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, 1992bJump To The Next Citation Point) 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 27View Image). Waves are allowed to propagate along the slab. The coordinate system introduced by Joarder and Roberts (1992bJump To The Next Citation Point) 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, 1991Jump To The Next Citation Point; Joarder and Roberts, 1992aJump To The Next Citation Point) is to consider it as a concentrated mass, M, suspended on an elastic string (representing the sagged magnetic field that supports the prominence; Figure 23View Imagea). 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, M g, is balanced by the upward component of the tension forces, 2T sin πœƒ, where T is the tension in one of the two strings and πœƒ is the angle made by the string and the horizontal. Small amplitude oscillations of the mass about this equilibrium state have a period

( ) 12 P = 2π L-tan πœƒ , (1 ) g
with 2L 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 (1991Jump To The Next Citation Point) noted that for typical parameter values (g = 274 m s–2, 2L = 50,000 km, and πœƒ between 3° and 30°), the period of these vertical oscillations is in the range 7 – 24 min, consistent with observationally reported values.
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Figure 23: Simple models of a prominence. (a) Mass suspended from an elastic string under the influence of gravity and the tension force. (b) Mass suspended from a taut string subject solely to the tension force (from Roberts, 1991Jump To The Next Citation Point). (c) Taut string with density ρc except for a central part with density ρp; gravity is also neglected (from Oliver et al., 1993Jump To The Next Citation Point). The size of the system (2β„“ in panels (a) and (b) and 2xc in panel (c)) is denoted by 2L in the text.

Roberts (1991) and Joarder and Roberts (1992bJump To The Next Citation Point) considered a second model of interest (Figure 23View Imageb), 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

( ) ωL- ωL- 2ρL- cstr tan cstr = M , (2 )
where again 2L is the distance between the anchor points, ρ is the mass density of the string (per unit length) and cstr is a natural wave speed of the string. To simplify matters one can assume that the mass of the string (2ρL) is negligible in comparison with M, that is, M ≫ ρL. 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 (2View Equation) reduces to a simple expression for the fundamental mode frequency,
( ) 1 -2T- 2 ω = M L , (3 )
where it has been taken into account that the tension force is 2 T = ρcstr. Although M has been considered a point mass, one can assume that it has a short, but finite, width 2xp. Then, the previous expression for the tension force applied to the prominence part of the structure is T = M c2proβˆ•(2xp), with cpro a natural prominence wave speed. Now, inserting this expression into Equation (3View Equation) we obtain for the period
(Lx )12 P = 2π ----p--. (4 ) cpro
For fast magnetoacoustic waves in a prominence (with transverse polarization of motions), c pro can be taken as the fast speed,
( ) c = v2 + c2 1βˆ•2, (5 ) f A s
with vA and cs the Alfvén and sound speeds, respectively. These quantities are given by
c2 = γRT-- (6 ) s &tidle;μ
and
B2 v2A = ---, (7 ) μρ
with γ the ratio of specific heats, R the gas constant, μ&tidle; the mean atomic weight, μ the magnetic permeability of vacuum, and T, ρ, and B the temperature, density, and magnetic field strength. For Alfvén modes (also characterized by transverse displacements in this simplified model) cpro = vA. 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), cpro can be taken to be the cusp speed,
---vAcs----- cT = (v2+ c2)1βˆ•2. (8 ) A s

The fast speed in Equation (5View Equation) and the cusp speed in Equation (8View Equation) 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 vA = 28 km s–1, cs = 15 km s–1 and a prominence width equal to one tenth the length of magnetic field lines (i.e., 2xp = 2Lβˆ•10 = 5000 km), Equation (4View Equation) yields the periods Pfast = 26 min, PAlfven = 30 min, and Pslow = 63 min, all of them within the range of observed intermediate- to long-period oscillations in prominences.

Oliver et al. (1993Jump To The Next Citation Point) (see also Roberts and Joarder, 1994Jump To The Next Citation Point) modified the mass loaded string model of Figure 23View Imageb by replacing the point mass M by a denser central string of width 2xp (Figure 23View Imagec). To solve the wave equation it is necessary to impose the continuity of the displacement and its spatial derivative at the joints x = ±xp. Here the x-axis is placed along the string with x = 0 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

ωx ( ρ ) 12 ω(L − x ) tan --p-= --c cot -------p--, (9 ) cpro ρp ccor
whereas the dispersion relation for odd solutions can be written as
( ) 12 cot ωxp-= − ρc- cot ω(L-−-xp-). (10 ) cpro ρp ccor
In these formulas cpro and ccor represent the natural wave speeds of the prominence and coronal parts of the string and ρp and ρc 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 cpro and ccor in the following expressions, although it must be understood that these two speeds need to be substituted by their corresponding cf, vA, or cT to derive the frequencies of the fast, Alfvén, and slow solutions.

Equations (9View Equation) and (10View Equation) 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 (xp β‰ͺ L) and that the prominence density is much larger than the coronal one (ρp ≫ ρc) (Joarder and Roberts, 1992bJump To The Next Citation Point; Roberts and Joarder, 1994Jump To The Next Citation Point). Further assuming that ρcβˆ•ρp β‰ͺ xpβˆ•L β‰ͺ 1 the following expression for the period of the fundamental mode can be obtained from Equation (9View Equation)

cpro ω = -----1-. (11 ) (Lxp )2
It is not surprising that the period corresponding to this frequency is just the one given by Equation (4View Equation). Other solutions to Equation (9View Equation) can be obtained by simply assuming ρcβˆ•ρp β‰ͺ 1. They come in two sets (Joarder and Roberts, 1992bJump To The Next Citation Point)
cpro ω = nπ x--, n = 1,2, 3,..., (12 ) p
and
ccor ω = nπ -------, n = 1,2, 3,... (13 ) L − xp
On the other hand, Equation (10View Equation) has no low-frequency solution analogous to that of Equation (11View Equation). Instead, it has just two sets of solutions: one of them is identical to Equation (13View Equation) and the other one is similar to that given by Equation (12View Equation), namely
π-cpro- ω = (2n + 1)2 xp , n = 0,1, 2,... (14 )

To understand the standing solutions supported by the string of Figure 23View Imagec, we now concentrate on their spatial distribution. Figures 24View Imagea and b display the two lowest frequency solutions of Equation (9View Equation), while Figures 24View Imagec and d show the two lowest frequency solutions of Equation (10View Equation). Their frequencies are approximately given by Equations (11View Equation), (12View Equation) with n = 1, (14View Equation) with n = 0, and (13View Equation) with n = 1, respectively. Let us refer to the parts of the string with density ρc and ρp as the external and internal regions. The eigenfunction in Figure 24View Imaged 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, 1992bJump To The Next Citation Point). One then is tempted to call the other three solutions in Figure 24View Image internal modes, but a simple experiment will prove this to be wrong (Oliver et al., 1993Jump To The Next Citation Point). Let us gradually reduce the size of the internal part of the string (by reducing xp). Then internal mode frequencies (cf. Equations (12View Equation) and (14View Equation)) tend to infinity and in the limit x → 0 p 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 24View Imagea, 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 xp → L), the central part of the string can be progressively expanded so that we end up with a uniform density ρ p. This makes external mode frequencies grow unbounded (cf. Equation (13View Equation)) and for xp → L only internal modes remain. In this process the mode of Figure 24View Imagea 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. (1993Jump To The Next Citation Point) labelled this solution a hybrid mode. The hybrid mode frequency is approximately given by Equation (11View Equation), the internal mode frequencies by Equations (12View Equation) and (14View Equation) and the external mode frequencies by Equation (13View Equation).

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Figure 24: Spatial distribution of some normal modes of the string shown in Figure 23View Imagec. (a) Hybrid mode, (b) first internal even harmonic, (c) first internal odd harmonic, (d) first external odd harmonic. The spatial coordinate is given in units of L and the string of density ρp is in the range − 0.1 ≤ x ≤ 0.1. The wave speeds are cpro = 15 km s–1 and ccor = 166 km s–1, representative of prominence and coronal sound speeds, and the density ratio is ρpβˆ•ρc = 11.25.

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, 1992bJump To The Next Citation Point; Oliver et al., 1993Jump To The Next Citation Point; Roberts and Joarder, 1994Jump To The Next Citation Point). Their respective frequencies are given by Equations (11View Equation), (12View Equation), (13View Equation) and (14View Equation) with cpro and ccor, substituted by the prominence and coronal fast speeds. Something similar can be said about Alfvén and slow modes.

Anzer (2009Jump To The Next Citation Point) performed some simple estimates of the main oscillatory periods of a prominence using the Kippenhahn–Schlüter model (Kippenhahn and Schlüter, 1957Jump To The Next Citation Point) 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 25View Image, 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 (2009Jump To The Next Citation Point) bears some resemblance to that of Figure 23View Imagea: the coronal magnetic field is almost uniform and makes an angle πœƒ with the horizontal direction. Hence, tan πœƒ = Bz1βˆ•Bx, with Bz1 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 23View Imagec.

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Figure 25: Sketch of a prominence configuration based on the Kippenhahn and Schlüter (1957Jump To The Next Citation Point) model. In the text the system size (2Δ) and the prominence width (D) are denoted by 2L and 2x p, respectively. Except for the filed line curvature, this configuration is identical to that of Figure 27View Image (from Anzer, 2009Jump To The Next Citation Point).

Instead of solving the MHD equations, Anzer (2009Jump To The Next Citation Point) took the clever approach of making educated guesses for the restoring forces acting over the prominence (F (ξ)) and then solving the equation

2 M d-ξ-= F(ξ), (15 ) dt2
where ξ is the plasma displacement and M is the prominence column mass. For magnetically driven oscillations in the x-direction, caused by the magnetic pressure gradient, it is postulated that
Bz1 g F (ξ) = − M ------ξ, (16 ) Bx L
so that the corresponding oscillatory period is
( ) 12 ( ) 12 P = 2π Bx-L- = 2π --L---- . (17 ) Bz1 g gtan πœƒ

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

-Bx-g- F (ξ) = − M Bz1 L ξ, (18 )
and the oscillatory period is
( ) ( )1 Bz1-L- L- 2 P = 2 π B g = 2π g tan πœƒ . (19 ) x
It does not come as a surprise that this formula is identical to Equation (1View Equation).

Anzer (2009Jump To The Next Citation Point) noted that the field line inclination is expected to be very small and, therefore, B βˆ•B β‰ͺ 1 z1 x. 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 (2009Jump To The Next Citation Point) 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

c2 F (ξ) = − M --s-ξ, (20 ) Lxp
and consequently the period of this mode is
1 P = 2π (Lxp-)2. (21 ) cs
This result coincides with that obtained from the simple string models of Figures 23View Imageb and c; see Equation (4View Equation).

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, 1992bJump To The Next Citation Point; 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 23View Image and 25View Image. Instead, the prominence magnetic field makes an angle Ο•, 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 2L by a factor 1βˆ• sin Ο• ≈ 3. 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 1βˆ•sinΟ•. 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. (2006Jump To The Next Citation Point).


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