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 $ \Pi 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 $ \Pi 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 ) = \alpha R T 4$, where $ \alpha R = 8 \pi 5 k B 4 / 15 c 3 h 3$, with

*c*denoting the speed of light,

*h*representing Planck's constant, and

*k*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 $ \lambda p \u226a L$, where

_{B}*λ*is the photon's mean free path, and

_{p}*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 \u223c \lambda p / c$, is significantly less than the fluid convection time, $ t \u223c L / U$, satisfying the condition $ \lambda p U / L c \u226a 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 $ \chi \xaf$, 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 \u2212 3\u2009 g \u2009cm \u2212 3$).

^{5}The observation of a high terminal velocity of stellar wind ( $ 10 3 \u2009km s \u2212 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 Knölker^{18} 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 Mihalas^{19} 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

*e*and enthalpy

*h*can be written as follows:

**E**(dimensionless) is given by Ohm's law for ideal MHD

*L*is the characteristic length-scale of the system). The normalized velocity (

**U**), density (

*ρ*), magnetic field (

_{m}**B**), and temperature (

*T*) are given by

*T*

_{0,}and

*B*

_{0}are the reference values for density, temperature, and magnetic field, respectively. The radiation parameter $ \Pi 0$ is given as follows: $ \Pi 0 = \alpha R T 0 4 / \rho m 0 V s 2$, where setting $ \Pi 0 = 0$ results in the non-radiative MHD set of dynamic equations.

^{28}The Alfvén velocity is given by $ V A = ( B 0 / \mu 0 \rho m 0 ) V \u0302 A$, where $ V \u0302 A$ is a unit vector in the direction of $ B 0$ and $ V s = ( \Gamma p 0 / \rho m 0 ) 1 / 2$ is the acoustic speed. Here,

*p*

_{0}is the pressure defined as $ p 0 = R \rho m 0 T 0$, where

*R*is the specific gas constant. The plasma beta $ \beta = 2 \mu 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 / ( \Gamma \beta )$. While our set of dynamic equations (1)–(3) 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

^{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. (9) 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.

*κ*, as dissipation is conveyed through radiation diffusion. Specifically, $ Im [ \omega ( k ) ]$ vanishes only when either $ \Pi 0 = 0$ or

*κ*= 0. Setting

*κ*= 0 (or $ \Pi 0 = 0$) in Eq. (B16) results in the phase speed $ Re [ \omega ( k ) ] / k$ in Eq. (9) becoming independent of the wave number

*k*. As a consequence, the resulting linear wave will propagate without dispersion. By comparing Eqs. (9) and (10), it becomes evident that the complex number $ S ( k ) V s$ represents the modified acoustic speed in the presence of radiation pressure and energy density. In the limit $ \Pi 0 \u2192 0 , \u2009 A ( k )$ approaches $ \Gamma \u2212 1$ according to Eq. (B16). Utilizing this limiting value of $ A ( k )$ in Eq. (B15), we obtain the following limit: $ S ( k ) \u2192 1$, resulting in the recovery of the sound speed for a non-radiative plasma, i.e., $ S ( k ) V s \u2192 V s$. When strong coupling between radiation and plasma is present ( $ 4 \Pi 0 / 3 \u226b 1 / \Gamma / ( \Gamma \u2212 1 )$), and in the long-wavelength limit where $ k \u226a k l$, with

*k*defined as

_{l}*p*as

_{r}*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

### A. Alfvén mode

### B. Slow magnetosonic RMHD mode

*ω*of Eq. (14) gives the slow magnetosonic mode in the RMHD plasma. Consider the following two cases in Eq. (14), (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 (14) transforms into the Alfvén mode at $ \theta = 0 \xb0$ and disappears at $ \theta = 90 \xb0$. However, when $ Re [ S ( k ) ] < V A 2 / V s 2$, the slow magnetosonic mode transforms into an acoustic mode at $ \theta = 0 \xb0$ and diminishes at $ \theta = 90 \xb0$. In the short-wavelength limit, i.e., $ k \u2192 \u221e$, Eq. (14) yields the frequency $ \omega s \u221e$ as follows:

_{s}### C. Fast magnetosonic RMHD mode

*ω*[Eq. (15)] 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 (15) transforms into the acoustic mode when propagation parallel to the magnetic field ( $ \theta = 0 \xb0$) and to magnetoacoustic in the perpendicular direction ( $ \theta = 90 \xb0$). 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 ( $ \theta = 0 \xb0$) and to magnetoacoustic at $ \theta = 90 \xb0$. In the short-wavelength limit, Eq. (15) yields the fast mode frequency $ \omega f \u221e$ as follows:

_{f}*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. (17) and (20) 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

*k*) is at its minimum value of $ k \u2192 0$. As the wave number increases, specifically as

*k*approaches the limit of infinity ( $ k \u2192 \u221e$), $ Im [ \omega 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

*et al.*

^{8}in the context of MHD waves in radiative plasma within the solar corona. The real part of

*ω*in Fig. 2 exhibits a linear behavior near $ k \u2243 0$ until it starts to curve. Within the plateau observed in the curve of $ Re [ \omega f ]$ in Fig. 2, the group velocity $ \u2202 Re [ \omega s ] / \u2202 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.

_{f}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 \u2243 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-Ghafri^{24} 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.

*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 ( $ \kappa \u2192 \u221e$). For higher values of $\kappa $, diffusive effects become significant for long wavelengths ( $ k \u2192 0$ as $ \kappa \u2192 \u221e$) only, as depicted in Fig. 3. However, the limit $ \kappa \u2192 \u221e$ 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 $ \lambda > \lambda p$ is satisfied, where

^{2}

### A. Changes in radiation parameter $ \Pi 0$

In Fig. 4, the linear dispersion relation of fast MHD waves in a radiative plasma is shown for two different values of $ \Pi 0$. In Fig. 4(a), the linear dispersion relation of fast MHD waves in a radiative plasma is depicted for two distinct values of $ \Pi 0$. The broken curve corresponds to ( $ \Pi 0 = 20$), and the solid curve corresponds to ( $ \Pi 0 = 10$). As the value of $ \Pi 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 \u2192 0$) limit increases with an increasing value of $ \Pi 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 $ \Pi 0$. As the value of $ \Pi 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 $ \Pi 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 $ \Pi 0$.

Similar to the case of fast waves, the dispersion in slow mode enhances with an increase in the value of $ \Pi 0$ as depicted Fig. 4(c). In Fig. 4(d), the damping of slow MHD waves in a radiative plasma ( $ Im [ \omega s ]$) is illustrated for two different values of $ \Pi 0$. The damping in the linear mode increases with an increasing value of $ \Pi 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 \u2192 0$), the damping rate of the slow wave decreases with increasing $ \Pi 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 \u2212 \Pi 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 $ \Pi 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 \u2192 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 $ \kappa \u221d \lambda 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 $ \u223c \lambda 3 = ( 2 \pi / k ) 3$ in the fluid. Therefore, linear waves with wavelengths much smaller than *λ _{p}* remain unaffected by the process of radiation diffusion.

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 [ \omega f ]$, but the minima in $ Im [ \omega 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 $ \kappa = 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 \u2212 \kappa $ 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*.

### 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 $ \beta = 0.45$ and solid curve for $ \beta = 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.

### 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).

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

*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 $ \theta = 90 \xb0$; however, at $ \theta = 0 \xb0$, 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 $ \theta = 0 \xb0$ 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 $ \theta = 90 \xb0$, but at $ \theta = 0 \xb0$, it will convert to the linear acoustic mode, which is a compressive MHD mode with non-zero damping at $ \theta = 0 \xb0$ [shown in Fig. 9(d)]. In Figs. 9(c)–9(f),

*ω*and

_{f}*ω*discontinuously convert into each other near $ \theta = 30 \xb0$. The reason for this specific behavior can be explained as follows: In the given range of values, Eqs. (15) and (14) can together be written in a compact form as follows:

_{s}*F*is defined as

*F*can be written as follows:

*ω*in (27) becomes double-valued due to the square root term $ F$ (which has two ± values with the phase difference $ e i \pi , \u2009 F$ is not the analytic function at $ r e i \phi = 0$, see Appendix C). Negative real axis is the branch cut for

*F*. Whenever we cross the branch cut, $ F$ will get multiplied by $ e i \pi $ (and mode crossing will happen; the fast mode will transform into the slow mode and vice versa). With these two distinct ± values for $ F$, Eq. (27) represents two different modes: fast and slow. In Fig. 11, we depict the dependence of $ \phi , \u2009 Re [ F ]$, and $ Im [ F ]$ on angle

*θ*. It is observed that as $\phi $ crosses the negative real axis (near $ \theta = 30 deg$), the changes in $ Re [ F ]$ and $ Im [ F ]$ are not smooth ( $ Re [ F ]$ is not differentiable and $ Im [ F ]$ is discontinuous). For larger

*k*[such as in Figs. 9(g) and 9(h), where the curves are continuous functions of angle

*θ*], $\phi $ 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 Kumar

^{31}in a stratified environment.

## 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 ( $ \Pi 0 \u2260 0$). Dispersion and damping of fast and slow magnetosonic modes in a radiative plasma are affected by the changes in diffusivity *κ*, radiation parameter $ \Pi 0$, and plasma beta *β*. For strong coupling between radiation and plasma ( $ \Pi 0 \u226b 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 $ \Pi 0$ and diffusivity *κ* are non-zero, stationary points {where the wave stops propagating and the group velocity is zero ( $ \u2202 Re [ \omega s ] / \u2202 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

^{2}

^{,}

*λ*.

_{p}^{32,33}Diffusivity

*κ*is also defined as $ \kappa = ( 1 / 3 \chi t ) c$, where $ \chi t = \chi a + \chi s$ is the total opacity and $ \chi a ( \chi s )$ is the absorption (scattering) opacity.

^{34}The viscous terms in the radiation stress-energy tensor $ T \u0302 R$ defined in Eq. (A2) are ignored because they are much smaller than the isotropic components of pressure.

*ε*is the total energy density, which includes magnetic, kinetic, and radiative energy density, $ F \epsilon $ is the flux of total energy

*ε*and $ T \u0302 stress$ is the total stress. The magnetic stress-energy tensor is given as follows:

^{28}

*ρ*is the fluid mass density,

_{m}*e*is the specific internal energy,

**U**is the fluid velocity, and

*h*is enthalpy. Hence, $ T \u0302 stress$ defined in Eq. (A4) is

*h*of the plasma is defined in terms of the specific internal energy

*e*and the plasma pressure

*p*as follows:

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

*T*

_{1}gives

*T*

_{1}from Eq. (B11) in Eq. (B7) and after simplification yields the following three equations for three components of the fluid velocity vector $ U 1$, i.e.,

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

*f*such that

*z*= 0, which is called the branch point. Writing

*z*and $ f ( z )$ in polar coordinates, $ z = r e i \phi , \u2009 f ( z ) = \rho e i \Phi $, gives

*z*(with arguments $\phi $ or $ \phi + 2 \pi $), there are two distinct values for $ f ( z )$ [with angle $\Phi $ equal to $ \phi / 2$ and $ ( \phi / 2 ) + \pi $)] in Eq. (C2).

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 $\phi $ will then be restricted to the following value $ \u2212 \pi \u2264 \phi < \pi $, which means we will never cross the line $ \phi = \pi $. Across branch cut, the imaginary part of *f* will be discontinuous.

## REFERENCES

*Physics of Shock Waves and High Temperature Hydrodynamic*

*Foundations of Radiation Hydrodynamics*

*Radiative Processes in Astrophysics*

*Introduction to Plasma Physics with Space and Laboratory Applications*

*Radiation Gas Dynamics*

*Mathematical Methods for Physicists*