5.2 Damping of oscillations by ion-neutral collisions

Since the temperature of prominences is typically of the order of 104 K, the prominence plasma is only partially ionized. The exact ionization degree of prominences is unknown and the reported ratio of electron density to neutral hydrogen density (see, e.g., Patsourakos and Vial, 2002) covers about two orders of magnitude (0.1 – 10). Partial ionization brings the presence of neutrals in addition to electrons and ions, thus collisions between the different species are possible. Because of the occurrence of collisions between electrons with neutral atoms and ions, and more importantly between ions and neutrals, Joule dissipation is enhanced when compared with the fully ionized case. A partially ionized plasma can be represented as a single-fluid in the strong coupling approximation, which is valid when the ion density in the plasma is low and the collision time between neutrals and ions is short compared with other time-scales of the problem. Using this approximation it is possible to describe the very low frequency and large-scale fluid-like behaviour of plasmas (Goossens, 2003).

Partial ionization affects the induction equation, which contains additional terms due to the presence of neutrals and a non-zero resistivity (Soler et al., 2009dJump To The Next Citation Point). These terms account for the processes of ohmic diffusion, with coefficient η; ambipolar diffusion, with coefficient ηA; and Hall’s magnetic diffusion, with coefficient η H. They govern collisions between the different plasma species. Ohmic diffusion is mainly due to electron-ion collisions and produces magnetic diffusion parallel to the magnetic field lines; ambipolar diffusion is mostly caused by ion-neutral collisions and Hall’s effect is enhanced by ion-neutral collisions since they tend to decouple ions from the magnetic field, while electrons remain able to drift with the magnetic field (Pandey and Wardle, 2008Jump To The Next Citation Point). The ambipolar diffusivity can be expressed in terms of Cowling’s coefficient, ηC, that accounts for diffusion perpendicular to magnetic field lines, as

ηC-−-η- ηA = B2 , (28 )
with B the magnetic field strength. For a fully ionized plasma, ηC = η and there is no ambipolar diffusion, so magnetic diffusion is isotropic. Due to the presence of neutrals, ηC ≫ η, which means that perpendicular magnetic diffusion is much more efficient than longitudinal magnetic diffusion in a partially ionized plasma. It is important to note that ηC ≫ η even for a small relative density of neutrals.

5.2.1 Homogeneous and unbounded prominence medium

Several studies have considered the damping of MHD waves in partially ionized plasmas of the solar atmosphere (De Pontieu et al., 2001; James et al., 2003; Khodachenko et al., 2004; Leake et al., 2005Jump To The Next Citation Point). In the context of solar prominences, Forteza et al. (2007Jump To The Next Citation Point) derived the full set of MHD equations for a partially ionized, one-fluid hydrogen plasma and applied them to the study of the time damping of linear, adiabatic fast and slow magnetoacoustic waves in an unbounded prominence medium. This study was later extended to the non-adiabatic case by including thermal conduction by neutrals and electrons and radiative losses (Forteza et al., 2008Jump To The Next Citation Point). The main effects of partial ionization on the properties of MHD waves manifest through a generalized Ohm’s law, which adds some extra terms in the resistive magnetic induction equation, in comparison to the fully ionized case. Forteza et al. (2007Jump To The Next Citation Point) considered a uniform and unbounded prominence plasma and found that ion-neutral collisions are more important for fast waves, for which the ratio of the damping time to the period is in the range 1 to 105, than for slow waves, for which values between 104 and 108 are obtained. Fast waves are efficiently damped for moderate values of the ionization fraction, while in a nearly fully ionized plasma, the small amount of neutrals is insufficient to damp the perturbations.

A hydrogen plasma was considered in the above studies, but 90% of the prominence chemical composition is hydrogen and the remaining 10% is helium. The effect of including helium in the model of Forteza et al. (2008Jump To The Next Citation Point) was assessed by Soler et al. (2010bJump To The Next Citation Point). The species present in the medium are electrons, protons, neutral hydrogen, neutral helium (He i) and singly ionized helium (He ii), while the presence of He iii is neglected (Gouttebroze and Labrosse, 2009).

The hydrogen ionization degree is characterized by &tidle;μH, which varies between 0.5 for fully ionized hydrogen and 1 for fully neutral hydrogen. The helium ionization degree is characterized by ξHeII δHe = ξHeI, where ξHeII and ξHe I denote the relative densities of single ionized and neutral helium, respectively. Figure 48View Image displays τd∕P as a function of the wavenumber, k, for the Alfvén, fast and slow waves, and the results corresponding to several helium abundances are compared for hydrogen and helium ionization degrees of &tidle;μH = 0.8 and δHe = 0.1, respectively. We can observe that the presence of helium has a minor effect on the results.

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Figure 48: Wave damping by ion-neutral effects in a uniform medium. (a) – (c) Ratio of the damping time to the period, τd∕P, versus the wavenumber, k, corresponding to the Alfvén wave, fast wave and slow wave, respectively. (d) Damping time, τ d, of the thermal wave versus the wavenumber, k. The different linestyles represent the following abundances: ξHeii = 0% (solid line), ξHeii = 10% (dotted line), and ξHeii = 20% (dashed line). In all computations, &tidle;μH = 0.8 and δHe = 0.1. The results for ξHe ii = 10% and δHe = 0.5 are plotted by means of symbols for comparison. The shaded regions correspond to the range of typically observed wavelengths of prominence oscillations. In all the figures the angle between the wavevector and the magnetic field is π∕4 (from Soler et al., 2010b).

The thermal mode is a purely damped, non-propagating disturbance (ωR = 0), so only the damping time, τd, is plotted (Figure 48View Imaged). We observe that the effect of helium is different in two ranges of k. For k > 10 −4 m− 1, thermal conduction is the dominant damping mechanism, so the larger the amount of helium, the shorter τ d because of the enhanced thermal conduction by neutral helium atoms. On the other hand, radiative losses are more relevant for − 4 −1 k < 10 m. In this region, the thermal mode damping time grows as the helium abundance increases. Since these variations in the damping time are very small, we again conclude that the damping time obtained in the absence of helium does not significantly change when helium is taken into account. Therefore, the inclusion of neutral or single ionized helium in partially ionized prominence plasmas does not modify the behaviour of linear, adiabatic or non-adiabatic MHD waves already found by Forteza et al. (2007Jump To The Next Citation Point) and Forteza et al. (2008).

5.2.2 Cylindrical filament thread model

Soler et al. (2009cJump To The Next Citation Point) applied the equations derived by Forteza et al. (2007Jump To The Next Citation Point) to the study of MHD waves in a partially ionized filament thread modeled as an infinite cylinder with radius a embedded in the solar corona (see Figure 32View Image). As in Forteza et al. (2007Jump To The Next Citation Point), the one-fluid approximation for a hydrogen plasma was considered. The internal and external media are characterized by their densities, temperatures, and their own relative densities of neutrals, ions and electrons. The contribution of the electrons is neglected. The coronal medium is considered as fully ionized, while the ionization fraction in the prominence plasma, &tidle;μp, is allowed to vary.

In their analysis, Soler et al. (2009cJump To The Next Citation Point) neglected Hall’s term since it can be ignored when the plasma is magnetized, i.e., when ions and electrons are tightly bound to the magnetic field. The condition to neglect Hall’s term can be written in terms of the ion-gyrofrequency (ωi) and the ion-neutral collision time (τ) as ωiτ ≫ 1, which once expanded gives,

∘ ------- eB- -πmi-----1-- ≫ 1, (29 ) mi 16kBT nn Σin
where Σin is the ion-neutral cross-section and nn the neutrals number density (Leake et al., 2005; Pandey and Wardle, 2008). Using prominence conditions (ρ = 5 × 10−11 kg m −3, B = 10 G, T = 8000 K, μ&tidle; = 0.8 p), we obtain the numerical value ω τ = 467 i, which fully justifies the neglect of Hall’s term. Parallel and perpendicular magnetic diffusion can be evaluated by defining the corresponding Reynolds numbers as Rm ∥ = cspa∕η and Rm ⊥ = 4π2csp∕ηCk2za, where the typical velocity scale has been associated to the sound speed in the prominence, csp. The parallel Reynolds number is independent of the wavenumber, while the relative importance of Cowling’s diffusion increases with kz, the longitudinal wavenumber. In the range of observed wavelengths (− 3 −1 kza ∼ [10 –10 ]) both Cowling’s and ohmic diffusion could therefore be important. Soler et al. (2009cJump To The Next Citation Point) analyzed separately the effect(s) of partial ionization in Alfvén, fast kink and slow waves.

For torsional Alfvén waves, Soler et al. (2009cJump To The Next Citation Point) found that wave propagation is constrained between two critical wavenumbers (top panels of Figure 49View Image). These critical wavenumbers are, however, outside the range of the observed wavelengths, in which τd∕P is in the range 10 – 100 and so is considerably larger than the observed damping rate. Nevertheless, a prominence ionization fraction larger than the maximum one considered here (namely μ&tidle;p > 0.95) can yield τd∕P = 1 − 10, in agreement with observations. For short wavenumbers, the values of the damping time over the period are independent of the ionization degree, while for large wavenumbers they become smaller for larger values of μ&tidle;p. This behaviour is explained in Soler et al. (2009cJump To The Next Citation Point) by considering solutions to the dispersion relation in which one of the two possible damping mechanisms, i.e., partial ionization or ohmic dissipation, is neglected. Soler et al. (2009cJump To The Next Citation Point) observed that ohmic diffusion dominates for small wavenumbers. Nevertheless, for large wavenumbers Cowling’s diffusion dominates over ohmic dissipation and so a larger number of neutrals decreases the damping time: the larger &tidle;μ in the thread, the shorter τd and, consequently, the smaller τd∕P.

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Figure 49: Wave damping by ion-neutral effects in an infinitely long prominence thread. Dimensionless phase speed (left panels) and ratio of the damping time to the period (right panels) as a function of k a z for Alfvén waves (top panels), kink waves (middle panels), and slow waves (bottom panels). The different linestyles represent different ionization degrees: &tidle;μp = 0.5 (dotted), &tidle;μp = 0.6 (dashed), &tidle;μp = 0.8 (solid), and &tidle;μp = 0.95 (dash-dotted). Symbols are the approximate solution given by Equation (36) in Soler et al. (2009cJump To The Next Citation Point) for μ&tidle;p = 0.8. The shaded zones correspond to the range of typically observed wavelengths of prominence oscillations. The Alfvén speed in the thread, vAf, the kink speed, ck, and the cusp speed in the thread, cTp, have been used to compute the dimensionless phase speed (adapted from Soler et al., 2009cJump To The Next Citation Point).

The presence of critical wavenumbers is also found in the case of transverse kink waves (middle panels of Figure 49View Image). Within the range of observed wavelengths, the phase speed closely corresponds to its ideal counterpart, ck = ω∕kz, so non-ideal effects are irrelevant for wave propagation. The behaviour of the damping rate as a function of wavelength and ionization fraction is seen to closely resemble the result obtained for Alfvén waves, with τd∕P > 10 in the range of observed wavelengths. Therefore, neither ohmic diffusion nor ion-neutral collisions seem to provide damping times as short as those observed for kink waves in filament threads. Only for an almost neutral plasma, with &tidle;μp > 0.95, the obtained damping rates are compatible with the observed time-scales. Just like for Alfvén waves, ohmic diffusion dominates for small wavenumbers, while ion-neutral collisions are the dominant damping mechanism for large wavenumbers.

Regarding slow waves (bottom panels of Figure 49View Image), Soler et al. (2009c) concentrated their analysis on the radially fundamental mode with m = 1, since the behaviour of the slow mode is weakly affected by the value of the azimuthal wavenumber. Slow wave propagation is constrained by only one critical wavenumber, that strongly depends on the ionization fraction, in such a way that for k z below this critical wavenumber the wave is totally damped. More importantly, for large enough values of the ionization fraction, the corresponding critical wavelength lies in the range of observed wavelengths of filament oscillations. As a consequence, the slow wave might not propagate in filament threads under certain circumstances. As for the damping rate, it is found that ion-neutral collisions are a relevant damping mechanism for slow waves, since very short damping times are obtained, especially close to the critical wavenumber. By comparing the particular effects of ohmic diffusion and ion-neutral collisions, the slow mode damping is seen to be completely dominated by ion-neutral collisions. Ohmic diffusion is found to be irrelevant, since the presence of the critical wavenumber prevents slow wave propagation for small wavenumbers, where ohmic diffusion would start to dominate.

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