Joarder and Roberts (1992b) considered the purely transverse magnetic field of Figure 27. From the characteristic wavenumbers of the solutions in the x-direction, Joarder and Roberts (1992b) created the distinction between internal and external modes (see Section 4.1 for a discussion of the features of these solutions). According to these authors, the former group of modes arises principally from the magnetoacoustic properties of the plasma slab, although these modes are somewhat influenced by the external material because of the presence of free interfaces between the prominence and corona. External modes are present, on the other hand, even in the absence of the prominence plasma but are modified because of the introduction of this cool, dense slab. The dispersion diagrams of kink and sausage modes are shown in Figure 28, where and are the sound and Alfvén speeds in the prominence, while and are their coronal counterparts. Moreover, Joarder and Roberts (1992b) also removed propagation along the prominence by setting . The mode frequencies are then those on the vertical axes of Figure 28. In this case the dispersion relations of kink and sausage modes are those discussed for a string with densities and (Figure 23c), namely Equations (9) and (10). Joarder and Roberts (1992b) gave the approximate solutions of Equations (11) to (14), which are in very good agreement with the results of Figure 28 for .
Oliver et al. (1993) provided more insight into the nature of internal and external modes while using the non-isothermal Kippenhahn–Schlüter solution represented in Figure 25. These authors followed the evolution of fast and slow modes in the dispersion diagram when the prominence is slowly removed by taking . They noted that the frequency of internal modes, both slow and fast, progressively grows until the modes disappear from the dispersion diagram and, therefore, only external modes remain. The presence of the prominence region thus provides physical support for the existence of internal modes. The same is true for external modes when the corona is gradually removed by making . A clear distinction then arises between the two types of modes, although it turns out that the fundamental mode is internal and external at the same time, since it survives both when the prominence and the corona are eliminated. For this reason, this mode with mixed internal and external properties was called hybrid by Oliver et al. (1993) and later string by Joarder and Roberts (1993b) because it arises in the string analogy. Nevertheless, internal and external modes are also present in the string analogy (Section 4.1), so perhaps hybrid mode is a better denomination for this solution.
From Oliver et al. (1993) it also appears that the amplitude of perturbations in the prominence is rather small for external modes, a feature that is also present in the string solutions of Figure 24. For this reason it was postulated that they would probably be difficult to detect in solar prominences and that the reported periodic variations are produced by the hybrid and internal modes. In addition, the frequency of internal modes is shown to depend on prominence properties only, while that of hybrid and external modes depends on other physical variables such as the length of field lines. This is in agreement with the approximate Equations (11) to (14).
The essential difference between the equilibrium models in Joarder and Roberts (1992b) and in Oliver et al. (1993) is that gravity is neglected in the former, which results in straight magnetic field lines, while it is a basic ingredient in the later, which results in the curved shape of field lines characteristic of the Kippenhahn–Schlüter equilibrium model. Despite the different shape of field lines, the main features of the oscillatory spectrum are similar and so the influence of gravity and field line shape on the properties of the MHD modes is not too relevant in this kind of configurations.
A study of the oscillatory modes of the Kippenhahn and Schlüter (1957) prominence model was undertaken by Oliver et al. (1992). The equilibrium model is represented in Figure 25 although the corona is omitted. This implies that this work only provides a restricted account of the MHD modes of a slab prominence since there are no hybrid and external solutions in the absence of the corona. Oliver et al. (1992) noted that the three MHD modes possess different velocity orientations. The fast mode is characterized by vertical motions. The Alfvén mode by motions along the filament long axis, and the slow mode by plasma displacements parallel to the equilibrium magnetic field, which in this configuration is practically horizontal and transverse to the prominence. The immediate consequence of this association between modes and velocity polarization is that periodic variations in the Doppler shift are more likely to be detected in filaments near the disk centre for fast modes and in limb prominences for Alfvén and slow modes, depending on the orientation of the prominence with respect to the observer. These features of the MHD modes are retained in other models in which the equilibrium magnetic field is assumed perpendicular to the filament axis (Joarder and Roberts, 1992b, 1993a; Oliver et al., 1993; Oliver and Ballester, 1995, 1996). Nevertheless, the distinction between the three MHD modes is lost when the observed longitudinal magnetic field component is taken into account (Joarder and Roberts, 1993b). Probably, there are no characteristic oscillatory directions associated to the various modes (unfortunately, the issue of velocity polarization in a skewed magnetic equilibrium model has not yet been addressed in the context of prominence oscillations). The actual velocity field in prominences can be substantially more complex than that indicated by investigations based on models with magnetic field purely transverse to the prominence slab.
It is well-known (Leroy, 1988, 1989) that magnetic lines are actually oriented at a rather small angle (around 20°) with the prominence axis. Joarder and Roberts (1993b) took this observational fact into account by adding a longitudinal magnetic field component to the equilibrium model used in Joarder and Roberts (1992b); see Figure 29. Now, the xy-plane is defined to contain the assumed horizontal magnetic field. Then, it is not possible to place the z-axis parallel to the wavenumber along the prominence axis, so now the and components must be considered. The assumptions of transverse field and propagation of perturbations in the z-direction made in the works discussed above simplify the MHD wave equations since the Alfvén mode is decoupled from the slow and fast modes, which can be studied separately with a subsequent reduction in complexity of the mathematical problem. The problem considered by Joarder and Roberts (1993b) contains coupled fast, Alfvén and slow modes. The resulting dispersion diagram (Figure 30) displays a very rich mode structure with plenty of mode couplings, which anticipates the complex nature of actual prominence MHD modes. Unfortunately, the physical properties of perturbations (velocity polarization, importance of the various restoring forces, perturbations of the equilibrium variables, …) for the modes in the dispersion diagram have not been examined yet. Oscillation periods up to 4 h (for the slow hybrid mode) are present in this configuration.
The previous results rely on models in which the prominence and coronal temperatures are uniform, with a sharp jump of this physical variable from the cool to the hot region at an infinitely thin interface. A smoothed temperature transition between the two domains, representing the prominence-corona transition region (PCTR), was used by Oliver and Ballester (1996) to investigate the MHD modes of a more realistic configuration. Despite the presence of the PCTR in the equilibrium model, internal, external, and hybrid modes are still supported, just like in configurations with two uniform temperature regions. Nevertheless, the PCTR results in a slight frequency shift and in the modification of the spatial velocity distribution so as to decrease the oscillatory amplitude of internal modes inside the prominence. Hybrid modes are not so much affected by the presence of the PCTR because their characteristic wavelength is much longer than the width of the PCTR. Then, the conclusion is that the PCTR may influence the detectability of periodic prominence perturbations arising from internal modes.
Some two-dimensional equilibrium models were considered by Galindo Trejo (1987, 1989a,b, 1998, 2006). The focus of these works was in the stability properties of prominence equilibrium configurations (using the MHD energy principle of Bernstein et al., 1958) and for this reason the author concentrated in the lowest eigenvalue squared. This means that information about all other modes of the system is absent. Galindo Trejo (1987) considered four prominence models, namely those by Kippenhahn and Schlüter (1957), Dungey (1953), Menzel (1951), and Lerche and Low (1980). All these models are isothermal, i.e., they do not incorporate the corona around the prominence plasma. This implies that the important hybrid modes are absent in the analysis. In spite of this, some interesting results were obtained by Galindo Trejo (1987). Here we only mention the most relevant ones. For example, the fundamental mode of the Kippenhahn–Schlüter configuration, whose period is 16 min, has motions polarized mainly across the prominence slab, so it can be associated with the internal slow mode. On the other hand, the fundamental mode of Dungey’s model has horizontal motions mostly along the prominence axis (such as corresponds to Alfvén waves) which are more important at the top of the prominence than at the bottom. The oscillatory period ranges from 55 to 80 min. In the case of Menzel’s model, the lowest frequency eigenmode has a period of 40 min and motions whose amplitude increases with height and oriented across the prominence. The eigenmode of Lerche & Low’s solution presents a greater range of periods (17 – 50 min) and, once more, with horizontal plasma displacements transverse to the prominence axis. Two improvements of this elaborated work can be done: the inclusion of the coronal plasma and the consideration of the oscillatory properties of other modes.
In two subsequent papers the stability of the prominence model of Low (1981) was investigated. In the first one (Galindo Trejo, 1989a) a uniform magnetic field component along the prominence axis was used, whereas in the second one (Galindo Trejo, 1989b) this quantity is not uniform. The author concluded that, as long as this magnetic field component is weak, these different choices of the magnetic configuration do not influence much the period of the fundamental mode, which is in the range 3 – 7 min. The spatial distribution of motions is similar to that found by Galindo Trejo (1987) for Menzel’s and Lerche & Low’s equilibrium models.
The following paper of this series (Galindo Trejo, 1998) is concerned with the prominence model of Osherovich (1985), which is characterized by a surrounding horizontal magnetic field connected with the prominence field. Different values of the equilibrium parameters were used and as a result the fundamental mode has periods that range from 4 to 84 min. Galindo Trejo (1998) found that for small values of the longitudinal magnetic field component large velocity amplitudes predominate in the upper part of the prominence, while the opposite happens for a stronger longitudinal component. The magnetic field shear is also relevant: for a moderate (and hence non-uniform) shear, the fundamental eigenmode is in the intermediate-period range and for a uniform shear long periods are obtained.
Galindo Trejo (2006) investigated the equilibrium solution of Osherovich (1989), that is characterized by an external vertical magnetic field that allows the prominence to be placed on the boundary between two regions of opposite photospheric magnetic polarity. A wide range of periods was obtained in this work (9 – 73 min). Also, horizontal oscillatory motions either along the prominence or almost across it were found. Therefore, it seems that in most configurations studied by Galindo Trejo the fundamental oscillatory mode is a slow mode.
Living Rev. Solar Phys. 9, (2012), 2
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