Dispersion and damping of magnetohydrodynamic modes in radiative plasmas

Sadiq, Safeer; Mahmood, Shahzad

doi:10.1063/5.0169211

The presented work introduces a theoretical model for radiative magnetohydrodynamics (RMHD) in the equilibrium diffusion limit, focusing on the dynamics of radiation energy. For small amplitude waves, the basic set of dynamic equations is perturbed to derive the dispersion relation for three fundamental modes: fast, intermediate, and slow magnetosonic waves in RMHD plasmas. The study reveals that both fast and slow magnetosonic waves exhibit dispersion and damping in RMHD plasma. It is also revealed that mode conversion between fast and slow RMHD waves occurs at specific values of the wavenumber and propagation angle. The investigation extends to exploring the influence of various parameters characterizing radiative plasma, such as radiation pressure, plasma beta, and radiation diffusivity, on the dispersion and damping of magnetosonic modes (both fast and slow) in RMHD plasma. The findings are elucidated through numerical illustrations. The proposed model finds application in scenarios involving optically thick regions within stars, specifically in their inner atmosphere and interior region. In these regions, the transport of radiation adheres to equilibrium diffusion, and radiation pressure and energy density reach magnitudes comparable to thermal energy and pressure.

I. INTRODUCTION

Radiation energy density constitutes a significant portion of the total energy density in the environment and interior regions of stars. Additionally, radiative transfer of energy and momentum, as highlighted by Zel'dovich and Raizer,¹ emerges as the most efficient mechanism. At any temperature, matter emits energy in the form of electromagnetic waves with frequencies dependent on the radiating body's temperature.

The parameter $Π_{0} = p_{r} / p$ is introduced, signifying the relative importance of radiation energy density and pressure compared to the thermal energy and pressure of the fluid. O-type stars exemplify instances where radiation plays a crucial role in the stellar atmosphere, with $Π_{0}$ estimated to be on the order of 10 or even higher.² Wave motion in both the outer and inner regions of stars is influenced by radiation, coupled with the driving mechanism of stellar wind, which is strongly affected by radiation pressure.

In thermal equilibrium, the radiation energy density e_r is temperature-dependent, governed by Stefan's law: $e_{r} (T) = α_{R} T^{4}$ ⁠, where $α_{R} = 8 π^{5} k_{B}^{4} / 15 c^{3} h^{3}$ ⁠, with c denoting the speed of light, h representing Planck's constant, and k_B is the Boltzmann constant. The isotropy of radiation energy in thermal equilibrium requires that radiation energy density is proportional to radiation pressure, expressed as $p_{r} (T) = e_{r} (T) / 3$ ⁠. A local thermodynamic equilibrium (LTE) is established in optically thick materials, characterized by the condition $λ_{p} ≪ L$ ⁠, where λ_p is the photon's mean free path, and L is the characteristic length.^1–4

In the stellar interior, local thermodynamic equilibrium between plasma and radiation is achieved when an emitted photon is absorbed within the same local environment before encountering different conditions. Quantitatively, the mean free time of a photon, $t_{p} \sim λ_{p} / c$ ⁠, is significantly less than the fluid convection time, $t \sim L / U$ ⁠, satisfying the condition $λ_{p} U / L c ≪ 1$ ⁠, where U represents the fluid speed. Plasma and radiation in the stellar interior exist in equilibrium, forming a composite system with two main components: plasma and radiation. The transport of radiation energy primarily occurs through the diffusion process under these equilibrium conditions.²

In the diffusion limit, the complex opacity function is replaced by the mean Rosseland opacity $\bar{χ}$ ⁠, a single averaged quantity. Using the Rosseland opacity is convenient for studying linear and nonlinear dynamics in stellar interiors. In the equilibrium diffusion regime, matter and radiation are strongly coupled, establishing thermal equilibrium and rendering radiation pressure isotropic. The addition of radiation pressure and energy density to material pressure and energy density allows the calculation of total pressure and energy density for the composite system.²

The evolution of massive stars, resulting in type II supernovae, produces strong shocks. In the radiative precursor of the shock, the radiative energy density surpasses the energy density of matter by ten to a hundred orders of magnitude (mass density ρ_m on the order of $10^{- 3} g {cm}^{- 3}$ ⁠).⁵ The observation of a high terminal velocity of stellar wind (⁠ $10^{3} km s^{- 1}$ for O-type stars) can be explained by considering the high momentum input from the intense radiation field.⁶ Radiation pressure and energy density play a crucial role in the fast damping of kink oscillations of extreme ultraviolet (EUV) loops.⁷

Extensive studies on linear and nonlinear wave dynamics in the stellar interior and atmosphere are found in the literature.^8–17

Bogdan and Knölker¹⁸ obtained the linear dispersion relation in a radiating magnetized plasma by employing the Eddington, LTE, and gray atmosphere approximations. In their study, they considered both optically thick and Newtonian cooling (optically thin) approximations. Previously, Mihalas and Mihalas¹⁹ investigated linear waves in an unmagnetized, LTE, gray plasma that adheres to the Eddington approximation. Mikhalyaev et al.⁷ investigated the linear dispersion relation of magnetohydrodynamic (MHD) waves in the optically thin solar corona. Similar to other authors,^13,20–27 they considered the influence of radiation by incorporating heating and cooling functions. However, our presentation of the dynamic model differs from theirs. In this article, we present the radiative MHD (RMHD) model in the equilibrium diffusion limit, with a focus on understanding the dispersive and dissipative properties of MHD waves in a radiative plasma. We explore how these waves behave in different regimes of frequency/wave number. In our present work, we are exploring the impact of radiative pressure, energy density, and radiative heat transfer on wave characteristics within various wave number limits. Fast and slow waves undergo a shift in their characteristics at specific values of angle and wave number, highlighting an intriguing behavior of radiative plasma that warrants further study.

The manuscript is organized as follows: in Sec. II, model equations of RMHD are presented. In Sec. III, the dispersion relation for linear RMHD modes is derived. Section IV presents different RMHD modes, including slow, intermediate, and fast modes. Section V illustrates numerical plots. Section VI concludes our findings.

II. MODEL EQUATIONS

The dimensionless form of the equations for the conservation of mass, momentum, and total energy (in the equilibrium diffusion limit) for RMHD plasma is given below. The equation for the conservation of mass is

\frac{\partial ρ_{m}}{\partial t} + \nabla \cdot (ρ_{m} U) = 0.

(1)

From Eqs. and in Appendix A, the conservation equations for momentum and energy are written as follows:

\begin{matrix} \frac{\partial}{\partial t} (ρ_{m} U) + \nabla \cdot [ρ_{m} U U + \frac{ρ_{m} T}{Γ} \hat{I} \\ + \frac{B^{2}}{2} \frac{V_{A}^{2}}{V_{s}^{2}} \hat{I} - B B \frac{V_{A}^{2}}{V_{s}^{2}} + \frac{Π_{0} T^{4}}{3} \hat{I}] = 0, \end{matrix}

(2)

\begin{matrix} \frac{\partial}{\partial t} (\frac{1}{2} ρ_{m} U^{2} + ρ_{m} e + \frac{V_{A}^{2}}{V_{s}^{2}} \frac{B^{2}}{2} + Π_{0} T^{4}) \\ + \nabla \cdot [\frac{1}{2} ρ_{m} U^{2} U + h U + \frac{V_{A}^{2}}{V_{s}^{2}} (E \times B) + \frac{4 Π_{0}}{3} T^{4} U] \\ = Π_{0} \nabla \cdot [\nabla (κ T^{4})] . \end{matrix}

(3)

The dimensionless equation of state (EOS) is given by

p = \frac{ρ_{m} T}{Γ} .

(4)

Using the above equation of state, the specific internal energy e and enthalpy h can be written as follows:

e = c_{v} T = \frac{T}{Γ (Γ - 1)},

(5)

h = c_{p} T = \frac{T}{(Γ - 1)} .

(6)

The induced electric field E (dimensionless) is given by Ohm's law for ideal MHD

E = - U \times B .

(7)

Taking the curl of the above equation and using Faraday's law of induction, we obtain

\frac{\partial B}{\partial t} = \nabla \times U \times B .

(8)

In the dimensionless form, the spatial coordinates are normalized as

r \to r / L

and time as

t \to V_{s} t / L

(where L is the characteristic length-scale of the system). The normalized velocity (U), density (ρ_m), magnetic field (B), and temperature (T) are given by

U \to \frac{U}{V_{s}}, κ \to \frac{κ}{L V_{s}}, ρ_{m} \to \frac{ρ_{m}}{ρ_{m 0}}, B \to \frac{B}{B_{0}}, T \to \frac{T}{T_{0}},

where

ρ_{m 0}

⁠, T_0, and B₀ are the reference values for density, temperature, and magnetic field, respectively. The radiation parameter

Π_{0}

is given as follows:

Π_{0} = α_{R} T_{0}^{4} / ρ_{m 0} V_{s}^{2}

⁠, where setting

Π_{0} = 0

results in the non-radiative MHD set of dynamic equations.²⁸ The Alfvén velocity is given by

V_{A} = (B_{0} / \sqrt{μ_{0} ρ_{m 0}}) {\hat{V}}_{A}

⁠, where

{\hat{V}}_{A}

is a unit vector in the direction of

B_{0}

and

V_{s} = {(Γ p_{0} / ρ_{m 0})}^{1 / 2}

is the acoustic speed. Here, p₀ is the pressure defined as

p_{0} = R ρ_{m 0} T_{0}

⁠, where R is the specific gas constant. The plasma beta

β = 2 μ_{0} p_{0} / B_{0}^{2}

⁠, which is defined as the ratio of thermal to magnetic pressure, is related to

V_{A}^{2} / V_{s}^{2}

and Γ by the following expression:

V_{A}^{2} / V_{s}^{2} = 1 / (Γ β)

⁠. While our set of dynamic equations are presented in a different form, it can be equivalently transformed into a set of dynamic equations introduced by Pai.²⁹

III. LINEAR WAVES DISPERSION RELATION

Linearization and Fourier analysis of Eqs. (with the geometry of the wavevector and magnetic field as illustrated in Fig. 1) gives the following dispersion relation for linear radiative MHD wave modes (see Appendix B for detailed calculations):

\begin{matrix} (ω^{2} - k^{2} V_{A}^{2} {cos}^{2} θ) [ω^{4} - \\ ω^{2} (S (k) V_{s}^{2} + V_{A}^{2}) k^{2} + S (k) V_{s}^{2} V_{A}^{2} k^{4} {cos}^{2} θ] = 0. \end{matrix}

(9)

FIG. 1.

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Geometry of MHD waves in a radiative plasma.

In the following limit

Π_{0} \to 0

(non-radiative plasma),

S (k) \to 1

⁠, and Eq. gives

\begin{matrix} (ω^{2} - k^{2} V_{A}^{2} {cos}^{2} θ) [ω^{4} - \\ ω^{2} (V_{s}^{2} + V_{A}^{2}) k^{2} + V_{s}^{2} V_{A}^{2} k^{4} {cos}^{2} θ] = 0 \end{matrix},

(10)

which is the dispersion relation of MHD waves in the non-radiative plasmas.²⁸ The linear dispersion relation in radiatively cooled plasma was derived by Nakariakov et al.,²⁰ with the objective of investigating the damping of MHD waves in an optically thin solar corona. They utilized the optically thin plasma model, incorporating radiation cooling and heating terms in the energy equation. Their study revealed that MHD waves exhibit dispersion and damping in radiative plasma. Notably, the investigation by Nakariakov et al.²⁰ did not account for the effects of radiation pressure and energy on wave damping and dispersion, given the optically thin nature of the solar corona. The dispersion relation, as described by Eq. (34) in Nakariakov et al.,²⁰ shares similarities with our own dispersion relation represented by Eq. in the sense that both exhibit dispersion and damping. Kumar et al.²¹ studied the linear dispersion relation for a radiatively cooled and heated plasma with thermal conduction. They derived the dispersion relation and solved it numerically to investigate the time damping of linear waves. Zavershinskii et al.²² also derived the linear wave dispersion relation for slow waves in an optically thin solar corona, considering the cooling and heating imbalance. The radiation effect in the above theoretical models^20–22 manifests primarily as a cooling term in the energy equation, attributed to the low density and high-temperature conditions of the solar corona. Consequently, the influence of radiation pressure, energy density, and diffusivity can be safely ignored without compromising the validity of the theoretical model. However, in our equilibrium diffusion model, we take into account densities and temperatures such that plasma radiation must be treated as an isotropic black body field, and radiation diffusion becomes the predominant mechanism for heat conduction.

Frequency

ω (k)

in Eq. is complex, with

Re [ω (k)]

representing the dispersive effects and

Im [ω (k)]

represents the damping in the system. Both dispersion and damping in the radiative plasma result from non-zero values of the radiative parameter

Π_{0}

and diffusivity κ, as dissipation is conveyed through radiation diffusion. Specifically,

Im [ω (k)]

vanishes only when either

Π_{0} = 0

or κ = 0. Setting κ = 0 (or

Π_{0} = 0

⁠) in Eq. results in the phase speed

Re [ω (k)] / k

in Eq. becoming independent of the wave number k. As a consequence, the resulting linear wave will propagate without dispersion. By comparing Eqs. and , it becomes evident that the complex number

\sqrt{S (k)} V_{s}

represents the modified acoustic speed in the presence of radiation pressure and energy density. In the limit

Π_{0} \to 0, A (k)

approaches

Γ - 1

according to Eq. . Utilizing this limiting value of

A (k)

in Eq. , we obtain the following limit:

\sqrt{S (k)} \to 1

⁠, resulting in the recovery of the sound speed for a non-radiative plasma, i.e.,

\sqrt{S (k)} V_{s} \to V_{s}

⁠. When strong coupling between radiation and plasma is present (⁠

4 Π_{0} / 3 ≫ 1 / Γ / (Γ - 1)

⁠), and in the long-wavelength limit where

k ≪ k_{l}

⁠, with k_l defined as

k_{l} = \frac{1}{2} \sqrt{\frac{1}{Π_{0} κ} (\frac{1}{Γ (Γ - 1)} + \frac{4 Π_{0}}{3})},

(11)

one obtains

S (k) \to 4 Π_{0} / 3

from Eq. , allowing the expression of the acoustic speed in this limit in terms of the radiation pressure p_r as

Re [\sqrt{S (k)} V_{s}] \to \sqrt{Γ_{R} p_{r} / ρ_{m 0}},

(12)

where

Γ_{R} = 4 / 3

is defined as the adiabatic index for the photon gas, and the above Eq. employs the definition

Π_{0} = p_{r} / p_{0}

⁠. In the aforementioned limit of wave number k, the wave propagation occurs with the restoring force provided by the radiation pressure of the photon gas, while the required inertia is mainly contributed by the plasma, particularly the ion component.

IV. MODIFIED MHD MODES IN A RADIATIVE PLASMA

The dimensionless form of RMHD wave dispersion relation is a cubic equation in

ω^{2}

⁠, and it gives the following three pairs (left and right propagating linear waves) of roots, i.e.:

\frac{ω_{A}^{2}}{k^{2}} = \frac{V_{A}^{2}}{V_{s}^{2}} {cos}^{2} θ,

(13)

\begin{matrix} \frac{ω_{s}^{2}}{k^{2}} = \frac{1}{2} (S (k) + \frac{V_{A}^{2}}{V_{s}^{2}}) - \frac{1}{2} \sqrt{{(S (k) - \frac{V_{A}^{2}}{V_{s}^{2}})}^{2} + 4 S (k) \frac{V_{A}^{2}}{V_{s}^{2}} {sin}^{2} θ} \end{matrix},

(14)

and

\begin{matrix} \frac{ω_{f}^{2}}{k^{2}} = \frac{1}{2} (S (k) + \frac{V_{A}^{2}}{V_{s}^{2}}) + \frac{1}{2} \sqrt{{(S (k) - \frac{V_{A}^{2}}{V_{s}^{2}})}^{2} + 4 S (k) \frac{V_{A}^{2}}{V_{s}^{2}} {sin}^{2} θ .} \end{matrix}

(15)

The three roots given by Eqs. are the so-called Alfvén, slow, and fast MHD modes in a radiative plasma. These modes are discussed below.

A. Alfvén mode

Equation provides the linear transverse Alfvén mode. It can be observed that the linear Alfvén mode is not influenced by radiation pressure. For transverse Alfvén waves (or shear Alfvén wave), the perturbations in magnetic field and velocity field are perpendicular to the ambient magnetic field

B_{0}

⁠, i.e.,

k \cdot B_{1} = k \cdot U_{1} = B_{0} \cdot B_{1} = 0.

(16)

Therefore, Alfvén waves do not compress the fluid, and as a result, fluid pressure and radiation pressure have no effect on the linear Alfvén mode. Since the dispersion relation for Alfvén waves in our case is the same as in the case of a non-radiative plasma, we will not further discuss Alfvén waves.

B. Slow magnetosonic RMHD mode

The second root ω_s of Eq. gives the slow magnetosonic mode in the RMHD plasma. Consider the following two cases in Eq. , (i)

Re [S (k)] > V_{A}^{2} / V_{s}^{2}

and (ii)

Re [S (k)] < V_{A}^{2} / V_{s}^{2}

⁠. In the case where

Re [S (k)] > V_{A}^{2} / V_{s}^{2}

⁠, the slow magnetosonic mode equation transforms into the Alfvén mode at

θ = 0 °

and disappears at

θ = 90 °

⁠. However, when

Re [S (k)] < V_{A}^{2} / V_{s}^{2}

⁠, the slow magnetosonic mode transforms into an acoustic mode at

θ = 0 °

and diminishes at

θ = 90 °

⁠. In the short-wavelength limit, i.e.,

k \to \infty

⁠, Eq. yields the frequency

ω_{s \infty}

as follows:

\begin{matrix} \frac{ω_{s \infty}^{2}}{k_{\infty}^{2}} = \frac{1}{2} (\frac{1}{Γ} + \frac{V_{A}^{2}}{V_{s}^{2}}) - \frac{1}{2} \sqrt{{(\frac{1}{Γ} - \frac{V_{A}^{2}}{V_{s}^{2}})}^{2} + \frac{4}{Γ} \frac{V_{A}^{2}}{V_{s}^{2}} {sin}^{2} θ} \end{matrix},

(17)

where

Re [S (k_{\infty})] \to 1 / Γ

and

Im [S (k_{\infty})] \to 0

in the short-wavelength limit. For a magnetized plasma with

β < 1

⁠, such that

V_{A}^{2} / V_{s}^{2} > 1 / Γ

if the wave propagation angle

θ \to 0 °

⁠, Eq. reduces to

\frac{ω_{s \infty}}{k_{\infty}} = \frac{1}{Γ} .

(18)

In another case

θ \to 90 °

⁠, Eq. gives

\frac{ω_{s \infty}}{k_{\infty}} = 0.

(19)

Hence, in the short-wavelength limit, the slow wave transforms into the acoustic mode in the parallel direction, while in the perpendicular direction, the slow mode disappears.

In the case of an intensely radiating plasma, where the radiation energy density

e_{r 0} = 4 Π_{0} / 3

significantly exceeds the plasma energy density

e_{0} = 1 / [Γ (Γ - 1)]

⁠, i.e.,

e_{r 0} ≫ e_{0}

and under the conditions of long wavelength, i.e.,

k ≪ k_{l}

⁠, the limiting value of the slow mode

ω_{s 0}

is given by

\begin{matrix} \frac{ω_{s 0}^{2}}{k_{0}^{2}} = \frac{1}{2} (\frac{4 Π_{0}}{3} + \frac{V_{A}^{2}}{V_{s}^{2}}) - \frac{1}{2} \sqrt{{(\frac{4 Π_{0}}{3} - \frac{V_{A}^{2}}{V_{s}^{2}})}^{2} + \frac{16 Π_{0}}{3} \frac{V_{A}^{2}}{V_{s}^{2}} {sin}^{2} θ .} \end{matrix}

(20)

C. Fast magnetosonic RMHD mode

The third root ω_f [Eq. ] represents the fast magnetosonic mode in the RMHD plasma. In cases where

Re [S (k)] > V_{A}^{2} / V_{s}^{2}

⁠, the fast magnetosonic mode equation transforms into the acoustic mode when propagation parallel to the magnetic field (⁠

θ = 0 °

⁠) and to magnetoacoustic in the perpendicular direction (⁠

θ = 90 °

⁠). However, for

Re [S (k)] < V_{A}^{2} / V_{s}^{2}

⁠, the fast magnetosonic mode converts to the Alfvén mode when propagating parallel to the magnetic field (⁠

θ = 0 °

⁠) and to magnetoacoustic at

θ = 90 °

⁠. In the short-wavelength limit, Eq. yields the fast mode frequency

ω_{f \infty}

as follows:

\begin{matrix} \frac{ω_{f \infty}^{2}}{k_{\infty}^{2}} = \frac{1}{2} (\frac{1}{Γ} + \frac{V_{A}^{2}}{V_{s}^{2}}) + \frac{1}{2} \sqrt{{(\frac{1}{Γ} - \frac{V_{A}^{2}}{V_{s}^{2}})}^{2} + \frac{4}{Γ} \frac{V_{A}^{2}}{V_{s}^{2}} {sin}^{2} θ} \end{matrix},

(21)

where

Re [S (k_{\infty})] \to 1 / Γ

and

Im [S (k_{\infty})] \to 0 °

in the short-wavelength limit. In the case where

V_{A}^{2} / V_{s}^{2} > 1 / Γ

⁠, in the parallel direction (⁠

θ = 0 °

⁠), Eq. reduces to

\frac{ω_{f \infty}}{k_{\infty}} = \frac{V_{A}}{V_{s}} .

(22)

However, in the perpendicular direction (⁠

θ = 90 °

⁠), Eq. yields

\frac{ω_{f \infty}}{k_{\infty}} = \sqrt{\frac{1}{Γ} + \frac{V_{A}^{2}}{V_{s}^{2}}},

(23)

which represents the magnetoacoustic speed in a radiative plasma in the short-wavelength limit.

In the case of an intensely radiating plasma, i.e.,

e_{r 0} ≫ e_{0}

and under the long-wavelength limit, i.e.,

k ≪ k_{l}

⁠, the fast wave frequency

ω_{f 0}

is given by

\begin{matrix} \frac{ω_{f 0}^{2}}{k_{0}^{2}} = \frac{1}{2} (\frac{4 Π_{0}}{3} + \frac{V_{A}^{2}}{V_{s}^{2}}) + \frac{1}{2} \sqrt{{(\frac{4 Π_{0}}{3} - \frac{V_{A}^{2}}{V_{s}^{2}})}^{2} + \frac{16 Π_{0}}{3} \frac{V_{A}^{2}}{V_{s}^{2}} {sin}^{2} θ .} \end{matrix}

(24)

Zavershinskii et al.²² have discussed the low-frequency and high-frequency limits of the linear dispersion relation, along with the corresponding phase speeds, for slow waves in radiative MHD plasma, considering the radiative cooling/heating imbalance. They observed that both low and high-frequency MHD waves do not exhibit dispersion. However, their phase speed expressions differ from those in Eqs. and because they have modeled an optically thin plasma. In their model, only radiation loss or gain terms in the energy equation are considered, and radiation does not affect the fluid dynamics through radiation pressure and energy density terms. In contrast, our model involves an optically thick plasma, where heat is conducted through radiation diffusion. The fluid pressure (energy density) is influenced by radiation pressure (energy density), as the photon mean free path does not exceed the gradients produced by the perturbations in the fluid. Moreover, Morton et al.²⁷ have investigated the influence of time-dependent temperature on the damping of MHD waves in a radiatively cooled plasma.

V. RESULTS AND DISCUSSION

In this section, we present numerical plots of the dispersion relation given by Eqs. (14) and (15).

The real and imaginary parts of the dispersion relation are illustrated in Fig. 2. In Fig. 2, the damping factor, denoted as

Im [ω_{f}]

⁠, is observed to be zero when the wave number (k) is at its minimum value of

k \to 0

⁠. As the wave number increases, specifically as k approaches the limit of infinity (⁠

k \to \infty

⁠),

Im [ω_{f}]

experiences a progressive increase. The shaded gray region within the figure demarcates the range of normalized wave numbers (k) wherein the damping factor attains its maximum value. This shaded region serves to highlight the specific range of wavelengths for which the damping effect is most pronounced in the system under consideration. The quality factor

Q = - \frac{| Re [ω_{f}] |}{2 Im [ω_{f}]},

(25)

which quantifies the damping of waves, decreases significantly when we approach

k \to 0

or

\infty

and attains its maximum value within the shaded area in Fig. 2. A similar behavior has been observed by Carbonell et al.⁸ in the context of MHD waves in radiative plasma within the solar corona. The real part of ω_f in Fig. 2 exhibits a linear behavior near

k ≃ 0

until it starts to curve. Within the plateau observed in the curve of

Re [ω_{f}]

in Fig. 2, the group velocity

\partial Re [ω_{s}] / \partial k = 0

⁠, indicating non-propagating oscillations that are subject to damping. In the long-wavelength limit, fluid elements are coupled through radiation pressure, acting as the restoring force that determines the frequency of any compressive mode. However, as the wavelength decreases, steeper temperature gradients form in the system, promoting dissipative processes due to the radiation diffusion term. As a result, a decrease in wavelength is not directly proportional (as observed in both long and short-wavelength limits) to increase in wave frequency.

FIG. 2.

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Example showing the damping of fast waves [real and imaginary parts of Eq. (15)] in a radiative plasma. Wave damping is significant in the shaded area, and it gets weaker in the short-wavelength limit although it is not equal to zero. Slow waves [real and imaginary parts of Eq. (14)] also qualitatively show a similar behavior (where κ = 1, $β = 0.45, θ = 60 °$ and $Π_{0} = 20$ ⁠).

Within a radiative plasma, both the fast and slow magnetosonic modes display dispersion, contrary to the non-radiative MHD fast and slow waves that propagate without dispersion in the linear wave limit. The emergence of dispersive effects in fast and slow magnetosonic modes is attributed to the influence of the radiation diffusion term, which diminish in the absence of diffusion (κ = 0) as depicted in Fig. 3. The large phase speed (a steeper straight line) of linear waves near $k ≃ 0$ is attributed to the prominent influence of radiation pressure and energy density in the long-wavelength limit. In the case of short wavelengths, however, both fast and slow magnetosonic linear modes exhibit no dispersion, represented by a straight-line plot. This (short wavelength) behavior is analogous to MHD waves in a non-radiative plasma, as depicted in Fig. 3. This result is consistent with the conclusion drawn by Al-Ghafri²⁴ for slow waves in the solar corona, considering radiative cooling and heating. Waves having both very short and long wavelengths asymptotically approach a non-dispersive mode of propagation.

FIG. 3.

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The dispersion relation of fast waves [real part of Eq. (15)] is depicted. The solid straight (thin) line corresponds to κ = 0, intersecting with the solid (thicker) curve in the short-wavelength limit. The broken (dashed) straight line represents the short-wavelength limit (⁠ $k \to \infty$ ⁠) [Eq. (21)], coinciding with the large diffusivity limit (⁠ $κ \to \infty$ ⁠) for Eq. (15). The thick solid curve is plotted for $Π_{0} = 20$ ⁠, $θ = 60 °$ and κ = 1.

The broken line in Fig. 3 shows the asymptotic value of

ω_{f \infty}

as given by Eq. . In the long-wavelength limit (⁠

k \to 0

⁠), the phase speed of fast wave approaches

ω_{f 0} / k

as given by Eq. . The slow wave demonstrates identical behavior. The solid thick line is plotted for

Π_{0} = 20

and κ = 1. In the long-wavelength limit depicted in Fig. 3, the case with κ = 0 (solid thin line) intersects with the solid red curve. The dashed curve, representing the short-wavelength limit, corresponds to the case of large diffusivity, specifically when κ approaches infinity (⁠

κ \to \infty

⁠). For higher values of

κ

⁠, diffusive effects become significant for long wavelengths (⁠

k \to 0

as

κ \to \infty

⁠) only, as depicted in Fig. 3. However, the limit

κ \to \infty

is contradictory with the equilibrium diffusion. The contradiction arises because, on the scale of wavelength λ, the plasma will behave as an optically thick medium if the following condition

λ > λ_{p}

is satisfied, where

λ_{p} = 3 V_{s} κ / c

(26)

(in normalized units). For wavenumber

k ≪ 2 π / λ_{p}

⁠, the wave disturbance is optically thick, whereas, in the opposite limit, i.e.,

k ≫ 2 π / λ_{p}

⁠, it is optically thin. Since

λ_{p} \to \infty

when

κ \to \infty

⁠, the radiation diffusion theory is not properly applicable (or flux-limited).²

A. Changes in radiation parameter $Π_{0}$

In Fig. 4, the linear dispersion relation of fast MHD waves in a radiative plasma is shown for two different values of $Π_{0}$ ⁠. In Fig. 4(a), the linear dispersion relation of fast MHD waves in a radiative plasma is depicted for two distinct values of $Π_{0}$ ⁠. The broken curve corresponds to (⁠ $Π_{0} = 20$ ⁠), and the solid curve corresponds to (⁠ $Π_{0} = 10$ ⁠). As the value of $Π_{0}$ increases, the dispersion in the linear mode also increases as depicted in Fig. 4(a). The phase speed of the fast wave in the long wavelength (⁠ $k \to 0$ ⁠) limit increases with an increasing value of $Π_{0}$ ⁠. In Fig. 4(b), the damping rate [imaginary part of Eq. (15)] of fast MHD waves in a radiative plasma is drawn for two distinct values of $Π_{0}$ ⁠. As the value of $Π_{0}$ increases, damping in the linear mode also increases as depicted in Fig. 4(b). The non-radiative case can be recovered from relations [(14) and (15)] by letting $Π_{0}$ approach zero, resulting in the disappearance of damping and dispersion in that particular scenario. In Fig. 4(c), the linear dispersion relation of slow MHD waves in a radiative plasma is presented for two distinct values of $Π_{0}$ ⁠.

FIG. 4.

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Dispersion relations and damping rates for fast [real and imaginary parts of Eq. (15)] and slow [real and imaginary parts of Eq. (14)] waves are shown for distinct values of radiation parameter $Π_{0}$ ⁠, for solid curves (⁠ $Π_{0} = 10$ ⁠) and for dashed curves (⁠ $Π_{0} = 20$ ⁠): (a) dispersion of fast waves, (b) damping rate of fast waves, (c) dispersion relation of slow waves, and (d) damping rate of slow waves (remaining parameters are fixed as follows: $β = 0.45$ ⁠, κ = 1 and $θ = 60 °$ ⁠).

Similar to the case of fast waves, the dispersion in slow mode enhances with an increase in the value of $Π_{0}$ as depicted Fig. 4(c). In Fig. 4(d), the damping of slow MHD waves in a radiative plasma (⁠ $Im [ω_{s}]$ ⁠) is illustrated for two different values of $Π_{0}$ ⁠. The damping in the linear mode increases with an increasing value of $Π_{0}$ ⁠, as illustrated in Fig. 4(d). This behavior is similar to the fast MHD waves in the short-wavelength limit, as demonstrated in Fig. 4(b). Contrary to the fast wave mode depicted in Fig. 4(b), in the long-wavelength limit (⁠ $k \to 0$ ⁠), the damping rate of the slow wave decreases with increasing $Π_{0}$ ⁠, as indicated in Fig. 4(d).

Contour plots of the real parts of fast [Eq. (15)] and slow [Eq. (14)] MHD waves are presented in the $k - Π_{0}$ space in Figs. 5(a) and 5(b), respectively. The corresponding imaginary parts of the fast and slow waves are depicted in Figs. 5(c) and 5(d). These contour plots illustrate the variation of both the real and imaginary parts of slow and fast frequencies across a range of changes in k and $Π_{0}$ ⁠.

B. Changes in diffusivity parameter κ

The linear dispersion relation of fast waves in a radiative plasma is depicted in Fig. 6 for two different values of κ. The dispersion in the fast MHD wave decreases with an increase in diffusivity κ. The phase speed of the fast wave in the long-wavelength limit (⁠ $k \to 0$ ⁠) remains unaffected by changes in the value of κ. However, decreasing κ allows radiation pressure and energy density to extend their influence on relatively shorter wavelengths. Since $κ \propto λ_{p}$ ⁠, a reduction in diffusivity leads to a decrease in the mean free path of photons, confining radiation within a smaller volume. Conversely, an increase in κ enables photons to escape without interaction within a volume of size $\sim λ^{3} = {(2 π / k)}^{3}$ in the fluid. Therefore, linear waves with wavelengths much smaller than λ_p remain unaffected by the process of radiation diffusion.

FIG. 5.

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Contour plots of the real and imaginary parts of the fast [Eq. (15)] [(a) and (c), first column] and slow [Eq. (14)] [(b) and (d), second column] RMHD modes are shown in $k - Π_{0}$ space (with $θ = 60 °, β = 0.45$ and κ = 1).

FIG. 6.

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Dispersion relations and damping rates for fast [real and imaginary parts of Eq. (15)] and slow [real and imaginary parts of Eq. (14)] waves are shown for different values of diffusivity κ, where solid curves (⁠ $κ = 0.5$ ⁠) and dashed curves (κ = 1) (a) dispersion of fast waves, (b) damping rate of fast waves, (c) dispersion relation of slow waves, and (d) damping rate of slow waves (where $β = 0.45, θ = 60 °$ ⁠, and $Π_{0} = 20$ ⁠).

In Fig. 6(b), the damping rate [imaginary part of Eq. (15)] of fast magnetosonic modes in RMHD plasma is presented for two different values of diffusivity κ. Increasing κ raises the minimum value of $Im [ω_{f}]$ ⁠, but the minima in $Im [ω_{f}]$ itself shifts to longer wavelengths as illustrated in Fig. 6(b).

In Figs. 6(c) and 6(d), the linear dispersion relation of the slow magnetosonic mode in a radiating MHD plasma and its damping rate are shown for two different values of diffusivity κ (broken curve for κ = 1 and solid curve for $κ = 0.5$ ⁠). Slow and fast magnetosonic waves exhibit qualitatively similar behavior concerning variations in κ.

Contour plots of the real part of fast [Eq. (15)] and slow [Eq. (14)] waves are shown in $k - κ$ space in Figs. 7(a) and 7(b), respectively. The corresponding imaginary parts of fast and slow MHD waves are shown in Figs. 7(c) and 7(d). These contour plots illustrate the variation of real and imaginary parts of both fast and slow waves over a range of changes in k and κ.

FIG. 7.

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Contour plots of the real and imaginary parts of the fast [Eq. (15)] [(a) and (c), first column] and slow [Eq. (14)] [(b) and (d), second column] RMHD modes are shown in $k - κ$ space [with $θ = 60 °, β = 0.45$ and $Π_{0} = 20$ ⁠).

C. Changes in plasma beta β

In Fig. 8(a), the linear dispersion relation of fast MHD waves in a radiative plasma is shown for two different values of plasma beta β (broken curve for $β = 0.45$ and solid curve for $β = 0.1$ ⁠). The phase speed of the linear fast wave decreases with an increase in plasma beta. In Fig. 8(b), the damping rate of fast MHD waves in a radiative plasma is shown for two different values of β. Damping of fast waves increases with an increase in plasma beta. In Fig. 8(c), the linear dispersion relation of slow MHD waves in a radiative plasma is drawn for two different values of plasma beta. The phase speed of linear slow waves decreases with an increased value of plasma beta. In Fig. 8(d), the damping rate of slow MHD waves in a radiative plasma is shown for two different values of plasma beta. In contrast to fast waves, the damping of slow waves decreases with an increased value of plasma beta, both in the long and short-wavelength limits.

FIG. 8.

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Dispersion and damping of fast [real and imaginary parts of Eq. (15)] and slow [real and imaginary parts of Eq. (14)] waves are shown for different values of plasma beta β, where solid curves are plotted for (⁠ $β = 0.1$ ⁠) and dashed curves are plotted for (⁠ $β = 0.45$ ⁠): (a) dispersion of fast waves, (b) damping of fast waves, (c) dispersion of slow waves, and (d) damping rate of slow waves (where κ = 1, $θ = 60 °$ ⁠, and $Π_{0} = 20$ ⁠).

D. Changes in direction angle θ

In Figs. 9(a)–9(h), the real and imaginary parts of Eqs. (14) (dashed) and (15) (solid) as a function of θ are drawn at distinct values of the wavenumber (k = 1.8, 3.8, 4.8, and 7.8).

FIG. 9.

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Variation of the real and imaginary parts of ω_f [Eq. (15), solid curve] and ω_s [Eq. (14), broken curve] with angle θ are shown. (a) Real and (b) imaginary parts of slow and fast wave frequencies are shown at k = 1.8, (c) real and (d) imaginary parts of slow and fast wave frequencies are shown at k = 3.8, (e) real and (f) imaginary parts of slow and fast wave frequencies are shown at k = 4.8, and (g) real and (h) imaginary parts of slow and fast wave frequencies are shown at k = 7.8 (where, κ = 1, $β = 0.45$ ⁠, and $Π_{0} = 20$ ⁠).

In Fig. 9, solid curves represent the fast mode and the dotted curves represent the slow mode.

For further illustration,

Re [S (k)]

is drawn in Fig. 10. The critical value of the wave number {at which

Re [S (k)] = V_{A}^{2} / V_{s}^{2}

} in Fig. 10 is located near k = 2.48. Below this critical value of k,

Re [S (k)] > V_{A}^{2} / V_{s}^{2}

⁠, and the slow mode will vanish at

θ = 90 °

⁠; however, at

θ = 0 °

⁠, it will transform to the linear Alfvén mode, which is a non-compressive MHD mode and we observe zero damping in the slow mode at

θ = 0 °

and 90° [as shown in Fig. 9(b)]. For k > 2.48,

Re [S (k)] < V_{A}^{2} / V_{s}^{2}

in Fig. 10 and the slow mode will still disappear at

θ = 90 °

⁠, but at

θ = 0 °

⁠, it will convert to the linear acoustic mode, which is a compressive MHD mode with non-zero damping at

θ = 0 °

[shown in Fig. 9(d)]. In Figs. 9(c)–9(f), ω_f and ω_s discontinuously convert into each other near

θ = 30 °

⁠. The reason for this specific behavior can be explained as follows: In the given range of values, Eqs. and can together be written in a compact form as follows:

\frac{ω^{2}}{k^{2}} = \frac{1}{2} (S (k) + \frac{V_{A}^{2}}{V_{s}^{2}}) + \frac{1}{2} \sqrt{F},

(27)

where F is defined as

F = {(S (k) - \frac{V_{A}^{2}}{V_{s}^{2}})}^{2} + 4 S (k) \frac{V_{A}^{2}}{V_{s}^{2}} {sin}^{2} θ .

(28)

FIG. 10.

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Real part S(k) [Eq. (B15)] is shown as a function of k (solid curve). The dotted line represents the value of $V_{A}^{2} / V_{S}^{2}$ ⁠, and the dashed line represents $1 / Γ$ (where $Π_{0} = 20$ ⁠, κ = 1, and $β = 0.45$ ⁠).

In the polar form, the complex number F can be written as follows:

F = r e^{i φ} .

(29)

Frequency ω in becomes double-valued due to the square root term

\sqrt{F}

(which has two ± values with the phase difference

e^{i π}, \sqrt{F}

is not the analytic function at

r e^{i φ} = 0

⁠, see Appendix C). Negative real axis is the branch cut for F. Whenever we cross the branch cut,

\sqrt{F}

will get multiplied by

e^{i π}

(and mode crossing will happen; the fast mode will transform into the slow mode and vice versa). With these two distinct ± values for

\sqrt{F}

⁠, Eq. represents two different modes: fast and slow. In Fig. 11, we depict the dependence of

φ, Re [\sqrt{F}]

⁠, and

Im [\sqrt{F}]

on angle θ. It is observed that as

φ

crosses the negative real axis (near

θ = 30 deg

⁠), the changes in

Re [\sqrt{F}]

and

Im [\sqrt{F}]

are not smooth (⁠

Re [\sqrt{F}]

is not differentiable and

Im [\sqrt{F}]

is discontinuous). For larger k [such as in Figs. 9(g) and 9(h), where the curves are continuous functions of angle θ],

φ

does not cross the branch cut, and hence there is no jump in the real and imaginary parts of fast and slow waves. Mode conversion between slow and fast waves occurs at any surface where there is an equipartition between acoustic and Alfvénic energies, represented by the condition V_s = V_A.³⁰ Mode conversion is also discussed by Kumar and Kumar³¹ in a stratified environment.

FIG. 11.

Real (dotted, blue) and imaginary (solid, red) parts of F [Eq. (27)] and the argument φ (dashed, black) of F are drawn as functions of θ (remaining parameters are fixed at the following values: k = 2.8 , Π 0 = 20, κ = 1, and β = 0.45).

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Real (dotted, blue) and imaginary (solid, red) parts of $\sqrt{F}$ [Eq. (27)] and the argument $φ$ (dashed, black) of F are drawn as functions of θ (remaining parameters are fixed at the following values: $k = 2.8, Π_{0} = 20$ ⁠, κ = 1, and $β = 0.45$ ⁠).

VI. CONCLUSION

In conclusion, we have formulated a radiative model for a magnetized fluid in a manner that facilitates easier integration. Both slow and fast magnetosonic waves in RMHD plasma show dispersion and damping. Dispersion in fast and slow magnetosonic modes is essential for forming solitary structures in a radiative plasma. Dispersion and damping in fast and slow magnetosonic waves are caused by the non-zero value of the radiative parameter (⁠ $Π_{0} \neq 0$ ⁠). Dispersion and damping of fast and slow magnetosonic modes in a radiative plasma are affected by the changes in diffusivity κ, radiation parameter $Π_{0}$ ⁠, and plasma beta β. For strong coupling between radiation and plasma (⁠ $Π_{0} ≫ 1$ ⁠), both fast and slow magnetosonic modes sustain a long wavelength radiative mode in which radiative pressure plays the dominant role. When both radiative parameter $Π_{0}$ and diffusivity κ are non-zero, stationary points {where the wave stops propagating and the group velocity is zero (⁠ $\partial Re [ω_{s}] / \partial k = 0$ ⁠)} exist in the dispersion relation for fast and slow magnetosonic modes. These oscillations are damped for both fast and slow magnetosonic modes. Mode crossing between fast and slow waves occurs at specific values of the physical parameters. This phenomenon refers to a point where the characteristics of fast and slow waves are interchanged with each other. Our findings are applicable to wave propagation in the inner atmosphere and interior region of stars where radiation pressure and radiation diffusion are important.²

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Safeer Sadiq: Conceptualization (lead); Formal analysis (lead); Writing – original draft (lead). Shahzad Mahmood: Conceptualization (supporting); Formal analysis (supporting); Writing – review & editing (lead).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

APPENDIX A: CALCULATIONS OF MODEL EQUATIONS

In the equilibrium diffusion model, the dimensional form of the radiation stress-energy tensor,

{\hat{T}}_{R}

⁠, in the co-moving frame of reference is defined as follows:²^,

{\hat{T}}_{R} = (\begin{matrix} α_{R} T^{4} & S_{R} \\ S_{R} & \frac{α_{R} T^{4}}{3} \hat{I} \end{matrix}) .

(A1)

Also, tensor

{\hat{T}}_{R}

is re-defined in the inertial frame of reference as follows:

{\hat{T}}_{R} = (\begin{matrix} α_{R} T^{4} & S_{R} + \frac{4 α_{R} T^{4}}{3} U \\ S_{R} + \frac{4 α_{R} T^{4}}{3} U & \frac{α_{R} T^{4}}{3} \hat{I} \end{matrix}),

(A2)

where

\hat{I}

is the identity matrix and

(α_{R} T^{4} / 3) \hat{I}

gives the stress part of the tensor

{\hat{T}}_{R}

and the following term

4 α_{R} T^{4} U / 3

is equal to the sum of work done on plasma by radiation pressure and convection of radiation energy

α_{R} T^{4}

out of the controlled volume element. In the local thermodynamical equilibrium of radiation with plasma, the conductive energy flux

S_{R}

is proportional to

\nabla T^{4}

⁠, i.e.,

S_{R} = - κ \nabla (α_{R} T^{4}),

(A3)

where the proportionality constant

κ = (1 / 3) c λ_{p}

is the diffusivity of radiation field (in the dimensional form) and it is defined in the diffusion approximation with the assumption that the gradient of radiation energy density is small compared to the mean free length of the photon λ_p.^32,33 Diffusivity κ is also defined as

κ = (1 / 3 χ_{t}) c

⁠, where

χ_{t} = χ_{a} + χ_{s}

is the total opacity and

χ_{a} (χ_{s})

is the absorption (scattering) opacity.³⁴ The viscous terms in the radiation stress-energy tensor

{\hat{T}}_{R}

defined in Eq. are ignored because they are much smaller than the isotropic components of pressure.

In order to model the radiative MHD fluid in the equilibrium diffusion limit, we simply add the stress tensors of plasma

{\hat{T}}_{P}

⁠, the stress tensor of the magnetic field

{\hat{T}}_{H}

⁠, and the stress tensor of the radiation field

{\hat{T}}_{R}

⁠, which gives the total stress-energy tensor

{\hat{T}}_{S}

as follows:

\begin{matrix} {\hat{T}}_{S} = {\hat{T}}_{P} + {\hat{T}}_{H} + {\hat{T}}_{R} . \\ = (\begin{matrix} ε & F_{ε} \\ F_{ε} & {\hat{T}}_{stress} \end{matrix}), \end{matrix}

(A4)

where ε is the total energy density, which includes magnetic, kinetic, and radiative energy density,

F_{ε}

is the flux of total energy ε and

{\hat{T}}_{stress}

is the total stress. The magnetic stress-energy tensor is given as follows:²⁸

{\hat{T}}_{H} = (\begin{matrix} \frac{B^{2}}{2 μ_{0}} & S_{Poy} \\ S_{Poy} & - \frac{BB}{μ_{0}} + \frac{B^{2} \hat{I}}{2 μ_{0}} \end{matrix}),

(A5)

where the Poynting vector

S_{Poy} = E \times B / μ_{0}

gives the flux of the electromagnetic energy. The plasma stress-energy tensor

{\hat{T}}_{P}

is given as follows:

{\hat{T}}_{P} = (\begin{matrix} \frac{1}{2} ρ_{m} U^{2} + ρ_{m} e & (\frac{1}{2} ρ_{m} U^{2} + h) U \\ (\frac{1}{2} ρ_{m} U^{2} + h) U & p \hat{I} \end{matrix}) .

(A6)

Here, ρ_m is the fluid mass density, e is the specific internal energy, U is the fluid velocity, and h is enthalpy. Hence,

{\hat{T}}_{stress}

defined in Eq. is

{\hat{T}}_{stress} = - \frac{BB}{μ_{0}} + (p + \frac{B^{2}}{2 μ_{0}} + \frac{α_{R} T^{4}}{3}) \hat{I} .

(A7)

The conservation of momentum can be written as follows:

\frac{\partial (ρ_{m} U)}{\partial t} + \nabla \cdot {\hat{T}}_{stress} = 0

(A8)

and the conservation of energy is

\frac{\partial ε}{\partial t} + \nabla \cdot F_{ε} = 0,

(A9)

where

ε = ρ_{m} U^{2} / 2 + ρ_{m} e + B^{2} / 2 μ_{0} + α_{R} T^{4}

is the total energy density. The enthalpy h of the plasma is defined in terms of the specific internal energy e and the plasma pressure p as follows:

h = ρ_{m} e + p .

(A10)

APPENDIX B: LINEARIZATION OF EQUATIONS AND DERIVATION OF THE DISPERSION RELATION

Consider an ideal MHD (where resistive term is ignored in comparison with the conductive term) homogeneous fluid. Next, we consider the propagation of small amplitude waves in the said medium (the geometry of linear waves in a radiative plasma is shown in Fig. 1). The relevant equations of continuity , momentum , plasma energy equation , and Faraday's law are linearized by assuming the first order dynamic variables as perturbation to the time (space) independent zeroth order quantities,

U = U_{1}, B = 1 + B_{1}, p = 1 / Γ + p_{1}, ρ_{m} = 1 + ρ_{m 1}

and

T = 1 + T_{1}

⁠. The perturbed set of dynamic equations is obtained as follows:

\frac{\partial ρ_{m 1}}{\partial t} + \nabla \cdot U_{1} = 0,

(B1)

\begin{matrix} \frac{\partial U_{1}}{\partial t} + \nabla \cdot [(\frac{T_{1}}{Γ} + \frac{ρ_{m 1}}{Γ} + \frac{4 Π_{0} T_{1}}{3}) \hat{I} + {\hat{V}}_{A} . B_{1} \frac{V_{A}^{2}}{V_{s}^{2}} \hat{I} \\ - 2 B_{1} {\hat{V}}_{A} \frac{V_{A}^{2}}{V_{s}^{2}}] = 0, \end{matrix}

(B2)

\begin{matrix} \frac{\partial}{\partial t} {(\frac{1}{Γ (Γ - 1)} + 4 Π_{0}) T_{1} + \frac{V_{A}^{2}}{V_{s}^{2}} ({\hat{V}}_{A} . B_{1})} \\ + \nabla . {(\frac{1}{1 - Γ} + \frac{4 Π_{0}}{3}) U_{1} + \frac{V_{A}^{2}}{V_{s}^{2}} (- U_{1} \times {\hat{V}}_{A}) \times {\hat{V}}_{A}} \\ = \nabla . [\nabla (κ T_{1})], \end{matrix}

(B3)

\frac{\partial B_{1}}{\partial t} = \nabla \times U_{1} \times {\hat{V}}_{A},

(B4)

E_{1} = - U_{1} \times {\hat{V}}_{A} .

(B5)

Here,

E_{1}

in Eq. is replaced by

(- U_{1} \times {\hat{V}}_{A})

by virtue of relation .

Fourier analysis of differential equations is carried out by replacing the differential operators

\partial / \partial t

and

\nabla

by

- i ω

and

i k

(⁠

i = \sqrt{- 1}

⁠), respectively, since the perturbed quantities are of the form

\sim e^{i (k \cdot r - ω t)}

⁠. After replacing the operators, the resulting set of algebraic equations is written as follows:

- i ω ρ_{m 1} + i k \cdot U_{1} = 0,

(B6)

\begin{matrix} - i ω U_{1} + i k \cdot [{(\frac{1}{Γ} + \frac{4 Π_{0}}{3}) T_{1} + \frac{ρ_{m 1}}{Γ}} \hat{I} \\ + {\hat{V}}_{A} \cdot B_{1} \frac{V_{A}^{2}}{V_{s}^{2}} \hat{I} - {\hat{V}}_{A} B_{1} \frac{V_{A}^{2}}{V_{s}^{2}}] = 0, \end{matrix}

(B7)

\begin{matrix} - i ω {(\frac{1}{Γ (Γ - 1)} + \frac{4 Π_{0}}{3}) T_{1} + \frac{ρ_{m 1}}{Γ (Γ - 1)} + \frac{V_{A}^{2}}{V_{s}^{2}} ({\hat{V}}_{A} \cdot B_{1})} \\ + i k \cdot {(\frac{1}{(Γ - 1)} + \frac{4 Π_{0}}{3}) U_{1} - \frac{V_{A}^{2}}{V_{s}^{2}} (U_{1} \times {\hat{V}}_{A}) \times {\hat{V}}_{A}} \\ = - 4 Π_{0} κ (k \cdot k) T_{1}, \end{matrix}

(B8)

- i ω B_{1} = (i k \times U_{1}) \times {\hat{V}}_{A} .

(B9)

The following expression for the divergence of a dyad is used in the above set of equations, i.e.,

\nabla \cdot ({\hat{V}}_{A} B_{1}) = (\nabla \cdot {\hat{V}}_{A}) B_{1} + ({\hat{V}}_{A} \cdot \nabla) B_{1} = ({\hat{V}}_{A} \cdot \nabla) B_{1},

(B10)

where

\nabla \cdot {\hat{V}}_{A} = 0

because

{\hat{V}}_{A}

is a constant unit vector. Using the expression of

ρ_{m 1}

from Eq. and

B_{1}

from Eq. in Eq. and later solving Eq. for T₁ gives

T_{1} = \frac{(\frac{1}{Γ} + \frac{4 Π_{0}}{3}) k \cdot U_{1} + \frac{1}{μ_{0}} ({\hat{V}}_{A} \cdot (k \times U_{1}) \times {\hat{V}}_{A} - k \cdot (U_{1} \times {\hat{V}}_{A}) \times {\hat{V}}_{A})}{ω (\frac{1}{Γ (Γ - 1)} + \frac{4 Π_{0}}{3} + 4 ι Π_{0} κ k^{2}) .}

(B11)

Now, using the expression of

ρ_{m 1}

from Eq. and T₁ from Eq. in Eq. and after simplification yields the following three equations for three components of the fluid velocity vector

U_{1}

⁠, i.e.,

\begin{matrix} {ω^{2} - S (k) k^{2} {sin}^{2} θ - k^{2} \frac{V_{A}^{2}}{V_{s}^{2}}} U_{1 x} - S (k) k^{2} cos θ sin θ U_{1 z} = 0, \end{matrix}

(B12)

ω^{2} U_{1 y} - k^{2} \frac{V_{A}^{2}}{V_{s}^{2}} {cos}^{2} θ U_{1 y} = 0,

(B13)

\begin{matrix} - S (k) k^{2} cos θ sin θ U_{1 x} + {ω^{2} - S (k) k^{2} {cos}^{2} θ} U_{1 z} = 0, \end{matrix}

(B14)

where

S (k) = A (k) {\frac{1}{Γ} + \frac{4 Π_{0}}{3}} + \frac{1}{Γ}

(B15)

and

\begin{matrix} A (k) = \frac{(\frac{1}{Γ} + \frac{4 Π_{0}}{3}) {\frac{1}{Γ (Γ - 1)} + \frac{4 Π_{0}}{3}}}{{{(\frac{1}{Γ (Γ - 1)} + \frac{4 Π_{0}}{3})}^{2} + 16 Π_{0}^{2} κ^{2} k^{4}}} \\ + \frac{4 i Π_{0} κ k^{2} (\frac{1}{Γ} + \frac{4 Π_{0}}{3})}{{{(\frac{1}{Γ (Γ - 1)} + \frac{4 Π_{0}}{3})}^{2} + 16 Π_{0}^{2} κ^{2} k^{4}}} \end{matrix}

(B16)

are defined.

Relations can be written in the matrix form as follows:

\hat{H} (ω, k) \cdot U_{1} = 0,

(B17)

where

\hat{H} (ω, k)

is a 3 × 3 matrix defined as follows:

\hat{H} (ω, k) = [\begin{matrix} ω^{2} - S (k) k^{2} {sin}^{2} θ - \frac{V_{A}^{2}}{V_{s}^{2}} k^{2} & 0 & - S (k) k^{2} cos θ sin θ \\ 0 & ω^{2} - \frac{V_{A}^{2}}{V_{s}^{2}} k^{2} {cos}^{2} θ & 0 \\ - S (k) k^{2} cos θ sin θ & 0 & ω^{2} - S (k) k^{2} {cos}^{2} θ \end{matrix}] .

(B18)

Finally, for a non-trivial solution of Eq. , the determinant of the matrix

\hat{H} (ω, k)

must be equal to zero, which gives the following dispersion relation:

\begin{matrix} | \hat{H} (ω, k) | = (ω^{2} - k^{2} \frac{V_{A}^{2}}{V_{s}^{2}} {cos}^{2} θ) [ω^{4} - ω^{2} (S (k) + \frac{V_{A}^{2}}{V_{s}^{2}}) k^{2} + S (k) \frac{V_{A}^{2}}{V_{s}^{2}} k^{4} {cos}^{2} θ] = 0. \end{matrix}

(B19)

APPENDIX C: BRANCH POINT AND BRANCH CUT FOR COMPLEX SQUARE ROOT FUNCTION

Consider the following complex function, which maps a complex number

z = x + i y

to another complex number f such that

f (z) = \sqrt{z} .

(C1)

f (z)

is an analytic function in the whole complex plane except at z = 0, which is called the branch point. Writing z and

f (z)

in polar coordinates,

z = r e^{i φ}, f (z) = ρ e^{i Φ}

⁠, gives

ρ e^{i Φ} = \sqrt{r} e^{i φ / 2}

(C2)

for each single z (with arguments

φ

or

φ + 2 π

⁠), there are two distinct values for

f (z)

[with angle

Φ

equal to

φ / 2

and

(φ / 2) + π

⁠)] in Eq. .

We can make f single valued by introducing the concept of branch cut.³⁵ For Eq. (C1), the negative real axis is the branch cut, which extends from origin to negative infinity. Angle $φ$ will then be restricted to the following value $- π \leq φ < π$ ⁠, which means we will never cross the line $φ = π$ ⁠. Across branch cut, the imaginary part of f will be discontinuous.

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Dispersion and damping of magnetohydrodynamic modes in radiative plasmas

I. INTRODUCTION

II. MODEL EQUATIONS

III. LINEAR WAVES DISPERSION RELATION

IV. MODIFIED MHD MODES IN A RADIATIVE PLASMA

A. Alfvén mode

B. Slow magnetosonic RMHD mode

C. Fast magnetosonic RMHD mode

V. RESULTS AND DISCUSSION

A. Changes in radiation parameter $Π_{0}$

B. Changes in diffusivity parameter κ

C. Changes in plasma beta β

D. Changes in direction angle θ

VI. CONCLUSION

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: CALCULATIONS OF MODEL EQUATIONS

APPENDIX B: LINEARIZATION OF EQUATIONS AND DERIVATION OF THE DISPERSION RELATION

APPENDIX C: BRANCH POINT AND BRANCH CUT FOR COMPLEX SQUARE ROOT FUNCTION

REFERENCES

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Dispersion and damping of magnetohydrodynamic modes in radiative plasmas

I. INTRODUCTION

II. MODEL EQUATIONS

III. LINEAR WAVES DISPERSION RELATION

IV. MODIFIED MHD MODES IN A RADIATIVE PLASMA

A. Alfvén mode

B. Slow magnetosonic RMHD mode

C. Fast magnetosonic RMHD mode

V. RESULTS AND DISCUSSION

A. Changes in radiation parameter Π 0

B. Changes in diffusivity parameter κ

C. Changes in plasma beta β

D. Changes in direction angle θ

VI. CONCLUSION

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: CALCULATIONS OF MODEL EQUATIONS

APPENDIX B: LINEARIZATION OF EQUATIONS AND DERIVATION OF THE DISPERSION RELATION

APPENDIX C: BRANCH POINT AND BRANCH CUT FOR COMPLEX SQUARE ROOT FUNCTION

REFERENCES

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Submit your article

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A. Changes in radiation parameter $Π_{0}$