4.4 Fine structure oscillations (infinitely long thread limit)

Prominence models considered in Sections 4.1, 4.2, and 4.3 are very simplified representations of solar prominences. They provide us with information about a prominence global oscillatory behaviour, but high resolution observations (see Sections 3.4 and 3.6.4) have given us detailed information about the local oscillatory behaviour of the fine, internal structure of filaments. This has prompted the study of thread oscillations. Two situations can be considered: the simplest one is that of short waves propagating along a thread. By short we mean that the wavelength is much shorter than the thread length, so we refer to this problem as the infinitely long thread limit. On the other hand, the second situation includes propagating waves whose wavelength is comparable to or larger than the length of the thread and standing modes, whose wavelength is of the order of the length of the supporting magnetic tube and thus much larger than the thread length; the works concerned with this second kind of problem are presented in Section 4.5. Other important ingredients uncovered by observations (Sections 3.4 and 3.6.4) are the collective behaviour and the presence of flows in some oscillating threads. These features have been incorporated into some investigations and will be also discussed here.

4.4.1 Individual thread oscillations

A simple thread model consists of an infinitely long cylinder filled with cold, dense plasma and embedded in the hotter and less dense corona; field line curvature is neglected. The magnetic field is parallel to the cylinder axis and uniform everywhere (Figure 31View Image).

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Figure 31: Sketch of an infinitely long thread immersed in the solar corona (from Lin et al., 2009Jump To The Next Citation Point).

The MHD modes of this structure have been extensively studied in the context of coronal and photospheric magnetic tube oscillations (Spruit, 1982; Edwin and Roberts, 1983; Cally, 1986). The mode of interest here is the kink mode because it is the only one that produces a significant transverse displacement of the thread, which is the observed behavior of oscillating threads. In the absence of mass flows and assuming that the thread radius is much smaller than the wavelength, the kink frequency is given by

┌│ -------------- ∘ ------ │∘ ρfv2Ap-+-ρcv2Ac -2ζ--- ωk = kz ρp + ρc = kzvAp 1 + ζ , (22 )
with kz the axial wavenumber, ρp and ρc the prominence thread and coronal densities, ζ = ρp∕ρc the density contrast, and vAp,c = B ∕√ μρp,c the prominence thread and coronal Alfvén velocities. In terms of the density contrast, the period of kink oscillations with wavelength λ = 2 π∕k z can then be written as
√ -- ( )1 ∕2 P = --2-λ-- 1-+-ζ- . (23 ) 2 vAp ζ
Note that the factor containing the density contrast varies between √ -- 2 and 1 when ζ is allowed to vary between a value slightly larger that 1 (extremely tenuous thread) and ζ → ∞. This defines a narrow range of Alfvén speed values when the inverse problem is solved for plasma diagnostic purposes (see Section 6). One can plug typical parameter values into Equation (23View Equation) and periods ranging from 30 s to a few minutes are obtained. This result is in agreement with the observed periods of traveling waves in threads (see Section 3.6.4).

This formula for P is based on some assumptions, namely that the thread is much longer than the wavelength, which in turn is much larger than the thread radius (this last approximation is also know as the thin tube limit). Short-wavelength propagating waves in threads have been detected by Lin et al. (2007Jump To The Next Citation Point) (see Section 3.6.4 and Figure 22View Image). The length of the fine structure is around 20 arcsec, the reported wavelength is 3.8 arcsec, and the radius of threads is typically between 0.1 and 0.15 arcsec. We can appreciate that the assumptions made to derive Equation (23View Equation) are satisfied in this event.

Nakariakov and Roberts (1995) studied the magnetosonic modes of a magnetic slab when flows are present, while Soler et al. (2008Jump To The Next Citation Point) considered non adiabatic waves and included a mass flow parallel to the magnetic field in the thread model of Figure 32View Image, which is identical to that of Figure 31View Image except for the inclusion of plasma flows. Without loss of generality the flow speed in the corona was neglected in this last work, while typical values observed in prominences were taken for the flow speed in the thread (namely Up ≤ 30 km s−1). In the absence of flow, the complex oscillatory frequencies for a fixed, real and positive wavenumber kz appear in pairs, ω = ω + iω 1 R I and ω = − ω + iω 2 R I. The solution ω 1 corresponds to a wave propagating towards the positive z-direction (parallel to magnetic field lines). The ω2 solution corresponds to a wave that propagates toward the negative z-direction (antiparallel to magnetic field lines). Both solutions have exactly the same physical properties in the absence of flows. In the presence of flow, the frequencies are Doppler shifted. In addition, the symmetry between parallel and antiparallel propagation is broken. For instance, for strong enough flows, slow waves can only propagate parallel to the flow direction, antiparallel propagation being forbidden. Figure 33View Image presents the period of the slow, fast and thermal modes as a function of the flow speed in the thread. For Up ⁄= 0 the fast and slow waves acquire different periods that diverge as Up is increased. For Up ∼ 8.5 km s−1 the antiparallel slow wave becomes a backward wave, which causes its period to grow dramatically near this flow velocity. The influence of the flow on the fast mode is not so severe, while the thermal mode has a finite period that takes very large values.

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Figure 32: Sketch of an infinitely long thread immersed in the solar corona. The respective flow speeds in the thread and the corona are denoted by U p and U c (from Soler et al., 2008Jump To The Next Citation Point).
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Figure 33: Period of the fundamental oscillatory modes of an infinitely long thread versus the mass flow, Up. The top, middle, and bottom panels correspond to the slow, kink, and thermal modes. Different line styles correspond to waves propagating in the absence of flow (dotted), parallel waves (solid), and anti-parallel waves (dashed). The wavenumber is given by k a = 10−2 z, which is consistent with the wavelength of observed propagating waves in prominences (Section 3.6); a is the thread radius (from Soler et al., 2008Jump To The Next Citation Point).

4.4.2 Collective thread oscillations

Some authors have reported that groups of threads oscillate in unison (e.g., Yi et al., 1991) and that large areas of a prominence present in-phase oscillations (e.g., Terradas et al., 2002Jump To The Next Citation Point; Lin et al., 2007Jump To The Next Citation Point), which may be also taken as a sign of collective thread behaviour (see Sections 3.4, 3.6.3, and 3.6.4). Similar collective oscillations have been observed in coronal loops (Verwichte et al., 2004) and their properties have been studied by, e.g., Murawski (1993), Luna et al. (2008Jump To The Next Citation Point), Van Doorsselaere et al. (2008b), and Robertson and Ruderman (2011). To model this situation, an equilibrium model made of two homogeneous and infinitely long prominence threads embedded in the coronal medium has been considered (see Figure 34View Image).

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Figure 34: Sketch of an equilibrium model made of two infinitely long threads embedded in the solar corona (from Luna et al., 2008Jump To The Next Citation Point).

When identical threads are considered, the system exhibits four kink-like transverse oscillatory modes (Luna et al., 2008; Soler et al., 2009aJump To The Next Citation Point). These modes are denoted by Sx, Ax, Sy and Ay. The S and A denote symmetry or antisymmetry of the total pressure perturbation with respect to the yz-plane. The subscript describes the main direction of polarization of motions, that is in the xy-plane; the choice of the coordinate axes is shown in Figure 34View Image and the spatial distribution of the modes is displayed in Figure 35View Image. In addition to the kink-like modes, Soler et al. (2009aJump To The Next Citation Point) studied the collective slow modes and obtained only two fundamental collective solutions, one symmetric and the other antisymmetric with respect to the yz-plane, with motions mainly polarized along the z-direction (Figure 35View Image).

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Figure 35: Fundamental normal modes of two parallel and infinitely long threads (Figure 34View Image). Total pressure perturbation field (contour plot in arbitrary units) and transverse Lagrangian displacement vector field (arrows) in the xy-plane for the wave modes (a) S x, (b) A x, (c) S y, (d) A y, (e) Sz, and (f) Az for a separation between threads d = 4a and a longitudinal wavenumber −2 kza = 10, where a is the thread radius. The prominence thread boundaries are denoted by dotted circles (from Soler et al., 2009aJump To The Next Citation Point).

A measure of the interaction between threads is the frequency of their normal modes. If the modes have frequencies similar to that of the isolated cylinder, then the threads oscillate independently from one another. If the frequencies are significantly different, the threads oscillate in a collective manner. The left panel of Figure 36View Image displays the real part of the frequency of the four kink-like solutions as a function of the distance between cylinders. For large separations, i.e., for a distance between threads larger than about 6 or 7 radii, the collective kink mode frequencies are almost identical to the individual kink frequency. This is a signature of a weak interaction between threads, which behave as independent structures. On the other hand, for short thread separations the four frequencies separate in two branches as a consequence of a strong interaction between the cylinders. Therefore, the collective behaviour of oscillations becomes stronger when the threads are closer. In the case of slow modes the interaction between threads is almost negligible and as a result the frequencies of the Sz and Az modes are almost identical to the individual slow mode frequency (cf. right panel of Figure 36View Image) in the whole range of thread separations. This is in agreement with the fact that transverse motions (responsible for the interaction between threads) are not significant for slow modes in comparison with their longitudinal motions. Therefore, the Sz and Az modes essentially behave as individual slow modes, contrary to kink-like modes, which display a more significant collective behaviour.

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Figure 36: Fundamental normal modes of two parallel and infinitely long threads (Figure 34View Image). Left: Ratio of the frequency of the four collective kink-like modes, ω, to the frequency of the individual kink mode, ωk, as a function of the normalized distance between strand axes. Meaning of symbols: S x (solid line), A x (dotted line), S y (triangles), and A y (diamonds). Right: Ratio of the frequency of the two collective slow modes, ω, to the frequency of the individual slow mode, ωs for the Sz (solid line) and Az (dotted line) (from Soler et al., 2009aJump To The Next Citation Point).
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Figure 37: Cross-section of a two-thread model analogous to that of Figure 34View Image with the addition of mass flows along the cylinders (from Soler et al., 2009aJump To The Next Citation Point).

Soler et al. (2009aJump To The Next Citation Point) assessed the effect of material flows along two threads on the behaviour of collective modes (see Figure 37View Image for a sketch of the model). Arbitrary flows U1 and U2 were assumed in both cylinders, while coronal flows were neglected. The first main conclusion of this work is that the flows do not eliminate wave modes with collective dynamics (i.e., those that produce significant perturbations in the two threads), even in the case U1 ⁄= U2. Nevertheless, the requisite for retaining the collective dynamics is that the Doppler-shifted individual frequencies of the threads must be very similar. In the case of kink-like modes the Doppler-shifted frequencies are given by

Ωk1 = ωk1 + U1kz, (24 )
Ωk2 = ωk2 + U2kz, (25 )
where ωk1 and ωk2 are the kink frequencies of each thread, which are not equal if the thread densities differ. In the limit λ ≫ a, with a the tube radius, these frequencies are given by Equation (22View Equation). Now, the requirement for the two threads to oscillate in phase rather than independently is Ωk1 ∼ Ωk2. Using Equation (22View Equation) and making the reasonable assumption that the density contrast in both cylinders is much larger than one, Soler et al. (2009aJump To The Next Citation Point) obtained
√ -- U1 − U2 ≈ ± 2 (vA2 − vA1), (26 )
where the + sign is for parallel waves and the − sign is for anti-parallel propagation. A similar analysis can be performed for slow modes to obtain,
U − U ≈ ± (c − c ). (27 ) 1 2 s2 s1
which points out that the coupling between slow modes occurs at different flow velocities than the coupling between kink modes. Therefore, the simultaneous existence of collective slow and kink-like solutions in systems of non-identical threads is difficult. In the above equations, v A1,2 and c s1,2 correspond to Alfvén and sound speeds in both threads, respectively.

Soler et al. (2009aJump To The Next Citation Point) extracted another conclusion from Equations (26View Equation) and (27View Equation): the difference between the Alfvén (sound) speed of the threads determines the difference of the flow speeds for the existence of collective behaviour of kink (slow) modes. Therefore, when flows are present in the equilibrium, collective motions can be found even in systems of non-identical threads for very specific combinations of the two flow velocities. These velocities are within the observed values in prominences if threads with not too different temperatures and densities are considered. However, since the flow velocities required for collective oscillations must take very particular values, such a special situation may rarely occur in prominences. This conclusion has important repercussions for future prominence seismological applications, given that if collective oscillations are observed in large areas of a prominence, threads in such regions should possess very particular combinations of temperatures, densities, magnetic field strengths and flows.

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