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An Eigenvalue Analysis of Damping in Optical Thin Plasmas

In this work, the behavior of magnetohydrodynamic waves in optically thin plasmas considering dissipative processes, thermal and magnetic diffusion, a given ionization, and the heating and cooling functions are investigated for several particular cases. A numerical eigenvalues analysis of the dimensionless secular equations according to various cases is performed for the entire set of MHD equations.


I. INTRODUCTION
In recent years analytical as well as numerical tools for working out non linear partial differential equation, in particular, those governing general fluids have been enormously improved, nonetheless the linear problem resulting from analyzing these equations remains to be very important for many reasons: • The associated eigenvalue problem describes the behavior of magnetohydrodynamic waves (MHD) and other waves, say for instance thermal and radiation waves.
• Understanding the behavior of linear waves allows to understand many physical aspects of non linear problems, for instance the onset of the turbulence as well as its closed relation with it (Dennis and Chandran 2005, Burkert 2006, Hennebelle and Passot 2006, Pan and Padoan 2009, Braun and Schmidt 2012.
• The linear approach is closely related to the problem of stability of different flows and gas structures in different physical fields, in particular in astrophysical problems: planetary atmospheres, Earth oceans, stellar interiors (Hennebelle and Passot 2006, Parker 1979, Heiles and T. 2005) stellar atmospheres, say for instance the solar atmosphere, interstellar medium, intracluster media (Kegel andTraving 1976, Kippenhahn andWeigert 1991).
However, the present work is limited to the analysis of some aspects of MHD wave propagation in optically thin plasmas of interest in astrophysics. This limited problem however is of great importance and has been extensively worked out in: • The solar atmosphere (Ibáñez 1985, Ibáñez and Sanchez 1992, Ibáñez and Escalona 1993, Dahlburg and Mariska 1988, Field 1965, Giaretta 1979, Landau and Lifshitz 1960, Mihalas 1978, R. Rosner and Vaiana 1978, Spitzer 1962, Stix 1992, Mariska 1986, J. Vesecky and Underwood 1979, N. Kumar and Kumar 2006, Roberts 2000.
There are several aspects which have not been considered previously in the Alfvèn wave damping analysis and in the magnetosonic wave analysis and the associated eigenvalue problem for optically thin plasmas, as will be seen and discussed at the present work.
In section II, the set of MHD equations is linearized, leading to two independent cases where each matrix generates a dispersion relation whose roots for the case of Alfvèn waves are a complex equation. This relationship led to a deduction of the Landau damping expression in a different way to the one presented by classical texts.
In section III, for the linear approximation both modes are studied for the thermal and magneto-acoustic cases. They are damped by thermal conduction, viscosity and the influence of the cooling-heating function. The complex eigen-equation is described for several asymptotic cases: • The case when only one dissipative process is taken into account.
• The case with only magnetic diffusion term (ν m ).
In section IV, in the energy equation the dissipative terms were neglected, but the effects of the heat/loss were accounted, because of its great importance in many astrophysical as well as laboratory plasma applications.
Finally, in section V, the kinetic coefficients in a magnetic field, for the case of a recombining hydrogen plasma are discussed.

II. GENERAL SET OF MAGNETOHYDRODYNAMIC EQUATIONS
If dissipative effects are accounted for a recombining gas, for an an optically thin and heat conducting plasma, the well known basic MHD equations can be written as (Landau and Lifshitz 1960, Stix 1992, Ibañez 2004, Fajardo 2016) and, where H i and v i are the i-esime components of the magnetic field and velocity, respectively. jki is the permutation symbol δ ik is Kronecker delta symbol, c the light speed and ρ, p, T , ξ, c v ,N 0 , k B and µ(ξ) respectively are mass density, pressure, temperature, ionization degree, specific heat at constant volume, the Avogadro number, the Boltzmann constant and the mean molecular weight of the gas.
X(ρ, T, ξ) is the net ionization rate and L(ρ, T, ξ) is the heat-loss function defined as energy losses minus energy gains per unit mass and time, which can be written as L(ρ, T, ξ) = Additionally, κ ik and σ ik are the thermal conduction and the viscous stress tensor, respectively.
Strictly speaking the induction equation becomes rather complicated, in particular, the electrical conductivity σ is also a tensor, however, for sake of simplicity and taking into account that σ /σ ⊥ = 1.96, this quantity will be assumed as a scalar of magnitude σ and the induction equation will be assumed in the simplified form given by Eq.
This set of equations reduces to the known MHD equations (Landau and Lifshitz 1960) when the heat/loss term is neglected.

HYDRODYNAMIC WAVES
For an inert plasma if all dissipative processes are neglected the Eqs. (1), (3) and (7) hold and the Eqs.
(2)-(6) simplify,i.e the set of ideal MHD equations can be written as (Fajardo For small disturbances superposed to an steady flow with velocity V 0 , magnetic field H 0 , pressure p 0 and mass density ρ 0 Lifshitz 1960, Ibáñez 2009) Where v , h, p and ρ are functions of (x, y, z, t). Therefore, Eqs. (8)-(11), up to the first order, become ∂p ∂t Furthermore, for disturbances of the form ∼ exp[i(k · r−ωt)], Eqs. (12)-(16) reduce to where ρ = p /u 2 0 + (∂ρ/∂s) p s and u 2 0 = (∂p/∂ρ) s . Eq. (17) implies that h is perpendicular to k, therefore, from Eq. (21) follows that p = 0 and Eqs. (19) and (20) reduce to Without lost of generality, V 0 and H 0 are assumed to be on the x−y plane. The above relations define an entropy vortex wave which is carried along with the flow and is independent on other linear modes which correspond to the solutions These modes are defined by the eigen equations and where u = ω/k is the phase velocity. Herein after, the sub index 0 indicating equilibrium values will be omitted, except for u 0 , that is, the adiabatic sound speed. k is taken here to be along the x-axis.
In the particular case of a plasma initially at rest, the compatibility condition for the Equations (23) and (24) respectively become and As it is well known, Eqs. (23 and (25) define the Alfvèn modes, and Eqs. (24) and (26) define the fast and slow magnetosonic modes (Alfvèn suggested the existence of hydromagnetic waves in 1942, Lifshitz 1960, Ibáñez 2009).
In the general case of a plasma flowing with an initial constant velocity V 0 the dispersion relations are modified accordingly but the nature of the wave modes remains.
In conclusion, as far as the linear approximation concerns, there are three kind of waves in a plasma flow, and which are independent each other: • The entropy-vortex modes.
Te entropy entropy-vortex modes were worked out in (Ibáñez 2016, Ibañez and Nuñez

IV. DISSIPATIVE PROCESSES IN MAGNETO HYDRODYNAMIC WAVES WITH A GIVEN IONIZATION AND HEAT/LOSS EFFECTS
For a plasma with a given ionization and taking into account dissipative and heat/loss effects, the linearization of Eqs.
(1)- (7) give, as in the ideal case, two sets of equations independent each other, that is (Ibáñez and Conde 2011) where The coefficients of viscosity appearing into the viscous stress tensor σ ij are tensors due to the anisotropy introduced by the magnetic field, the ratio between the parallel and perpendicular kinematic viscosity becomes η /η ⊥ ≈ 1.98 (Spitzer 1962), therefore, this coefficient as well as the bulk viscosity ζ can be assumed as scalars of magnitude η and ζ, respectively, in the equations (27) and (28).
Additionally, the strong anisotropy inherent in the thermal conduction tensor κ ij (η ⊥ /η ≈ 10 −12 ) has been taken into account assuming the heat flux vector to be where n and n ⊥ are unit vectors along and perpendicular to H 0 , respectively. Therefore, In dimensionless form Eq. (28) can be written as A. Numerical results for the Alfvèn wave damping The corresponding dimensionless secular equation of the system of equations (27) becomes The roots of Eq. (34) are complex, that is,k =k r +ik i wherek r andk i are real quantities.
Due to the fact thatν as well asν m (≈l/λ 1, wherel is the mean free path and λ the Alfvèn wave length (Landau and Lifshitz 1960), the quartic term of Eq. (34) can be neglected and the resulting quadratic equation has the solution sought.
Because the disturbance has been taken in the form ∼ exp(k · r−iωt), therefore k i = ωk i /a x becomes the absorption coefficient.
One must remark that the expression (35) holds as far as the damping per wave length is very small. This expression was first obtained by (Landau and Lifshitz 1960) in a different way.
Strictly speaking if both coefficientsν andν m are different from zero, Eq.(34) has two roots fork 2 , however only one of the roots fulfils the conditionk i 1 for which the present approximation holds.
For this physical meaningful mode, the velocities (v), the damping coefficient (k i ), the damping per unit wave length (l d /λ = k r /2πk i ) and the ratio h z /v z √ πρ have been plotted The condition of compatibility of the system of equations (32) can be written as (a 0r + i a 0i )k 8 + (a 1r + i a 1i )k 6 + (a 2r + i a 2i )k 4 + (a 3r + i a 3i )k 2 + 1 + iL T = 0, (36) the coefficients a kl are defined as Generally speaking, the parameters defining the coefficients of the fourth order polynomial ink 2 (36) depend on two thermodynamic quantities, say ρ and T and two quantities defining the magnetic field, i.e. H and θ. Therefore, these parameters define the corresponding four wave modes resulting from the equation (36).
The square root ±k represents two waves propagating in opposite directions each other.
The angle ranges between 0 ≤ θ ≤ π/2, but the ranges for ρ, T and H where the dispersion relation (36) holds is rather wide.
Therefore, here only a few asymptotic cases will be discussed and the solution of the full polynomial (36) will serve only for specific applications.
If all dissipative mechanisms as well as the heat/input effects are neglected and θ = π 2 , the dispersion relation reduces to a quadratic polynomial fork 2 (26) corresponding to the undamped fast and slow magnetosonic waves (mw) (Landau and Lifshitz 1960) but when θ = π 2 only the fast magnetosonic mode remains. If the only one dissipative process taken into account is the thermal conductivity and θ = π 2 , Eq.(36) reduces to a cubic polynomial the roots of which correspond to two damped magnetosonic waves sw and a thermal wave T hw.
The another two roots withk = 1 are a damped magnetosonic wave sw and an over damped thermal wave T hw for which p = |ρωv x /k|, all of which all plotted in Fig. 2.
In Fig. 2, the phase velocity (a), the damping coefficient (b) and the damping per unit wave length (c) are plotted for three different values of β = 0.2 (red lines), 1 (blue lines), 2 (green lines) as function of κ 0 .
Note that the maximum damping of the magnetosonic wave (red mw line, occurs at the same value of κ 0 at which the maximum damping of the thermal wave occurs for the three β T hw values in Fig. 2(b) (Ibáñez 1985, Ibáñez andEscalona 1993).
If θ = π 2 the dispersion equation reduces to a quadratic equation, one root becomes a damped thermal wave and the another one a damped magnetosonic wave for whichṽ y = 0, h y /ṽ x = k , and |p/ṽ x | = β 2 (1 −k 2 )/k , see Fig. 3 where the above two wave modes are plotted for β = 0.2 (red lines), β = 1 (blue lines), β = 2 (green lines) for both mw and T hw modes.
Here one must emphasize that in the above figures the wave parameters have been plotted as function ofκ ⊥ = (κ ⊥ ω/ρc v a 2 ) instead ofκ , i.e. the scales involved here are quite different (by a factor of the order of 10 12 ) from those involved in Fig. 2.
In this figure however, the wave parameters have been plotted as function ofκ ⊥ = κ ⊥ ω/ρc v a 2 instead ofκ .
The thermal waves show a minimum of |p/ṽ x | at the value ofκ 0 at which the maximum damping per unit wave length l d /λ occurs.
The magnetosonic wave showing its minimum of l d /λ at lower value ofκ 0 presents an increasing value of |p/ṽ x | whenκ 0 increases, but the magnetosonic wave with its minimum occurring at a larger value ofκ 0 has a decreasing ratio |p/ṽ x | whenκ 0 increases.
If only the magnetic diffusion (ν m ) is accounted for, there should be no thermal waves because only the magnetic terms are considered. In this case the dispersion equation (36) reduces to a quadratic equation fork 2 for which |p/ṽ x | = 1.
When the magnetic energy density is of the order or larger than the kinetic energy in the wave β ≤ 1, there is not crossing of slow and fast modes, but mode crossing occurs when β = 1, see Figs 5a and 6a for instance.
The damping coefficient for the slow mode is a decreasing function ofν m (∼ ω) but that for the fast mode shows a maximum at a value ofν m depending on the value of β (Figs. 5b and 6b) and for the damping per wave length there is a corresponding minimum (Figs 5c and 6c). This minimum occurs at the mode crossing point when β = 1.
The case when only thermal conduction and heat/loss effects are accounted for in the equations, but neglecting the viscosities and the magnetic diffusion as well as the above asymptotic cases, but neglecting the anisotropy effects of the thermal conduction coefficient, have been analyzed in a previous work (Ibáñez and Escalona 1993). The case when in the energy equation the dissipative terms are neglected but the effects of the heat/loss are accounted for deserve further analysis, because this particular case is of great importance in many astrophysical as well as laboratory plasma.
In this case the Eq.(36) reduces to a quadratic equation ink 2 (if θ = π/2) corresponding to two magnetosonic waves modified by the heat loss input.
For θ = 0 one root becomesk = 1 corresponding to an undamped Alfvèn wave for which h y /ṽ y = 1 and the other root corresponding to the magnetosonic wave becomes For θ = π/2, this is the only one root, but in this case, the magnetosonic wave hasṽ y = 0.
As a first approximation, the heat/loss function can be can be parameterized to the form where T i and η are parameters depending on the interval of temperature under consideration (see Table 1, J. Vesecky and Underwood 1979).
Additionally, the parameters C 0 , a and b depend on the heating processes considered. In particular: 1. For a constant per unit volume heating a = 0 and b = 0. From Eq. (39) follows thatL The cooling function φ i (T ) has been plotted as a function of temperature in Fig. 7(a) in magenta color. The derivatives ωL T (ρ, T ) (red thick line) and ωL ρ (ρ, T ) (blue thin line) as function of T have been plotted in Fig. 7(b) for a constant heating per unit volume (case 1), for a completely ionized gas (ξ = 1) and a particle density n = ρN 0 µ = 1. The intervals of temperature whereL T (ρ, T ) > 0 are indicated with the red label +, The plots corresponding to the cases (2) to (5) also are shown: Fig. 8(a) for a constant per unit mass heating heating, fig. 8(b) shows the heating by coronal current dissipation, fig. 8(c) plots the heating by Alfvèn mode/mode conversion, the heating by Alfvèn mode/anomalous conduction damping is shown in fig. 8(d).
Due to the fact that the cooling term in Eq.(39) as well as its derivatives respect to temperature and density are ∼ ρ, for other densities, the corresponding values simply must line) as function of T , that correspond to cases (2) to (5) are shown in figures (a) to (d) for a completely ionized gas (ξ = 1) and a particle density n ρN 0 µ = 1 be multiply by the factor n.
On the other hand and as a first approximation the total dissipative coefficient for magnetosonic waves can be writing as where χ = κ/ρc p is the thermometric conductivity and ν m the magnetic diffusion (Landau and Lifshitz 1987, Lifshitz and Pitaevskii 1981, Conde 2010. Note that at the present approximation ν(T ), ν m (T ), and χ(n, T, ξ) explicitly depend on the particular form of the rate function X(n, T, ξ) and the wave frequency (Ibañez 2004, Conde 2010, Ibáñez and Contreras 2019. In (Ibáñez and Contreras 2019) the problem of reacting gases and the bulk viscosity has been discussed to some extent.
Note that χ ∼ n 2 , therefore, the effect of increasing (decreasing) the density is to increase (decrease) the respective values of χ. The value of ν m 10 10 cm −2 s −1 in the range of T under consideration has not been plotted.
However, ν m parallel and perpendicular to the magnetic field can become of the order or greater than of 4 3 ν and (γ − 1)χ for high densities (n ≥ 10 10 cm −3 ) and strong magnetic fields H ≥ 1 G, say for instance in the solar low atmosphere and photosphere.
For context, in Fig. 9(b) all dissipation coefficients are shown for H = 1 G and n = 10 15 cm −3 ; from where it is apparent that the magnetic dissipation parallel (ν m ) as well as perpendicular (ν m⊥ ) to the magnetic field becomes dominant in range of temperatures depending on the particular values of the ionization degree as well as the particle density.
The parallel magnetic diffusion (ν m ) is also plotted is Fig 9(b) (dash black line).
The present work was aimed at investigating the behavior and propagation of MHD waves in optically thin plasmas, with ionization and dissipative effects. The results are summarized in four sections.
In section II, the set of MHD equations was linearized, leading to two independent cases where each matrix generates a dispersion relation whose roots for the case of Alfvèn waves are a complex equation, this relationship led to the deduction of the Landau damping expression in a different way to the one presented in classical texts.
In section III, for the linear approximation it was observed that both, thermal and magneto-acoustic modes are damped by the thermal conduction, viscosity and the influence of the cooling-heating function. The complex eigen-equation was described with some detail, and several asymptotic cases of the full polynomial solutions were discussed (36): • The case when the only dissipative process taken into account is the thermal conductivity was discussed for several values of θ in Eq.(36). We found eigenvalues corresponding to two damped magnetosonic waves, and a thermal wave. We also found a small jump in the phase velocity for magnetosonic modes for the case of β = 1 which is reflected also in the amplitude.
• In the case with only the magnetic diffusion term (ν m ), the dispersion equation (36) reduces to a quadratic equation fork 2 for which |p/ṽ x | = 1 and lacks the thermal mode. It was found that if the magnetic energy density is of the order or larger than the kinetic energy in the wave for β = 1, a crossing of slow and fast magnetosonic modes was observed.
In section IV, in the energy equation the dissipative terms were neglected, but the effects of the heat/loss were accounted, because of its great importance in many astrophysical as well as laboratory plasma applications. In this case the Eq.(36) reduces to a quadratic equation ink 2 corresponding to two magnetosonic waves modified by the heat loss input.
We described in this section five heating processes for a thin optical plasma.
Finally, in section V, the kinetic coefficients in a magnetic field, for the case of a recombining hydrogen plasma were briefly discussed. It was found that the magnetic dissipation parallel (ν m ) as well as perpendicular (ν m⊥ ) to the magnetic field become dominant in a range of temperatures depending on the particular values of the ionization degree ξ as well as the particle density n.