Issue 
A&A
Volume 551, March 2013



Article Number  A86  
Number of page(s)  11  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201220576  
Published online  27 February 2013 
Effect of partial ionization on wave propagation in solar magnetic flux tubes
^{1}
Solar Physics Group, Departament de Física, Universitat de les
Illes Balears, 07122
Palma de Mallorca,
Spain
email:
roberto.soler@uib.es
^{2}
Instituto de Astrofísica de Canarias, 38200, La Laguna, Tenerife, Spain
^{3}
Departamento de Astrofísica, Universidad de La
Laguna, 38206, La
Laguna, Tenerife,
Spain
^{4}
Centre for Mathematical Plasma Astrophysics, Department of
Mathematics, KU Leuven,
Celestijnenlaan 200B, 3001
Leuven,
Belgium
Received:
16
October
2012
Accepted:
22
January
2013
Observations show that waves are ubiquitous in the solar atmosphere and may play an important role for plasma heating. The study of waves in the solar corona is usually based on linear ideal magnetohydrodynamics (MHD) for a fully ionized plasma. However, the plasma in the photosphere and the chromosphere is only partially ionized. Here we theoretically investigate the impact of partial ionization on MHD wave propagation in cylindrical flux tubes in a twofluid model. We derive the general dispersion relation that takes into account the effects of neutralion collisions and the neutral gas pressure. We assumed the neutralion collision frequency to be an arbitrary parameter. Specific results for transverse kink modes and slow magnetoacoustic modes are shown. We find that the wave frequencies only depend on the properties of the ionized fluid when the neutralion collision frequency is much lower that the wave frequency. For high collision frequencies that realistically represent the solar atmosphere, ions and neutrals behave as a single fluid with an effective density corresponding to the sum of densities of fluids plus an effective sound velocity computed as the average of the sound velocities of ions and neutrals. The MHD wave frequencies are modified accordingly. The neutral gas pressure can be neglected when studying transverse kink waves but it has to be included for a consistent description of slow magnetoacoustic waves. The MHD waves are damped by neutralion collisions. The damping is most efficient when the wave frequency and the collision frequency are on the same order of magnitude. For high collision frequencies slow magnetoacoustic waves are more efficiently damped than transverse kink waves. In addition, we find the presence of cutoffs for certain combinations of parameters that cause the waves to become nonpropagating.
Key words: Sun: oscillations / Sun: atmosphere / waves / magnetic fields / magnetohydrodynamics (MHD)
© ESO, 2013
1. Introduction
Observations show that magnetohydrodynamic (MHD) waves are ubiquitous in the solar atmosphere. Waves are routinely observed in the corona (e.g.., Tomczyk et al. 2007; Tomczyk & McIntosh 2009; McIntosh et al. 2011), the chromosphere (e.g.., Kukhianidze et al. 2006; De Pontieu et al. 2007; Zaqarashvili et al. 2007; He et al. 2009; Okamoto & De Pontieu 2011), and the photosphere (e.g., Jess et al. 2009; Fujimura & Tsuneta 2009). In addition, waves and oscillations in thin threads of solar prominences have been reported (e.g., Okamoto et al. 2007; Lin et al. 2007, 2009; Ning et al. 2009). These recent observations have motivated a number of theoretical works aiming at understanding the properties of the observed waves using the MHD theory (e.g., Van Doorsselaere et al. 2008; Pascoe et al. 2010, 2012; Terradas et al. 2010; Verth et al. 2010; Soler et al. 2011a,b, 2012a). It is believed that the observed waves may play an important role for the heating of the atmospheric plasma (see, e.g., Erdélyi & Fedun 2007; De Pontieu et al. 2007; McIntosh et al. 2011; Cargill & de Moortel 2011).
The study of waves in the solar corona is usually based on the linear ideal MHD theory for a fully ionized plasma (see Priest 1984; Goedbloed & Poedts 2004). In this context, the paper by Edwin & Roberts (1983) on wave propagation in a magnetic flux tube has been used as a basic reference for subsequent works in this field (see also the papers by, e.g., Wentzel 1979; Cally 1986; Goossens et al. 2009, 2012). The assumption of full ionization is adequate for the coronal plasma. However, the lower temperatures of the chromosphere and the photosphere cause this assumption to become invalid. Therefore, it is necessary to determine the impact of partial ionization on the wave modes discussed in Edwin & Roberts (1983) to perform a realistic theoretical modeling of wave propagation in partially ionized magnetic wave guides.
There are many papers that have studied the effect of neutralion collision on MHD waves (see, e.g., Braginskii 1965; De Pontieu et al. 2001; Khodachenko et al. 2004, 2006; Leake et al. 2005; Forteza et al. 2007). Since the present paper deals with wave propagation in a magnetic cylinder, we discuss here those works that studied waves in structured media. In solar plasmas the typical frequency of the observed waves is lower than the expected value of the neutralion collision frequency. For this reason the singlefluid theory is usually adopted as a first approximation. The singlefluid approximation assumes a strong coupling between ions and neutrals, so that both fluids behave in practice as a single fluid. Soler et al. (2009a) studied wave propagation in a partially ionized flux tube in the singlefluid approximation. The results of Soler et al. (2009a) show that the waves discussed in Edwin & Roberts (1983) are damped due to neutralion collisions. Although partial ionization may be relevant for plasma heating (see Khomenko & Collados 2012), the damping by neutralion collisions is in general too weak to explain the observed rapid attenuation. In addition, Soler et al. (2009a) found critical values for the longitudinal wavelength that constrains wave propagation. However, Zaqarashvili et al. (2012) have pointed out that these critical wavelengths may not be physical and may be an artifact of the singlefluid approximation. This is so because the singlefluid approximation misses important effects when short length scales and/or high frequencies are involved. In these cases, the general multifluid description is a more suitable approach (see, e.g., Zaqarashvili et al. 2011). In a multifluid description no restriction is imposed on the relative values of the wave frequency and the neutralion collision frequency. In this more general treatment the mathematical complexity of the equations is substantially increased compared to the singlefluid case.
A particular form of the multifluid theory is the twofluid theory in which ions and electrons are considered together as an ionelectron fluid, i.e., the plasma, while neutrals form another fluid that interacts with the plasma by means of collisions. This approach was followed by Kumar & Roberts (2003), who studied surface wave propagation in a Cartesian magnetic interface. In the study by Kumar & Roberts (2003) the gas pressure of neutrals was neglected, meaning that the dynamics of neutrals was only governed by the friction force with ions. Recently, Soler et al. (2012a) used the twofluid theory to study resonant Alfvén waves in flux tubes. Soler et al. (2012a) focused on transverse kink waves and, consequently, they also neglected gas pressure.
In the present paper we go beyond these previous studies. We derive the general dispersion relation for waves in cylindrical flux tubes in the twofluid theory. To do so, we consider a consistent description of the neutral fluid dynamics that includes the effect of neutral gas pressure. This is necessary for a realistic description of magnetoacoustic waves. The use of cylindrical geometry and the consideration of neutral pressure are two significant improvements with respect to the Cartesian model of Kumar & Roberts (2003). The dispersion relation derived here is the twofluid generalization of the wellknown dispersion relation of Edwin & Roberts (1983) for a fully ionized singlefluid plasma. The collision frequency between ions and neutrals and the ionization degree are two important parameters of the model. The impact of these two parameters on the waves discussed by Edwin & Roberts (1983) in the fully ionized case is investigated.
This paper is organized as follows. Section 2 contains a description of the equilibrium configuration and the basic equations. In Sect. 3 we follow a normal mode analysis and derive the general dispersion relation for the wave modes. The case in which neutral gas pressure is neglected is explored in Sect. 4, both analytically and numerically. We compare our results with the previous findings of Kumar & Roberts (2003). Then, we incorporate the effect of neutral gas pressure in Sect. 5. Finally, we discuss the implications of our results and give our main conclusions in Sect. 6.
2. Model and twofluid equations
We study waves in a partially ionized medium composed of ions, electrons, and neutrals. We use the twofluid theory in which ions and electrons are considered together as an ionelectron fluid, while neutrals form another fluid that interacts with the ionelectron fluid by means of collisions (see, e.g., Zaqarashvili et al. 2011; Soler et al. 2012a,b; Díaz et al. 2012). Throughout, the subscripts “ie” and “n” refer to ionelectrons and neutrals.
We assume a cylindrically symmetric equilibrium and, for convenience, we use cylindrical coordinates, namely r, ϕ, and z for the radial, azimuthal, and longitudinal coordinates. The equilibrium is made of a straight magnetic flux tube of radius R embedded in an unbounded environment. The equilibrium magnetic field is straight and constant along the zdirection, namely B = B ẑ. Gravity is ignored. As a consequence of the force balance condition, the gradients of the equilibrium gas pressures of the different species are zero. We also assume a static equilibrium so that there are no equilibrium flows. We restrict ourselves to the study of linear perturbations superimposed on the equilibrium state. Hence, the governing equations are linearized (see the general expressions in, e.g., Zaqarashvili et al. 2011). We adopt some additional simplifications that enable us to tackle the problem of wave propagation analytically. We neglect collisions of electrons with neutrals because of the low momentum of electrons. We also drop from the equations the nonadiabatic mechanisms and the magnetic diffusion terms since our purpose here is to determine the impact of neutralion collisions only. Thus, the set of coupled differential equations governing linear perturbations from the equilibrium state are where v_{i} and v_{n} are the velocities of ions and neutrals, p_{ie} and p_{n} are the pressure perturbations of ionelectrons and neutrals, b is the magnetic field perturbation, ρ_{i} and ρ_{n} are the equilibrium densities of ions and neutrals, P_{ie} and P_{n} are the equilibrium gas pressures of ionelectrons and neutrals, μ is the magnetic permittivity, γ is the adiabatic index, and ν_{ni} is the neutralion collision frequency. For our following analysis it is useful to define the ionization degree as χ = ρ_{n}/ρ_{i}. This parameter ranges from χ = 0 for a fully ionized plasma to χ → ∞ for a neutral gas. Also for convenience, in the following expressions we replace the velocities of ions and neutrals by their corresponding Lagrangian displacements, ξ_{i} and ξ_{n}, given by (6)
3. Normal modes
We assume that the equilibrium densities ρ_{i} and ρ_{n} are functions of r alone, so that the equilibrium is uniform in both azimuthal and longitudinal directions. The temperatures of ionelectrons and neutrals vary with r accordingly to keep the equilibrium gas pressures of both fluids constant. Hence we can write the perturbed quantities proportional to exp(imϕ + ik_{z}z), where m and k_{z} are the azimuthal and longitudinal wavenumbers, respectively. We express the temporal dependence of the perturbations as exp(−iωt), with ω the frequency. Now we combine Eqs. (1)–(5) and arrive at four coupled equations for the radial components of the Lagrangian displacements of ions and neutrals, ξ_{r,ie} and ξ_{r,n}, the ionelectron total pressure Eulerian perturbation, P′ = p_{ie} + Bb_{z}/μ, and the neutral pressure Eulerian perturbation, p_{n}, namely (7)with the coefficients , , , and defined as where , , and are the squares of the modified frequency, the Alfvén frequency, and the cusp frequency given by In addition, is the square of the Alfvén velocity and and are the squares of the sound velocities of ionelectrons and neutrals, respectively, computed as (18)Equations (7)–(10) are valid for all density profiles in the radial direction. For c_{ie} = c_{n} = 0 and p_{n} = 0 they revert to the equations discussed in Soler et al. (2012a). Equations (7)–(10) are singular when . The positions of the singularities are mobile and depend on the frequency, ω. The whole set of frequencies satisfying at some r form two continua of frequencies called the Alfvén and cusp (or slow) continua (see, e.g., Appert et al. 1974). A wave whose frequency is within any of the two continua is damped by resonant absorption (see, e.g., Goossens et al. 2011). Resonantly damped waves in partially ionized magnetic cylinders were studied by Soler et al. (2009b) in the singlefluid approximation and by Soler et al. (2012a) in the twofluid formalism.
3.1. Piecewise constant equilibrium
In the present paper we are not interested in studying the resonant behavior of the waves. We refer to Soler et al. (2009b, 2012a) for studies of resonant waves in partially ionized plasmas. Instead, here we focus on the modification of the wave frequencies due to neutralion collisions. For this reason, we avoid the presence of the resonances by choosing both ρ_{i} and ρ_{n} to be piecewise constant functions, namely (19)where R is the radius of the cylinder with constant densities ρ_{i,0} and ρ_{n,0} embedded in an external environment with constant densities ρ_{i,ex} and ρ_{n,ex}. This is the density profile adopted by Edwin & Roberts (1983) in the fully ionized case. Hereafter the indices “0” and “ex” represent the internal and external media. Because the magnetic field and equilibrium pressure of both fluids are constant, the rest of equilibrium quantities are piecewise constants as well. For simplicity we also take ν_{ni} as a constant free parameter.
For a piecewise constant equilibrium we can combine Eqs. (7) and (10) to eliminate ξ_{r,ie} and ξ_{r,n} and obtain two coupled equations involving P′ and p_{n} only, namely with Equations (20) and (21) are the basic equations of this investigation. They are two coupled Besseltype differential equations representing the coupled behavior of ionelectrons and neutrals due to neutralion collisions. Note that the terms with q_{1} and q_{2} are the ones that couple the equations. These terms vanish in fully ionized and fully neutral cases. We need to solve Eqs. (20) and (21) to find the general dispersion relation.
For later use we also give the expressions of ξ_{r,ie} and ξ_{r,n} in terms of P′ and p_{n}, namely Note that for ν_{ni} = 0 ions and neutrals are decoupled and the magnetic field has no influence on the dynamics of neutrals.
3.2. Strong thermal coupling and characteristic velocities
Up to here, no restriction on the values of the characteristic Alfvén and sound velocities has been made. As a result of the pressure balance condition at r = R, the six characteristic velocities in the equilibrium are related by (28)Now for simplicity we assume a strong thermal coupling between the ionized and neutral fluids, so that both fluids have the same temperature locally. However, the temperature can still be different in the internal and external media. Using the ideal gas law for both species, this requirement has the consequence that the local sound velocity of the ionized and neutral fluids are related by . Hence we can define an effective sound velocity of the whole plasma, c_{s,eff}, as (29)so that the parameter space of characteristic velocities can be reduced from six to four different velocities, namely c_{A,0}, c_{s,eff,0}, c_{A,ex}, and c_{s,eff,ex}.
3.3. Coupled solutions
We continue the mathematical study of the normal modes. We look for solutions to the coupled Eqs. (20) and (21). In the internal medium, i.e., r ≤ R, physical solutions imply that both P′ and p_{n} are regular at r = 0. The general solutions of P′ and p_{n} satisfying this condition in the internal medium are where J_{m} is the Bessel function of the first kind of order m, A_{1} and A_{2} are constants, and k_{1,0} and k_{2,0} are the two possible values of the radial wavenumber, k, in the internal medium given by the solution of the following equation (32)Equivalently, in the external medium, i.e., r > R, the solutions of P′ and p_{n} are of the form where is the Hankel function of the first kind of order m, A_{3} and A_{4} are constants, and k_{1,ex} and k_{2,ex} are the two possible values of the radial wavenumber, k, in the external medium also given by the solution of Eq. (32) but now using external values for the parameters. The expressions of P′ and p_{n} for r > R are written generally in terms of Hankel functions instead of the usual modified Bessel functions, K_{m}, used to represent trapped waves. Thus, our formalism takes into account the possibility of wave leakage, i.e., wave radiation in the external medium. Although we do not explore leaky waves in the present work, the obtained dispersion relation will also be valid for leaky waves. For ideal trapped waves, the external radial wavenumber is purely imaginary and the function consistently reverts to the modified Bessel function of the second kind, K_{m}, used by Edwin & Roberts (1983). In the partially ionized case the radial wavenumber is a complex quantity and the use of the function causes us to be very careful when choosing the branch of the external radial wavenumber. To avoid nonphysical energy propagation coming from infinity, we need to select the appropriate branch of the external radial wavenumber so that the condition for outgoing waves is satisfied (see details in Cally 1986; Stenuit et al. 1999).
To understand why there are two different values of the radial wavenumber, k, it is instructive to assume a weak coupling between the species, i.e., ν_{ni} ≪ ω , so that the quadratic terms in ν_{ni} can be neglected in Eq. (32). The neglected terms involve the coupling coefficients q_{1} and q_{2}. Then, the two independent wavenumbers in Eq. (32) simplify to (35)where we have dropped the indices “0” or “ex” because the same result is valid in both internal and external media. The two values of k for weak coupling reduce to the wavenumbers of the ionized and neutral fluids, respectively. For high collision frequencies, these two wavenumbers contain terms with q_{1} and q_{2} that couple the two fluids.
3.4. Dispersion relation
To find the dispersion relation we match the internal solutions to the external solutions by means of appropriate boundary conditions at r = R. The boundary conditions at the interface between two media in the twofluid formalism are discussed in Díaz et al. (2012). In the absence of gravity, these boundary conditions reduce to the continuity of P′, p_{n}, ξ_{r,ie}, and ξ_{r,n} at r = R. After applying the boundary conditions at r = R we find a system of four algebraic equations for the constants A_{1}, A_{2}, A_{3}, and A_{4}. The condition that the system has a nontrivial solution provides us with the dispersion relation. The dispersion relation is (36)with the full expression of given in Appendix A. This dispersion relation is the twofluid generalization of the dispersion relation of Edwin & Roberts (1983) for a fully ionized singlefluid cylindrical flux tube. In the following sections we discuss the corrections due to neutralion collisions on the wave modes described by Edwin & Roberts (1983) in the fully ionized case.
4. Pressureless neutral fluid
Before exploring the solutions of the general dispersion relation we first consider the case that neutral pressure is neglected and only the gas pressure of the ionized fluid is taken into account. In the absence of neutral pressure the only force acting on the neutral fluid is the friction force due to neutralion collisions. This is the situation studied by Kumar & Roberts (2003) in planar geometry. We set P_{n} = 0 so that c_{n} = 0. After some algebraic manipulations the general dispersion relation simplifies to (37)
with k_{0} and k_{ex} given by In Eq. (37) we have used the modified Bessel function of the second kind, K_{m}, instead of the Hankel function of the first kind, , in order to write Eq. (37) in the same form as the dispersion relation of Edwin & Roberts (1983). Indeed, the comparison between Eq. (37) and the dispersion relation of Edwin & Roberts (1983) reveals that both equations are exactly the same if we replace by ω^{2} in Eq. (37). We can take advantage of this result to easily study the modifications due to neutralion collisions of the solutions of the fully ionized case.
4.1. Analytic approximate study
4.1.1. Standing waves
We focus on standing waves so that we fix the longitudinal wavenumber, k_{z}, to a real value. Due to temporal damping by neutralion collisions ω is complex. i.e., ω = ω_{R} + iω_{I}, where ω_{R} and ω_{I} are the real and imaginary parts of the frequency. Therefore the wave amplitude is damped in time due through the exponential factor exp(−ω_{I}t).
We assume that ω = ω_{0} is a trapped solution of the dispersion relation of Edwin & Roberts (1983) in the fully ionized case. Here we only consider trapped waves so that ω_{0} is real. For leaky waves see, e.g., Cally (1986). Then is a solution of Eq. (37). Using the expression of (Eq. (15)) we expand the equation as (40)The study of the modifications due to neutralion collisions of the modes of Edwin & Roberts (1983) reduces to the study of the solutions of Eq. (40). This study is general and is independent of the particular mode considered since we have not specified what mode ω_{0} corresponds to. All modes are affected in the same way by neutralion collisions when neutral pressure is absent.
Equation (40) is a cubic equation, hence it has three solutions. To determine the nature of the solutions we perform the change of variable ω = −is, so that Eq. (40) becomes a cubic equation in s with all its coefficients real. We compute the discriminant, Δ, of the resulting equation as (41)The discriminant, Δ, is defined so that (i) Eq. (40) has one purely imaginary zero and two complex zeros when Δ < 0; (ii) Eq. (40) has a multiple zero and all its zeros are purely imaginary when Δ = 0; and (iii) Eq. (40) has three distinct purely imaginary zeros when Δ > 0. This criterion points out that oscillatory solutions of Eq. (40) are only possible when Δ < 0.
To determine when oscillatory solutions are not possible we set Δ = 0 and find the corresponding relation between ω_{0} and ν_{ni} in terms of χ as (42)\\Since ν_{ni}/ω_{0} must be real, Eq. (42) imposes a condition on the minimum value of χ which allows Δ = 0. This minimum value is χ = 8 and the corresponding critical ν_{ni}/ω_{0} is . When χ < 8 oscillatory solutions are always possible. When χ > 8 we have to take into account the + and the − signs in Eq. (42), so that Eq. (42) defines a range of values of ν_{ni}/ω_{0} in which oscillatory solutions are not possible. We call this interval the cutoff region. To our knowledge, Kulsrud & Pearce (1969) were the first to report on the existence of a cutoff region of MHD waves in a partially ionized plasma, although Kulsrud & Pearce (1969) only investigated Alfvén waves in a homogeneous medium. The existence of cutoff regions for MHD waves in structured media is also deduced from the plots of Kumar & Roberts (2003), but these authors did not explore this phenomenon in detail.
We look for approximate analytic expressions of the frequency of the oscillatory solutions. To do this, we assume that ν_{ni}/ω_{0} is outside the cutoff interval defined by Eq. (42), meaning that Eq. (40) has two complex (oscillatory) solutions and one purely imaginary (evanescent) solution. First we focus on the oscillatory solutions. We write ω = ω_{R} + iω_{I} and insert this expression into Eq. (40). We assume weak damping and take  ω_{I} ≪ ω_{R} . Hence we neglect terms with and higher powers. It is crucial for the validity of this approximation that ν_{ni}/ω_{0} is not within or close to the cutoff region in which ω_{R} = 0. After some algebraic manipulations we derive approximate expressions for ω_{R} and ω_{I}. For simplicity we omit the intermediate steps and give the final expressions, namely A mode with the same ω_{I} but with ω_{R} of opposite sign is also a solution. When ν_{ni} = 0, so that neutralion collisions are absent, we find ω_{R} = ω_{0} and ω_{I} = 0. Hence we recover the ideal undamped modes of Edwin & Roberts (1983). Now these modes are damped due to neutralion collisions and the frequency has an imaginary part. In addition, the real part of the frequency is also modified.
It is instructive to study the limit values of ν_{ni}. For a low collision frequency, . Equations (43) and (44) become Thus, the real part of the frequency when is just the frequency found in the fully ionized case (Edwin & Roberts 1983) and it does not depend on the amount of neutrals. The imaginary part of the frequency is independent of ω_{0}, meaning that it is the same for all wave modes. On the other hand, for a high collision frequency, and Eqs. (43) and (44) simplify to Now the expression of ω_{R} involves χ in the denominator, so that the larger the amount of neutrals, the lower ω_{R} compared to the frequency in the fully ionized case. The frequency is reduced by the factor compared to the fully ionized value. This is the same factor as found by Kumar & Roberts (2003) and is equivalent to replace the ion density by the sum of densities of ions and neutrals or, equivalently, to replace ρ_{i} by (1 + χ)ρ_{i}. The imaginary part of the frequency depends now on ω_{0}, meaning that the various modes have different damping rates.
In addition to the oscillatory solutions, Eq. (40) has one purely imaginary solution whose approximation is given by (49)\\The perturbations related to these modes are evanescent in time. There exists a different evanescent solution related to each mode, ω_{0}, of the fully ionized case. Purely imaginary solutions were also found by Zaqarashvili et al. (2011), who studied waves in a partially ionized homogeneous medium. These modes are not present in the fully ionized case. These evanescent perturbations may be relevant during the excitation of the waves, since part of the energy used to excite the waves may go to these modes instead of being used to excite the oscillatory solutions. This could be investigated by going beyond the present normal mode analysis and solving the initial value problem.
4.1.2. Propagating waves
Now we turn to propagating waves. In the fully ionized case the study of propagating waves is equivalent to that of standing waves. Here we shall see that neutralion collisions break this equivalence and propagating waves are worth being studied separately. Propagating waves were not investigated by Kumar & Roberts (2003). For propagating waves, we fix the frequency, ω, to a real value and solve the dispersion relation for the complex k_{z}, i.e., k_{z} = k_{z,R} + ik_{z,I}, where k_{R} and k_{I} are the real and imaginary parts of k_{z}. Therefore the wave amplitude is damped in space due to the exponential factor exp(−k_{z,I}z).
Equation (37) also holds for propagating waves. We assume that k_{z} = k_{z,0}, with k_{z,0} real, is a solution of the ideal dispersion relation of Edwin & Roberts (1983). Then, (50)\\is a solution of Eq. (37) following the same argument as explained for standing waves. We write k_{z} = k_{z,R} + ik_{z,I} and insert this expression in Eq. (50). We obtain exact expressions for k_{z,R} and k_{z,I}, namely The same values of k_{z,R} and k_{z,I} but with opposite signs are also solutions. In principle, Eq. (51) allows two different values of k_{z,R} because of the ± sign. However, the solution with the − sign is not physical since it corresponds to k_{z,R} imaginary, which is an obvious contradiction. For this reason we discard the solution with the − sign and take the + sign in Eq. (51). Now we realize that (53)Thus we can approximate k_{z,R} and k_{z,I} as These expressions are equivalent to Eqs. (43) and (44) obtained for standing waves. However, contrary to Eqs. (43) and (44), Eqs. (54) and (55) are valid for all values of the ratio ν_{ni}/ω. There is no cutoff region for propagating waves because k_{z,R} never vanishes. There are no purely imaginary solutions of k_{z}, i.e., all the solutions always have an oscillatory behavior in z. This is an important difference to standing waves.
4.2. Comparison with numerical results
Here we numerically solve the dispersion relation (Eq. (37)). For simplicity we focus on standing waves and show the results for transverse kink (m = 1) waves only, although we have checked that equivalent results are found for other modes as, e.g., slow magnetoacoustic modes. In the fully ionized case, the frequency of the transverse kink waves in the thin tube (TT) limit, i.e., k_{z}R ≪ 1, is ω = ω_{k}, with ω_{k} the kink frequency given by (56)Figure 1 displays the real and imaginary parts of the frequency of the kink mode as functions of the ratio ν_{ni}/ω_{k} for a particular choice of parameters given in the caption of the figure. We compare the numerical results with the analytic approximations. In Fig. 1 we have considered χ < 8 so that no cutoff region is present. Regarding the real part of the frequency (Fig. 1a), we obtain that for ν_{ni}/ω_{k} ≪ 1 the results are independent of the ionization degree, with ω_{R} ≈ ω_{k} as in the fully ionized case. When the ratio ν_{ni}/ω_{k} increases, plasma and neutrals are more coupled and the ionization degree becomes a relevant parameter, so that ω_{R} decreases until de value is reached. This behavior is consistent with the analytic Eq. (43) although the details of the transition are not fully captured by the approximation. Regarding the imaginary part of the frequency (Fig. 1b), we find that ω_{I} tends to zero in the limits ν_{ni}/ω_{k} ≪ 1 and ν_{ni}/ω_{k} ≫ 1. The damping is most efficient when ν_{ni} and ω_{k} are, approximately, on the same order of magnitude. This result is also consistent with the analytic Eq. (44), although the approximation underestimates the actual damping rate when ω_{I} is minimal. This is so because Eq. (44) was derived in the weak damping approximation, i.e.,  ω_{I} ≪ ω_{R} , while  ω_{I}  and  ω_{R}  are on the same order when ω_{I} is minimal, meaning that the damping is strong.
Fig. 1 a)ω_{R}/ω_{k} and b) ω_{I}/ω_{k} versus ν_{ni}/ω_{k} for the transverse kink (m = 1) mode with k_{z}R = 0.1, ρ_{i,0}/ρ_{i,ex} = 3, c_{ie,0}/c_{A,0} = 0.2, and χ = 4. The solid line is the result obtained by numerically solving the dispersion relation in the absence of neutral pressure (Eq. (37)). The symbols are the approximate analytic results in the weak damping approximation (Eqs. (43) and (44)). 
Now we take χ > 8 and repeat the previous computations. The results are shown in Fig. 2. We notice the presence of a cutoff region for a certain range of ν_{ni}/ω_{k}. In this cutoff region ω_{R} = 0. The location of the cutoff region agrees very well with the range given by Eq. (42). This zone exists for relatively low ν_{ni}/ω_{k}. In the solar atmosphere the expected value of the collision frequency (see, e.g., De Pontieu et al. 2001, Figs. 1 and 2) is much higher than the frequency of the observed waves (e.g., De Pontieu et al. 2007; Okamoto & De Pontieu 2011). The minimum value of the neutralion collision frequency in the solar atmosphere is on the order of 10 Hz, while the dominant frequency in the observations of chromospheric kink waves by Okamoto & De Pontieu (2011) is around 22 mHz. Therefore, the cutoff region found here is not in the range of ν_{ni}/ω_{k} consistent with solar atmospheric parameters and observed wave frequencies. However, in other astrophysical situations the presence of the cutoff region may be relevant (see Kulsrud & Pearce 1969). As expected, the analytic approximations (Eqs. (43) and (44)) completely miss the presence of the cutoff region, although they are reasonably good far from the location of the cutoff.
Fig. 2 Same as Fig. 1 but with χ = 20. The shaded area denotes the cutoff region given by Eq. (42). 
It is worth mentioning again that the effect of resonant absorption on the damping of transverse kink waves is absent because we used a piecewise constant density profile (Ruderman & Roberts 2002; Goossens et al. 2002). Soler et al. (2012a) showed that damping by resonant absorption is more efficient than damping by neutralion collisions when ν_{ni}/ω_{k} ≫ 1. Hence, the actual damping of transverse kink waves would be stronger than the damping shown here if resonant absorption were taken into account (see details in Soler et al. 2012a).
Fig. 3 a)ω_{R}/ω_{c,0} and b) ω_{I}/ω_{c,0} versus ν_{ni}/ω_{c,0} for the slow magnetoacoustic mode with χ = 4 and the rest of parameters the same as in Fig. 1. The solid line is the full result taking neutral pressure into account, while the dashed line is the result when neutral pressure is absent. 
5. Role of neutral pressure
We incorporate the effect of neutral pressure. We anticipate that neutral pressure is relevant for slow magnetoacoustic waves, while its effect on transverse kink waves is minor. Goossens et al. (2009, 2012) showed that transverse kink waves in flux tubes have the typical properties of surface Alfvén (or Alfvénic) waves. In thin tubes, i.e., k_{z}R ≪ 1, the main restoring force of these waves is magnetic tension and the gas pressure force is negligible. We computed the kink mode frequency in the presence of neutral pressure and found no significant differences with the result in the absence of neutral pressure. Hence, here we focus on the study of slow magnetoacoustic waves and explore the modification of the slow mode frequency due to neutralion collisions when neutral pressure is taken into account. In the fully ionized case (Edwin & Roberts 1983), slow magnetoacoustic waves in the TT limit have frequencies ω ≈ ω_{c,0}, with ω_{c,0} the internal cusp frequency given by (57)This result is independent of the azimuthal wavenumber, m. In a lowβ plasma, i.e., when the magnetic pressure is much more important than the gas pressure, ω_{c,0} ≈ k_{z}c_{ie,0}. Due to the complexity of the general dispersion relation (Eq. (36)) it is not possible to obtain simple analytic approximations of the slow mode frequency. For this reason we perform this investigation in a numerical way.
Figure 3 shows the numerically obtained real and imaginary parts of the slow mode frequency as functions of the ratio ν_{ni}/ω_{c,0} for χ < 8. The rest of parameters are given in the caption of the figure. For comparison, we overplot the results in the absence of neutral pressure, which are obtained by solving Eq. (37) with the same parameters. First we discuss the behavior of the ω_{R} (Fig. 3a). When ν_{ni}/ω_{c,0} ≪ 1 we recover the result in the fully ionized case, i.e., ω_{R} ≈ ω_{c,0}. When ν_{ni}/ω_{c,0} increases, the real part of the slow mode frequency decreases until a plateau is reached at the value . This is equivalent to performing the replacements and in Eq. (57) and is the same behavior as that found in the absence of neutral pressure, so that both results are superimposed in Fig. 3a. However, as the ratio ν_{ni}/ω_{c,0} continues to increase, the real part of the frequency rises again until a second plateau is finally reached. The presence of this second plateau of ω_{R} for high collision frequencies owes its existence to the effect of neutral pressure and is absent when neutral pressure is neglected.
We now analyze in detail the second plateau of ω_{R} at high collision frequencies. Since we have no simple analytic expressions for the slowmode frequency we performed a parameter study. We considered various values of c_{ie} and numerically computed the slow mode ω_{R} at the second plateau. The result of this parameter study (not shown here for simplicity) points out that the value of ω_{R} at the second plateau is, approximately, (58)This expression can be obtained from Eq. (57) by performing the replacements and , with the expression of the effective sound velocity, c_{s,eff}, given in Eq. (29). In a lowβ plasma, and ω_{c,0} ≈ k_{z}c_{ie,0} so that Eq. (58) simplifies to (59)Hence, for practical purposes we obtain that at high collision frequencies the sound velocity of the ionized fluid has to be replaced by the effective sound velocity in the expressions of the frequency. The effective sound velocity takes into account the sound velocity of neutrals in addition to the sound velocity of ions. In contrast, for intermediate collision frequencies it is enough to replace c_{ie} by , so that the sound velocity of neutrals can be ignored.
We turn to the imaginary part of the frequency (Fig. 3b). As for the real part of the frequency, there are striking differences between the results with and without neutral pressure. The full result shows the presence of two different minima of ω_{I}, whereas only the first minimum is present in the absence of neutral pressure. The second minimum takes place in the range of high collision frequencies compared to the ideal cusp frequency. As a consequence of the presence of this second minimum for high collision frequencies, the efficiency of neutralion collisions for the damping of the slow mode is higher than for transverse kink modes when ν_{ni} approaches realistic values. The stronger damping of the slow modes due to neutralion collisions compared to the weak damping of the transverse kink modes was already obtained by Soler et al. (2009a) in the singlefluid approximation.
Now we consider χ > 8 and repeat the previous computations (see Fig. 4 for χ = 20). As in the case without neutral pressure, we notice the presence of a cutoff region that agrees well with the range given by Eq. (42). The existence of the cutoff region is not affected by neutral pressure, i.e., the location of the cutoff region is the same as when neutral pressure is neglected. For the set of parameters considered in Fig. 4 the cutoff region appears in the vicinity of ν_{ni}/ω_{c,0} ≈ 10^{1}. We increase the ionization fraction to χ = 100 (Fig. 5). In addition to the cutoff region described above, now we find a second cutoff region. The second cutoff region appears for relatively high collision frequencies, ν_{ni}/ω_{c,0} ≈ 10^{1} for the set of parameters used in Fig. 5. This new cutoff region is only present when neutral pressure is taken into account. The result in the absence of neutral pressure completely misses this second cutoff region. The existence of a second cutoff region is a result of slow modes only, since it is not found for transverse kink modes. Unfortunately, unlike for the first cutoff region, we have no analytic expression that gives us the location of the second cutoff region. Instead, out numerical study informs us that the second cutoff appears when χ ≳ 24, although this value may be affected by the particular choice of parameters used in the model.
As for the transverse kink modes, it is worth mentioning that some mechanisms not considered in the present analysis may produce damping of the slow mode comparable to or even stronger than that due to neutralion collisions. This is the case for nonadiabatic mechanisms, e.g., thermal conduction and radiative losses (see, e.g., De Moortel & Hood 2003; Soler et al. 2008). Partial ionization increases the efficiency of thermal conduction due to the additional contribution of neutral conduction (see Forteza et al. 2008; Soler et al. 2010), so that the combined effect of thermal conduction and partial ionization produces a very efficient damping of the slow mode (see, e.g., Khodachenko et al. 2006; Soler 2010).
6. Discussion
In this paper we have explored the impact of partial ionization on the properties of the MHD waves in a cylindrical magnetic flux tube using the twofluid formalism. Unlike previous works (e.g., Kumar & Roberts 2003; Soler et al. 2012a), we considered a consistent description of the dynamics of the neutral fluid that takes gas pressure into account. We derived the dispersion relation for the wave modes, that is, the twofluid generalization of the dispersion relation of Edwin & Roberts (1983). Instead of performing particular applications, we considered the neutralion collision frequency as an arbitrary parameter. The modifications of the wave frequencies due to neutralion collisions were explored.
First, we investigated the case that neglects neutral pressure. This case was previously investigated for surface waves in a Cartesian interface by Kumar & Roberts (2003). For ν_{ni}/ω ≪ 1 we recovered the frequencies in fully ionized plasma (Edwin & Roberts 1983). As the ratio ν_{ni}/ω  increases, both ionelectrons and neutrals become more and more coupled. For ν_{ni}/ω ≫ 1 ionelectrons and neutrals behave as a single fluid. Compared to the fully ionized case, when ν_{ni}/ω ≫ 1 the wave frequencies are lower and depend on the ionization ratio, χ. The frequencies are reduced by the factor (χ + 1)^{−1/2} compared to their values in the fully ionized case. This result is the same for all wave modes and is equivalent to replacing the density of ions by the sum of densities of ions and neutrals. The same conclusion was reached in the previous work by Kumar & Roberts (2003). Concerning the damping by neutralion collisions, we found that damping is weak when ν_{ni}/ω ≪ 1 unless the plasma is very weakly ionized, i.e., χ ≫ 1. Damping is most efficient when the wave frequency is on the same order of magnitude as the collision frequency. Damping is again weak when ν_{ni}/ω ≫ 1. This last result is always true regardless of the value of χ. Again, our results agree well with the previous findings of Kumar & Roberts (2003).
Then we incorporated the effect of neutral gas pressure. Since transverse kink waves are largely insensitive to the sound velocity, we obtained the same results as in the absence of gas pressure. However, the consideration of neutral pressure has dramatic consequences for slow magnetoacoustic modes. For high collision frequencies the slowmode frequency is higher than the value obtained in the absence of neutral pressure. This is so because the slowmode frequency depends on an effective sound velocity that corresponds to the weighted average of the sound velocities of ionselectrons and neutrals. When χ ≫ 1, the effective sound velocity tends to be like the sound velocity of neutrals. Concerning the imaginary part of the frequency, the results with and without neutral pressure also show significant differences for high collision frequencies. The imaginary part of the slowmode frequency has an additional minimum at high collision frequencies when neutral pressure is included. This causes the slow mode to be more efficiently damped by collisions when neutral pressure is taken into account.
Here we go back to the discussion given in the introduction about the necessity of using a multifluid theory instead of the simpler singlefluid MHD approximation. In the solar atmosphere the expected value of the neutralion collision frequency (see De Pontieu et al. 2001, Figs. 1 and 2) is much higher than the frequency of the observed waves (e.g., De Pontieu et al. 2007; Okamoto & De Pontieu 2011). In this case, the results of this paper point out that for practical purposes, it is enough to use the singlefluid MHD approximation, but taking into account the following simple recipe to adapt the results of fully ionized models to the partially ionized case. First of all, the ion density has to be replaced by the total density, i.e., the sum of densities of ions and neutrals. Second, the effective sound velocity should be used instead of the sound velocity of the ionized fluid. This effective sound velocity is the weighted average of the sound velocities of ionelectrons and neutrals. This approach is appropriate for works that are not interested in the details of the interaction between ions and neutrals, but only in the result of this interaction on the wave frequencies. Hence, this recipe may be useful to perform magnetoseismology of chromospheric spicules and other partially ionized structures as prominence threads. For example, in their seismological analysis of transverse kink waves in spicules, Verth et al. (2011) considered a definition of the kink velocity that incorporates the density of neutrals. Although our recipe provides a good approximation to obtain the wave frequencies, it misses the effect of damping due to neutralion collisions, which might have an impact on the seismological estimates using the amplitude of the waves. Taking into account the range of observed wave frequencies, the damping of kink waves due to neutralion collisions may be negligible compared to other effects such as resonant absorption (Soler et al. 2012a). However, the damping of slow magnetoacoustic waves may be important even for high collision frequencies. Other damping effects such as resonant absorption and nonadiabatic mechanisms should be considered in addition to neutralion collisions to explain the damping of MHD waves in the solar atmosphere (see, e.g., Khodachenko et al. 2006; Soler et al. 2012a).
On the other hand, a relevant result obtained in this paper is the presence of cutoffs for certain combinations of parameters (see Kulsrud & Pearce 1969). This result cannot be obtained in the singlefluid MHD approximation. At the cutoffs the real part of the frequency vanishes, meaning that perturbations are evanescent in time instead of oscillatory. For transverse kink waves we found a cutoff region for low values of ν_{ni}/ω_{k} when χ > 8. For slow magnetoacoustic waves there are two cutoff regions. The first one is the same as that found for transverse kink waves, whereas the second one is only present in the case of slow modes when neutral pressure is considered. The second cutoff region takes place when χ ≳ 24 and relatively high collision frequencies. The relevance of these cutoffs for wave propagation in the solar atmosphere should be investigated in forthcoming works.
Finally, we are aware that the model used in this paper is simple and misses effects that may be of importance in reality. The model should be improved in the future by considering additional ingredients not included in the present analysis. Among these effects, the variation of physical parameters, e.g., density, ionization degree, etc., along the magnetic wave guide, ionization and recombination processes, and the influence of a dynamic background are worth being explored in the near future.
Acknowledgments
We acknowledge the anonymous referee for his/her constructive comments. We thank T. V. Zaqarashvili for reading an early draft of this paper and for giving helpful comments. RS, JLB, and MG acknowledge support from MINECO and FEDER funds through project AYA201122846. RS and JLB acknowledge support from CAIB through the “Grups Competitius” scheme and FEDER funds. MG acknowledges support from KU Leuven via GOA/2009009. AJD acknowledges support from MINECO through project AYA20101802.
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Appendix A: expression of the dispersion relation
The dispersion relation is , with given by the solution of the following determinant, (A.1)with (A.5)(A.6)(A.7)(A.8)(A.9)(A.10)(A.11)(A.12)
All Figures
Fig. 1 a)ω_{R}/ω_{k} and b) ω_{I}/ω_{k} versus ν_{ni}/ω_{k} for the transverse kink (m = 1) mode with k_{z}R = 0.1, ρ_{i,0}/ρ_{i,ex} = 3, c_{ie,0}/c_{A,0} = 0.2, and χ = 4. The solid line is the result obtained by numerically solving the dispersion relation in the absence of neutral pressure (Eq. (37)). The symbols are the approximate analytic results in the weak damping approximation (Eqs. (43) and (44)). 

In the text 
Fig. 2 Same as Fig. 1 but with χ = 20. The shaded area denotes the cutoff region given by Eq. (42). 

In the text 
Fig. 3 a)ω_{R}/ω_{c,0} and b) ω_{I}/ω_{c,0} versus ν_{ni}/ω_{c,0} for the slow magnetoacoustic mode with χ = 4 and the rest of parameters the same as in Fig. 1. The solid line is the full result taking neutral pressure into account, while the dashed line is the result when neutral pressure is absent. 

In the text 
Fig. 4 Same as Fig. 3 but with χ = 20. The shaded area denotes the cutoff region given by Eq. (42). 

In the text 
Fig. 5 Same as Fig. 4 but with χ = 100. 

In the text 
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