5.1 Damping of oscillations by thermal mechanisms

In a seminal paper, Field (1965) studied the thermal instability of a dilute gas in mechanical and thermal equilibrium. Using this approach, the time damping of magnetohydrodynamic waves in bounded Kippenhahn–Schlüter and Menzel prominence models was studied by Terradas et al. (2001Jump To The Next Citation Point). Similar studies using prominence slabs embedded in the solar corona were undertaken by Soler et al. (2007Jump To The Next Citation Point) and Soler et al. (2009bJump To The Next Citation Point).

5.1.1 Non-adiabatic magnetoacoustic waves in prominence slabs

Terradas et al. (2001) studied the radiative damping of quiescent prominence oscillations. They adopted a relatively simple non-adiabatic damping mechanism, by including a radiative loss term based on Newton’s law of cooling with constant relaxation time. The influence of this type of radiative dissipation on the normal modes of Kippenhahn–Schlüter and Menzel quiescent prominence models was analyzed. The normal modes of these configurations had previously been investigated by Oliver et al. (1992) and Joarder and Roberts (1993a); cf. Section 4.2. In a Kippenhahn–Schlüter prominence model, the fundamental slow mode is unaffected by radiation, but its harmonics are strongly damped. On the other hand, in a Menzel prominence configuration all slow modes are characterized by short damping times. The damping time depends on the curvature of field lines, in such a way that more curved models produce larger damping times. In both prominence models, fast modes are practically unaffected by radiative losses and have very long damping times.

A more involved analysis was performed by Soler et al. (2007Jump To The Next Citation Point) by including thermal conduction, optically thin or thick radiation, and heating in the energy equation. The prominence was modeled as a plasma slab embedded in an unbounded corona and with a magnetic field oriented along the direction parallel to the slab axis (see Figure 26View Image); this is the equilibrium configuration of Joarder and Roberts (1992a), whose normal modes have been discussed in Section 4.2. Soler et al. (2007Jump To The Next Citation Point) found that radiation losses have a different effect on magnetoacoustic waves depending on their wavenumber. For typical values of observed wavelengths, the internal slow mode is attenuated by radiation from the prominence plasma, the fast mode by the combination of prominence radiation and coronal conduction and the external slow mode by coronal conduction. This study highlights the relevance of the coronal physical properties on the damping properties of fast and external slow modes, whereas this aspect does not affect the internal slow modes at all. For thin slabs, representing a fine thread, Soler et al. (2007Jump To The Next Citation Point) found that the fast mode is less attenuated, whereas both internal and external slow modes are not affected by non-adiabatic damping mechanisms.

Damping of magnetoacoustic waves in slab prominence models with a transverse magnetic field (see Figure 27View Image and Section 4.2 for a description of the normal modes) were studied by Soler et al. (2009b). The most relevant damping processes are coronal thermal conduction and radiative losses from the prominence plasma. In terms of the spatial distribution of the studied normal modes, it was found that both mechanisms govern together the attenuation of hybrid modes, whereas prominence radiation is responsible for the damping of internal modes and coronal conduction essentially dominates the attenuation of external modes. In terms of the different magnetohydrodynamic wave types, slow modes were found to be efficiently damped, with damping times compatible with observations. On the contrary, fast modes are less attenuated by non-adiabatic effects and their damping times are several orders of magnitude larger than those observed. The inclusion of the coronal medium in the analysis causes a decrease of the damping times compared to those of an isolated prominence slab, but this effect is still insufficient to obtain fast mode damping times compatible with observations.

5.1.2 Non-adiabatic magnetoacoustic waves in a single thread with mass flows

Soler et al. (2008Jump To The Next Citation Point) investigated the effects of both mass flow and non-adiabatic processes on the oscillations of an individual prominence thread modeled as an infinite homogeneous cylinder Figure 32View Image). Thermal conduction and radiative losses were taken into account as damping mechanisms. For a discussion of the oscillatory features of this system, see Section 4.4.1.

The analysis of the damping time-scales for the different wave types shows that slow and thermal modes are efficiently attenuated by non-adiabatic mechanisms. On the contrary, fast kink modes are much less affected and their damping times are much larger than those observed. These results are compatible with the known damping properties of these waves in the absence of flows.

In addition, Soler et al. (2008Jump To The Next Citation Point) analyzed how mass flows affect these damping properties. Figure 46View Image shows the dependence of the period, damping time, and their ratio as a function of the flow velocity for the slow, fast and thermal modes (for a discussion of the thermal mode, see Carbonell et al., 2009). Note that the left column of this figure has been already presented in Figure 33View Image, but it is retained here to facilitate our explanation. Flow velocities in the range 0 – 30 km s–1, that correspond to the observed flow speeds in quiescent prominences, were considered. The damping time of slow and thermal modes is found to be independent of the flow velocity, but the attenuation of the fast kink mode is affected by the flow. The larger the flow velocity, the more attenuated the parallel fast kink wave, whereas the opposite occurs for the anti-parallel solution. This behaviour is due to the weak coupling of the fast modes to external slow modes (Soler et al., 2008Jump To The Next Citation Point).

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Figure 46: Wave damping by thermal effects in a uniform, infinitely long thread (Figure 32View Image). Period (left), damping time (center), and ratio of the damping time to the period (right) versus the flow velocity for the fundamental oscillatory modes. The upper, middle, and lower panels correspond to the slow, fast kink, and thermal modes, respectively. Different line styles represent parallel waves (solid line), anti-parallel waves (dashed line), and solutions in the absence of flow (dotted line) (from Soler et al., 2008Jump To The Next Citation Point).

Although the presence of steady mass flows improves the efficiency of non-adiabatic mechanisms on the attenuation of transverse kink oscillations for propagation parallel to the flow, its effect is still not enough to obtain damping times compatible with observations.

5.1.3 Non-adiabatic magnetoacoustic waves in a two-thread system with mass flows

The oscillatory properties, namely the frequency and spatial distribution, of fast and slow magnetoacoustic waves in a system made of two infinite threads with mass flows are described in Section 4.4.2; see Figure 37View Image for a sketch of the equilibrium configuration. Soler et al. (2009aJump To The Next Citation Point) evaluated the damping time-scales caused by non-adiabatic effects as a function of the distance between the thread axes. The left panel of Figure 47View Image shows that the ratio of the damping time to the period of the four kink modes is very large, so that dissipation by non-adiabatic mechanisms is not efficient enough to damp these modes. Hence, the collective nature of the transverse oscillations in a system of two identical threads does not change the conclusion about the irrelevance of thermal mechanisms to account for the damping of fast modes already obtained for one thread.

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Figure 47: Wave damping by thermal effects in a two-thread system. Left: Ratio of the damping time to the period versus the distance between the thread axes of the Sx (solid line), Ax (dotted line), Sy (triangles), and Ay (diamonds) kink-like modes. Right: The same for the Sz (solid line) and A z (dotted line) slow wave modes (from Soler et al., 2009aJump To The Next Citation Point).

As concluded in Section 5.1.2, slow wave damping can be explained by thermal mechanisms. The right panel of Figure 47View Image shows the damping ratios of the Sz and Az solutions versus the distance between the two threads. Slow modes in a threaded prominence are efficiently attenuated by non-adiabatic mechanisms. Note that τd∕P is almost independent of the thread separation and the mode because the two threads oscillate independently in the Sz and Az modes. Time-scales τd∕P ≈ 5 are obtained, which is in agreement with previous studies (Soler et al., 2007, 2008) and consistent with observations.

Soler et al. (2009a) concluded that collective slow modes are efficiently damped by thermal mechanisms, with damping ratios similar to those reported in observations, while collective fast waves are poorly damped. This is a key point since efficiently damped transverse oscillations have been observed, which could suggest that other attenuation mechanisms could be at work.

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