Issue 
A&A
Volume 529, May 2011



Article Number  A82  
Number of page(s)  9  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201016326  
Published online  07 April 2011 
Magnetohydrodynamic waves in solar partially ionized plasmas: twofluid approach
^{1}
Space Research Institute, Austrian Academy of Sciences,
Schmiedlstrasse 6,
8042
Graz,
Austria
email: teimuraz.zaqarashvili@oeaw.ac.at; maxim.khodachenko@oeaw.ac.at; rucker@oeaw.ac.at
^{2} Abastumani Astrophysical Observatory at Ilia State
University, Kazbegi ave. 2a, Tbilisi, Georgia
Received:
15
December
2010
Accepted:
5
February
2011
Context. Partially ionized plasma is usually described by a singlefluid approach, where the ionneutral collision effects are expressed by Cowling conductivity in the induction equation. However, the singlefluid approach is not valid for timescales less than ionneutral collision time. For these timescales the twofluid description is the better approximation.
Aims. We aim to derive the dynamics of magnetohydrodynamic (MHD) waves in twofluid partially ionized plasmas and to compare the results with those obtained under singlefluid description.
Methods. Twofluid equations are used, where ionelectron plasma and neutral particles are considered as separate fluids. Dispersion relations of linear waves are derived for the simplest case of homogeneous medium. Frequencies and damping rates of waves are obtained for different parameters of background plasma.
Results. We found that two and singlefluid descriptions give similar results for lowfrequency waves. However, the dynamics of MHD waves in the twofluid approach is significantly changed when the wave frequency becomes comparable with or higher than the ionneutral collision frequency. Alfvén and fast magnetoacoustic waves attain their maximum damping rate at particular frequencies (for example, the peak frequency equals 2.5 times the ionneutral collision frequency for 50% of neutral hydrogen) in the wave spectrum. The damping rates are reduced for the higher frequency waves. The new mode of slow magnetoacoustic wave appears for higher frequency branch, which is connected to neutral hydrogen fluid.
Conclusions. The singlefluid approach perfectly deals with slow processes in partially ionized plasmas, but fails for timescales shorter than ionneutral collision time. Therefore, the twofluid approximation should be used for the description of relatively fast processes. Some results of the singlefluid description should be revised in future such as the damping of highfrequency Alfvén waves in the solar chromosphere due to ionneutral collisions.
Key words: Sun: atmosphere / Sun: oscillations
© ESO, 2011
1. Introduction
Astrophysical plasmas often are partially ionized. Neutral atoms may change the plasma dynamics through collisions with charged particles. The ionneutral collisions may lead to different new phenomena in the plasma, for example the damping of magnetohydrodynamic (MHD) waves (Khodachenko et al. 2004; Forteza et al. 2007). The solar photosphere, the chromosphere, and the prominences contain a significant amount of neutral atoms, therefore the complete description of plasma processes requires the consideration of partial ionization effects.
Braginskii (1965) gave the basic principles of transport processes in plasma including the effects of partial ionization. Since this review, numerous papers addressed the problem of partial ionization in the different regions of solar atmosphere. Khodachenko & Zaitsev (2002) studied the formation of the magnetic flux tube in a converging flow of the solar photosphere, while Vranjes et al. (2008) studied the Alfvén waves in weakly ionized photospheric plasma. Leake & Arber (2005) and Arber et al. (2007) studied the effect of partially ionized plasma on emerging magnetic flux tubes and concluded that the chromospheric neutrals may transform the magnetic tube into forcefree configuration. Haerendel (1992), De Pontieu & Haerendel (1998), James & Erdélyi (2002), and James et al. (2004) considered the damping of Alfvén waves through ionneutral collision as a mechanism of spicule formation. Khodachenko el al. (2004) and Leake et al. (2006) studied the importance of ionneutral collisions in the damping of MHD waves in the chromosphere and prominences. Forteza et al. (2007; 2008), Soler et al. (2009a; 2009b; 2010), and Carbonell et al. (2010) studied the damping of MHD waves in partially ionized prominence plasma with and without plasma flow.
All these papers considered the singlefluid MHD approach when the inertial terms in the momentum equation of the relative velocity between ions and neutrals are neglected. The partially ionized plasma effects are described by a generalized Ohm’s law with Cowling conductivity, which leads to the modified induction equation (Khodachenko el al. 2004). Ambipolar diffusion is more pronounced during the transverse motion of plasma with regard to the magnetic field, therefore the Alfvén and fast magnetoacoustic waves are more efficiently damped. The slow magnetoacoustic waves are weakly damped in the low plasma beta case. Moreover, Forteza et al. (2007) found that the damping rate of slow magnetoacoustic waves derived through a normal mode analysis is different from that estimated by Braginskii (1965). The cause of the discrepancy between the normal mode analysis (Forteza et al. 2007) and the energy consideration (Braginskii 1965) is still an open question, and the present study attempts to shed light on it.
The singlefluid approach has been shown to be valid for the timescales that are longer than the ionneutral collision time. However, the approximation fails for the shorter timescales, therefore the twofluid approximation, which means the treatment of ionelectron and neutral gases as separate fluids, should be considered. The twofluid approximation is valid for timescales longer than the ionelectron collision time, which is significantly shorter because of the Coulomb collision between ions and electrons.
In this paper, we study MHD waves in twofluid partially ionized plasma. We pay particular attention to the wave damping through ionneutral collisions and compare the wave dynamics in single and twofluid approximations. We derive the twofluid MHD equations from initial threefluid equations and solve the linearized equations in the simplest case of a homogeneous plasma.
2. Main equations
We aim to study partially ionized plasma, which consists of electrons, ions, and neutral atoms. We assume that each species has a Maxwell velocity distribution, therefore they can be described as separate fluids. Below we first write the equations in threefluid description and then perform the consequent transition to twofluid and singlefluid approaches.
2.1. Threefluid equations
The fluid equations for each species can be derived from Boltzmann kinetic equations, which have the forms (Braginskii 1965; Goedbloed & Poedts 2004)where m_{a}, n_{a}, p_{a}, T_{a}, V_{a} are the mass, the density, the pressure, the temperature and the velocity of particles a, E is the electric field, B is the magnetic field strength, q_{a} is the heat flux density of particles a, R_{a} is the change of impulse of particles a through collisions with other sort of particles, Q_{a} is the heat production through collisions of particles a with other sort of particles, π_{a} is the offdiagonal pressure tensor of particles a, e = 4.8 × 10^{10} statcoul is the electron charge, c = 2.9979 × 10^{10} cm s^{1} is the speed of light and k = 1.38 × 10^{16} erg K^{1} is the Boltzmann constant. The double dot indicates that a double sum over the Cartesian components is to be taken. The plasma is supposed to be quasineutral, which means n_{e} = Zn_{i}. Below we consider hydrogen ions and hydrogen neutral atoms that imply Z = 1. The description of the system is completed by Maxwell equations, which have the forms (without displacement current) where (13)is the current density.
For a Maxwell distribution in each species, R_{a} and Q_{a} are expressed as (Braginskii 1965): where α_{ab} = α_{ba} are the coefficients of friction between particles a and b.
For timescales longer than the ionelectron collision time, the electron and ion gases can be considered as a single fluid. This significantly simplifies the equations, taking into account the smallness of electron mass with regard to the masses of ion and neutral atoms. Then the threefluid description can be changed to the twofluid description, where one component is the ionelectron gas and the second component is the gas of neutral atoms.
2.2. Twofluid equations
Summing of Eqs. (4) and (5), Eqs. (7) and (8), and first two equations of Eq. (10), we obtain (after neglecting the electron inertia and the viscosity effect expressed by the offdiagonal pressure tensor π_{a}) (20)where p_{ie} = p_{i} + p_{e} is the pressure of ionelectron gas and γ = C_{p}/C_{v} = 5/3 is the ratio of specific heats.
Ohm’s law is obtained from the electron equation (Eq. (4)) after neglecting the electron inertia (i.e. the lefthand side terms) and it has the form (26)The Maxwell equation (Eq. (11)) and Ohm’s law (Eq. (26)) lead to the induction equation (27)where (28)is the coefficient of the magnetic diffusion.
The coefficient of friction between ions and neutrals (if they have the same temperature) is calculated as (Braginskii 1965) (29)where m_{in} = m_{i}m_{n}/(m_{i} + m_{n}) is the reduced mass and is the collision cross section between ions and neutrals.
The collision frequency between ions and neutrals is then (30)where the atomic cross section cm^{2} is used and T is normalized by 1 K. The chromospheric temperature of 10^{4} K and hydrogen ion and neutral number densities of 2.3 × 10^{10} cm^{3} and 1.2 × 10^{10} cm^{3} (Fontenla et al. 1990, model FAL3) give the collision frequency as 4 s^{1}.
For timescales longer than ionneutral collision time (1/ν_{in}), the system can be considered as a single fluid (the full equations of singlefluid MHD including neutral hydrogen are presented in Appendix A). However, when the timescales are near to or shorter than the ionneutral collision time, the singlefluid description is not valid and the twofluid equations should be considered. Below, in what follows we study the linear MHD waves in a twofluid description.
3. Linear MHD waves
We consider the simplest case of static and homogeneous plasma with a homogeneous unperturbed magnetic field. Then the linearized twofluid equations following from Eqs. (20)–(25) and (27) are (neglecting the Hall term and the collision between neutrals and electrons i.e. α_{en} ≪ α_{in}) where () are perturbations of the ion (neutral) density, v_{i} (v_{n}) are the perturbations of ion (neutral) velocity, () are the perturbations of ionelectron (neutral) gas pressure, b is the perturbation of the magnetic field, and ρ_{i0} = m_{i}n_{i0},ρ_{n0} = m_{n}n_{n0},p_{ie},p_{n},B_{0} are their unperturbed values, respectively. Equations (31)–(32) and Eqs. (36)–(37) lead to the expressions (38)where and are the sound speeds of ionelectron and neutral gases, respectively.
Below we consider the unperturbed magnetic field, B_{z}, directed along the z axis and the wave propagation in xz plane i.e. ∂/∂y = 0. Then Eqs. (31)–(37) can be split into Alfvén and magnetoacoustic waves.
3.1. Alfvén waves
Let us assume that the Alfvén waves are polarized along y axis. We intend to study the damping of Alfvén waves through the collision between ions and neutrals. Therefore, we neglect the magnetic diffusion for simplicity. Then, Eqs. (31)–(37) give
Fourier analyses assuming disturbances to be proportional to exp [i(k_{z}z − ωt)] give the dispersion relation
(42)where (43)The same dispersion relation can be obtained from linear singlefluid equations by retaining the inertial term in Eq. (A.6). The dispersion relation of the Alfvén waves in linear singlefluid equations without the inertial term can be easily derived as (44)The solution of Eq. (44) is (45)which for gives the damping rate (46)in full coincidence with Braginskii (1965). In contrast, the condition in Eq. (46) retains only the imaginary part, which gives the cutoff wave number (47)The value of the cutoff wave number has been obtained recently by Barcélo et al. (2010). Hence, the waves with a higher wave number than k_{c} are evanescent. However, it might be an incomplete conclusion because the complete treatment requires the inclusion of inertial terms, and therefore dealing with Eq. (42) instead of Eq. (44). The first term in Eq. (42) is important for the highfrequency part of the wave spectrum and could not be neglected. We demonstrate it by the solutions of Eqs. (44) and (42).
Fig. 1 Alfvén wave frequency vs. normalized wave number. The top (bottom) panel shows the real (imaginary) part of the normalized frequency ϖ = ω/k_{z}v_{A} vs. the normalized wavenumber a = k_{z}v_{A}/ν_{in}. The blue line corresponds to the solution of the singlefluid dispersion relation, i.e. Eq. (44) and red asterisks are the solutions of the twofluid dispersion relation, Eq. (42). The values are calculated for 50% of neutral hydrogen, ξ_{n} = 0.5. 

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Figure 1 displays the solutions of the singlefluid (Eq. (44), blue lines) and twofluid (Eq. (42), red asterisks) dispersion relations for ξ_{n} = 0.5. We see that the frequencies and damping rates of Alfvén waves are the same in the singlefluid and twofluid approaches for the lowfrequency branch of the spectrum (small a). But the behavior is dramatically changed when the wave frequency becomes comparable with or higher than the ionneutral collision frequency, ν_{in}, i.e. for a > 1. The damping time linearly increases with a and the wave frequency becomes zero at some point in the singlefluid case (blue lines). The point where the wave frequency becomes zero corresponds to the cutoff wave number k_{c} of Barcélo et al. (2010). However, there is no cutoff wave number in solutions of twofluid dispersion relation (red asterisks): Eq. (42) always has a solution with a real part. Therefore, the occurrence of the cutoff wave number in the singlefluid description is the result of neglecting the inertial terms in the momentum equation of relative velocity between ions and neutrals. Therefore, Eq. (42) is the correct dispersion relation for the whole spectrum of waves. But the dispersion relation (44) is still a good approximation for the lower frequency part of spectrum. Another interesting point of the twofluid approach is that the damping rate (i.e. ω_{I}) attains its maximal value at some wavelengths for which a ≈ 2.5. The damping rate decreases for smaller and larger a. This means that the waves, which have the frequency in the interval ν_{in} < ω < 10 ν_{in}, have stronger damping than other harmonics of the spectrum. This is totally different from the singlefluid solutions, which show the linear increase of damping rate with increasing wave number (lower panel, blue line).
Figure 2 displays the same solutions as in the Fig. 1, but for ξ_{n} = 0.1. The solutions have basically the same properties as those with ξ_{n} = 0.5. However, the wave length with the maximal damping rate is now shifted to a ≈ 10.
3.2. Magnetoacoustic waves
Now let us turn to magnetoacoustic waves. We consider the waves and wave vectors polarized in xz plane. Then Eqs. (31)–(37) are written as (magnetic diffusion is again neglected)
Fig. 2 Same as in Fig. 1, but for 10% of neutral hydrogen, ξ_{n} = 0.1. 

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A Fourier analysis with exp [i(k_{x}x + k_{z}z − ωt)] and some algebra give the dispersion relation (57)where .
The dispersion relation (57) is a seventh order equation with ω, therefore it has seven different solutions. For smaller wave numbers (or lower frequencies) four of the solutions represent the usual magnetoacoustic waves, while three other solutions are purely imaginary and are probably connected to the vortex modes (with Re(ω) = 0) that damped through ion neutral collisions. The vortex modes are solutions of fluid equations and they correspond to the fluid vorticity. The vortex modes have zero frequency in the ideal fluid, but may gain a purely imaginary frequency if dissipative processes are evolved. The two vortex modes are transformed into oscillatory modes for shorter wavelengths (see the next paragraph). Then we have two fast magnetoacoustic modes, four slow magnetoacoustic modes, and one vortex solution with a purely imaginary part. Below we consider that the temperatures of all three species are equal i.e. T_{i} = T_{e} = T_{n}, which gives .
Fig. 3 Frequency and damping rate of different wave modes in twofluid MHD vs. the normalized wavenumber a = k_{z}v_{A}/ν_{in}. Frequencies and damping rates are normalized by kv_{A}. Red asterisks correspond to the fast magnetoacoustic mode and green diamonds correspond to the usual slow magnetoacoustic mode. The mode with the blue squares is the new sort of slow magnetoacoustic wave (“neutral” slow mode), which arises for larger wave numbers. This mode has only an imaginary frequency for small wave numbers, which is not shown in this figure. Frequencies and damping rates are calculated for the waves propagating along the magnetic field. The neutral hydrogen is taken to be 50% (ξ_{n} = 0.5). Here we consider c_{sn}/v_{A} = 0.5. 

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Fig. 4 Damping rate of the fast (upper panel) and slow (lower panel) magnetoacoustic waves, i.e. with the imaginary part of ω normalized by kv_{A}, vs. the normalized wavenumber a = k_{z}v_{A}/ν_{in}. The blue solid lines correspond to the solution of the singlefluid dispersion relation and the dashed line corresponds to the slow magnetoacoustic damping rate of Braginskii (the expressions used are from Forteza et al. 2007). Red asterisks are the solutions of the twofluid dispersion relation, Eq. (57). The values are calculated for 10% of neutral hydrogen, ξ_{n} = 0.1, and for c_{sn}/v_{A} = 0.1. The damping rates are calculated for waves propagating with a 45° angle with regard to the magnetic field. 

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Figure 3 displays all oscillatory solutions of the twofluid dispersion relation for ξ_{n} = 0.5 (only the modes with positive frequencies are shown). The wave propagation is parallel to the magnetic field and we use c_{sn}/v_{A} = 0.5, where c_{sn} is the sound speed of neutral hydrogen. For smaller wave numbers, k < 3.5 ν_{in}/v_{A}, there are two normal magnetoacoustic modes, fast (red asterisks) and slow (green diamonds). However, for larger wavenumbers, k > 3.5 ν_{in}/v_{A} one additional sort of slow magnetoacoustic mode with a strong damping rate (blue squares) arises. The “neutral” slow mode is connected with neutral atoms. For the higher frequency range, i.e. for a higher than ionneutral collision frequency, the neutral gas does not feel the ions, therefore it supports the propagation of the additional oscillatory wave mode. This mode obviously disappears for lower frequencies because the collisions couple ions and neutrals and they behave as a single fluid. In other words, for lower frequencies (or small wave numbers) this mode has a zero real part, but a nonzero imaginary part (not shown in the figure). Therefore, the more correct statement is that the oscillatory mode transforms into nonoscillatory vortex mode for smaller wave numbers. The fast magnetoacoustic modes decouple from the slow waves for the parallel propagation and show the same behavior as Alfvén waves. Therefore, the plot of fast magnetoacoustic waves is similar to that of Alfvén waves (see Fig. 1).
It is useful to compare the solutions of twofluid dispersion relation with those obtained in the singlefluid approach. The damping of fast and slow magnetoacoustic waves has been derived from the energy equation by Braginskii (1965), Khodachenko et al. (2004), Khodachenko & Rucker (2005), and through a normal mode analysis by Forteza et al. (2007). Damping rates are the same in both considerations for fast magnetoacoustic waves, but they disagree for slow magnetoacoustic waves (Forteza et al. 2007). Namely, slow magnetoacoustic waves show damping for purely parallel propagation in the case of Braginskii, while the damping is absent in Forteza et al. (2007). Our Figs. 4 and 5 show the damping rates of fast and slow magnetoacoustic waves vs. a (i.e. k) for the propagation angles of 45° and 0°, respectively. Red asterisks are the solutions of the twofluid dispersion relation – Eq. (57). The blue solid lines correspond to the solutions of the singlefluid dispersion relation from Forteza et al. (2007). The dashed line corresponds to the slow magnetoacoustic damping rate of Braginskii (1965). Here we use c_{sn}/v_{A} = 0.1 so the plasma β is small enough. The neutral hydrogen concentration is taken to be 10%. The fast magnetoacoustic waves have essentially the same dynamics as the Alfvén waves. For the smaller wave numbers (or lower frequencies) the twofluid and singlefluid approaches give the same results, but for the larger wave numbers the damping rate is decreased in the twofluid description as in the case of Alfvén waves. In contrast, the slow magnetoacoustic waves have similar damping rates in both approaches. There is a small discrepancy between the damping rates for the waves propagating with 45° degree about the magnetic field, but all three cases (twofluid waves, singlefluid waves, and energy consideration) yield similar results. The parallel propagation reveals an interesting result: the twofluid solutions are exactly the same as those obtained by Braginskii (the damping rate obtained by Forteza et al. 2007 is zero for the parallel propagation). Therefore, the discrepancy between the damping rates of the slow magnetoacoustic waves obtained by Forteza et al. 2007) and Braginskii (1965) is again caused by neglecting the inertial terms in the momentum equation of relative velocity (Eq. (A.6)). Braginskii (1965) used the energy equation to calculate the damping rate, therefore his solution agrees with that obtained in our twofluid approach.
Fig. 5 The same as in Fig. 4 but for the parallel propagation, i.e. k_{x} = 0. 

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Figure 6 shows the comparison of the damping rates in the twofluid approach and those obtained by Braginskii (1965) for ξ_{n} = 0.5, c_{sn}/v_{A} = 0.1 and parallel propagation. The fast magnetoacoustic waves have the same behavior as the Alfvén waves (see lower panel of Fig. 1), which is significantly different from Braginskii (1965) and Forteza et al. (2007). But the slow magnetoacoustic waves have the same damping rate as those of Braginskii (1965). On the other hand, the damping rate of slow magnetoacoustic waves becomes different from the solution of Braginskii (1965) for higher plasma β. Figure 7 shows the same as Fig. 6, but for c_{sn}/v_{A} = 0.5. The damping rate of the slow magnetoacoustic waves now begins to deviate from the solution of Braginskii for higher wave numbers. The behavior of fast magnetoacoustic waves remains the same.
Fig. 6 Damping rate (the imaginary part of the normalized frequency ϖ = ω/k_{z}v_{A}) of fast (upper panel) and slow (lower panel) magnetoacoustic waves vs. the normalized wavenumber a = k_{z}v_{A}/ν_{in}. Red asterisks are the solutions of the twofluid dispersion relation. Blue lines correspond to the solutions of Braginskii. The values are calculated for 50% of neutral hydrogen and c_{sn}/v_{A} = 0.1. The damping rates are calculated for the waves that propagate along the magnetic field. 

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4. Discussion
Some parts of the solar atmosphere contain a large number of neutral atoms: most of the atoms are neutral at the photospheric level, but the ionization degree rapidly increases with height owing to the increased temperature. Solar prominences also contain neutral atoms. Neutral atoms may change the dynamics of the plasma through collision with charged particles. For timescales longer than the ionneutral collision time, the partially ionized plasma can be considered as one fluid, because collisions between neutrals and charged particles lead to the rapid coupling of the two fluids. Then the equation of motion is written for the centerofmass velocity, and the motion of the species is considered as diffusion with a low velocity compared with the velocity of the centerofmass. The corresponding collision terms appear in the equation of motion for the relative velocity (between ions and neutrals) and in the generalized Ohm’s law. Neglecting the inertial term in the equation of motion for the relative velocity, one simplifies the equations, and a traditional induction equation with Cowling conductivity is obtained (Braginskii 1965; Khodachenko et al. 2004). The inertial terms (lefthand side terms in Eq. (A.6)) are smaller than the collision term (the last term in the same equation), but become comparable for timescales near the ionneutral collision time. Therefore, it can be neglected only for longer timescales.
But for the timescales of less than the ionneutral collision time, both fluids may behave independently and the singlefluid approximation is not valid any more. Accordingly the twofluid approximation, when ionelectron and neutral atom gases are treated as separate fluids, should be considered when one tries to model the processes in partially ionized plasmas.
The normal mode analysis of the twofluid partially ionized plasma shows that frequencies and damping rates of lowfrequency MHD waves agree well with those found in the singlefluid approach. However, the waves with higher frequency than the ionneutral collision frequency show a significantly different behavior. Alfvén and fast magnetoacoustic waves have maximal damping rates in a particular frequency interval, which peaks, for example, at the frequency ω = 2.5 ν_{in}, (ν_{in} is the ionneutral collision frequency) for ξ_{n} = 0.5 and at the frequency ω = 10 ν_{in} for ξ_{n} = 0.1. The damping rates are reduced for the higher frequency part of wave spectrum (note that the damping rates are linearly increased in the singlefluid approach). Therefore, the statements concerning the damping of highfrequency Alfvén waves in the solar chromosphere through ionneutral collisions should be revised. A careful analysis is needed to study the damping of highfrequency Alfvén waves for a realistic height profile of the ionization degree in the chromosphere.
Fig. 7 The same as in Fig. 6, but for c_{sn}/v_{A} = 0.5. 

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Another important point concerning the Alfvén waves in partially ionized plasma is the cutoff wavenumber, which appears in the singlefluid approach (Barcélo et al. 2010). Barcélo et al. (2010) found that the Alfvén waves with larger wave numbers than the cutoff value are evanescent in partially ionized and resistive plasmas. However, our twofluid analysis shows that there is no cutoff wave number owing to ambipolar diffusion (see Fig.1, red asterisks). Therefore, the appearance of a cutoff wave number in the singlefluid approach is the result of neglecting of the inertial term in the equation of motion for the relative velocity. It is possible that the cutoff wave number that arises because of an usual magnetic resistivity is caused by neglecting the electron inertia, therefore the cutoff may completely disappear in a threefluid approach. The cutoff wave numbers also appear for fast magnetoacoustic waves in partially ionized and resistive plasmas (Barcélo et al. 2010). We suggest that this may also be caused by neglecting the inertial term. However, this point needs further study.
The twofluid approach reveals two different slow magnetoacoustic modes when the slow wave timescale becomes shorter than the ionneutral collision time (Fig. 3). The different slow modes correspond to ionelectron and neutral fluids. But only one slow magnetoacoustic mode remains at the lower frequency range as in the commonly used singlefluid approach. This is easy to understand physically. If the wave frequency is lower than the ionneutral collision frequency, the two fluids are coupled through collisions and only one slow magnetoacoustic wave appears. The mode connected with the neutral fluid (“neutral” slow mode) has only an imaginary frequency in this range of the wave spectrum (not shown in Fig. 3). This means that any slow wavetype change (density, pressure) in the neutral fluid is damped faster than the wave period owing to collisions with ions. The “neutral” slow magnetoacoustic wave has similar properties as the ion magnetoacoustic waves.
The twofluid approach of partially ionized plasma clarifies the uncertainty concerning the damping rate of slow magnetoacoustic waves found in the singlefluid approach. We found that the normal mode analysis and energy consideration method (used by Braginskii 1965) leads to different expressions for the damping rate of slow waves (Forteza et al. 2007). We found that the damping rate obtained in the twofluid approach agrees well with the damping rate of Braginskii, which is derived from the energy treatment (see lower panels of Figs. 5–6). Therefore, it seems that the discrepancy is again caused by neglecting the inertial terms in the equation of motion for the relative velocity in the singlefluid approach. Braginskii (1965) used the general energy method for the estimation of damping rates, and this is probably the reason why his results agree with those found in the twofluid approach.
Here we have considered only neutral hydrogen as a component of partially ionized plasma. However, other neutral atoms, for example neutral helium, may have important effects in the MHD wave damping processes. Soler et al. (2010) made the first attempt to include the neutral helium in the singlefluid description of prominence plasma. They concluded that the neutral helium has no significant influence on the damping of MHD waves. However, the twofluid approach may give some more details about the effects of neutral helium on MHD waves, therefore it is important to study this point in the future.
5. Conclusions
Frequencies and damping rates of lowfrequency MHD waves in the twofluid description are similar to those obtained in the singlefluid approach. But highfrequency waves (with a higher frequency than the ionneutral collision frequency) have a completely different behavior.
Alfvén and fast magnetoacoustic waves have maximal damping rates at some frequency interval that peaks at a particular frequency. The peak frequency is 2.5 ν_{in}, where ν_{in} is the ionneutral collision frequency, for 50% of neutral hydrogen. For 10% of neutral hydrogen, the peak frequency is shifted to10 ν_{in}. The damping rate is reduced for higher frequencies, therefore the damping of highfrequency Alfvén waves in the solar chromosphere with a realistic height profile of the ionization degree needs to be revised in future.
There are two types of slow magnetoacoustic waves in the highfrequency part of the wave spectrum: one connected with the ionelectron fluid and another with the fluid of neutrals.
There is no cutoff frequency of Alfvén waves because of ambipolar diffusion. The cutoff frequency found in the singlefluid approach is caused by neglecting the inertial terms in the momentum equation of relative velocity.
The damping rate of slow magnetoacoustic waves is similar to Braginksii (1965) in low plasma β approximation. The deviation from the Braginskii formula found by the normal mode analysis in the singlefluid approach (Forteza et al. 2007) is probably caused by neglecting the inertial terms.
Acknowledgments
The work was supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (project P21197N16). T.V.Z. also acknowledges financial support from the Georgian National Science Foundation (under grant GNSF/ST09/4310).
References
 Arber, T. D., Haynes, M., & Leake, J. E. 2007, ApJ, 666, 541 [NASA ADS] [CrossRef] [Google Scholar]
 Barcélo, S., Carbonell, M., & Ballester, J. L. 2010, A&A, 525, A60 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Braginskii, S. I. 1965, Rev. Plasma Phys., 1, 205 [NASA ADS] [Google Scholar]
 Carbonell, M., Forteza, P., Oliver, R., & Ballester, J. L. 2010, A&A, 515, A80 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 De Pontieu, B., & Haerendel, G. 1998, A&A, 338, 729 [NASA ADS] [Google Scholar]
 Fontenla, J. M., Avrett, E. H., & Loeser, R. 1990, ApJ, 355, 700 [NASA ADS] [CrossRef] [Google Scholar]
 Forteza, P., Oliver, R., Ballester, J. L., & Khodachenko, M. L. 2007, A&A, 461, 731 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Forteza, P., Oliver, R., & Ballester, J. L. 2008, A&A, 492, 223 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Goedbloed, H., & Poedts, S. 2004, Principles of Magnetohydrodynamics (Cambridge University Press) [Google Scholar]
 Khodachenko, M. L., & Rucker, H. O. 2005, Adv. Space Res., 36, 1561 [NASA ADS] [CrossRef] [Google Scholar]
 Khodachenko, M. L., & Zaitsev, V. V. 2002, Astrophys. Space Sci., 279, 389 [NASA ADS] [CrossRef] [Google Scholar]
 Khodachenko, M. L., Arber, T. D., Rucker, H. O., & Hanslmeier, A. 2004, A&A, 422, 1073 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Haerendel, G. 1992, Nature, 360, 241 [NASA ADS] [CrossRef] [Google Scholar]
 James, S. P., & Erdélyi, R. 2002, A&A, 393, L11 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 James, S. P., Erdélyi, R., & De Pontieu, B. 2004, A&A, 406, 715 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Leake, J. E., & Arber, T. D. 2006, A&A, 450, 805 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Leake, J. E., Arber, T. D., & Khodachenko, M. L. 2005, A&A, 442, 1091 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Soler, R., Oliver, R., & Ballester, J. L. 2009a, ApJ, 699, 1553 [NASA ADS] [CrossRef] [Google Scholar]
 Soler, R., Oliver, R., & Ballester, J. L. 2009b, ApJ, 707, 662 [NASA ADS] [CrossRef] [Google Scholar]
 Soler, R., Oliver, R., & Ballester, J. L. 2010, A&A, 512, A28 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Vranjes, J., Poedts, S., Pandey, B. P., & de Pontieu, B. 2008, A&A, 478, 553 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
Appendix A: Singlefluid equations
We use the total velocity (i.e. velocity of center of mass) (A.1)the relative velocity (A.2)and the total density (A.3)Equation (20)–(25) and (27) lead to the system where p = p_{e} + p_{i} + p_{n}, ξ_{i} = ρ_{i}/ρ, ξ_{n} = ρ_{n}/ρ and α_{n} = α_{in} + α_{en}.
Ohm’s law is now (A.9)Neglecting the inertia terms, i.e. all left hand side terms in Eq. (A.6), we have
Then the induction equation takes the form (A.10)where ϵ = α_{en}/α_{n}, G = ξ_{n}∇p_{ei} − ξ_{i}∇p_{n} and (A.11)These equations are traditionally used for the description of partially ionized plasmas.
All Figures
Fig. 1 Alfvén wave frequency vs. normalized wave number. The top (bottom) panel shows the real (imaginary) part of the normalized frequency ϖ = ω/k_{z}v_{A} vs. the normalized wavenumber a = k_{z}v_{A}/ν_{in}. The blue line corresponds to the solution of the singlefluid dispersion relation, i.e. Eq. (44) and red asterisks are the solutions of the twofluid dispersion relation, Eq. (42). The values are calculated for 50% of neutral hydrogen, ξ_{n} = 0.5. 

Open with DEXTER  
In the text 
Fig. 2 Same as in Fig. 1, but for 10% of neutral hydrogen, ξ_{n} = 0.1. 

Open with DEXTER  
In the text 
Fig. 3 Frequency and damping rate of different wave modes in twofluid MHD vs. the normalized wavenumber a = k_{z}v_{A}/ν_{in}. Frequencies and damping rates are normalized by kv_{A}. Red asterisks correspond to the fast magnetoacoustic mode and green diamonds correspond to the usual slow magnetoacoustic mode. The mode with the blue squares is the new sort of slow magnetoacoustic wave (“neutral” slow mode), which arises for larger wave numbers. This mode has only an imaginary frequency for small wave numbers, which is not shown in this figure. Frequencies and damping rates are calculated for the waves propagating along the magnetic field. The neutral hydrogen is taken to be 50% (ξ_{n} = 0.5). Here we consider c_{sn}/v_{A} = 0.5. 

Open with DEXTER  
In the text 
Fig. 4 Damping rate of the fast (upper panel) and slow (lower panel) magnetoacoustic waves, i.e. with the imaginary part of ω normalized by kv_{A}, vs. the normalized wavenumber a = k_{z}v_{A}/ν_{in}. The blue solid lines correspond to the solution of the singlefluid dispersion relation and the dashed line corresponds to the slow magnetoacoustic damping rate of Braginskii (the expressions used are from Forteza et al. 2007). Red asterisks are the solutions of the twofluid dispersion relation, Eq. (57). The values are calculated for 10% of neutral hydrogen, ξ_{n} = 0.1, and for c_{sn}/v_{A} = 0.1. The damping rates are calculated for waves propagating with a 45° angle with regard to the magnetic field. 

Open with DEXTER  
In the text 
Fig. 5 The same as in Fig. 4 but for the parallel propagation, i.e. k_{x} = 0. 

Open with DEXTER  
In the text 
Fig. 6 Damping rate (the imaginary part of the normalized frequency ϖ = ω/k_{z}v_{A}) of fast (upper panel) and slow (lower panel) magnetoacoustic waves vs. the normalized wavenumber a = k_{z}v_{A}/ν_{in}. Red asterisks are the solutions of the twofluid dispersion relation. Blue lines correspond to the solutions of Braginskii. The values are calculated for 50% of neutral hydrogen and c_{sn}/v_{A} = 0.1. The damping rates are calculated for the waves that propagate along the magnetic field. 

Open with DEXTER  
In the text 
Fig. 7 The same as in Fig. 6, but for c_{sn}/v_{A} = 0.5. 

Open with DEXTER  
In the text 
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