4.2 Oscillations of prominence slabs

In a series of three papers, Joarder and Roberts conducted analyses of the modes of oscillation of a magnetized prominence slab embedded in the corona. The influence of gravity was neglected and so the plasma variables (temperature, pressure, and density) are uniform both in the prominence and in the coronal region. In the first of these works (Joarder and Roberts, 1992aJump To The Next Citation Point) a purely longitudinal magnetic field was taken (see Figure 26View Image). The dispersion relation contains a variety of modes, which can be fast or slow, combined with kink or sausage and body or surface. Because of the strong difference of the prominence and coronal physical parameters, some eigensolutions are slow in the external medium and fast in the internal medium. Tabulated periods range from 9 h and 5 h to a few minutes. The first values had not been reported at the time this work was published, so emphasis was given by the authors to fast surface modes, with shorter periods around 1 h, and to 5-min and 3-min Pekeris and Love modes.
View Image

Figure 26: Sketch of the prominence slab model with longitudinal magnetic field used by Joarder and Roberts (1992aJump To The Next Citation Point). These authors assumed that the coronal environment in which the prominence is embedded extends infinitely in the x-direction. In this figure the width of the prominence is denoted by 2a, whereas in our text 2xp is used.
View Image

Figure 27: Schematic diagram of a prominence slab in a coronal environment. The magnetic field is perpendicular to the prominence axis and tied at the photosphere, represented by two rigid conducting walls at x = ± ℓ. Note that in the text the position of the photospheric walls is denoted by x = ±L (from Joarder and Roberts, 1992bJump To The Next Citation Point).

Joarder and Roberts (1992bJump To The Next Citation Point) considered the purely transverse magnetic field of Figure 27View Image. From the characteristic wavenumbers of the solutions in the x-direction, Joarder and Roberts (1992bJump To The Next Citation Point) 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 28View Image, where c sp and v Ap are the sound and Alfvén speeds in the prominence, while csc and vAc are their coronal counterparts. Moreover, Joarder and Roberts (1992bJump To The Next Citation Point) also removed propagation along the prominence by setting kz = 0. The mode frequencies are then those on the vertical axes of Figure 28View Image. In this case the dispersion relations of kink and sausage modes are those discussed for a string with densities ρc and ρp (Figure 23View Imagec), namely Equations (9View Equation) and (10View Equation). Joarder and Roberts (1992bJump To The Next Citation Point) gave the approximate solutions of Equations (11View Equation) to (14View Equation), which are in very good agreement with the results of Figure 28View Image for kz = 0.

View Image

Figure 28: Dispersion diagram of magnetoacoustic (a) kink modes and (b) sausage modes in the equilibrium model represented in Figure 27View Image. Meaning of labels at the right of each curve: modes are identified as fundamental (f), first harmonic (1h), second harmonic (2h), …; internal or external (I or E); and fast or slow (F or S). Here ω and kz are the frequency and the wavenumber along the slab, while vAe is the coronal Alfvén speed and L is half the length of the supporting magnetic field. Parameter values used: v Ap = 28 km s–1, c sp = 15 km s–1, vAc = 315 km s–1, csc = 166 km s–1 (from Joarder and Roberts, 1992bJump To The Next Citation Point).

Oliver et al. (1993Jump To The Next Citation Point) provided more insight into the nature of internal and external modes while using the non-isothermal Kippenhahn–Schlüter solution represented in Figure 25View Image. These authors followed the evolution of fast and slow modes in the dispersion diagram when the prominence is slowly removed by taking xp → 0. 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 xp → L. 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. (1993Jump To The Next Citation Point) and later string by Joarder and Roberts (1993bJump To The Next Citation Point) 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. (1993Jump To The Next Citation Point) 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 24View Image. 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 (11View Equation) to (14View Equation).

The essential difference between the equilibrium models in Joarder and Roberts (1992bJump To The Next Citation Point) and in Oliver et al. (1993Jump To The Next Citation Point) 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 (1957Jump To The Next Citation Point) prominence model was undertaken by Oliver et al. (1992Jump To The Next Citation Point). The equilibrium model is represented in Figure 25View Image 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. (1992Jump To The Next Citation Point) 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, 1992bJump To The Next Citation Point, 1993aJump To The Next Citation Point; Oliver et al., 1993; Oliver and Ballester, 1995, 1996Jump To The Next Citation Point). Nevertheless, the distinction between the three MHD modes is lost when the observed longitudinal magnetic field component is taken into account (Joarder and Roberts, 1993bJump To The Next Citation Point). 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 (1993bJump To The Next Citation Point) took this observational fact into account by adding a longitudinal magnetic field component to the equilibrium model used in Joarder and Roberts (1992bJump To The Next Citation Point); see Figure 29View Image. 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 ky and kz 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 (1993bJump To The Next Citation Point) contains coupled fast, Alfvén and slow modes. The resulting dispersion diagram (Figure 30View Image) 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.

View Image

Figure 29: Schematic diagram of a prominence slab in a coronal environment. The magnetic field makes an angle ϕ with the prominence axis and is tied at the photosphere, represented by two rigid, perfectly conducting walls (from Joarder and Roberts, 1993bJump To The Next Citation Point).
View Image

Figure 30: Dispersion diagram of magnetoacoustic modes in the equilibrium structure of Figure 29View Image. Meaning of labels at the right of each curve: modes are identified as string, fundamental (f), first harmonic (1h), second harmonic (2h), …; internal or external (I or E); and fast, Alfvén or slow (F, A or S) according to the mode’s nature for κ ≪ 1. Here ω and κ are the frequency and the wavenumber modulus, while vAe is the coronal Alfvén speed and ℓ is half the length of the supporting magnetic field. Parameter values used: vAp = 74 km s–1, csp = 15 km s–1, vAc = 828 km s–1, csc = 166 km s–1 (from Joarder and Roberts, 1993bJump To The Next Citation Point).

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 (1987Jump To The Next Citation Point, 1989aJump To The Next Citation Point,bJump To The Next Citation Point, 1998Jump To The Next Citation Point, 2006Jump To The Next Citation Point). 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 (1987Jump To The Next Citation Point) 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 (1987Jump To The Next Citation Point). 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, 1998Jump To The Next Citation Point) 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.


  Go to previous page Go up Go to next page