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. (1997) considered a thin thread with finite width and length in Cartesian geometry (Figure 38). 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- limit was taken) and consequently the slow mode is absent from their analysis. Joarder et al. (1997) 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.

Using the same two-dimensional configuration, Díaz et al. (2001) 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 39). Fast kink modes are associated to normal motions with respect to the thread length (i.e., in the x-direction; see Figure 38). 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.

Later on, Díaz et al. (2003) included wave propagation in the y-direction (see Figure 38) 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 (), 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 case (see Figure 40). 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.

Since cylindrical geometry is more suitable to model prominence threads, Díaz et al. (2002) considered a straight cylindrical flux tube with a cool region representing the prominence thread, which is confined by two symmetric hot regions (Figure 41). With this geometry the fundamental sausage mode (, with the azimuthal wavenumber) and its harmonics are always leaky. However, for all other modes (), 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 42). Therefore, an oscillating cylindrical thread is less likely to induce oscillations in its neighbouring threads than a Cartesian one.

To study the oscillations of the above mentioned configurations, Díaz et al. (2001, 2002) developed a very general, although cumbersome procedure. However, Dymova and Ruderman (2005) 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 (2005) 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. (2002) points out differences of the order of 1%. The only drawback of the method of Dymova and Ruderman (2005) 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. (2005) 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 (Figure 43). An inhomogeneous filament composed of five threads was constructed (Figure 44) 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 44 by a certain factor and the kink modes were computed. Their frequencies are displayed in Figure 45, where is a reference value representative of the separations between threads. When the separations are small, i.e., for , 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 45, i.e., , 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 , 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.

Díaz and Roberts (2006) studied the properties of the fast MHD modes of a periodic, Cartesian thread model (see Figure 1 of Díaz and Roberts 2006). This configuration provides a bridge between a structure with a limited number of threads (studied by Díaz et al., 2005, see Figure 43) and a homogeneous prominence with a transverse magnetic field (investigated by Joarder and Roberts, 1992b, see Figure 27). Díaz and Roberts (2006) 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 , with the period of the prominence slab and 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 (2006) 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. (2008) modeled the transverse oscillations of flowing prominence threads observed by Okamoto et al. (2007) 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 (2005) 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. (2007), Soler and Goossens (2011) have further studied the properties of kink MHD waves propagating in flowing threads. In good agreement with Terradas et al. (2008), 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 (1988) 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.

Living Rev. Solar Phys. 9, (2012), 2
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