1.2 T 15×103 K1.76 × 10−53 17.4 Prominence-1.3 T 15×103 K7.01 × 10−104 30 PCTR-2 15×103 K 8×105 K1.97 × 1024 −1.0 Klimchuk–Raymond-1 T 104.97 K3.91 × 109 2.0 Klimchuk–Raymond-2 104.97 105.67 K6.81 × 1018 0.0

Notes. Prominence-1.1, PCTR-1, PCTR-2, PCTR-3, and Corona-5 regimes are parameterizations from Hildner (1974). Prominence-1.2 and Prominence-1.3 regimes are taken from Milne et al. (1979) and Rosner et al. (1978), respectively. The three Prominence regimes represent different plasma optical thicknesses, Prominence- 1.1 corresponding to optically thin plasma, while Prominence-1.2 and Prominence-1.3 are for optically thick and very thick plasmas, respectively. The three Klimchuk–Raymond regimes are adapted from Klimchuk & Cargill (2001), where we have only taken into account the range of temperatures considered in our equilibrium.

∇· = D = ∂ ·∇ Finally, the heat-loss function L(T,ρ) accounts for the bal- along with the condition B 0, where Dt ∂t + v is the material derivative for time variations following the motion. ance between radiative cooling and heating. The determination In Equations (1)–(5), ρ, p, T, v, and B are the mass density, of an analytical function of the temperature and density that de- gas pressure, temperature, velocity vector, and magnetic field scribes radiative losses of the prominence plasma is a very diffi- vector, respectively. In addition, μ = 4π × 10−7 NA−2 is cult work that requires the numerical solution of nonlocal ther- the magnetic permeability, γ = 5/3 is the adiabatic index, modynamic equilibrium (NLTE) radiative transfer equations. R = 8.3 × 103 m2 s−2 K−1 is the ideal gas constant, L(T,ρ)is This is beyond the purpose and scope of the present study. One the heat-loss function, κ is the thermal conductivity tensor, η is reasonable semi-empirical approximation to an expression for the magnetic diffusivity, and μ˜ is the mean atomic weight. In a the radiative loss function was obtained by Hildner (1974). This fully ionized medium, μ˜ = 0.5. author assumed an optically thin plasma (e.g., Cox & Tucker 1969) and performed a piecewise fit of the radiative losses previ- 2.1. Nonideal Terms ously computed by several authors as a function of temperature. Here, we adopt Hildner’s approach. The functional expression The induction (Equation (3)) and energy (Equation (4)) equa- of L(T,ρ) considered by Hildner (1974)is tions contain several nonideal terms whose physical meaning is explained next. The term with the factor η in Equation (3) L(T,ρ) = ρχ∗T α − h, (8) corresponds to Ohm’s magnetic diffusion, while the equivalent ∗ term in Equation (4) accounts for Ohm’s heating. The other where χ and α are piecewise constants depending on the tem- nonideal terms in Equation (4) correspond to nonadiabatic ef- perature and h is an arbitrary heating function. The assumption fects, i.e., thermal conduction, radiative losses, and an arbitrary of an optically thin plasma seems a reasonable approximation heating input. for coronal and PCTR temperatures, whereas cool prominence As for thermal conduction, the thermal conductivity tensor, κ, plasmas may be considered optically thick. Some authors (e.g., can be expressed in terms of its parallel, κ, and perpendicular, Rosner et al. 1978; Milne et al. 1979) have proposed correc- tions to the values of χ ∗ and α in the range of cool prominence κ⊥, components to the magnetic field, namely, κ = κeˆB eˆB + ˆ temperatures, i.e., T<15,000 K, in order to represent ra- κ⊥(I −êB eˆB ), where eˆB = B/ |B| is the unit vector in the ˆ diation losses in optically thick plasmas using Equation (8). magnetic field direction and I is the identity tensor. The parallel In the literature, there are more recent parameterizations and perpendicular conductivities are dominated by the effect for χ ∗ and α which update Hildner’s values. In particular, of electrons and ions, respectively. The perpendicular electron the so-called Klimchuk–Raymond parameterization (see, e.g., conductivity and the parallel ion conductivity are negligible Klimchuk & Cargill 2001) may be a better representation of the in fully ionized plasmas. Expressions for the conductivities in radiative losses in the hotter part of the PCTR and in the so- MKS units for a hydrogen plasma are (see, e.g., Parker 1953; lar corona. On the contrary, the Klimchuk–Raymond function Spitzer 1962; Braginskii 1965) may not be adequate in the cool part of the thread. We use the

5/2 Klimchuk–Raymond function as an alternative to Hildner’s pa- −10 T κ = 1.8 × 10 , (6) rameterization in the hotter part of the equilibrium. The values ln Λ of the parameters χ ∗ and α for various temperature ranges and regimes are given in Table 1. Λ 2 −42 ln ρ κ⊥ = 1.48 × 10 , (7) 2.2. Equilibrium Configuration 2| |2 1/2 mi B T Equations (1)–(5) are applied to the following equilibrium where mi is the proton mass and ln Λ is the Coulomb logarithm, configuration. The prominence thread model is composed of a whose value is generally between 5 and 20 and has a weak straight and cylindrical magnetic flux tube embedded in a coro- dependence on temperature and density (see, e.g., Priest 1982). nal environment. We use cylindrical coordinates, namely, r, ϕ,

3 The Astrophysical Journal, 731:39 (13pp), 2011 April 10 Soler, Ballester, & Goossens

Figure 1. (a) Equilibrium temperature profiles, T0, normalized to the internal temperature, Tp, in the case l/R = 1. The different line styles correspond to the linear profile (solid), the sinusoidal profile (dashed), and the Gaussian profile (dot-dashed). (b) Corresponding density profiles, ρ0, normalized to the internal density, ρp. The vertical dotted lines in both panels denote the boundaries of the transitional zone. Note that the vertical axis in panel (b) is in the logarithmic scale. and z for the radial, azimuthal, and longitudinal coordinates, re- is denoted by R and is located at the center of the PCTR. The spectively. In this configuration, the equilibrium quantities are thickness of the PCTR is denoted by l and corresponds to the re- invariant in the azimuthal and longitudinal directions, so they gion R − l/2 R+ l/2 in order to avoid this small jump of the temperature when the T , if r R − l/2, p Gaussian profile is used. Figure 1 displays a plot of the three T (r) = T (r) , if R − l/2

4 The Astrophysical Journal, 731:39 (13pp), 2011 April 10 Soler, Ballester, & Goossens we restrict ourselves to the linear regime and Equations (1)–(5) ∂κ⊥ ∂κ⊥ ∂κ⊥ B0 · b1 are linearized. Since the equilibrium is invariant in the azimuthal κ˜⊥ = ρ1 + T1 + . ∂ρ ∂T ∂ |B0| |B0| and longitudinal directions, we write all perturbations propor- ρ0,T0 ρ0,T0 ρ0,T0 (24) tional to exp (st + imϕ − ikzz), where m and kz are the azimuthal and longitudinal wavenumbers, respectively, and s is the Equations (14)–(21) form an eigenvalue problem, with s growth (or damping) rate of the perturbation. Equations (1)–(5) being the eigenvalue. To eliminate one variable and simplify become matters, we have used the linearized version of Equation (5) to write the gas pressure perturbation, p1, in terms of the 1 m density, ρ , and the temperature, T , perturbations. In addition, to sρ =−ρ v − ρ v + v + v + k v , (14) 1 1 1 0 r 0 r r r r ϕ z z express Equations (14)–(21) in terms of real quantities, we have → → → performed the substitutions ivϕ vϕ, ivz vz, and ibr br . c2 ρ T ρ T v2 Note that the term related to Ohm’s heating in Equation (4) does =− s 1 1 − 0 − 0 A − svr + 2 ρ1 2 T1 + (kzbr bz), not contribute in the linear regime and so it is absent from the γ ρ0 T0 ρ0 T0 B0 (15) linearized energy equation (Equation (21)). In addition, since the heating function h represents an external, arbitrary heating 2 2 source, no perturbations of this function have been considered. cs m ρ1 T1 vA m svϕ = + − kzbϕ − bz , (16) We numerically solve Equations (14)–(21) and compute the γ r ρ0 T0 B0 r eigenvalues and their corresponding perturbations by means 2 of the PDE2D code (Sewell 2005) based on finite elements. cs ρ1 T1 svz = kz + , (17) The numerical integration of Equations (14)–(21) is performed γ ρ0 T0 = from the thread axis, r 0, to the finite edge of the numerical m m m2 domain, r = rmax, which is located far enough to obtain a good sb =−B k v + η b + b − b − k2b + k b , r 0 z r r ϕ r2 ϕ r2 r z r z z convergence of the solution and to avoid numerical errors. We (18) use a nonuniform grid with a large density of grid points within the inhomogeneous transitional layer, where the equilibrium properties vary by several orders of magnitude. The nonuniform sb = B k v grid also allows us to correctly describe the small spatial scales ϕ 0 z ϕ 1 1 m m m of the eigenfunctions within the transitional layer. The PDE2D + η b + b − b − b + b − k2b + k b code uses a collocation method, and the generalized matrix ϕ ϕ 2 ϕ r 2 r z ϕ z z r r r r r eigenvalue problem is solved using the shifted inverse power m 1 method. The output of the program is the closest eigenvalue to − η b − b − b , (19) r r ϕ r ϕ an initial provided guess and its corresponding perturbations. In the present investigation, we restrict ourselves to the study of thermal quasi-continuum eigenmodes. For real kz and m, these 1 m solutions correspond to real values of s. When the eigenmode sb =−B v + v + v z 0 r r r r ϕ is a stable, damped solution, s<0, while unstable thermal s> 1 m2 m k modes have 0. As shown by Van der Linden & Goossens + η b + b − b + k b − z b − k b (1991) and Ireland et al. (1992), some of the basic properties z z 2 z z ϕ r z r r r r r regarding the stability of the thermal quasi-continuum modes − + η (bz kzbr ), (20) can be deduced from the properties of the thermal continuum in the absence of perpendicular thermal conduction and magnetic diffusion. We perform this analysis in the next section. p p s 0 T − (γ − 1) 0 ρ = c2ρ v T 1 ρ 1 s 0 r 3. THE THERMAL CONTINUUM 0 0 2 p0 p0 m 2 In our equilibrium configuration, the full spectrum of ideal − (γ − 1) ω ρ + ω + κ⊥ + κk T ρ ρ 1 T T r2 z 1 eigenmodes, i.e., when nonideal effects are absent, contains 0 0 solutions related to two different continua, namely, the Alfven´ T − 1 − 0 and slow (or cusp) continua (see, e.g., Appert et al. 1974). These + (γ 1) κ⊥ + κ⊥ T1 + κ⊥T1 +(κ κ⊥) kzbr r B0 continua are modified or removed if nonideal effects are taken 1 into account. The effect of the different nonideal mechanisms + (γ − 1) T + T κ˜⊥ + T κ˜ , (21) 0 r 0 0 ⊥ on the continua has been studied by a number of authors whose relevant results for our investigation are briefly summarized where the prime denotes the derivative with respect to r, next. 2 = γp0 2 = B0 If magnetic diffusion is included but nonadiabatic terms are cs is the sound speed squared, vA is the Alfven´ ρ0 μρ0 neglected, both Alfven´ and slow continua are replaced by a set of speed squared, and the quantities ωρ , ωT , and κ˜⊥ are defined as follows: discrete modes with complex s and whose properties depend on the value of the diffusivity (see, e.g., Poedts & Kerner 1991). In the presence of radiation losses and parallel thermal conduction = ρ0 ∂L ωρ L (ρ0,T0) + ρ0 , (22) but for κ⊥ = η = 0, the Alfven´ continuum remains unaffected p0 ∂ρ ρ0,T0 and the slow continuum is only slightly modified becoming a nonadiabatic slow continuum with complex s. In addition, a new ρ ∂L ω = 0 T , (23) thermal continuum is present, which corresponds to real values T p 0 ∂T 0 ρ0,T0 of s (Van der Linden et al. 1991). When perpendicular thermal

5 The Astrophysical Journal, 731:39 (13pp), 2011 April 10 Soler, Ballester, & Goossens conduction is included, the thermal continuum is replaced by definitions of ωρ and ωT (Equations (22) and (23)) and take into a dense set of eigenmodes, i.e., a quasi-continuum (Van der account that the cooling function at equilibrium is L (ρ0,T0) = 0 Linden & Goossens 1991). Finally, if both perpendicular thermal when κ⊥ = 0, the instability criterion is conduction and magnetic diffusion are taken into account, the ∂L ρ ∂L three continua are removed and replaced by discrete eigenmodes 2 − 0 κkz + ρ0 < 0, (29) (Ireland et al. 1992). ∂T T0 ∂ρ To study the properties of the thermal continuum before which turns out to be the same instability criterion given by solving the full problem numerically, we fix κ⊥ = η = 0 in Equations (14)–(21). After a lengthy but straightforward Field (1965) in his Equation (25a), although Field’s criterion process, it is possible to combine Equations (14)–(21) to obtain was derived for thermal modes in a homogeneous medium and the two coupled, first-order differential equations of Van der our equilibrium quantities depend on r. Therefore, the stability Linden et al. (1991, Equations (2) and (3)), which depend on of the continuum also depends on r, and Equation (29)must the coefficient C defined as be computed for all r to assess the absolute instability of the 0 continuum. In addition, magnetic diffusion may play a role ρ3p for stability. This issue was addressed by Ireland et al. (1992) = 0 0 2 2 2 C0 r s + k v Ct , (25) who showed that, when magnetic diffusion is included, the T z A 0 contribution of Ohm’s heating to the equilibrium energy balance has to be taken into account. However, since the magnetic field with Ct being the following third-order polynomial in s: is homogeneous in our case, Ohm’s heating plays no role in 2 2 2 the energy balance of our equilibrium (Equation (9)). This fact cs + vA 3 cs 2 T0 2 p0 2 Ct = s + + v κk + ωT − ωρ s points out that, in our present application, Equation (29) remains γ − 1 γ A p z ρ 0 0 valid even when magnetic diffusion is present, and the thermal c2v2 c2 T s A 2 s 2 2 0 2 − continuum growth rates are not modified by diffusion. + kz s + vAkz κkz + ωT ωρ . (26) ∗ γ − 1 γ p0 Equation (29) can be rewritten in terms of parameters χ and α of the cooling function, namely, Van der Linden et al. (1991) showed that the roots of C = 0, 0 2 − 2 ∗ α−1 i.e., the singularities in their Equations (2) and (3), correspond κkz + (α 1) ρ0 χ T0 < 0. (30) to the three continua. Note that Equations (25) and (26)are κ = independent of the azimuthal wavenumber m, and so the three In the absence of parallel thermal conduction, i.e., 0, the α< continua do not depend on the value of m.Forafixedr,the instability criterion is satisfied for 1. According to the α< roots of C = 0 are two purely imaginary solutions given by different radiative regimes of Table 1, the condition 1 takes 0 T > s2 =−k2v2 , that correspond to the Alfven´ continuum, and the place for 0 80,000 K in Hildner’s parameterization, meaning z A that the thermal continuum is unstable in the PCTR-3, PCTR-4, solutions of Ct = 0. Since Ct is a third-order polynomial and the nonadiabatic terms are assumed to be small, the roots of and Corona-5 regimes. Considering the Klimchuk–Raymond fit, instability is present for T0 93,325 K. This result agrees with Ct = 0 are two complex conjugate solutions corresponding to the nonadiabatic slow continuum, and a real root corresponding the instabilities found by Van der Linden & Goossens (1991) to the thermal continuum. It is straightforward to check that in in the range of coronal temperatures studied by these authors. When parallel thermal conduction is present, the unstable part of the ideal case κ = ω = ω = 0, so C becomes ρ T t the continuum is stabilized by a critical value of the longitudinal ∗ s wavenumber, namely, kz , given by C = c2 + v2 s2 + c2v2 k2 . (27) t γ − 1 s A s A z 1 − α 1/2 ∗ = 2 ∗ α−1 kz ρ χ T . (31) c2v2 0 0 C = s2 =− s A k2 κ The roots of t 0 are then c2+v2 z , which correspond s A ∗ to the ideal slow continuum, while the thermal continuum Note that kz is also a function of r and Equation (31) only disappears and becomes the trivial solution s = 0. applies when α<1. The absolute stability of the continuum is ∗ guaranteed for kz larger than the maximum value of kz in the 3.1. Approximate Growth Rates and Stability Criterion equilibrium. An approximation to the thermal continuum growth rate 3.2. Properties of the Continuum can be obtained by neglecting the terms with s2 and s3 in Equation (26). Then, the equation Ct = 0 gives the approximate Figure 2 shows the thermal continuum computed from solution Equation (28) for our equilibrium configuration in the case of the sinusoidal temperature profile with l/R = 1, and consid- γ − 1 T ≈− 0 2 − ering Hildner’s parameterization for the radiative loss function. s κkz + ωT ωρ . (28) γ p0 The growth rate has been plotted in dimensionless form computed as s˜ = sR/csp, where s˜ is the dimensionless growth rate Equation (28) is equivalent to Equation (4.12) of Van der Linden and csp is the sound speed of the central, homogeneous part of & Goossens (1991). It is worth mentioning that Equation (28) the thread. At first sight, we notice the jumps of the growth rate is similar to the expression obtained by Carbonell et al. (2009) at the boundaries of the zones where different parameteriza- and Soler (2010) for the imaginary part of the frequency of tions of the cooling function are used. According to Hildner’s fit propagating thermal waves in a flowing medium. (see Table 1), the cooling function L (ρ,T) is continuous at the The sign of the constant term in Equation (26) provides us boundaries between the zones with different radiation regimes, with the instability criterion of the thermal continuum, i.e., the but the derivatives of L (ρ,T) with respect to density and tem- combination of parameters that causes s>0. When we use the perature are discontinuous. As Equation (28) depends on these

6 The Astrophysical Journal, 731:39 (13pp), 2011 April 10 Soler, Ballester, & Goossens

−2 −1 Figure 2. Normalized thermal continuum as a function of r/R in the case of the sinusoidal temperature proﬁle with l/R = 1for(a)kzR = 10 and (b) kzR = 10 . The vertical dotted lines denote the boundaries between the different parameterizations of the radiation function. Each zone is indicated by a number from 1 to 5 that corresponds to the different regimes of Hildner’s piecewise ﬁt (Table 1). The horizontal dotted line represents s˜ = 0. The different line styles in zone 1 correspond to the Prominence-1.1 (solid), Prominence-1.2 (dashed), and Prominence-1.3 (dot-dashed) parameterizations.

Figure 3. Maximum growth rate of the normalized thermal continuum as a Figure 4. Same as Figure 2(a) but using Klimchuk–Raymond’s fit for the function of kzR. The different line styles represent the result for the linear radiative loss function. (solid), sinusoidal (dashed), and Gaussian (dot-dashed) temperature profiles with l/R = 1. The vertical dotted line corresponds to the maximum value of ∗ maximum growth rate of the spectrum as a function of kzR.As the dimensionless critical wavenumber kz R for the linear temperature profile. Hildner’s parameterization for the radiative loss function has been used. expected, the maximum growth rate decreases as kzR increases, until absolute stability of the thermal continuum is achieved for a critical longitudinal wavenumber. This critical wavenumber derivatives, the thermal continuum has the apparent form of five is in perfect agreement with Equation (31). The result for the separated continuous spectra. If the actual cooling function of three different temperature profiles does not show significant prominences is known and is used here instead of the present differences. We must mention that the various parameterizations parameterization, the jumps would be replaced by continuous of the radiative regime Prominence-1, which aim to represent but very abrupt variations of the growth rate at the boundaries different optical thicknesses of the cool prominence plasma, are between the different regimes. We must point out that the jumps not relevant for the instability of the continuum. The thermal in the thermal spectrum do not affect the stability of the solu- spectrum in the cool part of the equilibrium, namely, zone 1, is tions (see Figures 1 and 3 of Van der Linden et al. 1991, where always stable independently of the considered parameterization this issue is also commented). An alternative radiative cooling (indicated by different line styles in Figure 2). function computed numerically (e.g., Schure et al. 2010) instead In Figure 4, we have plotted the thermal continuum growth −2 of the present piecewise parameterization might be considered rate for kzR = 10 using the Klimchuk–Raymond radiative in future applications. loss function. By comparing Figures 2(a) and 4, we see that −2 In Figure 2(a), corresponding to kzR = 10 , we see that slightly larger values of the growth rate are obtained using s>˜ 0 in zone 3 and part of zone 4, meaning that the continuum Klimchuk–Raymond’s fit with respect to the values for Hildner’s is unstable in these regions. This is consistent with the result fit. However, the qualitative behavior of the continuum is similar from the instability criterion (Equation (30)). For the particular is both cases. longitudinal wavenumber considered in Figure 2(a) parallel thermal conduction can suppress the instability in zone 5 and 4. PROPERTIES OF THE UNSTABLE THERMAL MODES part of zone 4. If the longitudinal wavenumber is increased −1 4.1. Effect of Perpendicular Thermal Conduction to kzR = 10 (see Figure 2(b)), the instability is also suppressed in zone 4 and part of zone 3. To completely suppress Here, we study how the unstable part of the thermal contin- the instability in the whole continuum, we must increase the uum is modified when perpendicular thermal conduction and ∗ longitudinal wavenumber until the maximum value of kz R, magnetic diffusion are taken into account. Unless otherwise given by Equation (31), is reached. Figure 3 displays the stated, Hildner’s parameterization for the radiative loss function

7 The Astrophysical Journal, 731:39 (13pp), 2011 April 10 Soler, Ballester, & Goossens

Figure 5. Temperature perturbation (in arbitrary units) of the four most unstable modes (from the left to the right) in the absence of magnetic diffusion. Computations −1 performed considering the sinusoidal temperature profile with l/R = 1, kzR = 10 ,andm = 0. The corresponding normalized growth rate, s˜, is displayed on top of each panel. is used in all the following computations. First, we set η = 0 and focus our investigation on the effect of perpendicular thermal conduction. We would like to stress that the real physical value of κ⊥ given by Equation (7) is used in the following computations. As stated by Van der Linden & Goossens (1991), the thermal continuum is replaced by a set of discrete modes when κ⊥ = 0. The eigenfunctions of these solutions display large variations in a region surrounding the position of the thermal continuum singularity for κ⊥ = 0. Figure 5 displays the temperature perturbation, T1, of the four most unstable modes of our equilibrium in the case of the sinusoidal temperature profile with l/R = 1, −1 kzR = 10 , and m = 0. For simplicity, Figure 5 shows the temperature perturbation only, because T1 is between one and two orders of magnitude larger than the other perturbations. This × means that the temperature perturbation is the dominant distur- Figure 6. Normalized growth rate of the 20 most unstable modes (symbol ) as a function of the position of their largest maximum or minimum. The solid bance related to the thermal modes, although these solutions line represents the thermal continuum in the case κ⊥ = 0. Results are computed −1 produce also velocity and magnetic field perturbations. Figure 5 for η = 0, kzR = 10 ,andm = 0. The sinusoidal temperature profile with focuses on the region where the eigenfunctions show significant l/R = 1 has been used. variations, i.e., the conductive layer described by Van der Linden (1993), whereas their amplitude outside the range plotted in Figure 5 is negligible. When Figure 5 is repeated for the linear Thus, the characteristic perpendicular spatial-scale decreases, and Gaussian temperature profiles, we find that the eigenfunc- causing the modes to cluster as r/R increases. tions are shifted toward smaller r/R for the linear profile, and To shed more light on the behavior of the eigenfunctions to larger r/R for the Gaussian profile. However, the form of the with κ⊥, we perform the substitution κ⊥ → λκ⊥ in the basic perturbations is very similar to those displayed in Figure 5 and, equations, with λ being an enhancing factor. We can artificially for simplicity, we do not plot the perturbations again. increase the value of κ⊥ by means of λ and assess its influence The order of the solutions can easily be identified by the on the eigenfunctions. We restrict ourselves to the most unstable number of extrema (maxima and minima) of their temperature mode. Figure 7(a) shows the temperature eigenfunction of the perturbation. Thus, the most unstable mode has one maximum most unstable mode for different values of λ.Asλ grows, only, the second most unstable mode has one maximum and the temperature eigenfunction becomes broader. To quantify one minimum, and so on. The position of the largest extremum this effect, we compute the width δ of the maximum of is shifted toward larger values of r/R as the order of the mode the temperature perturbation measured at its half height. The increases. In addition, the growth rate (indicated at the top of the parameter δ is related to the thickness of the conductive layer. panels of Figure 5) decreases with the order of the mode. These Following the method by Sakurai et al. (1991) originally used for two results are represented together in Figure 6, which displays Alfven´ and slow resonances, Van der Linden (1993) obtained the growth rate of the 20 most unstable modes as a function that the thickness of the conductive layer is proportional to 1/3 of the position of their largest extremum. In comparison with κ⊥ . In order to compare the analytical result of Van der Linden the thermal continuum, slightly smaller values of the growth rate (1993) with our numerical computations, Figure 7(b) shows δ/R are obtained for the quasi-continuum modes. The displacement versus the enhancing parameter λ. We see that the dependence of the largest extremum of the discrete modes toward larger of δ/R with λ is consistent with a scaling law of λ1/3, pointing values of r/R as their growth rate decreases is consistent with out that our numerical code works properly and recovers the the behavior of the thermal continuum. From Figure 6 we also behavior analytically predicted by Van der Linden (1993). see that the solutions are closer to each other as their order We have also computed the eigenfunctions for other values increases. This result can be explained by taking into account of the azimuthal wavenumber m. We do not have obtained vari- that the perpendicular thermal conductivity (Equation (7)) is ations of the spectrum of quasi-continuum modes for different ∼ 2 −1/2 = κ⊥ ρ0 T0 . As we move toward larger r/R, T0 increases and values of m. It is worth noting that for m 0, the perturbations vϕ ρ0 decreases in the equilibrium, meaning that κ⊥ gets smaller. and Bϕ are decoupled from the remaining perturbations, while

8 The Astrophysical Journal, 731:39 (13pp), 2011 April 10 Soler, Ballester, & Goossens

Figure 7. (a) Temperature perturbation (in arbitrary units) of the most unstable mode. The different line styles correspond to different values of the perpendicular conductivity enhancing factor: λ = 1 (solid), λ = 10 (dotted), and λ = 100 (dashed). (b) Width of the maximum of the normalized temperature perturbation, δ/R, corresponding to the most unstable mode (symbol ×) vs. the perpendicular conductivity enhancing factor λ. The dotted line corresponds to the dependence δ/R ∼ λ1/3. −1 Results are computed for η = 0, kzR = 10 ,andm = 0. The sinusoidal temperature proﬁle with l/R = 1 has been used.

Figure 8. (a) Ratio of the Alfven´ timescale, τA, to the thermal continuum timescale, τT . The vertical dotted lines denote the boundaries between the different parameterizations of the radiation function according to Table 1. (b) Normalized growth rate of the most unstable mode vs. the diffusivity enhancing factor .The sinusoidal temperature proﬁle with l/R = 1 has been used.

for m = 0 all perturbations are coupled. The temperature per- wavelength of the perturbation, namely, L = 2π/kz.Thisis turbation is independent of m. The effect of m on the growth done in Figure 8(a), which displays the ratio τA/τT versus r/R −1 rate is studied in Section 5. with kzR = 10 . We obtain that in the most unstable part of the continuum, i.e., the beginning of zone 3, τT and τA are of the 4.2. Effect of Magnetic Diffusion same order. Therefore, we are in a situation between the limit Here, we first consider magnetic diffusion and neglect per- cases studied by Ireland et al. (1992, 1998). We have numerically pendicular thermal conduction, i.e., κ⊥ = 0 and η = 0. Ireland checked that the growth rates are not modified by the presence of et al. (1992, 1998) studied the effect of magnetic diffusion on magnetic diffusion. In addition, by setting η = 0 and κ⊥ = 0in the thermal continuum. For their particular equilibrium, these the equations implemented in the PDE2D code, we have found authors studied two limit cases of the relative values of the no discrete thermal solutions. Therefore, the behavior of our thermal and Alfven´ timescales. When the thermal timescale is equilibrium is similar to that of the hot profile of Ireland et al. much longer than the Alfven´ timescale, Ireland et al. (1992, (1992, 1998), although in our case τT and τA are of the same 1998) found that the effect of diffusion is to replace the thermal order. continuum by a dense set of discrete modes whose growth rate A different issue is to determine how the previously described is displaced with respect to the values of the thermal continuum. modes for κ⊥ = 0 and η = 0 (Section 4.1) are affected when The displacement of the growth rate is independent of the value magnetic diffusion is included. Hence, we consider the general of the diffusivity. This is the case of the cool profile studied case κ⊥ = 0 and η = 0. As was done for the perpendicular by Ireland et al. (1992, 1998). On the other hand, if the thermal thermal conduction, we perform the substitution η → η in timescale is much shorter than the Alfven´ timescale, the thermal the basic equations, with being an enhancing factor. We can continuum is not affected by diffusion and the discrete modes artificially increase the value of η by means of and assess its are absent. This other possibility corresponds to the hot profile influence. As before, we restrict ourselves to the most unstable of Ireland et al. (1992, 1998). mode whose growth rate versus is displayed in Figure 8(b). To assess the effect of magnetic diffusion on the thermal For 50 the growth rate is constant and coincides with continuum of our equilibrium, let us compare the Alfven´ the value for η = 0, while the growth rate increases for timescale, τA = L/vA, with the thermal timescale, τT = 1/|s|, 50. For such large values of , magnetic diffusion governs where s is here the thermal continuum growth rate and L the behavior of the solutions. When the actual value of η is is a typical length scale. We relate L with the longitudinal considered, i.e., = 1, the growth rate is approximately the

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Figure 9. Temperature perturbation (in arbitrary units) of the most unstable mode for a diffusivity enhancing factor of (a) = 1, (b) = 50, (c) = 70, and (d) −1 = 100. Computations performed considering the sinusoidal temperature proﬁle with l/R = 1, kzR = 10 ,andm = 0. The corresponding normalized growth rate, s˜, is displayed on top of each panel.

−1 Figure 10. Normalized growth rate of the most unstable mode as a function of (a) the azimuthal wavenumber, m,forkzR = 10 , and (b) the dimensionless longitudinal wavenumber, kzR,form = 0. The results of both panels correspond to the sinusoidal temperature profile with l/R = 1. same as that obtained in the absence of magnetic diffusion. rate of the most unstable mode is s˜ ≈ 0.1994 using Hildner’s Thus, we need much larger, unrealistic values of η for the fit and s˜ ≈ 0.4370 using Klimchuk–Raymond’s fit. Apart from growth rate to be affected by diffusion. On the other hand, this difference in the growth rate, the results for both Hildner’s Figure 9 shows the evolution of the temperature perturbation and Klimchuk–Raymond’s parameterizations are equivalent. of the most unstable mode as increases. Again, the result for = 1 is almost indistinguishable from the eigenfunction for 5. PARAMETRIC STUDY OF THE GROWTH RATES η = 0 (compare Figures 5(a) and 9(a)). When is increased and so very large, unrealistic values of η are considered, the In this section, we study the growth rate of the most unstable eigenfunction is affected by an additional modulation caused by mode as a function of the dimensionless longitudinal wavenum- magnetic diffusion and develops smaller spatial scales as is ber, kzR, the azimuthal wavenumber, m, and the thickness of increased. On the basis of these results, we conclude that, for the transitional layer, l/R. Hildner’s fit for the radiative loss realistic values of the diffusivity, magnetic diffusion is irrelevant function is used in all cases. for both the growth rate and the eigenfunctions of the most First, we assess the effect of the azimuthal wavenumber, m, unstable modes. (see Figure 10(a)). Note that only positive integer values of m are considered in Figure 10(a) as the results are equivalent 4.3. Comparison between the Hildner and Klimchuk–Raymond when negative values are used. As for the thermal continuum, Radiative Loss Functions the value of m is irrelevant for the growth rate of the discrete thermal modes. The almost negligible variation of the growth We have performed some test computations using rate with m shown in Figure 10(a) might not be physical and Klimchuk–Raymond’s fit for the radiative loss function in order might be attributed to the accuracy of the numerical code. On to compare with the results obtained using Hildner’s fit (dis- the other hand, the dependence on the longitudinal wavenumber cussed in the previous subsections). Regarding the form of the is displayed in Figure 10(b). The maximum of the growth rate temperature perturbations, we find no significant differences. takes place for kzR ≈ 0.4, while s˜ decreases as kzR is increased The maximum of the temperature perturbation is shifted to- or reduced. For kzR 0.4, the growth rate decreases and the ward slightly larger values of r for Klimchuk–Raymond’s fit. thermal mode is stabilized by the effect of thermal conduction For the fundamental mode, the maximum of T1 takes place parallel to magnetic field lines, as commented in Section 3. at r/R ≈ 0.675 using Hildner’s fit and at r/R ≈ 0.69 us- The critical wavenumber for the stabilization is consistent with −1 ing Klimchuk–Raymond’s fit for kzR = 10 , l/R = 1, and the expression given in Equation (31) for the stabilization of m = 0. On the other hand, the growth rate is larger for the thermal continuum. The growth rate also decreases with Klimchuk–Raymond’s parameterization. This is in agreement respect to the maximum value when kzR 0.4. This behavior is with the behavior of the thermal continuum (see Figures 2(a) different from the behavior of the thermal continuum (compare and 4). For the same set of parameters as before, the growth Figures 3 and 10(b)). In addition, we find that for kzR → 0a

10 The Astrophysical Journal, 731:39 (13pp), 2011 April 10 Soler, Ballester, & Goossens magnetic field. The unstable discrete modes present in our model appear due to the effect of thermal conduction perpendicular to the magnetic field. Our results for the growth rate of the unstable thermal modes have direct implications for the stability and lifetime of prominence threads. In our analysis, we have restricted ourselves to the linear stage of the thermal instability. This stage represents the initial phase in which the unstable thermal mode grows after its excitation. The combined influence of the different unstable modes will affect the subsequent nonlinear evolution of the plasma within the PCTR, because many of the unstable modes have very similar growth rates and the evolution will not be dominated by a single mode. Significant changes in the equilibrium of the threads would take place during the subsequent nonlinear evolution of the Figure 11. Normalized growth rate of the most unstable mode vs. the thickness thermal instability, because of the large temperature and density of the transitional layer, l/R. The different line styles in the panel, almost superimposed, represent the linear (solid), sinusoidal (dashed), and Gaussian gradients generated, which may be detected by the observations. (dot-dashed) temperature profiles. All computations are performed with kzR = The growth rate of the linear phase provides us with a reasonable 10−1 and m = 0. estimation of the typical timescale on which the effect of the thermal instability in the PCTR of the threads would be rich collection of couplings between the different thermal modes observable and may affect the thread dynamics. takes place. This complex system of couplings is not visible on In the analysis of the previous sections, we have seen that the the scale of Figure 10(b). We do not explore these couplings in most relevant parameter affecting the instability growth rate is the longitudinal wavenumber, since it determines the efficiency detail here because they take place for very small kzR and far from the maximum value of the growth rate. of parallel thermal conduction as stabilizing mechanism. If un- Finally, the dependence on the thickness of the transitional stable thermal modes are excited in a prominence thread by an layer, l/R, is shown on Figure 11. In a real prominence thread, arbitrary disturbance, the instability would be dominated by the the thickness of the transitional layer is probably determined most unstable solution. We find that in our model the maximum ≈ by the coefficient of thermal conduction perpendicular to the of the growth takes place for kzR 0.4 and corresponds to a di- ˜ ≈ magnetic field. In fully ionized coronal plasmas, the perpen- mensionless growth rate of s 0.255 using Hildner’s radiative ˜ ≈ dicular thermal conductivity is very small. However, for cool loss function and s 0.467 using Klimchuk–Raymond’s ra- prominence temperatures the plasma is only partially ionized diative loss function. The corresponding instability timescale = and the perpendicular thermal conductivity increases by several is τT 1/s, with s being the dimensional growth rate =˜ orders of magnitude due to the effect of thermal conduction computed from its dimensionless value as s scsp/R.In by neutrals (see, e.g., Parker 1953;Ibańez˜ & Mendoza 1990; Figure 12,weplotτT as a function of the temperature at the Forteza et al. 2008). This effect might be important in the in- center of the thread, Tp, for different values of the thread radius, ner part of the PCTR. Unfortunately, since the ionization de- R. We find that, for realistic values of Tp and R, the instability gree in prominence threads is unknown (see recent estimations timescale is of the order of a few minutes. We obtain that τT by Gouttebroze & Labrosse 2009), the thickness of the transi- decreases as Tp gets higher and increases as R gets larger, mean- tional layer cannot be determined with confidence. For this rea- ing that cool and wide threads are more thermally stable than son, we take l/R as a free parameter. The growth rate is almost hot and thin threads. We also see that Klimchuk–Raymond’s constant for l/R 0.5, whereas s˜ decreases for l/R 0.5. parameterization produces smaller instability timescales than When l/R → 0, s˜ → 0 because the transitional layer is absent with Hildner’s parameterization. The values of τT indicated in and the thermal continuum disappears. As in previous computa- Figure 12 are of the same order of magnitude as the typically tions, the results for the different temperature profiles are almost observed lasting time of the threads in sequences of Hα im- identical, hence the form of the temperature profile is not rele- ages of solar filaments (e.g., Lin 2004; Lin et al. 2005, 2009). vant for the growth rate. This is an important result because it This result implies that the thermal instability may play a rel- means that our present conclusions apply for all possible tem- evant role for the dynamics and stability of prominence and perature profiles in the particular PCTR of prominence threads. filament fine structures, because the timescale related to the thermal instability is consistent with the observed timescale on which threads seem to suffer important changes in their 6. IMPLICATIONS FOR THREAD LIFETIMES equilibrium.

In this paper, we have shown that prominence threads, 7. DISCUSSION AND SUMMARY modeled as cool and dense magnetic flux tubes embedded in a much hotter and less dense coronal environment, are not stable In this paper, we have studied the properties of quasi- because unstable thermal modes are present in their particular continuum thermal modes in a prominence thread model which PCTR. This statement qualitatively agrees with the results from is transversely inhomogeneous. We have followed the method Karpen et al. (1989) who studied thermal instabilities in a of Van der Linden et al. (1991) and Van der Linden & Goossens two-dimensional spatially random magnetic field distribution (1991) to investigate, first, the stability of the thermal continuum and concluded that cool condensations cannot be in static and, later, the effect of cross-field thermal conduction and equilibrium with a hot exterior. However, in the work by magnetic diffusion. We have recovered the general results by Karpen et al. (1989) the surrounding of the condensations Van der Linden et al. (1991) and Van der Linden & Goossens simply continue to cool down because these authors did not (1991) in all cases studied in the present paper. In particular, we consider the effect of thermal conduction perpendicular to the have found that the thermal continuum in prominence threads is

11 The Astrophysical Journal, 731:39 (13pp), 2011 April 10 Soler, Ballester, & Goossens

Figure 12. Thermal instability timescale, τT , computed from the maximum growth of the most unstable mode vs. the temperature of the thread core, Tp,using(a) Hildner’s and (b) Klimchuk–Raymond’s parameterizations for the radiative loss function. The different line styles in both panels represent different values of the thread radius: R = 100 km (solid), 200 km (dotted), 400 km (dashed), and 800 km (dot-dashed). unstable for PCTR temperatures. In agreement with the results than 8 × 104 K, which is higher than the critical ionizing tem- of Van der Linden & Goossens (1991), the effect of cross- perature. This suggests that the ionization degree of the cool field thermal conduction is to replace the thermal continuum part of the fine structure is probably not relevant for the un- by discrete modes which retain the unstable character of the stable quasi-continuum modes, because they are mainly con- continuum. On the contrary, the role of magnetic diffusion is fined where hydrogen may be already fully ionized. However, negligible in our model (see Ireland et al. 1992, 1998). We a detailed investigation of this issue is needed for more robust have obtained the important result that the instability growth conclusions. rate of the most unstable mode is independent of the form of The present investigation is a first step and should be improved the temperature profile within the PCTR of the thread, and the in the future by taking into account relevant effects that have not instability timescale is consistent with the observed lifetime of been included in our model. Among these effects, the role of the threads in Hα observations of solar filaments (e.g., Lin et al. density and magnetic longitudinal inhomogeneity and the effect 2008, 2009). Considering our present results along with those of magnetic twist will be assessed in forthcoming works. We obtained by Van der Linden & Goossens (1991) and Van der have also neglected the effect of mass flows. Previous studies Linden (1993), we conclude that unstable thermal modes may on linear thermal modes in flowing homogeneous prominence play a relevant role for both the formation of the prominence fine plasmas (see, e.g., Soler et al. 2008; Carbonell et al. 2009, 2010; structure and the subsequent instability of the thin prominence Barceloetal.´ 2011) suggest that flows along magnetic field lines threads. do not affect the linear growth rate of the thermal instability. Here we have considered a fully ionized prominence. How- ever, due to the low temperature in the dense core of the threads, We thank Ronald Van der Linden for reading the manuscript the cool prominence plasma is expected to be partially ion- and for giving helpful comments. This work was started when ized (Gouttebroze & Labrosse 2009). If partial ionization is R.S. was a PhD student in UIB. R.S. is grateful to the CAIB for taken into account, several additional effects have to be con- financial support. R.S. acknowledges support from a postdoc- sidered, namely, thermal conduction by neutrals, ambipolar (or toral fellowship in Leuven within the EU Research and Training Cowling’s) diffusion caused by ion-neutral collisions, and a Network SOLAIRE (MTRN-CT-2006-035484). R.S. and J.L.B. modified radiation function (see, e.g., Forteza et al. 2008; Soler acknowledge the financial support received from the Spanish et al. 2010a). The role of both effects depends strongly on MICINN and FEDER funds (AYA2006-07637). R.S. and J.L.B. the plasma ionization degree. It requires complicated compu- also acknowledge discussion within ISSI Team on Solar Promi- tations of NLTE radiative transfer and statistical equilibrium nence Formation and Equilibrium: New data, new models. M.G. of atomic level populations to obtain the precise profile of the acknowledges support from K.U. Leuven via GOA/2009-009. ionization degree in prominence threads (see extensive details in, e.g., Gouttebroze & Labrosse 2009; Labrosse et al. 2010). REFERENCES Nevertheless, some relevant conclusions can be obtained if one reasonably assumes that the ionization degree in prominence Appert, K., Gruber, R., & Vaclavik, J. 1974, Phys. Fluids, 17, 1471 plasmas is mainly determined by the temperature and, to a Arregui, I., & Ballester, J. L. 2010, Space Sci. Rev., in press lesser extend, by the density. While the ionization degree of Ballester, J. L. 2006, Phil. Trans. R. Soc. A, 364, 405 the core of the thread is unknown, it is realistic to assume that Ballester, J. L., & Priest, E. R. 1989, A&A, 225, 213 the condition of full ionization occurs in the PCTR for tem- Barcelo,´ S., Carbonell, M., & Ballester, J. L. 2011, A&A, 525, A60 ∗ Berger, et al. 2008, ApJ, 676, L89 peratures higher than a critical temperature, namely, T .The Braginskii, S. I. 1965, Rev. 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