Magnetoacoustic waves in a partially ionized astrophysical plasma with the thermal misbalance: A two-fluid approach

Molevich, N. E.; Pichugin, S. Yu.; Riashchikov, D. S.

doi:10.1063/5.0201945

We consider the propagation of magnetoacoustic (MA) and acoustic waves of various frequency ranges in a partially ionized plasma at an arbitrary angle to the magnetic field, taking into account the influence of heating, radiative, and thermo-conductive cooling, as well as ion-neutral collisions. A dispersion equation that describes the evolution of nine modes was obtained in a compact mathematical form using the two-fluid model. The number and type of propagating waves (modified fast and slow MA waves, MA waves in the ion component, acoustic waves in the neutral component, as well as isothermal MA and isothermal acoustic waves) vary in different frequency ranges depending on the parameters of the medium. Analytical expressions are found for the speed and damping rates of all these propagating waves, and it is shown how dispersion and damping are formed by three processes: thermal misbalance, ion-neutral collisions, and thermal conductivity. Comparison of analytical calculations of the velocity and damping rates of MA waves with the numerical solution of the dispersion relation under conditions characteristic of the low solar atmosphere and prominences showed high accuracy of the obtained analytical expressions. The strong influence of thermal misbalance caused by gasdynamic perturbations on the speed and damping rate of modified magnetoacoustic waves in a strongly coupled region is shown as well.

I. INTRODUCTION

Plasma astrophysics studies plasma media beyond Earth's atmosphere and includes many different astrophysical environments such as the solar atmosphere, the interstellar medium, and molecular clouds. All these media consist of a mixture of neutral gas and charged particles (ions and electrons), which interact differently with magnetic fields. In this case, single-fluid models are not always able to fully describe the processes occurring in these media and the use of more complex multi-fluid models is required.^1,2 In particular, for time-scales comparable with or higher than the ion-neutral collision time, the two-fluid description is the better approximation than a single-fluid approach to describe the propagation features of magnetohydrodynamic (MHD) modes.^3–18 The two-fluid model assumes that the charged particles create a single ion-electron fluid that interacts with neutrals. Thus, two fluids that interact due to ion-neutral collisions are considered. One of them consists of charged particles (below in the article it is called ion fluid), and the other one consists of neutral particles. In a partially ionized plasma, the dispersion properties of magnetoacoustic (MA) waves change significantly. Moreover, the number of possible wave modes becomes larger, since, in addition to modes associated with charged particles, there are also acoustic modes associated with neutrals. In the strongly coupled ion-neutral fluid, the properties of MA waves differ from MA waves in a fully ionized plasma, since disturbances propagate in ion and neutral fluids simultaneously. We, following Refs. 1 and 7 call these waves modified MA waves (MMA). Their properties differ from classical MA waves, since they create perturbations not only in the ion fluid, but also in the neutral one. Unlike the one-fluid model, the two-fluid model also describes cutoffs and forbidden spectral intervals for the propagation of MA waves.^5,19

Various processes of heat release and radiative cooling also take place in astrophysical plasma. In theoretical studies, the presence of these processes is modeled by introducing a generalized heat-loss function into the energy equation.²⁰ This function represents the sum of heating and radiative cooling (in the thin layer radiation approximation) powers, depending on the parameters of the medium such as temperature and density. Under stationary conditions, this function is equal to zero, so one can say that heating and cooling balance each other. Due to the different functional dependence of heating and cooling on the medium parameters, gasdynamic perturbations will lead to a violation of the heat balance. The consequences of this thermal misbalance (TM) for a propagating acoustic wave can be as follows. If this TM leads to an increase in heat release in areas of increased temperature and density (high pressure areas) of acoustic perturbation, then acoustic (isentropic) instability of the medium occurs. If, on the contrary, the TM leads to cooling in the region of increased pressure, then the acoustic disturbance is attenuated.^20–22

Recently, there has been an explosive growth of papers devoted to the effect of TM on the dynamics of gasdynamic and magnetohydrodynamic (MHD) waves propagating in interstellar gas^{10,14,23–28} and the solar atmosphere.^29–45 One reason for the increased interest in this topic in astrophysical applications has been the development of the resolving power of measuring instruments, as well as the appearance of new spacecraft designed to analyze the dynamics of the solar atmosphere and interstellar gas. In particular, this made it possible to develop such a field as coronal seismology, which studies plasma properties in the solar corona using data on MHD waves and oscillations.^46–50 In this regard, it is interesting to know, first, how the TM caused by waves affects the conditions for their formation and propagation, and second, the diagnostic ability to determine the local heating mechanisms by the dispersion properties of these waves.

It has been shown, in particular, that the TM leads to dispersion of the acoustic,^22,23 acoustic-gravity,⁴⁴ and MA waves (see review Ref. 51), including changes in the low-frequency limit of the tube velocity,^38,52 which determines the velocity of slow MA waves in solar magnetic tubes, and is an important quantity for diagnosing the parameters (magnetic field and temperature) of these tubes.

It is also shown that the frequency-dependent damping rate of acoustic^22,23 and MA waves (see reviews Refs. 36 and 51) due to misbalance is proportional to the coefficient of the bulk (second) viscosity. In the case of isentropic instability, the coefficient of the bulk viscosity is negative, which leads to amplification of acoustic and MA waves with the subsequent formation of quasiperiodic and autosoliton-like structures.^{24,51,53–58}

In the case of isentropic stability, the coefficient of the bulk viscosity is positive, which leads to frequency-dependent damping of waves consistent with the observed frequency-dependent damping of slow MA waves in various coronal plasma structures.^32,47 In addition, it is shown that the TM leads to an additional phase shift^40,59 and mixing of acoustic and thermal modes,³⁹ as well as to the appearance of acoustic flows as a result of nonlinear interactions of Alfvén and magnetoacoustic waves.^35,38,60,61 All this makes it possible to create a new tool for probing the unknown coronal heating function by investigation MA waves in different solar structures.⁴⁶

When describing the role of TM in partially ionized astrophysical media, one-fluid models and their modifications were mainly used.^45,62–68 In Refs. 3 and 69, the two-fluid model was used to study the effect of partial ionization on the properties of the evanescent condensation (entropy) mode developed under TM conditions in interstellar media. In this paper, we investigate the role of TM on the dispersion properties of propagating MA waves in a partially ionized plasma in the framework of a two-fluid linear model that takes into account heating, radiative, and thermo-conductive cooling in both fluids (neutral and charged).

This paper is organized as follows. To start with, in Sec. II, we describe the model equations. Then, in Sec. III, the two-fluid dispersion equation is obtained and written in a compact analytical form. This equation allows us to obtain in Sec. IV simple analytical expressions for phase velocities and damping rates of oscillatory modes in strongly coupled and uncoupled limiting cases (Secs. IV A and IV B, respectively). In Sec. V, the results are applied to the description of MA waves under conditions typical of the low solar atmosphere and prominences. Finally, Sec. VI discusses the results and contains some concluding remarks.

Notations for the key parameters are given in Table I.

TABLE I.

Notation for the key parameters.

Parameter	Meaning	Equation
Ω_in	Collision rate coefficient
ν_in	Ion-neutral collision frequency	$ν_{i n} = Ω i n ρ_{n 0}$
ν_ni	Neutral-ion collision frequency	$ν_{n i} = Ω i n ρ_{i 0}$
ν₀	Total ion-neutral and neutral-ion frequency	$ν_{0} = ν_{i n} + ν_{n i}$
ν_av	Average collision frequency of ions and neutrals	$ν_{a v} = (ρ_{i 0} ν_{i n} + ρ_{n 0} ν_{n i}) / ρ_{0}$
c_A	Alfvén speed	$c_{A}^{2} = B_{0}^{2} / 4 π ρ_{i 0}$
V_A	Effective (modified) Alfvén speed	$V_{A}^{2} = B_{0}^{2} / 4 π ρ_{0}$
c_i	High-frequency ion sound speed	$c_{i}^{2} = γ P_{i 0} / ρ_{i 0}$
$c_{n}$	High-frequency neutral sound speed	$c_{n}^{2} = γ P_{n 0} / ρ_{n 0}$
$γ_{0 i, n}$	Low-frequency adiabatic index	(14)
$c_{0 i}$	Low-frequency ion sound speed	$c_{0 i}^{2} = c_{i}^{2} γ_{0 i} / γ$
$c_{0 n}$	Low-frequency neutral sound speed	$c_{0 n}^{2} = c_{n}^{2} γ_{0 n} / γ$
$c_{0 f, s}$	Low-frequency sound speed of fast/slow MMA waves	(19)
$c_{f, s}$	High-frequency sound speed of fast/slow MMA waves	(25)
$c_{i f, s}$	Velocities of fast/slow MA waves in ion gas	(31)
$c_{T i, n}$	Isothermal speed of sound in ion and neutral gas	$c_{T i, n}^{2} = c_{i, n}^{2} / γ$
$c_{T f, s}$	Velocities of fast/slow magnetoisothermal waves	(41)

Parameter	Meaning	Equation
Ω_in	Collision rate coefficient
ν_in	Ion-neutral collision frequency	$ν_{i n} = Ω i n ρ_{n 0}$
ν_ni	Neutral-ion collision frequency	$ν_{n i} = Ω i n ρ_{i 0}$
ν₀	Total ion-neutral and neutral-ion frequency	$ν_{0} = ν_{i n} + ν_{n i}$
ν_av	Average collision frequency of ions and neutrals	$ν_{a v} = (ρ_{i 0} ν_{i n} + ρ_{n 0} ν_{n i}) / ρ_{0}$
c_A	Alfvén speed	$c_{A}^{2} = B_{0}^{2} / 4 π ρ_{i 0}$
V_A	Effective (modified) Alfvén speed	$V_{A}^{2} = B_{0}^{2} / 4 π ρ_{0}$
c_i	High-frequency ion sound speed	$c_{i}^{2} = γ P_{i 0} / ρ_{i 0}$
$c_{n}$	High-frequency neutral sound speed	$c_{n}^{2} = γ P_{n 0} / ρ_{n 0}$
$γ_{0 i, n}$	Low-frequency adiabatic index	(14)
$c_{0 i}$	Low-frequency ion sound speed	$c_{0 i}^{2} = c_{i}^{2} γ_{0 i} / γ$
$c_{0 n}$	Low-frequency neutral sound speed	$c_{0 n}^{2} = c_{n}^{2} γ_{0 n} / γ$
$c_{0 f, s}$	Low-frequency sound speed of fast/slow MMA waves	(19)
$c_{f, s}$	High-frequency sound speed of fast/slow MMA waves	(25)
$c_{i f, s}$	Velocities of fast/slow MA waves in ion gas	(31)
$c_{T i, n}$	Isothermal speed of sound in ion and neutral gas	$c_{T i, n}^{2} = c_{i, n}^{2} / γ$
$c_{T f, s}$	Velocities of fast/slow magnetoisothermal waves	(41)

II. THE MODEL

The dynamics of MA waves propagating in partially ionized heat-releasing plasma can be described by the following system of equations:³

\frac{\partial B}{\partial t} = \nabla \times (V_{i} \times B),

(1)

\nabla B = 0,

(2)

\begin{matrix} ρ_{i} (\frac{\partial V_{i}}{\partial t} + (V_{i} \nabla) V_{i}) \\ = - C_{V i} (γ - 1) \nabla (T_{i} ρ_{i}) - α_{i n} (V_{i} - V_{n}) - \frac{1}{4 π} B \times (\nabla \times B), \end{matrix}

(3)

\frac{\partial ρ_{i}}{\partial t} + \nabla (ρ_{i} V_{i}) = 0,

(4)

\begin{matrix} C_{V i} ρ_{i} (\frac{\partial T_{i}}{\partial t} + V_{i} \nabla T_{i}) - C_{V i} (γ - 1) T_{i} (\frac{\partial ρ_{i}}{\partial t} + V_{i} \nabla ρ_{i}) \\ = - ρ_{i} W_{i} (ρ_{i}, T_{i}) + \nabla (K_{i} \nabla T_{i}), \end{matrix}

(5)

\begin{matrix} ρ_{n} (\frac{\partial V_{n}}{\partial t} + (V_{n} \nabla) V_{n}) = - C_{V n} (γ - 1) \nabla (T_{n} ρ_{n}) - α_{i n} (V_{n} - V_{i}), \end{matrix}

(6)

\frac{\partial ρ_{n}}{\partial t} + \nabla (ρ_{n} V_{n}) = 0,

(7)

\begin{matrix} C_{V n} ρ_{n} (\frac{\partial T_{n}}{\partial t} + V_{n} \nabla T_{n}) - C_{V n} (γ - 1) T_{n} (\frac{\partial ρ_{n}}{\partial t} + V_{n} \nabla ρ_{n}) \\ = - ρ_{n} W_{n} (ρ_{n}, T_{n}) + \nabla (K_{n} \nabla T_{n}) . \end{matrix}

(8)

In Eqs. (1)–(8), variables ρ, T, V, and B are density, temperature, velocity, and a magnetic field, respectively. The subscript i denotes the ion fluid, and the subscript n denotes the neutral one. $γ = C_{P i} / C_{V i} = C_{P n} / C_{V n}$ is the adiabatic index, where C_Vi, C_Vn, C_Pi, and C_Vi are heat capacities at constant volume and pressure of ion and neutral fluids, respectively. $α_{i n} = Ω_{i n} ρ_{i} ρ_{n}$ is the ion-neutral friction coefficient, where Ω_in is the collision rate coefficient per unit mass.³^, $K_{i}, K_{n}$ are thermal conductivity coefficients of ion and neutral fluids. $W_{n} (ρ_{n}, T_{n}), W_{i} (ρ_{i}, T_{i})$ are generalized heat-loss functions, defined as radiative energy losses minus energy gains per gram per second²⁰ in ion and neutral fluids. When writing Eqs. (1)–(8), the influence of dissipative processes caused by the presence of viscosity and finite electrical conductivity was neglected. In addition, as in Refs. 3, 4, and 7 a strong thermal coupling between the ion fluid and the neutral fluid, which results in the proximity of temperatures in both fluids, was assumed. Moreover, we assume that in the equilibrium state, the medium is motionless and homogeneous, thus, for both components, the energy losses are balanced by the heat gains, i.e., $W_{i} (ρ_{i 0}, T_{i 0}) = W_{n} (ρ_{n 0}, T_{n 0}) = 0$ ⁠. The subscripts 0 here and below correspond to equilibrium values. In real media, the temperature $T_{i 0}$ may differ from $T_{n 0}$ in a partially ionized medium. The influence of this difference is intended to be studied separately.

We consider small perturbations of the equilibrium in the following form:

\begin{matrix} ρ_{i} = ρ_{i 0} + {\tilde{ρ}}_{i}, ρ_{n} = ρ_{n 0} + {\tilde{ρ}}_{n}, V_{i} = {\tilde{V}}_{i}, V_{n} = {\tilde{V}}_{n}, T_{i} = T_{i 0} + {\tilde{T}}_{i}, T_{n} = T_{n 0} + {\tilde{T}}_{n}, B = B_{0} + \tilde{B}, \end{matrix}

(9)

where

{\tilde{ρ}}_{i} / ρ_{i 0} \sim {\tilde{ρ}}_{n} / ρ_{n 0} \sim {\tilde{T}}_{i} / T_{i 0} \sim {\tilde{T}}_{n} / T_{n 0} \sim \tilde{B} / B_{0} \sim {\tilde{V}}_{i} / c_{i} \sim {\tilde{V}}_{n} / c_{n} \sim ε

(c_i and c_n are high-frequency velocities of sound in ion and neutral fluids defined below) and ε is the magnitude of the first order of smallness. In addition, we will conduct research in the Cartesian coordinate system x, y, z under the following assumptions. In the equilibrium, the magnetic field vector

B_{0}

lies in the x – z plane at an angle θ to the z-axis. MA disturbances propagate along the z-axis.

If we substitute expressions in Eqs. and limit ourselves to terms of the first order of smallness, then one can obtain the following linear equations for small perturbations of

{\tilde{B}}_{y}, {\tilde{B}}_{x}

components of the magnetic field vector

\tilde{B}

⁠, as well as perturbations of the densities of ion and neutral fluids,

\frac{\partial}{\partial t} (\frac{\partial^{2} {\tilde{B}}_{y}}{\partial t^{2}} - c_{A}^{2} {cos}^{2} θ \frac{\partial^{2} {\tilde{B}}_{y}}{\partial z^{2}}) + ν_{0} (\frac{\partial^{2} {\tilde{B}}_{y}}{\partial t^{2}} - \frac{ρ_{i 0}}{ρ_{0}} c_{A}^{2} {cos}^{2} θ \frac{\partial^{2} {\tilde{B}}_{y}}{\partial z^{2}}) = 0,

(10)

\frac{\partial}{\partial t} (\frac{\partial^{2} {\tilde{B}}_{x}}{\partial t^{2}} - c_{A}^{2} {cos}^{2} θ \frac{\partial^{2} {\tilde{B}}_{x}}{\partial z^{2}}) + ν_{0} (\frac{\partial^{2} {\tilde{B}}_{x}}{\partial t^{2}} - \frac{ρ_{i 0}}{ρ_{0}} c_{A}^{2} {cos}^{2} θ \frac{\partial^{2} {\tilde{B}}_{x}}{\partial z^{2}}) = \frac{B_{0} sin θ}{ρ_{i 0}} (\frac{\partial^{3} {\tilde{ρ}}_{i}}{\partial t^{3}} + ν_{0} \frac{\partial^{2} {\tilde{ρ}}_{i}}{\partial t^{2}}),

(11)

\begin{matrix} C_{V i} \frac{\partial}{\partial t} (\frac{\partial^{2} {\tilde{ρ}}_{i}}{\partial t^{2}} - c_{i}^{2} \frac{\partial^{2} {\tilde{ρ}}_{i}}{\partial z^{2}} + ν_{i n} \frac{\partial {\tilde{ρ}}_{i}}{\partial t} - ν_{n i} \frac{\partial {\tilde{ρ}}_{n}}{\partial t}) + W_{i T} (\frac{\partial^{2} {\tilde{ρ}}_{i}}{\partial t^{2}} - c_{0 i}^{2} \frac{\partial^{2} {\tilde{ρ}}_{i}}{\partial z^{2}} + ν_{i n} \frac{\partial {\tilde{ρ}}_{i}}{\partial t} - ν_{n i} \frac{\partial {\tilde{ρ}}_{n}}{\partial t}) \\ - \frac{K_{i 0}}{ρ_{i 0}} \frac{\partial^{2}}{\partial z^{2}} (\frac{\partial^{2} {\tilde{ρ}}_{i}}{\partial t^{2}} - \frac{c_{i}^{2}}{γ} \frac{\partial^{2} {\tilde{ρ}}_{i}}{\partial z^{2}} + ν_{i n} \frac{\partial {\tilde{ρ}}_{i}}{\partial t} - ν_{n i} \frac{\partial {\tilde{ρ}}_{n}}{\partial t}) = \frac{B_{0} sin θ}{4 π} (C_{V i} \frac{\partial^{3} {\tilde{B}}_{x}}{\partial t \partial z^{2}} + W_{i T} \frac{\partial^{2} {\tilde{B}}_{x}}{\partial z^{2}} - \frac{K_{i 0}}{ρ_{i 0}} \frac{\partial^{4} {\tilde{B}}_{x}}{\partial z^{4}}), \end{matrix}

(12)

\begin{matrix} C_{V n} \frac{\partial}{\partial t} (\frac{\partial^{2} {\tilde{ρ}}_{n}}{\partial t^{2}} - c_{n}^{2} \frac{\partial^{2} {\tilde{ρ}}_{n}}{\partial z^{2}} - ν_{i n} \frac{\partial {\tilde{ρ}}_{i}}{\partial t} + ν_{n i} \frac{\partial {\tilde{ρ}}_{n}}{\partial t}) + W_{n T} (\frac{\partial^{2} {\tilde{ρ}}_{n}}{\partial t^{2}} - c_{0 n}^{2} \frac{\partial^{2} {\tilde{ρ}}_{n}}{\partial z^{2}} - ν_{i n} \frac{\partial {\tilde{ρ}}_{i}}{\partial t} + ν_{n i} \frac{\partial {\tilde{ρ}}_{n}}{\partial t}) \\ - \frac{K_{n 0}}{ρ_{n 0}} \frac{\partial^{2}}{\partial z^{2}} (\frac{\partial^{2} {\tilde{ρ}}_{n}}{\partial t^{2}} - \frac{c_{n}^{2}}{γ} \frac{\partial^{2} {\tilde{ρ}}_{n}}{\partial z^{2}} - ν_{i n} \frac{\partial {\tilde{ρ}}_{i}}{\partial t} + ν_{n i} \frac{\partial {\tilde{ρ}}_{n}}{\partial t}) = 0. \end{matrix}

(13)

In Eqs. , we have used the following notations.

ρ_{0} = ρ_{i 0} + ρ_{n 0}

is the equilibrium density of the plasma, and ν_ni, ν_in are the frequencies of neutral-ion and ion-neutral collisions, respectively,

\begin{matrix} ν_{n i} = Ω_{i n} ρ_{i 0}, ν_{i n} = Ω_{i n} ρ_{n 0}, \\ ν_{0} = ν_{n i} + ν_{i n} = Ω_{i n} (ρ_{i 0} + ρ_{n 0}) = Ω_{i n} ρ_{0} . \end{matrix}

K_{n 0}

and

K_{i 0}

are thermal conductivity coefficients of the neutral and ion-electron fluids along the wave propagation direction, respectively, where

K_{i ∥}

and

K_{i ⊥}

stand for thermal conductivity coefficients parallel and perpendicular to the magnetic field,

K_{n 0} = K_{n} (T_{n 0}), K_{i 0} = K_{i ∥} (T_{i 0}) {cos}^{2} θ + K_{i ⊥} (T_{i 0}) {sin}^{2} θ .

The Alfvén speed c_A in the ion fluid is determined by the following expression:

c_{A}^{2} = \frac{B_{0}^{2}}{4 π ρ_{i 0}} .

We also introduce the low-frequency (⁠

c_{0 i}, c_{0 n}

⁠) and high-frequency (c_i, c_n) sound velocities in the ion and neutral fluids by analogy with Refs. 23, 24, 31 and 36,

\begin{matrix} c_{i}^{2} = γ \frac{P_{i 0}}{ρ_{i 0}}, c_{n}^{2} = γ \frac{P_{n 0}}{ρ_{n 0}}, \\ c_{0 i}^{2} = \frac{γ_{0 i}}{γ} c_{i}^{2}, c_{0 n}^{2} = \frac{γ_{0 n}}{γ} c_{n}^{2}, \end{matrix}

where

γ_{0 i} = 1 - \frac{ρ_{i 0} W_{i ρ}}{T_{i 0} W_{i T}}, γ_{0 n} = 1 - \frac{ρ_{n 0} W_{n ρ}}{T_{n 0} W_{n T}} .

(14)

Here,

P_{i 0}, P_{n 0}

are the stationary pressures, and

γ_{0 i}, γ_{0 n}

are the low-frequency adiabatic exponents in the ion and neutral fluids, respectively. The values of the latter quantities are completely determined by the form of generalized heat-loss functions,²²

\begin{matrix} W_{i T} = \frac{\partial W_{i}}{\partial T_{i}} (T_{i 0}, ρ_{i 0}), W_{i ρ} = \frac{\partial W_{i}}{\partial ρ_{i}} (T_{i 0}, ρ_{i 0}), \\ W_{n T} = \frac{\partial W_{n}}{\partial T_{n}} (T_{n 0}, ρ_{n 0}), W_{n ρ} = \frac{\partial W_{n}}{\partial ρ_{n}} (T_{n 0}, ρ_{n 0}) . \end{matrix}

The examples of calculating the low-frequency adiabatic exponent are given in Refs. 31 and 36 for the conditions of the solar corona, and in Refs. 23 and 24 for the interstellar gas.

III. DISPERSION RELATIONS

Linear equations describe the propagation of small perturbations along the z-axis in a partially ionized heat-releasing plasma in an external magnetic field. Next, we use a Fourier analysis in plane waves and assume that the perturbations are

\begin{matrix} {\tilde{B}}_{y, x} = A_{y, x} exp (- i ω t + ikz), {\tilde{ρ}}_{i} = A_{i} exp (- i ω t + ikz), \\ {\tilde{ρ}}_{n} = A_{n} exp (- i ω t + ikz), \end{matrix}

where ω is the frequency and k is the wavenumber. Substitution of these expressions in Eqs. allows us to obtain two independent dispersion relations as follows:

ω (\frac{ω^{2}}{k^{2}} - c_{A}^{2} {cos}^{2} θ) + i ν_{0} (\frac{ω^{2}}{k^{2}} - \frac{ρ_{i 0}}{ρ_{0}} c_{A}^{2} {cos}^{2} θ) = 0,

(15)

\begin{matrix} D_{i} (ω, k) D_{n} (ω, k) = i \frac{ω}{k^{2}} ν_{i n} [D_{i} (ω, k) \frac{ρ_{i 0}}{ρ_{n 0}} + (c_{A}^{2} {cos}^{2} θ \frac{ω + i ν_{n i}}{ω + i ν_{0}} - \frac{ω^{2}}{k^{2}}) D_{n} (ω, k)] \\ + c_{A}^{2} {cos}^{2} θ \frac{i ν_{i n}}{ω + i ν_{0}} ({\tilde{c}}_{i}^{2} - \frac{ω^{2}}{k^{2}}) [D_{n} (ω, k) - i ν_{n i} \frac{ω}{k^{2}}], \end{matrix}

(16)

where

\begin{matrix} D_{i} (ω, k) = {\tilde{c}}_{i}^{2} (c_{A}^{2} {cos}^{2} θ - \frac{ω^{2}}{k^{2}}) + \frac{ω^{4}}{k^{4}} - \frac{ω^{2}}{k^{2}} c_{A}^{2}, D_{n} (ω, k) = {\tilde{c}}_{n}^{2} - \frac{ω^{2}}{k^{2}}, \\ {\tilde{c}}_{i}^{2} = \frac{c_{i}^{2}}{γ} \frac{γ_{0 i} W_{i T} - i ω C_{P i} + \frac{K_{i 0}}{ρ_{i 0}} k^{2}}{W_{i T} - i ω C_{V i} + \frac{K_{i 0}}{ρ_{i 0}} k^{2}}, {\tilde{c}}_{n}^{2} = \frac{c_{n}^{2}}{γ} \frac{γ_{0 n} W_{n T} - i ω C_{P n} + \frac{K_{n 0}}{ρ_{n 0}} k^{2}}{W_{n T} - i ω C_{V n} + \frac{K_{n 0}}{ρ_{n 0}} k^{2}} . \end{matrix}

Particularly, in the case θ = 0, we get from Eqs. the dispersion relation for Alfvén waves linearly polarized along the x-axis propagating along the z-axis and the dispersion relation for the acoustic waves,

\begin{matrix} ω (\frac{ω^{2}}{k^{2}} - c_{A}^{2}) + i ν_{0} (\frac{ω^{2}}{k^{2}} - \frac{ρ_{i 0}}{ρ_{0}} c_{A}^{2}) = 0, \\ (\frac{ω^{2}}{k^{2}} - {\tilde{c}}_{i}^{2}) (\frac{ω^{2}}{k^{2}} - {\tilde{c}}_{n}^{2}) = i \frac{ω}{k^{2}} ν_{0} [\frac{ρ_{i 0}}{ρ_{0}} {\tilde{c}}_{i}^{2} + (1 - \frac{ρ_{i 0}}{ρ_{0}}) {\tilde{c}}_{n}^{2} - \frac{ω^{2}}{k^{2}}] . \end{matrix}

(17)

The dispersion relation (15) is equivalent to the dispersion relation for linear Alfvén waves in a partially ionized plasma, obtained and analyzed, for example, in Refs. 4 and 70. It also describes a purely imaginary solution, which corresponds to a purely damped, nonpropagating vortex mode whose damping rate is given by the ion-neutral collision frequency.^70,71 As in a fully ionized plasma, TM has no effect on Alfvén waves in the linear regime. Therefore, we do not consider Alfvén waves in the following. Note that, unlike linear Alfvén waves, the effect of TM on the propagation of nonlinear Alfvén waves (i.e., waves of larger amplitude) can be quite significant.^33,72,73

The dispersion equation describes nine compressible modes, namely, MA and thermal (entropy) modes, in the heat-releasing two-fluid plasma. Its notation is compact and makes it easy to pass to the known dispersion relations obtained for other conditions of the plasma. Indeed,

D_{i} (ω, k) = 0

is the dispersion relation for MA waves in the fully ionized (⁠

ρ_{n 0} = 0

⁠) heat-releasing plasma,⁷⁴ and

D_{n} (ω, k) = 0

is the dispersion relation for acoustic waves in the neutral (⁠

ρ_{i 0} = 0

⁠) heat-releasing gas.³² Moreover, if we neglect both thermal conductivity and the generalized heat-loss functions (W_i = 0, W_n = 0), then we have

{\tilde{c_{i}}}^{2} = c_{i}^{2}, {\tilde{c_{n}}}^{2} = c_{n}^{2}

(i.e., coincidence with the speeds of sound in the ion and neutral fluids), and Eq. becomes the dispersion relation for MA waves in the two-fluid plasma without heat release, studied in detail in Ref. 7,

\begin{matrix} D_{i 0} (ω, k) D_{n 0} (ω, k) = i \frac{ω}{k^{2}} ν_{i n} [D_{i 0} (ω, k) \frac{ρ_{i 0}}{ρ_{n 0}} + (c_{A}^{2} {cos}^{2} θ \frac{ω + i ν_{n i}}{ω + i (ν_{i n} + ν_{n i})} - \frac{ω^{2}}{k^{2}}) D_{n 0} (ω, k)] + c_{A}^{2} {cos}^{2} θ \frac{i ν_{i n}}{ω + i (ν_{i n} + ν_{n i})} (c_{i}^{2} - \frac{ω^{2}}{k^{2}}) [D_{n 0} (ω, k) - i ν_{n i} \frac{ω}{k^{2}}], \\ D_{i 0} (ω, k) = c_{i}^{2} (c_{A}^{2} {cos}^{2} θ - \frac{ω^{2}}{k^{2}}) + \frac{ω^{4}}{k^{4}} - \frac{ω^{2}}{k^{2}} c_{A}^{2}, D_{n 0} (ω, k) = c_{n}^{2} - \frac{ω^{2}}{k^{2}} . \end{matrix}

IV. ANALYTICAL APPROXIMATION OF PHASE SPEED AND DECREMENT OF MA WAVES

This section is devoted to the analysis of dispersion properties of MA waves on the basis of relation (16). We will be looking for the values of the complex frequency $ω = ω_{R} + i ω_{I}$ ⁠, at which Eq. (16) is satisfied for given real wave numbers k, as was done in Refs. 4 and 7 Here, ω_R and ω_I are the real and imaginary parts of the frequency. Therefore, for $ω_{I} < 0$ ⁠, the perturbations are damped in time with the exponential factor $exp (- | ω_{I} | t)$ ⁠. On the contrary, under conditions of thermal instability, imaginary part of frequency $ω_{I} > 0$ ⁠. Hence, the wave amplitude grows in time due to $exp (| ω_{I} | t)$ ⁠.

Below, we will consider the case of stable thermal (entropy) modes only. This is due to a significant complication of analytical solutions in these regions. In particular, there will appear frequency ranges in which MA waves become nonpropagating (this effect is observed even in a fully ionized single-fluid plasma³⁹), which requires additional investigation. We further refer to ω_I as the damping rate, assuming that its value is negative. If in the case of isentropic instability $ω_{I} > 0$ ⁠, it should be understood as the amplification rate of MA waves.

Among the nine modes described by Eq. (16), there are both complex (oscillatory) solutions and purely imaginary (evanescent) solutions with $ω_{R} = 0$ ⁠. Moreover, an even number of oscillatory modes and an odd number of purely imaginary modes vary, depending on the plasma parameters and the considered spectral region. Purely imaginary modes were described in Ref. 69. In this article, we will be interested in the features of the dispersion characteristics of oscillatory modes only.

We determine phase velocities of MA waves and study their damping due to TM, thermal conductivity, and ion-neutral collisions. Each of these processes is characterized by its characteristic time. Namely, TM is characterized by times τ_wi, τ_wn,^31,32,36,39 collisions of ions and neutrals have the characteristic time τ_av, and the effect of thermal conductivity is determined by

τ_{Kif, s}

⁠, τ_Kn,

\begin{matrix} τ_{w i} = C_{V i} / W_{i T}, τ_{w n} = C_{V n} / W_{n T}, \\ τ_{a v} = ν_{a v}^{- 1}, ν_{a v} = (ρ_{i 0} ν_{i n} + ρ_{n 0} ν_{n i}) / ρ_{0} = 2 η (1 - η) ν_{0}, \\ τ_{Kif, s} = K_{i 0} / c_{f, s}^{2} ρ_{i 0} C_{V i}, τ_{K n} = K_{n 0} / c_{n}^{2} ρ_{n 0} C_{V n} . \end{matrix}

Here, ν_av is the average collision frequency of ions and neutrals,⁷

η = ρ_{i 0} / ρ_{0}

is a degree of ionization, and c_f. c_s are the velocities of the fast (f) and slow (s) MA waves.

For definiteness, we will consider the timescale as $τ_{Kif, s}, τ_{K n} < τ_{a v} ≪ τ_{w i}, τ_{w n}$ ⁠, which is typical for the conditions of prominences and the chromosphere.^63,75 For convenience, we have placed these characteristic times on the timescale (Fig. 1). In this figure, for simplicity, we assume that the characteristic times associated with the same physical process in ion and neutral fluids are of the same order of magnitude and are therefore combined with each other. Then, one can distinguish four regions, in each of which the dispersion properties of MA waves differ significantly. We number these regions from I to IV in order of decreasing time of wave processes (increasing wave frequency).

FIG. 1.

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Timescale with relative position of characteristic times of thermal conductivity, ion-neutral collisions, and thermal misbalance typical for the conditions of prominences and the chromosphere. These times divide the timescale into four regions, in each of which the contribution of one of the studied processes (thermal conduction, ion-neutral collisions, and thermal misbalance) to the dispersive properties of the waves changes significantly. The sign “+” means that it is necessary to take into account the process in the considered region, while “−” means that the contribution of this process can be neglected in comparison with the others.

In each of these regions, the phase velocities and damping rates may have their own value, which varies when passing from one region to another. Such regions of constant values with relatively sharp transitions between them form a stair-like structure with several steps. A detailed analysis of these “steps” is given in Sec. V.

The contribution of each of the physical processes to the damping rate of MA waves can be unequal. In some regions, the influence of some processes can be neglected in the calculation of the damping rate compared to others without significant loss of accuracy. The signs (+) in Fig. 1 opposite each of the effects mean that its contribution to the damping rate was taken into account in obtaining analytical expressions, and the sign (−) means that the influence of this effect was neglected.

For oscillatory modes with a given wave number k, the values of $ω_{I}, ω_{R}$ can be found using a numerical solution of the dispersion equation (16). Examples of such a solution for the conditions of prominences and the chromosphere are given in Sec. V. The damping rate of disturbances in subranges with characteristic wave periods far from the boundaries of regions I-IV is approximately equal to the sum of the damping rates caused by each physical process separately. Then, step by step, we will obtain in Secs. IV A and IV B analytical expressions for the phase velocities and damping rates of waves in each of such subranges of regions I-IV, depending on their wave numbers. The results for the case θ = 0 are summarized in Table II. In Sec. V, we will compare analytical expressions with numerical results.

TABLE II.

Sound speeds and damping rates of acoustic modes in the case θ = 0.

Regiona	Sound speeds	$- Im {(ω)}^{(W)}$	$- Im {(ω)}^{(F)}$	$- Im {(ω)}^{(K)}$
I	$c_{0 M}$	$- \frac{k^{2}}{2} [η c_{i}^{2} τ_{w i} (\frac{γ_{0 i}}{γ} - 1) + (1 - η) c_{n}^{2} τ_{w n} (\frac{γ_{0 n}}{γ} - 1)]$	$\frac{k^{2} (c_{0 i}^{2} - c_{0 M}^{2}) (c_{0 M}^{2} - c_{0 n}^{2})}{2 ν_{0} c_{0 M}^{2}}$	⋯
II	c_M	$- \frac{1}{2 c_{M}^{2}} [\frac{η c_{i}^{2}}{τ_{w i}} (\frac{γ_{0 i}}{γ} - 1) + \frac{(1 - η) c_{n}^{2}}{τ_{w n}} (\frac{γ_{0 n}}{γ} - 1)]$	$\frac{k^{2} (c_{i}^{2} - c_{M}^{2}) (c_{M}^{2} - c_{n}^{2})}{2 ν_{0} c_{M}^{2}}$	$(γ - 1) \frac{k^{2}}{2 c_{M}^{2}} [η \frac{K_{i 0} c_{i}^{2}}{ρ_{i 0} C_{P i}} + (1 - η) \frac{K_{n 0} c_{n}^{2}}{ρ_{n 0} C_{P n}}]$
${III}_{i}$	c_i	⋯	$\frac{(1 - η) ν_{0}}{2}$	$\frac{k^{2} K_{i 0} (γ - 1)}{2 ρ_{i 0} C_{P i}}$
${III}_{n}$	c_n	⋯	$\frac{η ν_{0}}{2}$	$\frac{k^{2} K_{n 0} (γ - 1)}{2 ρ_{n 0} C_{P n}}$
${IV}_{i}$	c_Ti	⋯	$\frac{(1 - η) ν_{0}}{2}$	$\frac{c_{T i}^{2} ρ_{i 0} C_{V i} (γ - 1)}{2 K_{i 0}}$
${IV}_{n}$	c_Tn	⋯	$\frac{η ν_{0}}{2}$	$\frac{c_{T n}^{2} ρ_{n 0} C_{V n} (γ - 1)}{2 K_{n 0}}$

^a

The index i next to the region number denotes the ion component, and the index n denotes the neutral component.

When deriving approximate expressions, it is assumed that there is low attenuation over the wave period, i.e. (⁠ $| ω_{I} | ≪ | ω_{R} |$ ⁠). Damping rates due to TM, collisions of ions with neutrals, and thermal conductivity will be indicated by the superscripts W, F, and K, respectively. Also, in regions I-IV, damping rates will be indicated by subscripts 1–4, respectively.

A. Strongly coupled case

Strongly coupled limit implies that ion-electrons and neutrals behave as a single fluid. In this case, the frequency of the waves ω_R is much less than the average collision frequency of ions and neutrals ν_av,⁷

ω_{R} ≪ ν_{a v} .

Dispersion equation in the strongly coupled limit reduces to the following relation:

\frac{ω^{4}}{k^{4}} - (V_{A}^{2} + {\tilde{c}}_{M}^{2}) \frac{ω^{2}}{k^{2}} + V_{A}^{2} {\tilde{c}}_{M}^{2} {cos}^{2} θ = i \frac{ω}{ν_{0}} A (ω, k),

(18)

where

\begin{matrix} A (ω, k) = \frac{k^{2}}{ω^{2}} [(1 - η) c_{A}^{2} {cos}^{2} θ [({\tilde{c}}_{i}^{2} - \frac{ω^{2}}{k^{2}}) (D_{n} (ω, k) + η \frac{ω^{2}}{k^{2}}) \\ + (1 - η) D_{n} (ω, k) \frac{ω^{2}}{k^{2}}] - D_{i} (ω, k) D_{n} (ω, k)], \\ V_{A}^{2} = η c_{A}^{2} = B_{0}^{2} / 4 π ρ_{0}, {\tilde{c}}_{M}^{2} = η {\tilde{c}}_{i}^{2} + (1 - η) {\tilde{c}}_{n}^{2} . \end{matrix}

Here,

V_{A}^{2}, {\tilde{c}}_{M}^{2}

are the squares of the effective Alfvén and complex sound velocities, respectively.

Oscillatory solutions of dispersion equation (18) correspond to fast and slow MMA waves propagating in the single ion-neutral fluid.

In the strongly coupled limit, the characteristic time $ω_{R}^{- 1}$ of waves is much greater than the characteristic time of ion-neutral collisions τ_av, so that $τ_{a v} ≪ ω_{R}^{- 1}$ ⁠. Therefore, if the characteristic times of TM τ_wi and τ_wn are of the same order, then one can distinguish two frequency subranges with significantly different dispersion properties: low-frequency (⁠ $ω_{R} τ_{w i, n} ≪ \sqrt{γ_{0 i, n} / γ}$ ⁠) and high-frequency (⁠ $ω_{R} τ_{w i, n} ≫ \sqrt{γ_{0 i, n} / γ}$ ⁠) ones determined by the characteristic times of TM.³¹ These subranges belong to regions I and II from Fig. 1, respectively.

1. Region I

In this region at

ω_{R} τ_{w i, n} ≪ \sqrt{γ_{0 i, n} / γ}

⁠, the quantities

{\tilde{c}}_{i}^{2}, {\tilde{c}}_{n}^{2}

from Eq. are reduced to the following forms:

\begin{matrix} {\tilde{c}}_{i}^{2} = c_{0 i}^{2} + i c_{i}^{2} ω_{R} τ_{w i} (γ_{0 i} / γ - 1), \\ {\tilde{c}}_{n}^{2} = c_{0 n}^{2} + i c_{n}^{2} ω_{R} τ_{w n} (γ_{0 n} / γ - 1) . \end{matrix}

Using these expressions, one can find from dispersion equation that phase velocities of the fast (f) and slow (s) MMA waves take form .

In this limit, only TM and collisions of ions with neutrals contribute to the total decrement (increment). The contribution of thermal conductivity is neglected due to very small characteristic times $τ_{Kif, s}$ ⁠, τ_Kn compared to the period of waves.

The total damping rate (22) in subrange (⁠ $ω_{R} τ_{w i, n} ≪ \sqrt{γ_{0 i, n} / γ}$ ⁠) will be approximately equal to the sum of two damping rates associated with TM (20) and ion-neutral friction (21).

The imaginary part of frequency determining the damping (amplification) of MMA waves due to TM (20) is determined by equating the right side of Eq. (18) to zero and zeroing the thermal conductivity coefficients.

To find the damping rate of MMA waves due to ion-neutral collisions , we use relation , in which we set

{\tilde{c}}_{i}^{2} = c_{0 i}^{2}, {\tilde{c}}_{n}^{2} = c_{0 n}^{2}

⁠,

c_{0 f, s}^{2} = \frac{1}{2} [(V_{A}^{2} + c_{0 M}^{2}) \pm {(V_{A}^{4} + c_{0 M}^{4} - 2 V_{A}^{2} c_{0 M}^{2} cos 2 θ)}^{1 / 2}],

(19)

- Im {(ω)}_{1 f, s}^{(W)} = - k^{2} Σ_{1 f, s} [η c_{i}^{2} \frac{τ_{w i}}{4} (\frac{γ_{0 i}}{γ} - 1) + (1 - η) c_{n}^{2} \frac{τ_{w n}}{4} (\frac{γ_{0 n}}{γ} - 1)],

(20)

- Im {(ω)}_{1 f, s}^{(F)} = \frac{k^{2} | (1 - η) c_{A}^{2} {cos}^{2} θ [η c_{0 f, s}^{2} (c_{0 i}^{2} - c_{0 n}^{2}) + c_{0 i}^{2} (c_{0 n}^{2} - c_{0 f, s}^{2})] - (c_{0 f, s}^{2} - c_{0 i f}^{2}) (c_{0 f, s}^{2} - c_{0 i s}^{2}) (c_{0 n}^{2} - c_{0 f, s}^{2}) |}{2 ν_{0} c_{0 f, s}^{2} (c_{0 f}^{2} - c_{0 s}^{2})},

(21)

Im {(ω)}_{1 f, s} = Im {(ω)}_{1 f, s}^{(W)} + Im {(ω)}_{1 f, s}^{(F)},

(22)

where

Σ_{1 f, s} = 1 \pm \frac{c_{0 M}^{2} - V_{A}^{2} cos 2 θ}{{(V_{A}^{4} + c_{0 M}^{4} - 2 V_{A}^{2} c_{0 M}^{2} cos 2 θ)}^{1 / 2}},

(23)

\begin{matrix} c_{0 i f, s}^{2} = \frac{1}{2} [(c_{A}^{2} + c_{0 i}^{2}) \pm {(c_{A}^{4} + c_{0 i}^{4} - 2 c_{A}^{2} c_{0 i}^{2} cos 2 θ)}^{1 / 2}], \\ c_{0 M}^{2} = η c_{0 i}^{2} + (1 - η) c_{0 n}^{2} . \end{matrix}

(24)

Note that sign (+) in Eqs. (23) and (24) corresponds to the fast wave, and sign (−) refers to the slow wave.

It is also worth mentioning that in the case of $τ_{w i, n} (\frac{γ_{0 i, n}}{γ} - 1) > 0$ ⁠, the value of $Im {(ω)}_{1 f, s}^{(W)}$ is positive. It corresponds to the condition of isentropic instability, under which the amplification of MA waves is possible.³¹

2. Region II

Let us now consider the high-frequency subrange of region II, where $ω_{R} τ_{w i, n} ≫ \sqrt{γ_{0 i, n} / γ}$ and $ω_{R} τ_{a v} ≪ 1, ω_{R} τ_{Kif, s}, ω_{R} τ_{K n} ≪ 1$ ⁠.

In this subrange, one can find

{\tilde{c}}_{i}^{2} = c_{i}^{2} + i \frac{c_{i}^{2}}{ω_{R} τ_{w i}} (\frac{γ_{0 i}}{γ} - 1), {\tilde{c}}_{n}^{2} = c_{n}^{2} + i \frac{c_{n}^{2}}{ω_{R} τ_{w n}} (\frac{γ_{0 n}}{γ} - 1) .

Then, the velocities of the MMA waves calculated in this subrange on the basis of Eq. (18) take form (25).

In the considered high-frequency subrange, the damping rate due to the TM (26) is determined by zeroing the right-hand side of Eq. (18) and neglecting thermal conductivity.

The damping rate of the MMA waves caused by thermal conductivity (27) can be found from expression (18), equating its right-hand side to 0 and setting $W_{i T} = W_{n T} = 0$ ⁠.

To find the damping rate of MMA waves caused by ion-neutral collisions (28), we use relation (18), in which, in the considered subrange, we set ${\tilde{c}}_{i}^{2} = c_{i}^{2}, {\tilde{c}}_{n}^{2} = c_{n}^{2}$ ⁠.

The damping rate of MMA waves , as before, is approximately equal to the sum of contributions due to several physical processes, namely, TM , thermal conductivity , and ion-neutral friction ,

c_{f, s}^{2} = \frac{1}{2} [(V_{A}^{2} + c_{M}^{2}) \pm {(V_{A}^{4} + c_{M}^{4} - 2 V_{A}^{2} c_{M}^{2} cos 2 θ)}^{1 / 2}],

(25)

- Im {(ω)}_{2 f, s}^{(W)} = - \frac{Σ_{2 f, s}}{c_{f, s}^{2}} [\frac{η c_{i}^{2}}{4 τ_{w i}} (\frac{γ_{0 i}}{γ} - 1) + \frac{(1 - η) c_{n}^{2}}{4 τ_{w n}} (\frac{γ_{0 n}}{γ} - 1)],

(26)

- Im {(ω)}_{2 f, s}^{(K)} = (γ - 1) \frac{k^{2}}{4 c_{f, s}^{2}} [η \frac{K_{i 0} c_{i}^{2}}{ρ_{i 0} C_{P i}} + (1 - η) \frac{K_{n 0} c_{n}^{2}}{ρ_{n 0} C_{P n}}] Σ_{2 f, s},

(27)

- Im {(ω)}_{2 f, s}^{(F)} = \frac{k^{2} | (1 - η) c_{A}^{2} {cos}^{2} θ [η c_{f, s}^{2} (c_{i}^{2} - c_{n}^{2}) + c_{i}^{2} (c_{n}^{2} - c_{f, s}^{2})] - (c_{f, s}^{2} - c_{i f}^{2}) (c_{f, s}^{2} - c_{i s}^{2}) (c_{n}^{2} - c_{f, s}^{2}) |}{2 ν_{0} c_{f, s}^{2} (c_{f}^{2} - c_{s}^{2})},

(28)

Im {(ω)}_{2 f, s} = Im {(ω)}_{2 f, s}^{(W)} + Im {(ω)}_{2 f, s}^{(K)} + Im {(ω)}_{2 f, s}^{(F)},

(29)

where

Σ_{2 f, s} = 1 \pm \frac{c_{M}^{2} - V_{A}^{2} cos 2 θ}{{(V_{A}^{4} + c_{M}^{4} - 2 V_{A}^{2} c_{M}^{2} cos 2 θ)}^{1 / 2}},

(30)

c_{i f, s}^{2} = \frac{1}{2} [(c_{A}^{2} + c_{i}^{2}) \pm {(c_{A}^{4} + c_{i}^{4} - 2 c_{A}^{2} c_{i}^{2} cos 2 θ)}^{1 / 2}],

(31)

c_{M}^{2} = η c_{i}^{2} + (1 - η) c_{n}^{2} .

(32)

When θ = 0 (c_f = V_A, c_s = c_M is the sound speed in the high-frequency subrange of region II), we find from Eq. the damping rate due to the ion-neutral collisions,

- Im {(ω)}_{2 s}^{(F)} (θ = 0) = \frac{k^{2} (c_{i}^{2} - c_{M}^{2}) (c_{M}^{2} - c_{n}^{2})}{2 ν_{0} c_{M}^{2}} .

(33)

We emphasize that according to the one-fluid model,⁷⁵ this damping rate is equal to zero, which does not coincide with the result of Refs. 4 and 76 and our two-fluid consideration (33). This shows the limitations of using single-fluid models even in a strongly coupled limit.

B. Uncoupled case

Let us now consider the uncoupled case, where

ω_{R} ≫ ν_{a v}

⁠. In this case, the following dispersion relation can be obtained from Eq. :

D_{i} (ω, k) D_{n} (ω, k) = i \frac{ν_{0}}{ω} [η \frac{ω^{2}}{k^{2}} D_{i} (ω, k) + (1 - η) (- \frac{ω^{4}}{k^{4}} + c_{A}^{2} {\tilde{c}}_{i}^{2} {cos}^{2} θ) D_{n} (ω, k) - i η (1 - η) \frac{ω ν_{0}}{k^{2}} c_{A}^{2} {cos}^{2} θ ({\tilde{c}}_{i}^{2} - \frac{ω^{2}}{k^{2}})] .

(34)

In the considered range, the contribution of the TM to the dispersion and damping rate of MA waves can be neglected. Here, two subranges can also be distinguished with significantly different dispersion properties: low-frequency (⁠ $ω_{R} τ_{Kif, s}, ω_{R} τ_{K n} ≪ 1$ ⁠) and high-frequency (⁠ $ω_{R} τ_{Kif, s}, ω_{R} τ_{K n} ≫ 1$ ⁠) ones determined by the characteristic times of thermal conduction. Within each subrange, the damping rate can be represented as the sum of the damping rates due to thermal conductivity and ion-neutral collisions, as shown below. These subranges belong to regions III and IV from Fig. 1, respectively.

1. Region III

In the low-frequency subrange of region III, where the conditions $ω_{R} τ_{Kif, s}, ω_{R} τ_{K n} ≪ 1$ are satisfied, we have from Eq. (16) ${\tilde{c}}_{i}^{2} \approx c_{i}^{2}, {\tilde{c}}_{n}^{2} \approx c_{n}^{2}$ ⁠.

In a completely uncoupled case, the right-hand side of Eq. (34) is equal to 0: $D_{i 0} (ω, k) D_{n 0} (ω, k) = 0$ ⁠. Therefore, Eq. (34) will describe the independent propagation of MA waves in the ion component [ $D_{i 0} (ω, k) = 0$ ] and the acoustic wave in the neutral one [ $D_{n 0} (ω, k) = 0$ ].

Here, the velocities of the fast and slow MA waves will have the values c_if and c_is, while the speed of the acoustic wave will be equal to c_n,

c = {\begin{matrix} c_{i f} & fast MA wave, \\ c_{i s} & slow MA wave, \\ c_{n} & acoustic wave . \end{matrix}

(35)

We took into account the influence of weak coupling [non-zero right-hand side of Eq. (34)] on the damping rates of MA waves using the method of successive approximations. The analytical expression for damping rates will be the sum of damping rates due to ion-neutral collisions and to thermal conductivity. As we mentioned above, the TM does not play a significant role in regions III, IV (Fig. 1).

Damping rates for MA waves in the low-frequency subrange of region III due to ion-neutral collisions can be found from Eq. by setting

K_{i 0} = K_{n 0} = 0

in it. Then, we get

- Im {(ω)}_{3 f, s}^{(F)} = \frac{ν_{0} (1 - η) | c_{A}^{2} c_{i}^{2} {cos}^{2} θ - c_{i f, s}^{4} |}{2 c_{i f, s}^{2} (c_{i f}^{2} - c_{i s}^{2})} .

(36)

The damping rate for an acoustic wave in the case

ν_{a v} ≪ ω_{R} ≪ ν_{0}

due to ion-neutral collisions can be found from Eq. by setting

{\tilde{c}}_{i}^{2} = c_{i}^{2}, {\tilde{c}}_{n}^{2} = c_{n}^{2}

in it,

- Im {(ω)}_{3 n}^{(F)} = \frac{η ν_{0}}{2} \frac{[(c_{i f}^{2} - c_{n}^{2}) (c_{i s}^{2} - c_{n}^{2}) - (1 - η) c_{A}^{2} {cos}^{2} θ (c_{i}^{2} - c_{n}^{2})] [(c_{i f}^{2} - c_{n}^{2}) (c_{i s}^{2} - c_{n}^{2}) - (1 - η) c_{A}^{2} {cos}^{2} θ (c_{i}^{2} - η c_{n}^{2})]}{{[(c_{i f}^{2} - c_{n}^{2}) (c_{i s}^{2} - c_{n}^{2}) - (1 - η) c_{A}^{2} {cos}^{2} θ (c_{i}^{2} - η c_{n}^{2})]}^{2} + {(1 - η)}^{2} \frac{ν_{0}^{2}}{ω_{R}^{2}} c_{n}^{4} {(η c_{A}^{2} {cos}^{2} θ - c_{n}^{2})}^{2}} .

(37)

The damping rates of MA waves and acoustic waves due to thermal conductivity in the considered low-frequency subrange of III can be found from expression by setting its right-hand side to zero and utilizing the expressions for

{\tilde{c}}_{i}^{2}

and

{\tilde{c}}_{n}^{2}

at

ω_{R} τ_{Kif, s}, ω_{R} τ_{K n} ≪ 1

⁠,

\begin{matrix} {\tilde{c}}_{i}^{2} = c_{i}^{2} [1 - i (γ - 1) k^{2} K_{i 0} / (ω_{R} ρ_{i 0} C_{P i})], \\ {\tilde{c}}_{n}^{2} = c_{n}^{2} [1 - i (γ - 1) k^{2} K_{n 0} / (ω_{R} ρ_{n 0} C_{P n})] . \end{matrix}

Then,

- Im {(ω)}_{3 f, s}^{(K)} = \frac{k^{2} c_{i}^{2} K_{i 0} (γ - 1)}{4 c_{i f, s}^{2} ρ_{i 0} C_{P i}} Σ_{3 f, s},

(38)

- Im {(ω)}_{3 n}^{(K)} = \frac{k^{2} K_{n 0} (γ - 1)}{2 ρ_{n 0} C_{P n}},

(39)

where

Σ_{3 f, s} = 1 \pm \frac{c_{i}^{2} - c_{A}^{2} cos 2 θ}{{(c_{A}^{4} + c_{i}^{4} - 2 c_{A}^{2} c_{i}^{2} cos 2 θ)}^{1 / 2}} .

(40)

The damping rates of MA and acoustic waves can be found as follows:

\begin{matrix} Im {(ω)}_{3 f, s} = Im {(ω)}_{3 f, s}^{(F)} + Im {(ω)}_{3 f, s}^{(K)}, \\ Im {(ω)}_{3 n} = Im {(ω)}_{3 n}^{(F)} + Im {(ω)}_{3 n}^{(K)} . \end{matrix}

2. Region IV

In the high-frequency subrange of IV, when

ω_{R} τ_{Kif, s}, ω_{R} τ_{K n} ≫ 1

⁠, we have from Eq. ,

\begin{matrix} {\tilde{c}}_{i}^{2} = c_{T i}^{2} [1 - i (γ - 1) ω_{R} ρ_{i 0} C_{V i} / (k^{2} K_{i 0})], \\ {\tilde{c}}_{n}^{2} = c_{T n}^{2} [1 - i (γ - 1) ω_{R} ρ_{n 0} C_{V n} / (k^{2} K_{n 0})], \end{matrix}

where c_Ti and c_Tn are the isothermal speeds of sound in ion and neutral components (⁠

c_{T i}^{2} = c_{i}^{2} / γ, c_{T n}^{2} = c_{n}^{2} / γ

⁠).

In a completely uncoupled case, Eq. will describe the independent propagation of magnetoisothermal waves with velocities c_Tf and c_Ts [Eq. ] in the ion component, as well as the isothermal sound wave with speed c_Tn in the neutral one,

c_{T f, s}^{2} = \frac{1}{2} [(c_{A}^{2} + c_{T i}^{2}) \pm {(c_{A}^{4} + c_{T i}^{4} - 2 c_{A}^{2} c_{T i}^{2} cos 2 θ)}^{1 / 2}] .

(41)

Damping rates of these waves, due to their damping due to thermal conductivity, will have the following forms:

- Im {(ω)}_{4 f, s}^{(K)} = \frac{c_{T i}^{2} ρ_{i 0} C_{V i} (γ - 1)}{4 K_{i 0}} Σ_{4 f, s},

(42)

- Im {(ω)}_{4 n}^{(K)} = \frac{c_{T n}^{2} ρ_{n 0} C_{V n} (γ - 1)}{2 K_{n 0}},

(43)

where

Σ_{4 f, s} = 1 \pm \frac{c_{T i}^{2} - c_{A}^{2} cos 2 θ}{{(c_{A}^{4} + c_{T i}^{4} - 2 c_{A}^{2} c_{T i}^{2} cos 2 θ)}^{1 / 2}} .

(44)

Damping rates of MA waves due to ion-neutral collisions are determined by expression with the velocities c_i, c_if, and c_is replaced by isothermal velocities

c_{T i}

⁠, c_Tf, and c_Ts,

- Im {(ω)}_{4 f, s}^{(F)} = \frac{ν_{0} (1 - η) | c_{A}^{2} c_{T i}^{2} {cos}^{2} θ - c_{T f, s}^{4} |}{2 c_{T f, s}^{2} (c_{T f}^{2} - c_{T s}^{2})} .

(45)

The damping rate for acoustic waves due to ion-neutral collisions can be found from Eq. by setting

{\tilde{c}}_{i}^{2} = c_{T i}^{2}, {\tilde{c}}_{n}^{2} = c_{T n}^{2}

in it,

- Im {(ω)}_{4 n}^{(F)} = \frac{η ν_{0}}{2} \frac{{(c_{T f}^{2} - c_{T n}^{2})}^{2} {(c_{T s}^{2} - c_{T n}^{2})}^{2} + {(1 - η)}^{2} \frac{ν_{0}^{2}}{ω_{R}^{2}} c_{A}^{2} {cos}^{2} θ (c_{T i}^{2} - c_{T n}^{2}) (c_{T i}^{2} c_{A}^{2} {cos}^{2} θ - c_{T n}^{4})}{{(c_{T f}^{2} - c_{T n}^{2})}^{2} {(c_{T s}^{2} - c_{T n}^{2})}^{2} + {(1 - η)}^{2} \frac{ν_{0}^{2}}{ω_{R}^{2}} {(c_{T i}^{2} c_{A}^{2} {cos}^{2} θ - c_{T n}^{4})}^{2}} .

(46)

Damping rates of MA and acoustic waves can be found as

\begin{matrix} Im {(ω)}_{4 f, s} = Im {(ω)}_{4 f, s}^{(F)} + Im {(ω)}_{4 f, s}^{(K)}, \\ Im {(ω)}_{4 n} = Im {(ω)}_{4 n}^{(F)} + Im {(ω)}_{4 n}^{(K)} . \end{matrix}

V. APPLICATION TO THE SOLAR CHROMOSPHERE

In the general case, dispersion relation (16) obtained in the two-fluid approximation does not have simple analytical solutions, but it is easily solved numerically by substituting the parameters of specific astrophysical media. In this section, we will illustrate the characteristic dependencies of the phase velocity of oscillatory modes $V = ω_{R} / k$ and their damping rates $| ω_{I} |$ on the wavenumber k in a partially ionized heat-releasing plasma for different values of ionization degree η.

Figures 2–4 show the calculated dependencies of the phase velocities V(k) and the modulus of

ω_{I} (k)

of fast and slow MA waves propagating at an angle

θ = π / 4

to the magnetic field at degrees of ionization

η = 0.01, 0.1, 0.5

⁠, respectively. For definiteness, calculations were carried out for conditions typical of the solar chromosphere and prominences previously used in single-fluid calculations in Ref. 63:

B_{0} = 10 G, T_{i 0} = T_{n 0} = T_{0} = 8000 K, ρ_{0} = 5 \times 10^{- 14} g {cm}^{- 3}

⁠. The total frequency of ion-neutral and neutral-ion collisions

ν_{0} = 12.8 s^{- 1}

was calculated based on the expressions given in Ref. 4. The thermal conductivity coefficients were assumed to be as follows:

\begin{matrix} K_{n 0} = 2.5 \times 10^{3} T_{0}^{1 / 2} (erg s^{- 1} {cm}^{- 1} K^{- 1}), \\ K_{i ∥ 0} = 1.84 \times 10^{- 5} T_{0}^{5 / 2} ln Λ_{C} (erg s^{- 1} {cm}^{- 1} K^{- 1}), \\ K_{i ⊥ 0} = 0, \end{matrix}

where

ln Λ_{C}

is the Coulomb logarithm. The heat-loss functions W_i and W_n were taken in our calculations in the same form and with the same parameters that were used in the calculations in Ref. 63,

\begin{matrix} W_{i} = 1.76 \times 10^{- 6} (ρ_{i} T_{i}^{7.4} - ρ_{i 0} T_{0}^{7.4}) (erg g^{- 1} s^{- 1}), \\ W_{n} = 1.76 \times 10^{- 6} \frac{η}{(1 - η)} (ρ_{n} T_{n}^{7.4} - ρ_{n 0} T_{0}^{7.4}) (erg g^{- 1} s^{- 1}) . \end{matrix}

In this case, the expression for the total heat-loss function

η W_{i} + (1 - η) W_{n}

will coincide with the expression used in the calculations in Ref. 63.

FIG. 2.

View large Download slide

Calculated dependencies on k of the velocities (a) and damping rates (b) of waves in a plasma with an ionization degree of 0.01. Red curve is the fast MMA wave, dashed red curve is the fast MA wave and fast magnetoisothermal wave, black curve is the slow MMA wave, acoustic wave, and isothermal acoustic wave, the green curve is the slow MA wave and the slow magnetoisothermal wave, and blue and yellow curves are evanescent waves.

FIG. 3.

View large Download slide

Calculated dependences on k of the velocities (a) and damping rates (b) of waves in a plasma with an ionization degree of 0.1. Red curve is the fast MMA wave, dashed red curve is the fast MA wave and fast magnetoisothermal wave, black curve is the slow MMA wave, acoustic wave, and isothermal acoustic wave, green curves are the slow MA wave and slow magnetoisothermal wave, and blue wave is the evanescent wave.

FIG. 4.

View large Download slide

Calculated dependencies on k of the velocities (a) and damping rates (b) of waves in a plasma with an ionization degree of 0.5. Red curve is the fast MMA wave, fast MA wave, and fast magnetoisothermal wave, black curve is the slow MMA wave, slow MA wave, and the slow magnetoisothermal wave, and green curve is the acoustic wave and isothermal acoustic wave.

For hydrogen plasma,

γ = 5 / 3, c_{i}^{2} = 2 c_{n}^{2}

⁠. Taking into account the above-mentioned functions W_i and W_n, one can find that

γ_{0 i} = γ_{0 n} = 0.865 < γ

⁠, i.e., there is no isentropic instability and MA waves are not amplified at any values of k (⁠

ω_{I} < 0

⁠). Moreover, taking into account the above-mentioned values of the magnetic field and plasma parameters, we find

V_{A} = 1.2 \times 10^{7} cm s^{- 1} ≫ c_{i}, c_{n}

⁠. Therefore, from Eqs. and , we find

\begin{matrix} c_{0 s} \approx c_{0 M} / \sqrt{2}, c_{s} \approx c_{M} / \sqrt{2}, c_{i s} \approx c_{i} / \sqrt{2} = c_{n}, \\ c_{T s} \approx c_{T i} / \sqrt{2} = c_{T n}, c_{0 f} \approx c_{f} \approx V_{A}, c_{i f} \approx c_{T f} \approx c_{A} . \end{matrix}

A. Dependencies of the phase velocities on the wavenumber

The left panels of Figs. 2–4 clearly show the characteristic dispersive steps in the velocities of slow MA waves, which are caused by the presence of TM, electron-neutral collisions, and thermal conductivity that are described in detail in Sec. IV. In contrast to the slow MA wave, the dispersion of the fast MA wave due to the TM and thermal conductivity is not significant. This is due to the high speed V_A compared to the speed of sound.

Under the given conditions of $V_{A} ≫ c_{n}$ ⁠, we should also focus on three additional features of the obtained dependencies.

First, with increasing wavenumber k, there is a continuous transition of the slow MMA wave, propagating in a single ion-neutral fluid with the speed c_s, into a wave of a different type. At low degrees of ionization η (Figs. 2 and 3), the MMA wave turns into the acoustic wave, propagating in the neutral fluid at the speed c_n. At high degrees of ionization (Fig. 4), the MMA wave turns into the slow MA wave, propagating in the ion fluid with the speed c_is. For the conditions under study, the boundary degree of ionization is $η \approx 0.3$ ⁠.

Second, as the wavenumber k grows, the fast MMA, propagating in a single ion-neutral fluid with the speed $c_{f} \approx V_{A}$ ⁠, transforms into a fast MA wave, propagating in the ion fluid with the larger speed $c_{i f} \approx c_{A}$ ⁠. Moreover, this transition occurs either continuously (Fig. 4) or with the cutoff region (Fig. 2). That is, there are such parameters of the medium under which only slow MMA waves remain as oscillating waves in a certain frequency range. This feature was mentioned earlier in Ref. 7. In the case under consideration, such a cutoff interval exists at degrees of ionization less than $η \approx 0.1$ ⁠.

Third, at low degrees of ionization, there are additional evanescent wave solutions (blue and yellow curves in Figs. 2 and 3), with decrement $| ω_{I} | \leq | ω_{R} |$ ⁠. These evanescent solutions have also been discovered in the interstellar medium.^11,14

It is useful to emphasize that from the resulting dispersion equation (16), it can also be obtained that in the inverse limit of the velocity ratio, that is $V_{A} ≪ c_{n}$ ⁠, at low degrees of ionization (for example, in the interstellar medium), on the contrary, the fast MMA will continuously transform into an acoustic wave, and the slow MMA will be cut off at some wavevector region.

B. Dependencies of damping rates of oscillating solutions on the wavenumber

First, let us make general comments on the obtained damping rates (right panels of Figs. 2–4).

For the strongly coupled fluids in the low-frequency subrange relative to the characteristic heat release times (⁠ $τ_{w i}, τ_{w n}$ ⁠) (Sec. IV A, region I), damping rates of fast and slow MMA waves increase proportionally to k² and approximately equal to the sum (22) of two damping rates associated with the TM (20) and ion-neutral friction (21).

For the conditions under study, it follows from Eqs. (20) and (23) that damping rates due to the heat release for both types of waves $Im {(ω)}_{1 f}^{(W)} \approx Im {(ω)}_{1 s}^{(W)}$ ⁠, since $Σ_{1 f} \approx Σ_{1 s} \approx 1$ ⁠. At the same time, the damping rate of the slow MMA wave due to ion-neutral collisions will be much less than the damping rate due to heat release [ $- Im {(ω)}_{1 s}^{(F)} ≪ - Im {(ω)}_{1 s}^{(W)}$ ]. Therefore, it can be ignored here. On the contrary, the damping rates of the fast MMA wave due to ion-neutral collisions and TM turned out to be of the same order.

For the strongly coupled fluids in the high-frequency subrange relative to the characteristic heat release times (⁠ $τ_{w i}, τ_{w n}$ ⁠) (Sec. IV A, region II), damping rates of fast and slow MMA waves equal to the sum (29) of three damping rates associated with the TM (26), ion-neutral friction (28), and thermal conductivity (27). For the conditions under study, for a slow MMA wave, damping due to TM is still dominant, but with a damping rate independent of the wave vector (26) [this is the lower “step” in the slow MMA damping rate in Figs. 2–4(b)]. On the contrary, for the fast MMA wave, the contribution to the damping rate from the TM in this subrange is practically not expressed, since $- Im {(ω)}_{2 f}^{(W)} ≪ - Im {(ω)}_{2 f}^{(F)}$ here. Thus, the step in this area is not formed.

In the strongly coupled case, the important role of taking into account the TM in slow MMA wave damping is clearly seen from the comparison of Figs. 4(b) and 5(b). Here, disregard for the TM [Fig. 5(b)] leads to an underestimation of the slow MMA damping rate by orders of magnitude. The lower “step” in the damping rate of the slow MMA wave in Fig. 5(b) is absent in this case.

FIG. 5.

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Calculated dependencies on k of the velocities (a) and damping rates (b) of waves in a plasma with an ionization degree of 0.5 without heat release (⁠ $W_{i} = W_{n} = 0$ ⁠). Red curve is the fast MMA wave, fast MA wave, and fast magnetoisothermal wave, black curve is the slow MMA wave, slow MA wave, and the slow magnetoisothermal wave, and green curve is the acoustic wave and isothermal acoustic wave.

The next steps are formed in the uncoupled fluids (Sec. IV B). Let us discuss their origin. Only ion-neutral collisions and thermal conductivity make a significant contribution to the damping rate in this case. Here, instead of fast and slow MMA waves, fast and slow MA waves propagate in the electron-ion fluid, as well as an acoustic wave in the neutral fluid. Since the speed of slow waves is significantly less than the speed of fast waves, the condition of unbound fluids $ω_{R} ≫ ν_{a v}$ for slow waves is realized at larger wave numbers (⁠ $k ≫ ν_{a v} / c_{i s}$ ⁠) than for fast waves (⁠ $k ≫ ν_{a v} / c_{i f}$ ⁠).

The damping rates of fast and slow MA waves due to ion-neutral collisions at $c_{A} ≫ c_{i}$ and $θ = π / 4$ are equal to each other and are determined from expressions (36) and (45) in the following form: $- Im {(ω)}_{3 f, s}^{(F)} \approx (1 - η) ν_{0} / 2$ ⁠.

Damping rates of fast and slow MA waves due to dissipation processes associated with thermal conductivity are determined by expressions (38) and (42) in regions III and IV, respectively. They are approximately the same for slow and fast waves.

For the conditions under study, unlike MA waves, the damping rate of an acoustic wave due to ion-neutral collisions [Eqs. (37) and (46)] in regions III and IV is proportional to the degree of ionization η. The damping rates of acoustic waves due to dissipation processes associated with thermal conductivity are determined by expressions (39) and (43) in regions III and IV, respectively.

In view of the foregoing, it is possible to determine the origin of the steps in Figs. 2–4(b). Tables III–V provide a comparison of analytical and numerical calculations of decrements corresponding to the “steps” in Figs. 2–4(b), respectively.

TABLE III.

Comparison of numerical solution of the full dispersion relation (16) and analytical calculations of $Im (ω) (s^{- 1})$ at $η = 0.01$ ⁠.

Region	I (⁠ $k = 10^{- 11} {cm}^{- 1}$ ⁠)	II (⁠ $k = 10^{- 8} {cm}^{- 1}$ ⁠)	III (⁠ $k = 10^{- 5} {cm}^{- 1}$ ⁠)	IV (⁠ $k = 10^{- 1} {cm}^{- 1}$ ⁠)
Slow wave—analytical calc.	$- 2.713 \times 10^{- 8}$	$- 3.305 \times 10^{- 4}$	$- 7.211$	$- 11.53$
Slow wave—numerical sol.	$- 2.713 \times 10^{- 8}$	$- 3.307 \times 10^{- 4}$	$- 7.470$	$- 11.56$
Fast wave—analytical calc.	$- 8.843 \times 10^{- 8}$	$- 6.111 \times 10^{- 2}$	$- 6.360$	$- 11.53$
Fast wave—numerical sol.	$- 8.667 \times 10^{- 8}$	$- 6.170 \times 10^{- 2}$	$- 6.360$	$- 11.52$

TABLE IV.

Comparison of numerical solution of the full dispersion relation (16) and analytical calculations of $Im (ω) (s^{- 1})$ at $η = 0.1$ ⁠.

Region	I (⁠ $k = 10^{- 10} {cm}^{- 1}$ ⁠)	II (⁠ $k = 10^{- 7} {cm}^{- 1}$ ⁠)	III (⁠ $k = 10^{- 4} {cm}^{- 1}$ ⁠)	IV (⁠ $k = 10^{- 1} {cm}^{- 1}$ ⁠)
Slow wave—analytical calc.	$- 3.419 \times 10^{- 10}$	$- 2.938 \times 10^{- 3}$	$- 14.29$	$- 57.43$
Slow wave—numerical sol.	$- 3.419 \times 10^{- 10}$	$- 2.939 \times 10^{- 3}$	$- 14.04$	$- 57.42$
Fast wave—analytical calc.	$- 9.002 \times 10^{- 7}$	$- 0.5557$	$- 5.788$	$- 57.51$
Fast wave—numerical sol.	$- 8.649 \times 10^{- 7}$	$- 0.6070$	$- 5.788$	$- 57.50$

TABLE V.

Comparison of numerical solution of the full dispersion relation (16) and analytical calculations of $Im (ω) (s^{- 1})$ at $η = 0.5$ ⁠.

Region	I (⁠ $k = 10^{- 10} {cm}^{- 1}$ ⁠)	II (⁠ $k = 10^{- 7} {cm}^{- 1}$ ⁠)	III (⁠ $k = 10^{- 4} {cm}^{- 1}$ ⁠)	IV (⁠ $k = 10^{- 1} {cm}^{- 1}$ ⁠)
Slow wave—analytical calc.	$- 1.311 \times 10^{- 7}$	$- 4.236 \times 10^{- 3}$	$- 4.908$	$- 260.57$
Slow wave—numerical sol.	$- 1.311 \times 10^{- 7}$	$- 4.152 \times 10^{- 3}$	$- 4.877$	$- 260.56$
Fast wave—analytical calc.	$- 1.943 \times 10^{- 7}$	$- 0.061 85$	$- 3.218$	$- 262.72$
Fast wave—numerical sol.	$- 1.931 \times 10^{- 7}$	$- 0.061 83$	$- 3.218$	$- 262.58$

Figure 2(b) corresponds to the degree of ionization $η = 0.01$ ⁠. The black curve corresponds to the slow MMA wave, which passes continuously into an acoustic wave (in a weakly coupled region III) and then into the isothermal acoustic wave (region IV). The lower step is determined by Eq. (26): $- Im {(ω)}_{2 s}^{(W)} = 10^{- 4} s^{- 1}$ ⁠. The middle step of the black curve is mainly determined by the damping rate (37): $- Im {(ω)}_{3 n}^{(F)} \approx η ν_{0} / 4 = 0.032 s^{- 1}$ ⁠. The upper step on the black curve is determined by the value of the high-frequency thermal conductivity damping rate (43): $- Im {(ω)}_{4 n}^{(K)} = 6 s^{- 1}$ ⁠. The dashed red curve corresponds to the fast MA wave. The green curve corresponds to the slow MA wave. The lower step in the green curve and dashed red curve corresponds to damping rate (36) $- Im {(ω)}_{3 f, s}^{(F)} \approx (1 - η) ν_{0} / 2 = 6.4 s^{- 1}$ ⁠. The upper step in the green and dashed red curves $- Im {(ω)}_{4 f, s} = 11.6 s^{- 1}$ is formed by the sum of the damping rate (45) $- Im {(ω)}_{4 f, s}^{(F)} = 6.4 s^{- 1}$ and thermal conductivity damping rate (42) equal to $- Im {(ω)}_{4 f, s}^{(K)} = 5.2 s^{- 1}$ ⁠. Here, MA waves are already isothermal.

Figure 3(b) corresponds to the degree of ionization $η = 0.1$ ⁠. Similar to Fig. 2(b), the black curve corresponds to the slow MMA wave, which passes continuously into the acoustic wave (in a weakly coupled region III) and then into the isothermal acoustic wave (region IV). The dashed red curve corresponds to the fast MA wave (region III) and the fast isothermal MA wave (region IV). The green curve corresponds to the slow MA wave (region III) and the slow isothermal MA wave (region IV). The lower step of the black curve is determined by Eq. (26): $- Im {(ω)}_{2 s}^{(W)} = 10^{- 3} s^{- 1}$ ⁠. The middle step here is not formed before the transition of the acoustic wave to an isothermal acoustic wave. The upper step in the black curve is determined by the value $- Im {(ω)}_{4 n}^{(K)} = 5.5 s^{- 1}$ ⁠. The bottom step in the dashed red curve corresponds to $- Im {(ω)}_{3 f, s}^{(F)} = 5.9 s^{- 1}$ ⁠. The upper step $- Im {(ω)}_{4 f, s} \approx 57.5 s^{- 1}$ in the dashed red and green curves is formed mainly by the high-frequency thermal conductivity damping rate $- Im {(ω)}_{4 f, s}^{(K)} = 51 s^{- 1}$ ⁠.

Figure 4(b) corresponds to the degree of ionization $η = 0.5$ ⁠.

The black curve now corresponds to the slow MMA wave (regions I, II), which passes continuously into the slow MA wave (region III) and the slow isothermal MA wave (region IV). Red curve corresponds to the fast MMA wave (regions I, II), which passes continuously into the fast MA wave (region III) and the fast isothermal MA wave (region IV). Green curve corresponds to the acoustic wave (region III) and isothermal acoustic wave (region IV). The middle step in the black curve is not formed, since the damping associated with thermal conductivity, growing quadratically with the wavenumber in region III, is of the same order as the k-independent ion-neutral friction in region III. Note that this fact (the closeness of two damping rates with an increase in the degree of ionization and the disappearance of a step) was also taken into account in the single-fluid model of Ref. 63. The upper step in the black and red curves (region IV) is determined by the values $- Im {(ω)}_{4 s} = 260.6 s^{- 1}$ and $- Im {(ω)}_{4 f} = 262.6 s^{- 1}$ ⁠. The bottom step in the red curve corresponds to $- Im {(ω)}_{3 f, s}^{(F)} = 3.2 s^{- 1}$ ⁠. The upper step of the green curve (acoustic wave damping rate) is determined by the damping rate $- Im {(ω)}_{4 n} = 6.2 s^{- 1}$ ⁠.

Thus, the application of analytical formulas obtained in Sec. IV to explain the features of numerically obtained dispersion curves makes it possible to better understand the reasons for their appearance and to conduct a comparative analysis of the importance of taking into account dissipative processes associated with TM, ion-neutral collisions, and thermal conductivity.

VI. CONCLUSION

We have considered a partially ionized plasma in a magnetic field, the stationary homogeneous state of which is formed by the balance of losses and energy gains. Perturbations of plasma parameters in MHD modes lead to a violation of this balance, since losses and energy gains depend differently on temperature and density. A TM arises, leading to the damping of MHD modes or their amplification.

In the framework of the two-fluid model, we considered the properties of oscillatory modes, taking into account both the partial ionization of the plasma and the friction between the neutral and charged components associated with this, and the presence of a TM. As a result, linear equations (10)–(13) were obtained, relating small perturbations of the transverse component of the magnetic field vector induction to perturbations of the densities of the charged and neutral components of the plasma.

Using these equations, we found and presented in a compact form the dispersion relation (16), which describes nine oscillatory and non-propagating modes in a two-fluid plasma, taking into account sources of heat release and thermal conductivity in ion and neutral fluids. The number of propagating modes depends on the parameters of the medium, propagation angle, and frequency range. In the limiting cases of a fully ionized plasma or a purely neutral gas, this equation goes over to the known forms $D_{i} (ω, k) = 0$ or $D_{n} (ω, k) = 0$ ⁠, respectively. If there is no TM, it goes to the known expression (17).

Based on the obtained dispersion relation, we performed analytical calculations of the velocities and damping coefficients of MA waves in different frequency ranges depending on the degree of ionization. It was shown that the dispersion of the phase velocities of waves and their damping rates are formed by three processes (TM, ion-neutral collisions and thermal conductivity). Since the characteristic times of these processes are significantly different, frequency regions with the dominance of a particular process were identified. Each process determines a damping rate that varies from a frequency-quadratic low-frequency damping rate to a frequency-independent high-frequency damping rate.

As an example of using the obtained dispersion relation, we calculated the phase velocities and damping rates of MA and acoustic waves propagating at an angle of 45 degrees to the direction of magnetic field in a partially ionized heat-releasing plasma for conditions typical of chromospheric plasma and prominences.^63,75 Depending on the frequency domain, two slow MMA modes and two fast MMA modes (strongly coupled region) propagate toward each other, or, in a weakly coupled region, two slow MA waves, two fast MA waves propagating in the ion component, and also two acoustic waves propagating in a neutral gas. In the limit of high frequencies, they transform into isothermal MA and isothermal acoustic waves. At low degrees of ionization, there are intermediate frequency regions of non-propagation of fast waves, as well as frequency regions of the existence of rapidly decaying low-frequency slow waves.

Previously, similar calculations under conditions typical of the chromosphere were performed in Refs. 63 and 75 using the single fluid model. Under the conditions considered, the single fluid model can be a good approximation in the strongly coupled region, but in the weakly coupled region, it describes only slow MA waves. The fast MA wave in this model exists only in the region $ω_{R} < ω_{a v}$ ⁠, and the acoustic wave does not exist. We also note that with the wave propagation angle equal to 0, the single fluid model,⁷⁵ unlike the two-fluid model, does not describe the actual damping associated with ion-neutral collisions at all. This contradiction of single fluid and two-fluid models was previously addressed in Ref. 4.

The dispersion equation (16) obtained in compact analytical form can also be used for conditions other than chromospheric plasma and prominences. It allows one to analyze the properties of oscillatory and condensation modes that are realized in astrophysical plasma in a wide range of parameters, including the degree of ionization, plasma density, and magnetic field strength.

ACKNOWLEDGMENTS

This study was supported by the Russian Science Foundation (Project No. 23-22-10008), https://rscf.ru/project/23-22-10008/.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

N. E. Molevich: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). S. Yu. Pichugin: Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). D. S. Riashchikov: Formal analysis (equal); Investigation (equal); Software (equal); Visualization (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Magnetoacoustic waves in a partially ionized astrophysical plasma with the thermal misbalance: A two-fluid approach

I. INTRODUCTION

II. THE MODEL

III. DISPERSION RELATIONS

IV. ANALYTICAL APPROXIMATION OF PHASE SPEED AND DECREMENT OF MA WAVES

A. Strongly coupled case

1. Region I

2. Region II

B. Uncoupled case

1. Region III

2. Region IV

V. APPLICATION TO THE SOLAR CHROMOSPHERE

A. Dependencies of the phase velocities on the wavenumber

B. Dependencies of damping rates of oscillating solutions on the wavenumber

VI. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Magnetoacoustic waves in a partially ionized astrophysical plasma with the thermal misbalance: A two-fluid approach Open Access

I. INTRODUCTION

II. THE MODEL

III. DISPERSION RELATIONS

IV. ANALYTICAL APPROXIMATION OF PHASE SPEED AND DECREMENT OF MA WAVES

A. Strongly coupled case

1. Region I

2. Region II

B. Uncoupled case

1. Region III

2. Region IV

V. APPLICATION TO THE SOLAR CHROMOSPHERE

A. Dependencies of the phase velocities on the wavenumber

B. Dependencies of damping rates of oscillating solutions on the wavenumber

VI. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Magnetoacoustic waves in a partially ionized astrophysical plasma with the thermal misbalance: A two-fluid approach