### 4.1 Oscillations of very simple prominence models

The aim of the works discussed in this section is to follow elementary arguments to derive approximations for the oscillatory period and the polarization of plasma motions of the main modes of oscillation of a prominence. Some of the obtained results correspond to MHD modes studied in more detail in other works (see Section 4.2). One of these works (Joarder and Roberts, 1992b) is concerned with a prominence treated as a plasma slab embedded in the solar corona and with a magnetic field perpendicular to the prominence main axis (Figure 27). Waves are allowed to propagate along the slab. The coordinate system introduced by Joarder and Roberts (1992b) has the x-axis pointing across the prominence (i.e., parallel to the magnetic field), the z-axis in the direction of wave propagation and the y-axis along the prominence. Three MHD modes exist in this configuration: the fast, Alfvén, and slow modes, with motions polarized in the z-, y- and x-directions, respectively. Some of the simple analogies discussed next allow us to derive approximations for the period of these modes.

A very simplified view of a prominence (Roberts, 1991; Joarder and Roberts, 1992a) is to consider it as a concentrated mass, , suspended on an elastic string (representing the sagged magnetic field that supports the prominence; Figure 23a). Such a model provides some insight into the period of the prominence oscillating vertically as a whole under the action of gravity and magnetic tension. The equilibrium state is simply one in which the gravitational force, , is balanced by the upward component of the tension forces, , where is the tension in one of the two strings and is the angle made by the string and the horizontal. Small amplitude oscillations of the mass about this equilibrium state have a period

with the separation distance between the two anchor points, which is analogous to the distance between the photospheric feet of the magnetic tube supporting the prominence plasma. Roberts (1991) noted that for typical parameter values ( = 274 m s–2,  = 50,000 km, and between 3° and 30°), the period of these vertical oscillations is in the range 7 – 24 min, consistent with observationally reported values.

Roberts (1991) and Joarder and Roberts (1992b) considered a second model of interest (Figure 23b), that resembles the previous one except that now gravity is ignored. In this configuration there are two possible types of oscillation: either longitudinal or transversal. The frequencies of oscillation are given by

where again is the distance between the anchor points, is the mass density of the string (per unit length) and is a natural wave speed of the string. To simplify matters one can assume that the mass of the string () is negligible in comparison with , that is, . Translating this inequality to prominences, it is equivalent to assuming that the mass of the cold plasma in a magnetic tube is much larger than the coronal mass in the same tube; this assumption seems most reasonable. Then, Equation (2) reduces to a simple expression for the fundamental mode frequency,
where it has been taken into account that the tension force is . Although has been considered a point mass, one can assume that it has a short, but finite, width . Then, the previous expression for the tension force applied to the prominence part of the structure is , with a natural prominence wave speed. Now, inserting this expression into Equation (3) we obtain for the period
For fast magnetoacoustic waves in a prominence (with transverse polarization of motions), can be taken as the fast speed,
with and the Alfvén and sound speeds, respectively. These quantities are given by
and
with the ratio of specific heats, the gas constant, the mean atomic weight, the magnetic permeability of vacuum, and , , and the temperature, density, and magnetic field strength. For Alfvén modes (also characterized by transverse displacements in this simplified model) . The Alfvén velocity is the group velocity but not the phase velocity for Alfvén waves except for parallel propagation. On the other hand, for slow magnetoacoustic waves (with longitudinal polarization of motions), can be taken to be the cusp speed,

The fast speed in Equation (5) and the cusp speed in Equation (8) are in general different from the phase speed and the group speed for fast and slow magnetoacoustic waves. Only for very specific directions of propagation are these quantities phase and/or group speeds. Using the same parameters as above together with  = 28 km s–1,  = 15 km s–1 and a prominence width equal to one tenth the length of magnetic field lines (i.e., ), Equation (4) yields the periods , , and , all of them within the range of observed intermediate- to long-period oscillations in prominences.

Oliver et al. (1993) (see also Roberts and Joarder, 1994) modified the mass loaded string model of Figure 23b by replacing the point mass by a denser central string of width (Figure 23c). To solve the wave equation it is necessary to impose the continuity of the displacement and its spatial derivative at the joints . Here the x-axis is placed along the string with the string centre. Then, upon imposing that the string is tied at its ends, the dispersion relation for even solutions about the centre of the string can be expressed as

whereas the dispersion relation for odd solutions can be written as
In these formulas and represent the natural wave speeds of the prominence and coronal parts of the string and and their respective densities. These expressions contain a rich array of solutions representing oscillatory modes of the system with different properties. And since there are three characteristic wave modes (fast, Alfvén, and slow, with their specific fast, Alfvén, and tube speeds in the prominence and corona), each set of modes is repeated three times. For the sake of simplicity, we here keep the parameters and in the following expressions, although it must be understood that these two speeds need to be substituted by their corresponding , , or to derive the frequencies of the fast, Alfvén, and slow solutions.

Equations (9) and (10) can be numerically solved to obtain the frequencies of the various solutions. Nevertheless, some simplifications can be done by taking into account that the prominence width is much shorter than the length of magnetic field lines () and that the prominence density is much larger than the coronal one () (Joarder and Roberts, 1992b; Roberts and Joarder, 1994). Further assuming that the following expression for the period of the fundamental mode can be obtained from Equation (9)

It is not surprising that the period corresponding to this frequency is just the one given by Equation (4). Other solutions to Equation (9) can be obtained by simply assuming . They come in two sets (Joarder and Roberts, 1992b)
and
On the other hand, Equation (10) has no low-frequency solution analogous to that of Equation (11). Instead, it has just two sets of solutions: one of them is identical to Equation (13) and the other one is similar to that given by Equation (12), namely

To understand the standing solutions supported by the string of Figure 23c, we now concentrate on their spatial distribution. Figures 24a and b display the two lowest frequency solutions of Equation (9), while Figures 24c and d show the two lowest frequency solutions of Equation (10). Their frequencies are approximately given by Equations (11), (12) with , (14) with , and (13) with , respectively. Let us refer to the parts of the string with density and as the external and internal regions. The eigenfunction in Figure 24d differs from the other three in that the displacement in the external region is an order of magnitude larger than in the internal region. For this reason it is termed an external mode, since its properties are dominated by the nature of the external part of the string (Joarder and Roberts, 1992b). One then is tempted to call the other three solutions in Figure 24 internal modes, but a simple experiment will prove this to be wrong (Oliver et al., 1993). Let us gradually reduce the size of the internal part of the string (by reducing ). Then internal mode frequencies (cf. Equations (12) and (14)) tend to infinity and in the limit internal modes disappear and only external modes remain. It turns out that this process of gradually removing the density enhancement in the central part of the string does not eliminate the mode of Figure 24a, which is transformed into the fundamental mode of the string. Thus, this is not an internal mode. By a similar process (i.e., by letting ), the central part of the string can be progressively expanded so that we end up with a uniform density . This makes external mode frequencies grow unbounded (cf. Equation (13)) and for only internal modes remain. In this process the mode of Figure 24a transforms into the fundamental mode of the string. Hence, this mode is not an external mode, either, and it owes its existence to the concurrent presence of both the internal and external parts of the string. For this reason Oliver et al. (1993) labelled this solution a hybrid mode. The hybrid mode frequency is approximately given by Equation (11), the internal mode frequencies by Equations (12) and (14) and the external mode frequencies by Equation (13).

This string analogy points out the basic nature of a prominence’s modes of oscillation. Because there are in general three MHD modes, there is a fast hybrid mode, an infinite number of internal fast modes, and an infinite number of external fast modes (Joarder and Roberts, 1992b; Oliver et al., 1993; Roberts and Joarder, 1994). Their respective frequencies are given by Equations (11), (12), (13) and (14) with and , substituted by the prominence and coronal fast speeds. Something similar can be said about Alfvén and slow modes.

Anzer (2009) performed some simple estimates of the main oscillatory periods of a prominence using the Kippenhahn–Schlüter model (Kippenhahn and Schlüter, 1957) modified so as to include the corona in which the prominence is embedded (for a general solution see Poland and Anzer, 1971). In this configuration, see Figure 25, a curved magnetic field provides support of the cold plasma against gravity. Field lines outside the prominence do not bend downwards and so the magnetic field in the coronal environment does not present the desired arcade shape. For this reason, the role of the dense photosphere is played by two vertical rigid walls. Note that the configuration used by Anzer (2009) bears some resemblance to that of Figure 23a: the coronal magnetic field is almost uniform and makes an angle with the horizontal direction. Hence, , with the magnetic field at the prominence boundary. A further similarity between the present model and the previous ones is that the density is analogous to that of Figure 23c.

Instead of solving the MHD equations, Anzer (2009) took the clever approach of making educated guesses for the restoring forces acting over the prominence () and then solving the equation

where is the plasma displacement and is the prominence column mass. For magnetically driven oscillations in the x-direction, caused by the magnetic pressure gradient, it is postulated that
so that the corresponding oscillatory period is

For oscillations in the y- and z-directions the restoring force is the magnetic tension of the stretched field lines. In both cases the restoring force is

and the oscillatory period is
It does not come as a surprise that this formula is identical to Equation (1).

Anzer (2009) noted that the field line inclination is expected to be very small and, therefore, . As a consequence, the period of x-oscillations will be much larger than that of the other two modes, polarized in the y- and z-directions.

Anzer (2009) also investigated perturbations driven by the gas pressure. He assumed that the coronal magnetic field is so strong that the prominence cannot distort it by a large amount. Further assuming that the magnetic field is horizontal, then the difference in gas pressure on either side of the prominence-corona interface can drive oscillations in the x-direction. The restoring force is approximated by

and consequently the period of this mode is
This result coincides with that obtained from the simple string models of Figures 23b and c; see Equation (4).

Four oscillatory modes can be identified from these elementary considerations, but the restoring forces in the x-direction act in unison to create a single mode, so we are left with the familiar three MHD modes: fast, Alfvén, and slow.

Some values of the periods given by Anzer (2009) are similar to those in previous works: 200 min for the magnetically dominated oscillations in the x-direction, 430 min for the gas pressure driven oscillations and 20 min for the transverse, magnetically driven oscillations.

A further refinement of the string analogy (Joarder and Roberts, 1992b; Roberts and Joarder, 1994) can be introduced by noting that the magnetic field of a prominence is not at 90° with the prominence axis, contrary to the simple models of Figures 23 and 25. Instead, the prominence magnetic field makes an angle , typically around 20°, with the long axis of the slab. This is not too important for the almost isotropic fast modes, but Alfvén and slow modes propagate mainly along field lines, which in a skewed magnetic configuration are longer than by a factor . Thus, the periods of these waves become larger by this same factor since the travel time needed for them to travel back and forth between the anchor points increases by . The result is that the hybrid Alfvén and slow modes can have periods up to 60 min and 5 h, respectively. It has been suggested that the last one may be the cause of the very long-period oscillations observed by Foullon et al. (2004); Pouget et al. (2006).