4.5 Fine structure oscillations (finite length threads)

Filament threads have been modeled as magnetic flux tubes anchored in the solar photosphere (Ballester and Priest, 1989Jump To The Next Citation Point; Rempel et al., 1999Jump To The Next Citation Point) which are stacked one on top of one another in the vertical and horizontal directions, giving place to the filament body.

Many observations of oscillatory events in threads (see Section 3.6.4) cannot be accounted for by the simple models of Section 4.4 because the obtained results rely on the assumption that the thread length is much larger than the wavelength. Exceptions to this hypothesis are standing waves and propagating waves whose wavelength is of the order of or larger than the thread length. In the models presented in this section a thread is envisaged as a cold, dense condensation that fills the central part of a magnetic tube containing hot coronal plasma and anchored in the solar photosphere. Although this structure has been modeled with some complexity (Ballester and Priest, 1989; Rempel et al., 1999), only oscillations of much simpler thread configurations have been investigated so far. Because the reported thread oscillations are transverse, we here concentrate on works that investigate this kind of motions.

Joarder et al. (1997Jump To The Next Citation Point) considered a thin thread with finite width and length in Cartesian geometry (Figure 38View Image). The thread is infinitely deep since the equilibrium configuration is invariant along the y-axis. The influence of the plasma pressure was neglected (i.e., the zero-β limit was taken) and consequently the slow mode is absent from their analysis. Joarder et al. (1997Jump To The Next Citation Point) obtained the dispersion relations for Alfvén and fast modes, and restricted their study to the oscillatory frequencies, omitting other properties that are also relevant for the understanding of oscillations such as the spatial structure and the polarization of perturbations.

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Figure 38: Sketch of the thread equilibrium model used by Joarder et al. (1997), Díaz et al. (2001Jump To The Next Citation Point), and Díaz et al. (2003Jump To The Next Citation Point). The blue zone of length 2W represents the cold part of the flux tube, i.e., the prominence thread. The length of the magnetic structure is 2L and the thread thickness (equivalent to its diameter) is 2b. The magnetic field is uniform and parallel to the z-axis, and the whole configuration is invariant in the y-direction (from Díaz et al., 2001Jump To The Next Citation Point).

Using the same two-dimensional configuration, Díaz et al. (2001Jump To The Next Citation Point) performed an analytical and numerical study of the behaviour of fast modes when a proper treatment of the boundary conditions at the different interfaces of this thread configuration is included. The main conclusion is that prominence threads can only support a few non-leaky modes of oscillation, those with the lowest frequencies. Also, for reasonable values of the thread length, the spatial structure of the fast fundamental even and odd kink modes is such that the velocity amplitude outside the thread takes large values over long distances (Figure 39View Image). Fast kink modes are associated to normal motions with respect to the thread length (i.e., in the x-direction; see Figure 38View Image). The fundamental kink mode (simply referred to as the kink mode) has a velocity maximum at the thread centre, while its first harmonic (that is, the fundamental odd kink solution) has a node in the same position.

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Figure 39: Kink mode normal velocity component across the axis of the Cartesian prominence thread depicted in Figure 38View Image. Solutions are symmetric about the thread axis (x = 0) and so they are only shown for x ≥ 0. The length of magnetic field lines is 2L = 200,000 km. (a) In a very thick thread (with a “radius” of 10,000 km) the perturbation is essentially confined to the thread itself, i.e., to 0 ≤ x∕L ≤ 0.1. (b) In an actual thread (with a “radius” of 100 km) the velocity displays a large amplitude beyond the thread boundary, at x∕L = 0.001. This means that wave energy spreads into the surrounding coronal medium (from Díaz et al., 2001Jump To The Next Citation Point).

Later on, Díaz et al. (2003Jump To The Next Citation Point) included wave propagation in the y-direction (see Figure 38View Image) making the model fully three-dimensional, and two important features appeared. The first is that the cut-off frequency, that separates confined and leaky modes, varies with the longitudinal wavenumber (ky), which allows the structure to trap more modes. The second one is that a much better confinement of the wave energy is obtained compared to the ky = 0 case (see Figure 40View Image). An interesting issue concerning these results obtained using Cartesian threads is that large velocity amplitudes are found in the corona, which seems to favour collective thread oscillations in front of individual oscillations.

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Figure 40: Normal velocity component (in arbitrary units) of the kink mode in the direction across the thread axis. The ratio of the thread “diameter” to the length of magnetic field lines is b∕L = 0.001, while the ratio of the thread length to the field lines length is W ∕L = 0.1. The solid, dotted, and dashed lines correspond to kyL = 0 (curve of Figure 39View Imageb), kyL = 3, and kyL = 20. All other parameter values are those of Figure 39View Image. The thread boundary is marked by a vertical dashed line (from Díaz et al., 2003).

Since cylindrical geometry is more suitable to model prominence threads, Díaz et al. (2002Jump To The Next Citation Point) considered a straight cylindrical flux tube with a cool region representing the prominence thread, which is confined by two symmetric hot regions (Figure 41View Image). With this geometry the fundamental sausage mode (m = 0, with m the azimuthal wavenumber) and its harmonics are always leaky. However, for all other modes (m > 0), at least the fundamental solution lies below the cut-off frequency. Hence, if any of these modes is excited the oscillatory energy in the prominence plasma does not vary in time after the initial transient has elapsed. Regarding the spatial structure of perturbations, in cylindrical geometry the modes are always confined to the dense part of the flux tube (Figure 42View Image). Therefore, an oscillating cylindrical thread is less likely to induce oscillations in its neighbouring threads than a Cartesian one.

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Figure 41: Sketch of the equilibrium configuration of a thread in a cylindrical coronal magnetic tube. The gray zone of length 2W represents the cold part of the flux tube, i.e., the prominence thread. The length of the magnetic structure is 2L and the thread radius is b. The magnetic field is uniform and parallel to the z-axis, and the whole configuration is invariant in the φ-direction (from Díaz et al., 2002Jump To The Next Citation Point).
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Figure 42: Cut of the normal velocity component in Cartesian geometry (dashed line, i.e., curve of Figure 39View Imageb) and the radial velocity component in cylindrical geometry (dotted line) in the direction across the thread axis. These solutions correspond to the (fundamental) kink mode in a prominence thread with the parameter values used in Figure 40View Image. The vertical long dashed line marks the thread boundary (from Díaz et al., 2002Jump To The Next Citation Point).

To study the oscillations of the above mentioned configurations, Díaz et al. (2001Jump To The Next Citation Point, 2002Jump To The Next Citation Point) developed a very general, although cumbersome procedure. However, Dymova and Ruderman (2005Jump To The Next Citation Point) considered the same problem and to simplify its study took advantage of the fact that the observed thickness of oscillating threads is orders of magnitude shorter than their length. Taking this into account, Dymova and Ruderman (2005Jump To The Next Citation Point) used the so-called thin flux tube (TT) approximation, that enables a simpler solution for the MHD oscillations of longitudinally inhomogeneous magnetic tubes. Once the partial differential equation for the total pressure perturbation is obtained, a different scaling (stretching of radial and longitudinal coordinates) of this equation inside the tube and in the corona can be performed. Following this procedure, two different equations for the total pressure perturbation inside and outside the flux tube, with well known solutions, are obtained. After imposing boundary conditions, the analytical dispersion relations for even and odd modes were derived and a parametric study was performed. A comparison between the numerical values of the periods obtained with this approach and that of Díaz et al. (2002Jump To The Next Citation Point) points out differences of the order of 1%. The only drawback of the method of Dymova and Ruderman (2005Jump To The Next Citation Point) is that it can be only applied to the fundamental mode with respect to the radial dependence.

Taking into account observations by, e.g., Lin (2005), which suggest in-phase oscillations of neighbouring threads in a filament, Díaz et al. (2005Jump To The Next Citation Point) studied multi-thread systems in Cartesian geometry. The equilibrium configuration consists of a collection of two-dimensional threads modeled as in Díaz et al. (2001) and separated by an adjustable distance 2c (Figure 43View Image). An inhomogeneous filament composed of five threads was constructed (Figure 44View Image) with thread density ratios thought to represent the density inhomogeneity of a prominence. The separations between threads were chosen randomly within a realistic range. The thread separations were then changed with respect to the values of Figure 44View Image by a certain factor and the kink modes were computed. Their frequencies are displayed in Figure 45View Image, where cref is a reference value representative of the separations between threads. When the separations are small, i.e., for cref∕L ≪ 1, there is a strong interaction between threads since the perturbed velocity in a given thread can easily extend over its neighbours. As a result, there is only one even non-leaky mode: the one producing in-phase oscillations of all threads. The other extreme of Figure 45View Image, i.e., cref∕L ≫ 1, corresponds to very large separations. In this situation all threads oscillate independently and the individual kink mode frequencies are recovered. Note that realistic thread separations correspond to cref∕L ∼ 10− 3–10 −2, for which only the kink mode mentioned before is supported by the system. Its frequency is lower than the individual kink mode frequencies. Although these results show some agreement with observations about the collective oscillations of threads, the use of Cartesian geometry favours this kind of combined behaviour and so a similar study based on a cylindrical model is also of interest.

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Figure 43: Sketch of a multi-thread equilibrium configuration. The grey zone represents the cold part of the magnetic tube, i.e., the prominence. The magnetic field is uniform and parallel to the z-axis, and the whole configuration is invariant in the y-direction (from Díaz et al., 2005Jump To The Next Citation Point).
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Figure 44: Sketch of the density profile in the direction z = 0 of an inhomogeneous multi-thread system. The density values of the are normalized to the coronal value. Between and under the threads the dimensionless separation, 2c ∕L, and “diameter”, 2b∕L, are given (from Díaz et al., 2005Jump To The Next Citation Point).
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Figure 45: Dimensionless frequency versus the dimensionless reference separation between threads in a multi-thread system. In this figure ca is the Alfvén speed in the corona (from Díaz et al., 2005Jump To The Next Citation Point).

Díaz and Roberts (2006Jump To The Next Citation Point) studied the properties of the fast MHD modes of a periodic, Cartesian thread model (see Figure 1 of Díaz and Roberts 2006Jump To The Next Citation Point). This configuration provides a bridge between a structure with a limited number of threads (studied by Díaz et al., 2005Jump To The Next Citation Point, see Figure 43View Image) and a homogeneous prominence with a transverse magnetic field (investigated by Joarder and Roberts, 1992b, see Figure 27View Image). Díaz and Roberts (2006Jump To The Next Citation Point) found that for thread separations of the order of their thickness the only confined modes are those in which large numbers of threads are constrained to oscillate nearly in phase. The spatial structure of these solutions is similar to that of the propagating modes of a homogeneous prominence, with small-scale deviations due to the presence of the dense threads. Their period is equal to √ -- f P, with P the period of the prominence slab and f the filling factor. The system with a limited number of threads has an even shorter period and a comparison between the different configurations considered by Díaz and Roberts (2006Jump To The Next Citation Point) gives periods of 23.6 min for the homogeneous prominence, between 12.1 and 19.3 min for the system of periodic threads and 5.3 for the four-thread configuration studied by Díaz et al. (2005). Hence, the main conclusion of Díaz and Roberts (2006) is that prominence fine structure plays an important role and cannot be neglected.

Terradas et al. (2008Jump To The Next Citation Point) modeled the transverse oscillations of flowing prominence threads observed by Okamoto et al. (2007Jump To The Next Citation Point) with HINODE/SOT (Section 3.6.4). The kink oscillations of a flux tube containing a flowing dense part, which represents the prominence material, were studied from both the analytical and the numerical point of view. In the analytical case, the Dymova and Ruderman (2005Jump To The Next Citation Point) approach with the inclusion of flow was used, while in the numerical calculations the linear ideal MHD equations were solved. The results point out that for the observed flow speeds there is almost no difference between the oscillation periods when static versus flowing threads are considered, and that the oscillatory period matches that of a kink mode. In addition, to obtain information about the Alfvén speed in oscillating threads, a seismological analysis as described in Section 6.6 was also performed. Also motivated by the observations by Okamoto et al. (2007Jump To The Next Citation Point), Soler and Goossens (2011Jump To The Next Citation Point) have further studied the properties of kink MHD waves propagating in flowing threads. In good agreement with Terradas et al. (2008Jump To The Next Citation Point), the period is seen to be slightly affected by mass flows. When the thread is located near the center of the supporting magnetic tube, and for realistic flow velocities, the effect of the flow on the period is estimated to fall within the error bars from observations. On the other hand, as the thread approaches the footpoint of the magnetic structure, flows introduce differences up to 50% in comparison to the static case. The variation of the amplitude of kink waves due to the flow is additionally analysed by Soler and Goossens (2011). It is found that the flow leads to apparent damping or amplification of the oscillations. During the motion of the prominence thread along the magnetic structure, the amplitude grows as the thread gets closer to the center of the tube and decreases otherwise. This effect might be important, since it would modify the actual observed attenuation, if any physical damping mechanisms is present.

Theoretical models described in this section have considered prominence plasmas as either slabs or cylindrical magnetic flux tubes. Slab models were intended to study the global oscillation properties of prominences, while flux tube models seem to be more appropriate for their application to the fine structure of prominences. Nevertheless, the properties of modes of oscillation like the kink mode have often been first studied in Cartesian geometry and then in cylindrical configurations. A few differences that arise are relevant when comparing the theoretical results to observations.

The theoretical frequencies for the kink mode in Cartesian geometry are above the value obtained for a cylindrical equivalent with the same physical properties. This has been shown by Arregui et al. (2007b), in the context of coronal loop oscillations. By assuming that a kink mode in a cylinder can be modeled in Cartesian geometry by adding a large perpendicular wavenumber, these authors show that in that limit the cylindrical kink mode frequency is recovered. A similar analogy was used by Hollweg and Yang (1988Jump To The Next Citation Point) who derived an expression for the damping time of a surface wave in Cartesian geometry and applied their result to coronal loops in the limit of large perpendicular wave number.

The spatial distribution of the eigenfunctions also differ when one compares, e.g., the kink mode properties in Cartesian and cylindrical geometry. The drop-off rate of the transverse velocity component is faster in cylindrical flux tubes than in slabs. A cylinder is a much better wave guide. For this reason, an oscillating cylindrical thread is less likely to induce oscillations in its neigboring threads than a Cartesian thread.


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